domain: math strands: counting: Counting & Number Sense placevalue: Place Value addsub: Addition & Subtraction multdiv: Multiplication & Division fractions: Fractions measure: Measurement & data (length, time, money, area/perimeter, angle, graphs) geometry: Geometry & spatial numbertheory: Number Theory integers: Integers & Negative Numbers ratio: Ratios, Rates & Percents algebra: Expressions & Equations nodes: - id: math.counting.count-sequence-100 title: Oral count sequence to 100 (by 1s and 10s), count on from any number strand: counting band: - 4 - 6 tier: L3 gates: L2: ≥90% on 10 fresh oral-counting prompts, untimed (say the sequence to 100; count on from a given number; count by 10s) L3: SM-2 spaced gate (3+ spaced reps, 4+ lifetime correct) across separate days requires: [] helps: - math.counting.skip-count-2-5-10 resources: - type: dimensions ref: Dimensions Math KA ch2 note: Numbers to 5 record: true - type: dimensions ref: Dimensions Math KA ch3 note: Numbers to 10 - type: dimensions ref: Dimensions Math KB ch7 note: Numbers to 20 - type: dimensions ref: Dimensions Math KB ch12 note: Numbers to 100 - type: dimensions ref: Dimensions Math 1A ch1 note: Numbers to 10 - type: dimensions ref: Dimensions Math 1A ch5 note: Numbers to 20 - type: singapore ref: PM2022 G2 ch1 §1A Count to 1,000 - type: beast ref: BA 2A ch2 Comparing - type: activity ref: count stairs / count to 100 aloud on a walk sources: - CC K.CC.1 - CC K.CC.2 - research/math-spine.md §3 notes: Counting words are a prereq to using them for counting (K2 p.4). Cora's zone; Penelope past this. instrument_gap: true representations: C: move: count real objects up a physical column — steps on the stairs, beads on a hundred-string, a tower of unifix cubes — saying each number word as you touch the next one; then start from a given rung (e.g. touch step 37) and count on from there without restarting at 1. check: start at 47 and count on to 60 — touch each step as you say it. N: move: 'make counting a lived count-out-loud on a walk or up the stairs: ''let''s see if we reach 100 before the mailbox''; count by 10s while dropping a coin in the jar each ten.' check: we're on step 68 — keep going to 80 while we walk. down: count small sets 1-to-1 first S: move: show the written numeral sequence on a number track/hundred chart and have her point-and-say, connecting the spoken word to the written digits, especially over decade bumps (29->30, 89->90). check: what number comes right after 79? after 99? - id: math.counting.count-objects-cardinality title: Count objects 1-to-1; last number = how many (cardinality); count out n strand: counting band: - 4 - 6 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: count a set 1-to-1 (line/array/circle/scattered), answer ''how many?'' with the last number, and count out a requested number of objects' L3: SM-2 spaced gate on mixed fresh sets across separate days requires: - math.counting.count-sequence-100 helps: [] resources: - type: dimensions ref: Dimensions Math KA ch2 note: Numbers to 5 — count 1-to-1, cardinality record: true - type: dimensions ref: Dimensions Math KA ch3 note: Numbers to 10 - type: activity ref: count and hand me N objects (blocks, crackers) - type: gumball ref: generator:fact_mc sources: - CC K.CC.4a - CC K.CC.4b - CC K.CC.5 - research/math-spine.md §3 notes: 'The ''count IS the answer'' error: pre-cardinality child re-counts instead of reporting the last number (K2 p.4). Arrangement difficulty: line→array→circle→scattered — one node, harder items gate L3.' representations: C: move: count a set of physical counters one-to-one, moving each into a 'counted' pile as she says its number; then STOP and ask 'how many?' — the answer is the last number said, no re-count. Also count OUT a requested number (give me 7 crackers) so cardinality runs both directions. check: count these 9 blocks, then without counting again tell me how many there are; now count out 6 for me. down: math.counting.count-sequence-100 G: move: circle a pictured group of objects and write the total; vary the arrangement (line -> array -> circle -> scattered) so 'how many' is the count regardless of shape. check: how many stars in this scattered group? — count once and write the number. N: move: 'count real belongings that matter to her — ''how many stickers do you have?'', ''set the table: count out one fork for each of the 5 chairs'' — where the last number genuinely answers a real ''how many''.' check: count out one napkin for every person at dinner — how many did you need? - id: math.counting.compare-numbers-20 title: Compare quantities and written numerals (greater/less/equal) within ~20 strand: counting band: - 5 - 7 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: compare two sets by matching+counting and compare two written numerals (>, =, <)' L3: SM-2 spaced gate across separate days requires: - math.counting.count-objects-cardinality helps: [] resources: - type: dimensions ref: Dimensions Math KA ch6 note: Comparing Numbers Within 10 record: true - type: dimensions ref: Dimensions Math 1B ch11 note: Comparing - type: singapore ref: PM2022 G2 ch1 §1C Compare and Order Numbers - type: beast ref: BA 2A ch2 Comparing sources: - CC K.CC.6 - CC K.CC.7 - research/math-spine.md §3 notes: '''looks like more'' ≠ ''is more'' — matching/counting can overturn a visual judgment (K2 p.5). < vs > symbol direction is a separate difficulty (K3 p.6): ''wide part next to the larger number''.' instrument_gap: true representations: C: move: line up two rows of counters and MATCH them one-to-one; the row that sticks out past the other is the greater set — this can overturn a 'looks like more' visual judgment (a spread-out small set vs a bunched large one). check: match these two rows — which has more, and how do you know it isn't just spread out? down: math.counting.count-objects-cardinality G: move: draw or shade two bars/ten-frames to the counted heights and compare their lengths; introduce the >, =, < symbols as a hungry mouth whose wide part opens toward the bigger number. check: put >, =, or < between 13 and 17 — which way does the wide part face? S: move: 'compare two written numerals directly (no objects): 8 __ 14, 20 __ 12, and read it aloud as a full statement (''fourteen is greater than eight'').' check: 'write the right symbol: 15 __ 15, then 19 __ 9.' N: move: compare real amounts she cares about — 'you have 12 marbles, your sister has 15, who has more and how many more?' — anchoring greater/less to fairness and quantity in the world. check: there are 8 red apples and 11 green apples — are there more green or more red? - id: math.counting.write-digits title: Write digits 0-9 and 1-2 digit numbers (tool skill; fluency ceiling) strand: counting band: - 5 - 9 tier: L4 fluency_aim: metric: digits_per_min_written target: 40 sustain: 2 of 3 separate days source: research/fluency-aims.md §3 (Rocket Math Writing Speed Test) gates: L2: ≥90% legible correct numerals on a fresh 0-20 write-the-numeral sheet, untimed L3: 'SM-2 spaced gate: correct legible numerals across separate days' L4: Writing Speed Test — write the (1-2 digit) numbers shown, 1 minute, count boxes; MEASURE the rate; this rate IS the ceiling for every written fact node requires: - math.counting.count-objects-cardinality helps: [] resources: - type: activity ref: Rocket Math Writing Speed Test (printed timed sheet, parent scores) - type: packet ref: analog-clocks-2026-07-12 note: part 8 draw/write practice reinforces numeral formation sources: - CC K.CC.3 - research/fluency-aims.md §3 - research/fluency-aims.md §6 notes: 'MOTOR-SPEED CEILING. Target 40 is [EXTRAPOLATED] (K-end ~40 → grade-1-end ~60, Rocket Math §3a); MEASURE with a real Writing Speed Test per term in the 7-9 steep-growth window — do NOT freeze the number. Every WRITTEN fact-fluency node requires this and sets its aim at 2/3 of the measured rate. Prefer oral aims to sidestep the ceiling. Confidence: literature-solid method / extrapolated value.' representations: C: move: form each numeral in a multisensory medium before pencil — trace the digit in a sand/salt tray or in the air with a big arm motion, following the start-dot and stroke arrows; this builds the motor path (the ceiling this node measures is MOTOR speed). check: make a 3 in the sand tray starting from the top dot — does it curve the right way, not backward? down: math.counting.count-objects-cardinality S: move: write the numerals to dictation from the printed model, then from a spoken/pictured quantity; run the timed Rocket-Math Writing Speed Test to MEASURE her digits-per-minute — this rate is the ceiling for every written-fact node. check: write the numbers I say — fourteen, forty-one, seven — watch the teen vs decade order (14 not 41). - id: math.counting.count-oral-skip title: Skip-count by 2s, 5s, 10s (oral fluency tool skill) strand: counting band: - 5 - 8 tier: L4 fluency_aim: metric: problems_per_min_oral target: 60 sustain: 2 of 3 separate days, ≤2 errors/min source: research/fluency-aims.md §8 (McDowell & Keenan skip-count 60-80/min) gates: L2: ≥90% on 10 fresh oral skip-count prompts (by 2s/5s/10s to 100), untimed L3: SM-2 spaced gate across separate days L4: 'oral 1-minute timing: skip-count numbers said/min, parent scores; 60-80/min range, lower number is pass' requires: - math.counting.count-sequence-100 helps: - math.counting.skip-count-2-5-10 resources: - type: dimensions ref: Dimensions Math 2A ch7 note: Multiplication and Division of 2, 5, and 10 — count-by foundation record: true - type: singapore ref: PM2022 G2 ch6 §6C Skip Count by 2s, 5s, and 10s - type: beast ref: BA 3A ch2 Skip-Counting - type: activity ref: count nickels/dimes aloud; count by 2s up the stairs sources: - CC 2.NBT.2 - research/fluency-aims.md §8 - research/singapore-beast.md §5.D notes: 'Oral (see/say) aim sidesteps the write-digits ceiling. Aim range 60-80/min; pass = 60. Feeds multiplication (skip-count → arithmetic sequences → ×, BA 3A ch2). Confidence: literature-solid.' instrument_gap: true representations: C: move: skip-count physical equal groups aloud at pace — drop nickels one at a time saying 5,10,15,20; step up the stairs two at a time saying 2,4,6,8 — the object drop sets the rhythm the oral rate rides on. check: count these dimes into the jar by 10s as fast as you can with no mistakes — how high did you get in a minute? down: math.counting.count-sequence-100 N: move: attach skip-counting to real fast-counting jobs — 'count the shoes by 2s', 'count how much money in nickels' — so the automaticity has a reason to be fast. check: count these 8 pairs of socks by 2s — how many socks total? S: move: show the skip-count sequence written (5,10,15,...) and have her say-the-next at a timed pace off a see/say sheet; the L4 gate is the 1-minute oral timing at 60-80/min. check: 'read this row out loud as fast as you can: 5, 10, 15, 20, 25, ... keep going to 60.' - id: math.counting.skip-count-2-5-10 title: Skip-count by 2s/5s/10s to 1000; count within 1000 (understanding, not rate) strand: counting band: - 6 - 8 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: continue a skip-count sequence by 2/5/10/100 from any start within 1000; fill missing terms' L3: SM-2 spaced gate across separate days requires: - math.counting.count-sequence-100 helps: - thinking.reasoning.pattern-rule resources: - type: dimensions ref: Dimensions Math 2A ch7 note: count-by 2/5/10 (equal-groups → ×) record: true - type: singapore ref: PM2022 G2 ch6 §6C Skip Count by 2s, 5s, and 10s - type: singapore ref: PM2022 G2 ch1 §1B Number Patterns - type: beast ref: BA 3A ch2 Skip-Counting sources: - CC 2.NBT.2 - research/math-spine.md §3 notes: Understanding rung (patterns, missing terms) distinct from the oral-rate rung count-oral-skip. Foundation for multiplication (arithmetic sequences, BA 3A ch2). instrument_gap: true representations: C: move: build the skip-count with equal stacks — lay down towers of 5 cubes and count 5,10,15 as each tower joins; leave a GAP and ask what number the missing tower holds (fill-the-missing-term with a physical stack). check: 'here are stacks of ten: 10, 20, __, 40 — build and name the missing stack.' G: move: hop along a number line in equal jumps (by 2s/5s/10s/100s) from any start, drawing the arcs; the landing points ARE the sequence, and a blank landing is a missing term to solve. check: 'draw hops of 5 from 35: land on 35, 40, __, 50 — what''s the blank?' S: move: continue and fill the written sequence from any start within 1000 (e.g. 320, 330, __, 350) and name the pattern rule ('add 10 each time'); connect to growing-pattern reasoning. check: 'fill the blanks: 250, __, 270, 280, __ — what is the rule?' down: math.counting.count-sequence-100 - id: math.placevalue.teen-numbers-place title: Teen numbers 11-19 as ten ones + some ones (10+n) strand: placevalue band: - 5 - 7 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: represent/write teen numbers as 10 + some more; e.g. 14 = 10 + 4' L3: SM-2 spaced gate across separate days requires: - math.counting.count-objects-cardinality helps: [] resources: - type: dimensions ref: Dimensions Math KB ch7 note: Numbers to 20 (teen numbers as 10+n) record: true - type: dimensions ref: Dimensions Math 1A ch5 note: Numbers to 20 - type: beast ref: BA 2A ch1 Place Value - type: activity ref: ten-frame + loose counters to build teen numbers sources: - CC K.NBT.1 - research/math-spine.md §4 notes: '''ten'' is unmarked in English number words; teen words say the ones digit first (''four-teen'' but 14) → reversal error (write 41 for fourteen) (K3 p.5-6). Fragile early node; hard prereq for make-a-ten.' instrument_gap: true representations: C: move: 'build a teen number on a ten-frame: fill one whole ten-frame (10) and put the extra loose counters beside it, then say ''14 is one full ten and 4 more'' — the completed ten made concrete fights the ''teen is just a big pile'' view.' check: show me 16 on a ten-frame — how many fill the frame, how many are left over? down: math.counting.count-objects-cardinality G: move: draw a filled ten-frame plus loose dots, and write it as 10 + n underneath; the picture and the 10 + 4 form sit together so 'ten and four' maps to 14. check: draw 13 as a full ten-frame plus extra dots, then write it as 10 + ___. S: move: 'write the teen as 10 + n and as the single numeral, saying it slowly to catch the reversal trap: fourteen SOUNDS like ''four'' first but is written 1 then 4, NOT 41.' check: write fourteen as a number — is it 14 or 41? write it as 10 + 4 to check. - id: math.placevalue.two-digit-place-value title: Two-digit numbers as tens + ones; compare; 10-more/10-less mentally strand: placevalue band: - 6 - 8 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: name/build a 2-digit number as tens+ones (67 = 6 tens 7 ones), compare two 2-digit numbers by tens then ones, and find 10 more / 10 less' L3: SM-2 spaced gate on mixed items across separate days requires: - math.placevalue.teen-numbers-place helps: [] resources: - type: dimensions ref: Dimensions Math 1B ch12 note: Numbers to 40 (tens & ones) record: true - type: dimensions ref: Dimensions Math 1B ch16 note: Numbers to 100 - type: dimensions ref: Dimensions Math KB ch12 note: Numbers to 100 (early) - type: singapore ref: PM2022 G2 ch1 Numbers to 1,000 - type: singapore ref: PMUS 2A u1 Numbers to 1000 - type: beast ref: BA 2A ch1 Place Value sources: - CC 1.NBT.2 - CC 1.NBT.3 - CC 1.NBT.5 - research/math-spine.md §4 notes: 'Unitize ten (view ten ones as one ''ten'') folded in here. Decade words are opaque (''twenty'' = 2 tens) — explicit teaching move (K3 p.6). Strict ridgeline: do NOT advance to 3-digit before this is L3.' instrument_gap: true representations: C: move: 'build a 2-digit number with base-ten blocks / place-value disks: 67 = 6 ten-rods and 7 unit-cubes; then ADD one ten-rod for ''10 more'' and REMOVE one for ''10 less'' so the mental ±10 is a physical swap of a single ten.' check: build 45 with ten-rods and ones; now show me 10 more — which piece did you add? down: math.placevalue.teen-numbers-place G: move: draw the number in a tens|ones place-value chart (sticks and dots, or block sketches) and compare two numbers by looking at the TENS column first, ones only to break a tie. check: in the chart, which is bigger, 52 or 48 — which column did you check first? S: move: 'write the number in expanded form 67 = 60 + 7 and name it as ''6 tens 7 ones''; unitize — the digit 6 in the tens place is worth 60, not 6 (decade words are opaque: ''twenty'' = 2 tens).' check: what is the 5 worth in 53? and the 3? write 53 as ___ + ___. N: move: use dimes and pennies as the ten|one anchor — 67 cents = 6 dimes + 7 pennies; trading a group of ten pennies for one dime is unitizing a ten in the real world. check: you have 4 dimes and 2 pennies — how many cents? now add a dime — how much now? - id: math.placevalue.three-digit-place-value title: Three-digit place value to 1000; compare; read/write/expanded form; ±10/100 strand: placevalue band: - 7 - 8 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: name a 3-digit number as hundreds+tens+ones, read/write in numerals and words and expanded form, compare two 3-digit numbers, mentally add/subtract 10 or 100' L3: SM-2 spaced gate on mixed items across separate days requires: - math.placevalue.two-digit-place-value helps: [] resources: - type: dimensions ref: Dimensions Math 2A ch1 note: Numbers to 1,000 record: true - type: singapore ref: PM2022 G2 ch1 Numbers to 1,000 - type: beast ref: BA 2A ch1 Place Value sources: - CC 2.NBT.1 - CC 2.NBT.3 - CC 2.NBT.4 - CC 2.NBT.8 - research/math-spine.md §4 notes: A hundred is a bundle of TEN TENS, a genuinely new unitizing step — don't assume it transfers from tens (K3 p.8). '500 as 5 hundreds' is not automatic even for a kid who says it after 499. instrument_gap: true representations: C: move: 'build a 3-digit number with base-ten blocks — hundred-flats, ten-rods, unit-cubes: 342 = 3 flats, 4 rods, 2 cubes; SHOW that one flat is ten ten-rods stacked (a hundred is a bundle of ten tens — a genuinely new unit, don''t assume it transfers from tens).' check: build 205 with flats, rods, and cubes — how many ten-rods make one flat? down: math.placevalue.two-digit-place-value G: move: record the number in a hundreds|tens|ones chart and sketch flats/rods/cubes; add/subtract 10 or 100 by changing exactly one column, and compare two 3-digit numbers hundreds-first. check: in the chart, add 100 to 462 — which column changes? which is bigger, 428 or 431? S: move: write in numerals, in words, and in expanded form (342 = 300 + 40 + 2); check that '500' is read as 5 hundreds, not automatic just because she counts past 499. check: write four hundred seven in numerals — is there a zero in the tens place? write it as ___ + ___ + ___. - id: math.placevalue.big-numbers-round title: Multi-digit place value to millions; rounding to any place strand: placevalue band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: read/write/compare numbers to 1,000,000; each digit = 10× the place to its right; round to nearest 10/100/any place' L3: SM-2 spaced gate on mixed items across separate days requires: - math.placevalue.three-digit-place-value helps: - thinking.reasoning.estimation-sanity-check resources: - type: singapore ref: PM2022 G3 ch1 Numbers to 10,000 - type: singapore ref: PM2022 G3 ch1 §1D Rounding Numbers - type: beast ref: BA 2D ch10 Big Numbers sources: - CC 3.NBT.1 - CC 4.NBT.1 - CC 4.NBT.2 - CC 4.NBT.3 - research/math-spine.md §4 notes: BA-L2 already teaches multi-digit place value (4.NBT.2) in Grade 2 (research/singapore-beast.md §5.A) — band set low, BA proves it. Rounding 'digit 9' carry is the wrinkle (MA 2.4). instrument_gap: true representations: C: move: 'extend the base-ten idea with grouped bundles / a place-value disk board out to thousands: each disk traded left is worth 10x the one to its right (ten 100-disks = one 1000-disk), making ''each place is 10x the place to its right'' a physical trade.' check: how many hundred-disks do you trade in to make one thousand-disk? down: math.placevalue.three-digit-place-value G: move: 'round on a number line: mark the two nearest ''landmark'' multiples (nearest 10, 100, etc.), plot the number, and see which landmark it''s closer to; the midpoint rounds up. Watch the digit-9 carry (a number just under a landmark).' check: on a number line to the nearest 100, is 347 closer to 300 or 400? what about 850? S: move: read/write/compare numbers to a million in a labeled place-value chart, then round to a named place by the underline-the-digit / look-right method; connect rounding to estimation (is this answer sensible?). check: round 4,682 to the nearest hundred; to the nearest thousand. Which digit did you look at each time? - id: math.addsub.represent-add-sub title: Represent & model add/sub within 10 (objects, fingers, drawings, equations) strand: addsub band: - 5 - 7 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: model an add or take-from situation within 10 and write a matching equation; solve by direct modeling (count all)' L3: SM-2 spaced gate on mixed fresh items across separate days requires: - math.counting.count-objects-cardinality helps: [] resources: - type: dimensions ref: Dimensions Math KB ch8 note: Number Bonds (part-whole) record: true - type: dimensions ref: Dimensions Math KB ch9 note: Addition - type: dimensions ref: Dimensions Math KB ch10 note: Subtraction - type: dimensions ref: Dimensions Math 1A ch3 note: Addition - type: dimensions ref: Dimensions Math 1A ch4 note: Subtraction - type: activity ref: story problems with counters - type: gumball ref: generator:fact_mc sources: - CC K.OA.1 - CC K.OA.2 - research/math-spine.md §5 notes: Level 1 strategy = Direct Modeling / Count All (K2 p.6). Foundation of the whole add/sub strand. representations: C: move: 'Set out the story with counters: for 4 + 3, count out a group of 4, then a group of 3, push them together and count ALL from one. For take-from (7 − 2) build 7, physically remove 2, count what''s left. Fingers work too.' check: Show me 5 and 3 more with counters — how many altogether? down: count objects 1-to-1 and say the last number as 'how many' G: move: 'Draw a part-part-whole (number-bond) picture: two circles for the parts, one for the whole, dots inside. The child SEES 4 dots and 3 dots joining into the 7-whole — the same picture works forward (add) and backward (take-from).' check: Draw the dots for 6 take away 2 — how many dots are left? S: move: 'After the counters/picture answer the problem, write the matching equation underneath — 4 + 3 = 7 — pointing at each part: ''this 4 is your first group, this 3 is the group you added, this 7 is all of them.'' Connect each numeral to a pile she can touch.' check: Write the number sentence that matches 5 apples and 2 more apples. N: move: 'Pose it as a lived situation before any symbols: ''You had 4 stickers, your friend gave you 3 more.'' Have her act it out with real objects, THEN answer. Rotate add-to and take-from stories so ''minus'' isn''t always ''the ones that leave.''' check: You have 8 grapes and eat 3. How many grapes now? - id: math.addsub.decompose-make-10 title: Decompose numbers ≤10 into pairs; find make-10 partners strand: addsub band: - 5 - 7 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: decompose a number ≤10 into pairs (5 = 2+3 = 4+1), and for any 1-9 name the partner that makes 10' L3: SM-2 spaced gate across separate days requires: - math.addsub.represent-add-sub helps: [] resources: - type: dimensions ref: Dimensions Math KB ch8 note: Number Bonds record: true - type: dimensions ref: Dimensions Math 1A ch2 note: Number Bonds - type: activity ref: ten-frame make-10 flash; number bonds - type: singapore ref: PM2022 G2 ch2 §2A Add Fluently Within 20 sources: - CC K.OA.3 - CC K.OA.4 - research/math-spine.md §5 notes: Make-10 partners is one of the three hard prereqs of make-a-ten (K2 p.15-16). Number bonds are the Singapore representation. instrument_gap: true representations: C: move: 'Ten-frame make-ten drill: drop 7 counters onto a ten-frame, ask ''how many empty cells?'' — the 3 empty cells ARE the partner. Repeat fast for every 1–9 so ''partner to 10'' becomes a look, not a count. Also break a pile of 6 different ways (5+1, 4+2, 3+3) to show a number has many decompositions.' check: I put 8 counters in the ten-frame — how many more to fill it? G: move: 'Number bond diagram: write 10 in the whole-circle, one part filled (say 6), the other part blank — she writes the missing part. Then flip it: give her a number and both blank parts to fill several correct ways (10 = 6+4 = 7+3 …).' check: 'Fill the number bond: whole is 10, one part is 4, what''s the other part?' down: math.addsub.represent-add-sub S: move: 'Record each partner pair as an equation, emphasizing the make-10 family: 6 + 4 = 10, 4 + 6 = 10, 10 − 6 = 4. Line the ten pairs up as a table so the pattern (0+10, 1+9, 2+8…) is visible — the child memorizes the SET, not ten isolated facts.' check: 9 + ? = 10. And what is 3 + ? = 10? - id: math.addsub.add-sub-fluency-5 title: Fluently add & subtract within 5 strand: addsub band: - 5 - 6 tier: L4 fluency_aim: metric: problems_per_min_oral target: 20 sustain: 2 of 3 separate days, ≤2 errors/min source: research/fluency-aims.md §8 (first fluency target; VanDerHeyden & Burns grade-band) gates: L2: ≥90% on 10 fresh add/sub-within-5 items, untimed L3: SM-2 spaced gate on mixed within-5 items across separate days L4: oral 1-minute timing (see/say), problems/min; pass 20, stretch higher; parent scores requires: - math.addsub.represent-add-sub helps: [] resources: - type: gumball ref: generator:fact_mc - type: activity ref: printed within-5 timed sheet (parent enters score) sources: - CC K.OA.5 - research/fluency-aims.md §8 notes: 'First CC fluency standard. Oral aim avoids the write-digits ceiling. Cora''s live L2/L3 zone (66% first-try within-5, still learning). Confidence: literature-solid. Target 20 = pass floor (a small first fluency rung), not the terminal PT range.' representations: C: move: 'Fast subitize/finger work: flash a hand showing 3 fingers, add 2 more, she says 5 WITHOUT recounting from one. Ten-frame flash cards (dots you show for one second) push recognition toward automatic. This is the see-then-say path that becomes the oral rate.' check: Flash — three dots and one dot — how many, quick? down: math.addsub.represent-add-sub S: move: 'Move to see/say fact cards (3 + 2, 5 − 1) with no counters, timing gently: she reads the fact and says the answer. Mix add and subtract so she''s retrieving, not running one procedure. The oral 1-minute sheet IS this at speed.' check: 4 + 1? … 5 − 2? … say them fast. - id: math.addsub.counting-on title: Count on to add/subtract (Level 2); subtraction as unknown-addend strand: addsub band: - 6 - 7 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: solve add/sub within 20 by counting on from the larger addend; solve a take-from problem by counting up (5+?=8)' L3: SM-2 spaced gate on mixed items across separate days requires: - math.addsub.add-sub-fluency-5 - math.counting.compare-numbers-20 helps: - math.measure.elapsed-time - math.measure.money-coins-bills sources: - CC 1.OA.4 - CC 1.OA.5 - research/math-spine.md §5 resources: - type: dimensions ref: Dimensions Math 1A ch6 note: Addition to 20 record: true - type: dimensions ref: Dimensions Math 1A ch7 note: Subtraction Within 20 - type: beast ref: BA 2A ch3 Addition - type: beast ref: BA 2B ch4 Subtraction notes: 'Counting-on is a genuine conceptual leap (first addend embedded in the total, seen as addend AND part, K2 p.14). Counting on for subtraction is EASIER than counting down and is the unknown-addend keystone (K2 p.15) — ''one of the essential understandings for middle school''. helps: count-up powers elapsed-time and count-up-change.' instrument_gap: true representations: C: move: 'Walk a floor/table number line: stand on the larger addend (8) and take 3 physical steps for +3, landing on 11 — she never recounts the 8. For subtraction as count-UP, stand on 5, step forward until you reach 8, count the STEPS (3) — this is easier than counting down and is the unknown-addend idea.' check: Start on 6, take 4 steps forward — where do you land? down: math.addsub.add-sub-fluency-5 G: move: 'Draw the number line and draw the hops: for 8 + 3, mark 8, draw three curved hops to 11, label each hop. For 5 + ? = 8, hop up from 5 to 8 and COUNT THE HOPS to find the missing addend. FADE the drawn hops on the last item (''no hop boxes this time, Agent'').' check: Draw hops from 7 up to 10 — how many hops? S: move: 'Write the count-on as an equation and name the move: 8 + 3, ''start at 8, count 9-10-11.'' For take-from, rewrite 8 − 5 as 5 + ? = 8 out loud so the child SEES subtraction as a missing-addend question, not a separate operation.' check: 5 + ? = 8. What number goes in the box? N: move: 'Count-up money and time make this concrete: ''you owe 8¢, you''ve counted out 5¢, keep going 6-7-8'' — that''s counting on. This is the exact skill under making change and finding elapsed time, so say so: same hop, different context.' check: You need 9 minutes. 6 have passed. Count up — how many more? - id: math.addsub.add-sub-fluency-10 title: Fluency for add & subtract within 10 strand: addsub band: - 6 - 7 tier: L4 fluency_aim: metric: problems_per_min_oral target: 25 sustain: 2 of 3 separate days, ≤2 errors/min source: research/fluency-aims.md §8 (within-10 facts base for make-a-ten) gates: L2: ≥90% on 10 fresh add/sub-within-10 items, untimed L3: SM-2 spaced gate on mixed within-10 items across separate days L4: oral 1-minute timing (see/say), problems/min; pass 25, stretch toward the add-facts-20 aim; parent scores requires: - math.addsub.add-sub-fluency-5 - math.addsub.counting-on helps: [] resources: - type: gumball ref: generator:fact_mc - type: singapore ref: PM2022 G2 ch2 §2A Add Fluently Within 20 - type: activity ref: printed within-10 timed sheet sources: - CC 1.OA.6 - research/fluency-aims.md §8 notes: 'Within-10 (from-memory-ish) is the base for make-a-ten; within-20 stays strategy-based until add-facts-memory-20. Target 25 is an intermediate rung between within-5 (20) and add-facts-20 (30-40). Confidence: literature-solid (rule) / extrapolated (exact interior number).' representations: C: move: 'Ten-frame + doubles/near-doubles flash: show 6+6 as two full rows minus, or 7+8 as a double-plus-one, so she has an anchor for the ones she can''t yet recall. Flash within-10 dot cards for see/say speed. Mix in subtraction (10 − 6) from the same ten-frame.' check: Flash — a full row of 5 and 3 more — how many? down: math.addsub.add-sub-fluency-5 S: move: See/say fact cards across the whole within-10 set, add and subtract mixed, lightly timed toward the oral rate. Group by strategy family (doubles, make-10 pairs, +1/+0) so retrieval has hooks rather than being 55 orphan facts. check: 7 + 2? … 9 − 4? … 6 + 3? — quick answers. - id: math.addsub.make-a-ten-strategy title: Add/sub within 20 via make-a-ten (Level 3) and derived facts; add 3 numbers strand: addsub band: - 6 - 8 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: use make-a-ten to add within 20 (8+6 = 8+2+4); apply commutative/associative; add three numbers with sum ≤20' L3: SM-2 spaced gate on mixed items across separate days requires: - math.addsub.add-sub-fluency-10 - math.addsub.decompose-make-10 - math.placevalue.teen-numbers-place helps: [] resources: - type: dimensions ref: Dimensions Math 1A ch6 note: Addition to 20 (make-ten within 20) record: true - type: dimensions ref: Dimensions Math 1B ch13 note: Addition and Subtraction Within 40 — lesson 3 Make the Next Ten (Penelope's live chapter) - type: singapore ref: PM2022 G2 ch2 §2A Add Fluently Within 20 - type: beast ref: BA 2A ch3 Addition - type: beast ref: BA 2C ch8 Strategies (+&−) sources: - CC 1.OA.2 - CC 1.OA.3 - CC 1.OA.6 - research/math-spine.md §5 notes: 'Level 3 = Convert to an Easier Problem (make-a-ten / derived facts, K2 p.6). Harder in English than East-Asian languages (irregular teen words, K2 p.16). Requires ALL THREE: within-10 fluency, decompose/make-10 partners, teen-numbers (K2 p.15) — don''t drop any.' instrument_gap: true representations: C: move: 'Two ten-frames, the classic bridge: build 8 + 6 as 8 on the first frame, 6 loose, then MOVE 2 of the six over to fill the first frame (now a full ten) leaving 4 — read it as 10 and 4 = 14. The physical move of two counters IS the make-a-ten step; do it slowly, then let her narrate it.' check: Show me 9 + 5 on two ten-frames — how many did you move to make the ten? down: math.addsub.decompose-make-10 G: move: 'Number-bond split under the equation: draw 6 splitting into 2 and 4, arrow the 2 up to the 8, write 10, then 10 + 4. The picture makes the decompose-then-recombine visible. This is Dimensions 1B ch13 ''Make the Next Ten'' — her actual current page.' check: '8 + 5: draw the bond that splits the 5 to make a ten with the 8.' S: move: 'Write the derived-fact chain: 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14, naming ''I took 2 from the 6 to make 10.'' Then show the same move works for 3-addend sums and for commuting to the larger addend first (6 + 8 → start at 8).' check: 'Finish it: 7 + 5 = 7 + 3 + ? = 10 + ? = ?' - id: math.addsub.equal-sign-unknown title: Meaning of =; true/false equations; find the unknown in any position strand: addsub band: - 6 - 8 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: judge equations true/false (8 = 8, 5+2 = 6 false), and find the unknown in an add/sub equation in any position (?+3=7, 5+?=8, 8-?=3)' L3: SM-2 spaced gate on mixed positions across separate days requires: - math.addsub.counting-on helps: [] resources: - type: beast ref: BA 2B ch5 Expressions - type: singapore ref: PMUS 2B u1 Finding the Missing Number sources: - CC 1.OA.7 - CC 1.OA.8 - research/math-spine.md §5 notes: 'Equal-sign misconception: ''='' means ''the answer comes next'', not ''same value as''. CCSSM attacks it early via 5=2+3 (total on LEFT, K2 p.10). Gates the start-unknown word problem. BA 2B ch5 pushes expressions/parentheses (5.OA) at grade 2 — early is fine.' instrument_gap: true representations: C: move: 'Pan balance (or a see-saw drawing with real weights): put 3 and 2 counters on one side, 5 on the other — it balances, so 3 + 2 = 5 AND 5 = 3 + 2. Physically show that ''='' means ''same weight on both sides'', directly attacking the ''= means the answer comes next'' error. Add/remove counters to make a false equation tip.' check: One pan has 4 + 3, the other has 8. Does it balance? Fix it. down: math.addsub.counting-on G: move: Draw the two sides of a balance/scale as boxes and put the expression in each. For a true/false judgment she shades or crosses. For a missing number she draws counters into the empty pan until both sides match — the unknown can sit on EITHER side (? + 3 = 7 and 8 = 5 + ?). check: Draw both pans for 6 = 2 + 4. Balanced or not? S: move: Present equations with the total on the LEFT and the unknown in every position (? + 3 = 7, 5 + ? = 8, 8 − ? = 3, 5 = 2 + ?), plus true/false items (8 = 8, 5 + 2 = 6). Read '=' aloud every time as 'is the same as', never 'makes' — the language carries the concept. check: 'True or false: 5 + 2 = 6. And solve: 8 − ? = 3.' - id: math.addsub.add-facts-memory-20 title: Know from memory all sums of two one-digit numbers (add/sub facts to 20) strand: addsub band: - 7 - 8 tier: L4 fluency_aim: metric: problems_per_min_oral target: 30 sustain: 2 of 3 separate days, ≤2 errors/min source: research/fluency-aims.md §8 (Penelope's live L4 frontier) gates: L2: ≥90% on 10 fresh mixed add/sub-within-20 fact items, untimed L3: SM-2 spaced gate on mixed fresh fact items across separate days (understanding gate — fires BEFORE the L4 rate gate) L4: oral (see/say) 1-minute timing, 30-40 problems/min, ≤2 errors; OR printed timed sheet at 2/3 × measured write-digits rate (≈20 DCPM grade-2 band). Aim met on 2 of 3 separate days. requires: - math.addsub.make-a-ten-strategy helps: [] resources: - type: dimensions ref: Dimensions Math 1A ch6 note: Addition to 20 (within-20 facts) record: true - type: dimensions ref: Dimensions Math 1A ch7 note: Subtraction Within 20 - type: gumball ref: generator:fact_mc - type: activity ref: printed 100-fact timed sheet (classic); parent enters DCPM - type: singapore ref: PM2022 G2 ch2 §2A Add Fluently Within 20 sources: - CC 2.OA.2 - research/fluency-aims.md §8 - research/math-spine.md §2 notes: 'PENELOPE''S LIVE L4 FRONTIER (spine §11.1). Oral aim 30-40 problems/min (pass 30, stretch toward Morningside 70-80 DCPM composite). Written aim = 2/3 × MEASURED write-digits rate (~20 DCPM grade-2, VanDerHeyden & Burns) — lower than it looks BY DESIGN (her hand is slow). Push the ORAL aim; let the written float up with handwriting. L4 gate fires only AFTER L3 understanding gate — never a shortcut (K2 p.19). 80-100 DCPM is the terminal/adult aim, NOT grade-2 — do not use it. Confidence: literature-solid (oral) / extrapolated (written number). She''s at ~L2 here: 88% add / 93% sub first-try untimed, no rate instrument yet.' representations: C: move: 'Only as a fallback anchor, not the drill: when a fact stalls, drop to the two-ten-frame make-a-ten move (7 + 8 → move 3, 10 + 5) so she DERIVES it, then re-flash it cold. The aim is to retire the counters — automaticity means the object is gone and the answer is just there.' check: 8 + 7 — if it's not instant, show the make-a-ten, then say it cold. down: math.addsub.make-a-ten-strategy S: move: See/say timed fact cards across the full within-20 set, add and subtract mixed, on the oral 1-minute sheet toward the 30–40/min aim. Sort into 'instant' vs 'still deriving' piles each session and drill only the second pile — spaced, not the whole deck every time. check: 'Rapid-fire: 9 + 6? 15 − 8? 7 + 8? 13 − 6? — no counting.' - id: math.addsub.wp-result-unknown title: 'Word problems: Add-To/Take-From/Put-Together, Result & Total Unknown' strand: addsub band: - 5 - 7 tier: L3 gates: L2: '≥90% on 10 fresh word problems, untimed: solve Add-To/Take-From Result-Unknown and Put-Together Total-Unknown (both-addends) within 20' L3: SM-2 spaced gate on mixed situation types across separate days requires: - math.addsub.represent-add-sub - math.addsub.decompose-make-10 helps: [] resources: - type: dimensions ref: Dimensions Math 1B ch11 note: Comparing — comparison stories / word problems record: true - type: singapore ref: PM2022 G2 ch4 §4B Part-Whole Model - type: beast ref: BA 2B ch6 Problem Solving sources: - CC K.OA.2 - CC 1.OA.1 - research/math-spine.md §5 notes: Easiest situation tier (CCSSM Table 1, K2 p.7). Bar-model / part-whole is the Singapore representation. 'Can add' ≠ 'can model a situation' — problem types are their own nodes. instrument_gap: true representations: N: move: 'Lead with the story and have her ACT IT OUT with objects before any bar or equation: ''Ben had 5 marbles, won 4 more'' (add-to result), ''there were 9, 3 rolled away'' (take-from), ''red and blue marbles, 6 total'' (put-together). Rotate the three situation types so she reads the ACTION, not a keyword.' check: There are 7 birds, 4 more land. How many birds now? G: move: 'Draw the Singapore bar model: for put-together, two labeled part-bars sitting end to end above a whole-bar; the ''?'' sits on the whole. For take-from, one long bar with a piece crossed off. The bar makes ''which number is the whole'' explicit before she picks an operation.' check: 'Draw a bar model for: 6 red and 3 blue marbles — how many marbles?' down: math.addsub.represent-add-sub S: move: Translate the bar into an equation with a box for the unknown (5 + 4 = ? ; 9 − 3 = ?), solve, then write the answer as a full sentence with the label ('There are 9 marbles'). Connect each number in the equation to a bar segment she drew. check: Write and solve the equation for 8 apples, eat 2 — how many left? - id: math.addsub.wp-change-addend-compare title: 'Word problems: Change-Unknown, Addend-Unknown, and basic Compare (difference)' strand: addsub band: - 6 - 8 tier: L3 gates: L2: '≥90% on 10 fresh word problems, untimed: solve Change-Unknown, Put-Together Addend-Unknown (sub as unknown addend), and Compare Difference-Unknown (''how many more/fewer'')' L3: SM-2 spaced gate on mixed situation types across separate days requires: - math.addsub.counting-on - math.addsub.wp-result-unknown helps: [] resources: - type: dimensions ref: Dimensions Math 1B ch11 note: Comparing (subtraction-as-comparison, difference) record: true - type: singapore ref: PM2022 G2 ch4 §4C Comparison Model - type: beast ref: BA 2B ch6 Problem Solving sources: - CC 1.OA.1 - research/math-spine.md §5 notes: 'Compare is the hardest situation type: the ''difference'' isn''t physically present (K2 p.12). Change-unknown needs counting-on. Bar-model comparison (Singapore ch4 §4C) makes the difference visible.' instrument_gap: true representations: N: move: 'Now the unknown moves off the result: change-unknown (''had 5, got some, now 12 — how many did she get?''), addend-unknown, and COMPARE (''Ana has 8, Ben has 5, how many more?''). Act out the compare with two lines of objects side by side so the DIFFERENCE — which isn''t physically present — becomes the un-matched objects.' check: Ana has 8 stickers, Ben has 5. How many more does Ana have? G: move: 'The comparison bar model is the key move: draw two bars of different length, one above the other, LEFT-aligned; the extra sticking-out piece is the difference, labeled ''?''. For change-unknown draw the start bar and the gap up to the total. This makes the invisible difference visible (Singapore ch4 §4C).' check: Draw two bars for 8 vs 5 — mark the piece that shows 'how many more'. down: math.addsub.wp-result-unknown S: move: 'Write the equation matching the bar, showing subtraction can find a MISSING ADDEND or a DIFFERENCE: 5 + ? = 12, or 8 − 5 = ?. Name why compare uses subtraction even though nothing was ''taken away'' — you''re finding the leftover gap between the bars.' check: 'Write an equation for: had 6, now has 10 — how many were added?' - id: math.addsub.wp-start-unknown-two-step title: 'Word problems: Start-Unknown, misleading-language Compare, and two-step' strand: addsub band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh word problems, untimed: solve Start-Unknown (algebraic), Bigger/Smaller-Unknown Compare with misleading language (''3 more... how many fewer''), and two-step problems within 100' L3: SM-2 spaced gate on mixed types across separate days requires: - math.addsub.equal-sign-unknown - math.addsub.wp-change-addend-compare - math.addsub.add-sub-fluency-100 helps: [] resources: - type: singapore ref: PM2022 G2 ch4 §4D Word Problems - type: singapore ref: PM2022 G3 ch2 §2D Word Problems - type: beast ref: BA 2D ch12 Problem Solving sources: - CC 2.OA.1 - research/math-spine.md §5 notes: 'DELIBERATE DEFERRAL to Grade 2: start-unknown and compare-with-misleading-language (''more'' suggests the wrong operation) held to Gr2 even though the arithmetic is within-20 (K2 p.14). Don''t optimize the deferral away — it prevents a specific misconception. Band kept 7-9 to honor it.' instrument_gap: true representations: N: move: The algebra-flavored tier, deliberately held to Gr2. Start-unknown ('had some, gave away 4, now has 6'), misleading-language compare ('Sam has 3 MORE than Kim, Sam has 9 — how many does KIM have?' where 'more' points the wrong way), and two-step. Have her restate the problem in her own words and flag the trap word BEFORE choosing an operation. check: Sam has 3 more than Kim. Sam has 9. How many does Kim have? (careful — 'more' is a trap here) G: move: 'Bar model with the unknown at the START: draw the whole and the known change, leave the start-bar blank, work backward. For two-step, draw the first bar, get an intermediate answer, then a second bar using it. For misleading compare, the bar shows WHO is longer regardless of the word ''more''.' check: 'Draw a bar model for: had some, spent 4, has 6 left — how many at the start?' down: math.addsub.wp-change-addend-compare S: move: Write start-unknown as an equation with the unknown FIRST (? − 4 = 6), solve as a missing-start using the inverse. For two-step, write two linked equations and label the intermediate quantity so the second step doesn't lose it. Check the answer back in the story. check: 'Solve for the start: ? − 4 = 6. Then: 12 − 5 = ?, then that answer + 3 = ?' - id: math.addsub.add-sub-within-100 title: Add/subtract within 100 (2-digit ± 1-digit, ± multiples of 10) strand: addsub band: - 6 - 8 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: add within 100 (2-digit + 1-digit and + multiple of 10) and subtract multiples of 10, using place-value reasoning; explain the tens/ones' L3: SM-2 spaced gate on mixed items across separate days requires: - math.placevalue.two-digit-place-value - math.addsub.make-a-ten-strategy helps: [] resources: - type: dimensions ref: Dimensions Math 1B ch13 note: Addition and Subtraction Within 40 (make-the-next-ten regrouping) record: true - type: dimensions ref: Dimensions Math 1B ch17 note: Addition and Subtraction Within 100 - type: singapore ref: PM2022 G2 ch2 §2B Add Tens or Hundreds - type: singapore ref: PM2022 G2 ch3 §3B Subtract Tens or Hundreds - type: beast ref: BA 2A ch3 Addition sources: - CC 1.NBT.4 - CC 1.NBT.6 - research/math-spine.md §5 notes: Bridge from within-20 facts to full within-100 fluency. Subtracting multiples of 10 kept separate from full within-100 subtraction (the non-multiple-of-10 case is deferred to Gr2 to prevent the subtract-smaller-from-larger generalization, K3 p.7). instrument_gap: true representations: C: move: 'Base-ten blocks: build 34 as 3 ten-rods + 4 units. For 34 + 20 add two ten-rods (the ones don''t change); for 34 + 5 add five units. Adding a multiple of 10 physically touches only the tens, adding ones only the ones — the child sees WHY the digits behave differently. Do a make-the-next-ten crossing (38 + 5 → trade 10 units for a rod).' check: Build 46, then add 30 with blocks — which pile grew? down: math.placevalue.two-digit-place-value G: move: 'Draw quick-tens (a line = 10, a dot = 1) instead of full blocks: sketch 34, draw 2 more tens-lines for +20. Or use an open number line: mark 34, jump +20 in one arc to 54, then +5 to 59. The jumps mirror the place-value reasoning without full re-drawing.' check: On an open number line, jump 52 + 30. Where do you land? S: move: 'Record the place-value reasoning symbolically: 34 + 20 → ''3 tens + 2 tens = 5 tens, 4 ones stay → 54''; 34 + 5 → ''add ones: 9 ones → 39.'' Keep tens and ones lined up in columns so ''add tens to tens, ones to ones'' is visible on the page.' check: 63 + 30 = ? and 63 + 4 = ? — say which digit changes each time. - id: math.addsub.add-sub-fluency-100 title: Fluently add & subtract within 100 (place-value strategies, with renaming) strand: addsub band: - 7 - 8 tier: L4 fluency_aim: metric: problems_per_min_oral target: 10 sustain: 2 of 3 separate days, ≤2 errors/min source: research/fluency-aims.md §2a (multidigit is slower; compound-computation rate) gates: L2: ≥90% on 10 fresh within-100 add/sub items incl. renaming/regrouping, untimed L3: SM-2 spaced gate on mixed items incl. the non-multiple-of-10 subtraction case, across separate days L4: 1-minute timing (oral or written); pass ~10 problems/min (compound computation is slower than facts); ≤2 errors; 2 of 3 separate days requires: - math.addsub.add-sub-within-100 helps: [] resources: - type: dimensions ref: Dimensions Math 1B ch17 note: Addition and Subtraction Within 100 record: true - type: dimensions ref: Dimensions Math 2A ch2 note: Addition and Subtraction — Part 1 - type: dimensions ref: Dimensions Math 2A ch3 note: Addition and Subtraction — Part 2 (regrouping) - type: singapore ref: PM2022 G2 ch2 §2C-2D Add Without/With Renaming - type: singapore ref: PM2022 G2 ch3 §3C-3D Subtract Without/With Renaming - type: beast ref: BA 2D ch11 Algorithms (+&−) sources: - CC 2.NBT.5 - research/fluency-aims.md §2a - research/math-spine.md §5 notes: 'PREREQ RELAXED (adj minor sweep): dropped the over-strong `three-digit-place-value` requirement. Fluency within 100 (2.NBT.5) operates only on 2-digit numbers — it needs 2-digit place value + the hundred-as-a-unit idea, NOT full 3-digit place value. Two-digit place value is satisfied transitively via `add-sub-within-100` (which requires two-digit-place-value); requiring 3-digit was gating a within-100 skill behind a to-1000 skill and would keep Penelope''s within-40/100 edge from surfacing. ''Fluently'' = strategy-based, NOT the standard algorithm yet (K3 p.11). THE canonical error: 82-45=43 from subtract-smaller-from-larger (K3 p.7). Full within-100 subtraction (both no-decompose and decompose cases together) lives HERE by design. Rate aim ~10/min: multidigit compound is 40-60 DCPM terminal (§2a) but a 2-digit problem is ~2-4 digits, so problems/min runs low — pass floor conservative, confidence extrapolated.' instrument_gap: true representations: C: move: 'Blocks for the regroup itself: 47 + 8 → 15 ones, TRADE ten ones for one ten-rod, leaving 5 ones and a new ten (55). Subtraction with renaming 52 − 7 → you can''t take 7 from 2 ones, so UNBUNDLE a ten-rod into 10 units first. The physical trade pre-empts the 82−45=43 subtract-smaller-from-larger error.' check: Subtract 34 − 8 with blocks — do you need to unbundle a ten? Show it. down: math.addsub.add-sub-within-100 G: move: 'Quick-tens drawing with an explicit trade: circle ten units and redraw them as one ten-line (compose) or slash a ten-line into ten units before removing (decompose). An open number line also works: 45 + 38 → +30 to 75, +5 to 80, +3 to 83, watching the ten-crossings.' check: Draw the trade for 28 + 15 — circle the ten you compose. S: move: Write the strategy vertically with the regroup shown honestly — the little '1' carried into the tens, or the crossed-out/renamed ten in subtraction — and cross-check with a place-value strategy (45 + 38 = 40+30 + 5+8). The point is fluent strategy use, not yet the silent compact algorithm. check: '82 − 45 = ? Show the rename. (Watch: it is NOT 43.)' - id: math.addsub.add-sub-within-1000 title: Add/subtract within 1000; explain strategies (place value) strand: addsub band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: add/subtract within 1000 with models & place-value strategies; add up to four 2-digit numbers; explain WHY the strategy works' L3: SM-2 spaced gate on mixed items across separate days requires: - math.addsub.add-sub-fluency-100 helps: [] resources: - type: dimensions ref: Dimensions Math 2A ch1 note: Numbers to 1,000 record: true - type: dimensions ref: Dimensions Math 2A ch2 note: Addition and Subtraction — Part 1 - type: dimensions ref: Dimensions Math 2A ch3 note: Addition and Subtraction — Part 2 - type: singapore ref: PM2022 G3 ch2 §2A-2B Add/Sub Within 1,000 / 10,000 - type: beast ref: BA 2D ch11 Algorithms (+&−) sources: - CC 2.NBT.6 - CC 2.NBT.7 - CC 2.NBT.9 - research/math-spine.md §5 notes: Recording newly-composed units in separate rows is a legitimate intermediate before the compact algorithm (K3 p.9). Explain-your-strategy (2.NBT.9) is the reasoning item on this node. instrument_gap: true representations: C: move: 'Add hundred-flats to the block set: build 234 as 2 flats + 3 rods + 4 units. Add/subtract 3-digit numbers by combining or trading at TWO boundaries (ten ones → a rod, ten rods → a flat). Trading up one place at a time on real blocks is the concrete model behind ''newly-composed units in separate rows.''' check: Build 156 + 47 with blocks — which trades do you make, and in what order? down: math.addsub.add-sub-fluency-100 G: move: 'Quick-place drawing (square=100, line=10, dot=1) or an open number line with hundred/ten/one jumps: 356 + 130 → +100, +30. Keep the recording expanded — 356 + 130 = (300+100)+(50+30)+6 — so she EXPLAINS why the strategy works (the 2.NBT.9 reasoning item), not just gets a number.' check: Explain with a drawing why 356 + 130 = 486 — where did each part go? S: move: 'Write partial-sums / expanded form as a legitimate intermediate before the compact algorithm: 356 + 275 = 500 + 120 + 11 = 631, recording each newly-composed unit on its own line. Then contrast with a subtraction that needs unbundling across the tens AND hundreds.' check: Use expanded form to add 428 + 145. Show the hundreds, tens, ones lines. - id: math.addsub.add-sub-fluency-1000 title: Fluently add & subtract within 1000 (strategies AND algorithms) strand: addsub band: - 8 - 9 tier: L4 fluency_aim: metric: problems_per_min_oral target: 9 sustain: 2 of 3 separate days, ≤2 errors/min source: research/fluency-aims.md §2a (compound within-1000; bridges to the algorithm) gates: L2: ≥90% on 10 fresh within-1000 add/sub items using both strategies and the algorithm, untimed L3: SM-2 spaced gate on mixed within-1000 items across separate days L4: 1-minute timing; pass ~9 problems/min (3-digit compound); ≤2 errors; 2 of 3 separate days requires: - math.addsub.add-sub-within-1000 helps: [] resources: - type: singapore ref: PM2022 G3 ch2 §2C Other Add/Sub Strategies - type: beast ref: BA 2D ch11 Algorithms (+&−) sources: - CC 3.NBT.2 - research/fluency-aims.md §2a - research/math-spine.md §2 notes: 'CC 3.NBT.2 ''fluently'' bridges strategies (Gr1-3) to the standard algorithm. BA moves this fluency DOWN to Level 2 (§5.C ''Will be included in Beast Academy Grade 2''). Rate aim conservative/extrapolated: 3-digit compound is a few DCPM below within-100. Distinct fluency rung from the compact algorithm.' instrument_gap: true representations: G: move: 'Bridge picture → algorithm: line up a quick-place drawing beside the vertical algorithm so each carried/borrowed unit in the written form maps to a traded block/line in the drawing. Once the mapping is solid she can drop the drawing and trust the columns.' check: For 463 + 279, point to where the carried 1 comes from in the drawing. down: math.addsub.add-sub-within-1000 S: move: Practice BOTH a place-value strategy and the standard algorithm on the same problem and confirm they agree — fluency here means choosing efficiently (e.g. 500 − 198 by adding-up 2 vs the algorithm). Lightly time toward the ~9/min aim; 3-digit compound is slower than facts by design. check: Solve 502 − 267 two ways — a strategy and the algorithm. Same answer? - id: math.addsub.add-sub-standard-algorithm title: Fluently add/subtract multi-digit via the STANDARD (compact) algorithm strand: addsub band: - 8 - 10 tier: L4 fluency_aim: metric: digits_per_min_written target: = 2/3 × her MEASURED digit-writing rate (write-digits node); pending Writing Speed Test sustain: 2 of 3 separate days, ≤2 errors/min source: 'research/fluency-aims.md §3 (motor-ceiling: written aim = 2/3 × measured digit-writing rate) + §2a (multidigit compound)' gates: L2: ≥90% on 10 fresh multi-digit add/sub items via the standard (compact) algorithm, untimed L3: SM-2 spaced gate on mixed multi-digit items across separate days L4: 1-minute WRITTEN timing, digits correct/min; pass = 2/3 × her measured write-digits rate (§3 motor ceiling); ≤2 errors; 2 of 3 separate days requires: - math.addsub.add-sub-fluency-1000 - math.counting.write-digits helps: [] resources: - type: dimensions ref: Dimensions Math 2A ch3 note: Addition and Subtraction — Part 2 (standard algorithm) record: true - type: singapore ref: PM2022 G3 ch2 Addition and Subtraction Within 10,000 - type: beast ref: BA 2D ch11 Algorithms (+&−) sources: - CC 4.NBT.4 - research/fluency-aims.md §3 - research/math-spine.md §2 - research/math-spine.md §5 notes: 'METRIC FIXED to digits_per_min_written (adj #5): the compact algorithm is executed PURELY ON PAPER — its fluency is written, not oral. Written aim = 2/3 × her MEASURED digit-writing rate (fluency-aims §3 motor-ceiling rule; PENDING a Rocket Math Writing Speed Test — the number is not frozen, it floats with her handwriting in the 7-9 steep-growth window). requires write-digits (adj #5): you structurally cannot write fast answers faster than you write digits — the tool-skill ceiling is now enforced by the DAG. The standard algorithm comes LAST on purpose (K3 p.9-11): strategies (Gr1-3) before the compact algorithm (4.NBT.4). Terminal add/sub fluency node. BA-L2 already reaches 4.NBT.4 (§5.A) — band 8-10 advisory not gating.' instrument_gap: true representations: C: move: 'Blocks make their LAST appearance here, only to justify each written step: as she writes ''13, carry the 1'', she physically trades ten units for a rod — proving the little carried 1 is a real ten, not a decoration. Retire the blocks once she trusts the notation; the algorithm is meant to run without them.' check: As you carry the 1 in the tens column, what did that 1 stand for in blocks? G: move: 'Expanded-form bridge to the compact form: write the addition in expanded columns first, then collapse it into the single-line carry notation, showing they''re the same computation compressed. A place-value chart keeps the digits aligned so a stray digit doesn''t slip into the wrong column.' check: Show 367 + 158 in expanded form, then rewrite it as the compact carry version. down: math.addsub.add-sub-fluency-1000 S: move: 'The algorithm IS the symbolic representation: teach the compact right-to-left procedure — align by place, add each column, carry/borrow the regrouped unit — as fluent PAPER execution. Rate is written (digits/min), capped at 2/3 of her measured digit-writing speed; drill accuracy first, speed only after the procedure is clean.' check: Set up and solve 604 − 358 with the standard algorithm, showing the borrow. - id: math.multdiv.equal-groups-arrays title: Equal groups & rectangular arrays as repeated addition; odd/even strand: multdiv band: - 7 - 8 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: express a rectangular array as repeated equal addends and as rows×columns; determine odd/even by pairing' L3: SM-2 spaced gate across separate days requires: - math.addsub.add-facts-memory-20 - math.geometry.array-rows-columns helps: [] resources: - type: dimensions ref: Dimensions Math 1B ch14 note: Grouping and Sharing (equal groups) record: true - type: singapore ref: PM2022 G2 ch6 §6A Add Equal Groups - type: singapore ref: PM2022 G2 ch6 §6B Even and Odd Numbers - type: beast ref: BA 3A ch2 Skip-Counting sources: - CC 2.OA.3 - CC 2.OA.4 - research/math-spine.md §6 notes: The bridge from add-facts to multiplication (K2 p.25). Requires array-rows-columns (spatial structuring) AND add-facts-memory-20. BA defers 2.OA.4 to BA-L3 (§5.B). instrument_gap: true representations: C: move: 'build equal groups with physical objects — 4 cups with 3 counters each — and count the total two ways: skip-count (3,6,9,12) and repeated addition (3+3+3+3); then arrange the SAME counters into a 4-by-3 rectangular array (rows of chairs) to show rows x columns is the same total.' check: make 3 groups of 5 with counters, then rebuild them as a 3-row array — is the total the same? down: math.geometry.array-rows-columns G: move: draw a dot array in rows and columns and label it as repeated addition AND as rows x columns; use pairing (two-by-two) on a number to show odd (one left over) vs even (none left over). check: 'draw a 2-by-6 array — write it as an addition (6+6) and count by pairs: is 12 even or odd?' S: move: 'write the array as repeated addition and connect it to the multiplication symbol: 3+3+3+3 = 4 x 3 = 12; name that 4 x 3 means ''4 groups of 3''.' check: write 5+5+5 as a multiplication sentence — how many groups, how big is each? N: move: frame arrays as real rectangular arrangements — rows of chairs, a muffin tin, tiles on a floor, an egg carton — and count total by rows-of-equal-size. check: a muffin tin has 3 rows of 4 cups — how many muffins fit? write it two ways. - id: math.multdiv.mult-div-meaning title: Interpret products & quotients; equal groups, arrays, sharing/grouping strand: multdiv band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: interpret 5×7 as 5 groups of 7 (unknown product); interpret 56÷8 as sharing OR grouping (group-size / number-of-groups unknown); write matching equations' L3: SM-2 spaced gate on mixed interpretations across separate days requires: - math.multdiv.equal-groups-arrays - math.counting.skip-count-2-5-10 helps: [] resources: - type: dimensions ref: Dimensions Math 1B ch14 note: Grouping and Sharing (early ×/÷ meaning) record: true - type: dimensions ref: Dimensions Math 2A ch6 note: Multiplication and Division - type: singapore ref: PM2022 G3 ch3 §3A Multiplication / §3H Division - type: beast ref: BA 3B ch4 Multiplication - type: beast ref: BA 3C ch8 Division sources: - CC 3.OA.1 - CC 3.OA.2 - research/math-spine.md §6 notes: 'Mult and div co-taught: division can be EASIER via count-by (24÷3: count by 3 until you hear 24, K2 p.25) — SOFT interleave, both meanings in one node. Three situation types escalate: equal groups → arrays → (Gr4) multiplicative compare.' instrument_gap: true representations: C: move: 'act out BOTH division meanings with counters: SHARING (deal 24 counters one at a time into 3 plates -> how many each) and GROUPING (make plates of 3 from 24 -> how many plates); show multiplication is the inverse — 3 plates of 8 rebuilds the 24.' check: share 20 counters fairly among 4 plates — how many each? Now make groups of 5 from 20 — how many groups? down: math.multdiv.equal-groups-arrays G: move: draw an equal-groups / bar picture for a product (5 groups of 7) and for a quotient (56 split into 8 equal parts, OR into parts of 8); write the matching x and division equations from the same picture. check: draw 56 divided into 8 equal groups — what's in each group? write it as a division and a multiplication. S: move: 'write both meanings as equations and connect them: 5 x 7 = 35, and 35 divided by 5 = 7 / 35 divided by 7 = 5; interpret the unknown''s position (product unknown vs group-size vs number-of-groups).' check: if 6 x 4 = 24, write the two division facts that go with it. N: move: use real sharing/grouping stories — 'we have 24 cookies and 3 kids, fair shares?'; 'how many 4-packs of juice from 20 boxes?' — and connect division-by-count-by (count by 3s until you hear 24). check: 18 crackers shared among 3 friends — how many each? Count by 3s to check. - id: math.multdiv.mult-div-word-problems title: Multiplication/division word problems; unknown in a mult/div equation strand: multdiv band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh word problems, untimed: solve equal-groups/array/measurement ×÷ word problems; find the unknown in a mult/div equation (8×?=48); understand ÷ as unknown-factor' L3: SM-2 spaced gate on mixed types across separate days requires: - math.multdiv.mult-div-meaning helps: [] resources: - type: dimensions ref: Dimensions Math 2A ch6 note: Multiplication and Division (word problems) record: true - type: singapore ref: PM2022 G3 ch3 §3G Word Problems - type: beast ref: BA 3C ch7 Variables sources: - CC 3.OA.3 - CC 3.OA.4 - CC 3.OA.6 - research/math-spine.md §6 notes: Division as unknown-factor is the keystone linking × and ÷ (K2 p.25). BA 3C ch7 (Variables) pushes 6.EE.7 (solve x+p=q, px=q) — a 6th-grade algebra standard taught at grade 3 (§5.C); early is fine for a strong reasoner. instrument_gap: true representations: N: move: read equal-groups / array / measurement x-and-division stories aloud, act out or model the situation before touching numbers, and ask which quantity is unknown (total, group size, or number of groups) — division reads as an unknown-factor question ('8 times WHAT is 48?'). check: 'a baker puts 6 rolls in each bag and fills 7 bags — how many rolls? Now: 48 rolls, 6 per bag — how many bags?' down: math.multdiv.mult-div-meaning G: move: draw a bar / equal-groups model for the word problem, boxing the known parts and marking the unknown with a ? so the picture selects the operation; use it especially for the unknown-factor (8 x ? = 48). check: draw a bar model for '8 times a number is 48' — where does the ? go, and what is it? S: move: translate the story into an equation with the unknown in the right place (8 x ? = 48, or 48 divided by 8 = ?) and solve using the mult<->div link (division = unknown factor). check: write an equation for '35 stickers shared equally on 5 pages' and solve it. - id: math.multdiv.mult-facts-easy title: Multiplication facts, easy-first (2s, 5s, 10s, 3s, 4s) via count-by & distributive strand: multdiv band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: recall/derive 2s, 5s, 10s facts (count-by), 3s and 4s (via distributive & commutative); use 3×6=6×3' L3: SM-2 spaced gate on mixed easy facts across separate days requires: - math.multdiv.mult-div-meaning helps: - math.counting.count-oral-skip resources: - type: dimensions ref: Dimensions Math 2A ch7 note: Multiplication and Division of 2, 5, and 10 record: true - type: dimensions ref: Dimensions Math 2B ch9 note: Multiplication and Division of 3 and 4 - type: singapore ref: PM2022 G3 ch3 §3B-3F Multiply by 2/5/10/3/4 - type: beast ref: BA 3B ch4 Multiplication - type: beast ref: BA 3B ch6 The Distributive Property - type: gumball ref: generator:fact_mc sources: - CC 3.OA.5 - CC 3.OA.7 - research/math-spine.md §6 notes: 'Facts built easy-first then mixed (K2 p.27): 2s/5s/10s (count-by) → distributive → 3s/4s. Commutative property (array rotation) + distributive (7×5=(6+1)×5) are named strategies. This is the ACCURACY rung feeding the L4 memory node; kept L3 (not its own L4 rung) per spine §11.1 recommendation. helps count-oral-skip (skip-count → facts).' representations: C: move: 'derive facts from physical structure: skip-count equal stacks for 2s/5s/10s; for a new fact like 3x6, build a 2x6 array and add one more row of 6 (distributive, made physical); rotate the array 90 degrees to SEE 3x6 = 6x3 (commutative).' check: 'build 4x5 from a known 2x5 array — how many more rows did you add? Now turn it: what is 5x4?' down: math.multdiv.mult-div-meaning G: move: use a dot-array / area sketch and split it to show distributive (7x5 = 6x5 + 1x5); mark the count-by ladder on a number line for 2s/5s/10s so the easy facts anchor the derivations. check: draw 6x5 and split off one row to get from 5x5 you already know — what does 6x5 equal? S: move: 'write the fact families and the strategy: 3x6 = (2x6) + (1x6), and 3x6 = 6x3; drill the 2s/5s/10s toward recall, keep 3s/4s strategy-derived. This is the ACCURACY rung feeding the L4 memory node.' check: use 5x8=40 to find 6x8 — write the step, then the answer. - id: math.multdiv.mult-facts-hard title: Multiplication facts 6s/7s/8s/9s via derived strategies (5+n, 9=10-1) strand: multdiv band: - 8 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: derive/recall 6/7/8/9 facts using 5+n pattern, 9=10−1, doubling, and known-fact anchors' L3: SM-2 spaced gate on mixed hard facts across separate days requires: - math.multdiv.mult-facts-easy helps: [] resources: - type: singapore ref: PM2022 G3 ch4 Multiplication and Division of 6,7,8,9 - type: beast ref: BA 3B ch5 Perfect Squares - type: gumball ref: generator:fact_mc sources: - CC 3.OA.7 - research/math-spine.md §6 notes: The 9s pattern (9=10−1) and 5+n pattern are named strategies (K2 p.26). Accuracy rung; the terminal automaticity (know-from-memory + rate) is mult-div-facts-memory-100. representations: G: move: 'picture the hard facts as area splits off known anchors: 7x8 as (5x8)+(2x8) shaded in two colors; 9x6 as (10x6)-(1x6) drawn as a full ten-rows block with one row removed (the 9=10-1 pattern made visual).' check: draw 9x7 as ten 7s with one 7 taken away — what does 9x7 equal? down: math.multdiv.mult-facts-easy S: move: 'write the derived-strategy chains for 6s/7s/8s/9s: 5+n pattern (7x6 = 5x6 + 2x6), 9=10-1 (9x4 = 10x4 - 4), doubling (8x7 = double 4x7), and known-square anchors (7x7 -> 7x8).' check: find 8x6 by doubling 4x6 — show the two steps. N: move: the 9s finger/pattern trick and real anchors — 'a week has 7 days, so 7x4 is 4 weeks'; make the derivation a quick mental habit tied to something memorable. check: how many days in 6 weeks? which fact is that? - id: math.multdiv.mult-div-facts-memory-100 title: Know from memory all products of two one-digit numbers; fluent ÷ within 100 strand: multdiv band: - 8 - 9 tier: L4 fluency_aim: metric: problems_per_min_oral target: 40 sustain: 2 of 3 separate days, ≤2 errors/min source: research/fluency-aims.md §8 (mult-facts oral 40-50 early → 60-90 fluent) gates: L2: ≥90% on 10 fresh mixed ×÷-within-100 fact items, untimed L3: SM-2 spaced gate on mixed fresh fact items across separate days (understanding gate — fires BEFORE the L4 rate gate) L4: oral (see/say) 1-minute timing; 40-50 problems/min early, stretch 60-90 fluent; ≤2 errors; 2 of 3 separate days. Written variant at 2/3 × measured write-digits rate. requires: - math.multdiv.mult-facts-hard - math.multdiv.mult-div-word-problems helps: [] resources: - type: gumball ref: generator:fact_mc - type: activity ref: printed times-table timed sheet (parent enters rate) - type: singapore ref: PM2022 G3 ch4 Multiplication and Division of 6,7,8,9 sources: - CC 3.OA.7 - research/fluency-aims.md §8 - research/math-spine.md §2 notes: 'PENELOPE''S NEXT-YEAR TRUNK (spine §11.1) — she has ~86% first-try small-factor multiplication accuracy in-app but ''fluent'' (fast/automatic) is NOT established and there''s no formal times-table assessment. Seed at L1/L2, NOT L3 (evidence-inventory contradiction, adjudicated: accuracy ≠ fluency). Math Academy floor = ×-tables-to-12 mastery — this node IS the graduation gate toward MA. L4 fires only after L3 understanding. Prefer oral aim. Confidence: literature-solid (oral) / extrapolated (written).' representations: S: move: mixed see/say recall to automaticity — flashcards / timed sheets across the whole 0-10 table and the matching division facts; the L4 gate is a 1-minute oral (see/say) timing at 40-50/min early, stretching to 60-90, but ONLY after the L3 understanding gate. check: 'quick, no counting: 7x8? 42 divided by 6? 9x7?' down: math.multdiv.mult-facts-hard G: move: keep a completed multiplication table visible and hunt PATTERNS (the diagonal of squares, symmetry across it = commutativity, the 5s/10s columns) so recall rests on structure, not rote-only. check: on the table, find two cells that both equal 24 — why are there two? - id: math.multdiv.mult-div-multidigit title: Multi-digit × 1-digit / 2×2-digit; multi-digit ÷ 1-digit with remainders strand: multdiv band: - 8 - 10 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: multiply up-to-4-digit × 1-digit and 2-digit × 2-digit (area model & place value); divide up-to-4-digit ÷ 1-digit with quotient & remainder; multiply by multiples of 10' L3: SM-2 spaced gate on mixed items across separate days requires: - math.multdiv.mult-div-facts-memory-100 - math.placevalue.three-digit-place-value helps: [] resources: - type: singapore ref: PM2022 G3 ch4 §4E-4F Multiply by Tens / 2-Digit × 1-Digit - type: beast ref: BA 3C ch8 Division sources: - CC 3.NBT.3 - CC 4.NBT.5 - CC 4.NBT.6 - research/math-spine.md §6 notes: MA splits this into ~13 topics by strategy × operand size (repeated-addition/place-value/area-model); we keep ONE node, strategy differences ride as assessment items + notes. Area model requires spatial structuring (helps from array-rows-columns, already upstream via facts). instrument_gap: true representations: C: move: 'build a multi-digit product with base-ten blocks / place-value disks: 4 x 23 as 4 groups of (2 tens 3 ones), gather 8 tens and 12 ones, then TRADE ten ones for a ten (regrouping made physical); for division, deal a 3-digit pile of blocks into equal groups and read the leftover as the remainder.' check: build 3 x 24 with ten-rods and ones — how many ones do you trade in for a ten? Now share 25 cubes into 4 groups — what is the remainder? down: math.multdiv.mult-div-facts-memory-100 G: move: 'draw the AREA MODEL: a rectangle split by place value — 23 x 14 as a 2x2 grid of partial products (20x10, 20x4, 3x10, 3x4) — then add the partials; for x by multiples of 10, show the rectangle scaling by a factor of ten.' check: draw the area model for 13 x 12 — label the four partial products and add them. What is 13 x 12? S: move: record the partial products / the division steps in the standard written layout, linking each written line back to its piece of the area model or the shared blocks; keep quotient-and-remainder explicit. check: solve 6 x 34 by partial products (6x30, 6x4) then add; solve 47 divided by 5 — give the quotient and remainder. - id: math.multdiv.multistep-4ops-patterns title: Multistep 4-operation problems (interpret remainders); factors, multiples, patterns strand: multdiv band: - 8 - 10 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: solve multistep problems with all four operations and interpret remainders in context; find factor pairs to 100, classify prime/composite, identify multiples; explain add/mult-table patterns; multiplicative comparison (35 = 5×7 = ''5 times as many as 7'')' L3: SM-2 spaced gate on mixed items across separate days requires: - math.multdiv.mult-div-multidigit - math.addsub.wp-start-unknown-two-step helps: [] resources: - type: beast ref: BA 3D ch11 Estimation - type: beast ref: BA 3B ch6 The Distributive Property sources: - CC 3.OA.8 - CC 3.OA.9 - CC 4.OA.1 - CC 4.OA.2 - CC 4.OA.3 - CC 4.OA.4 - research/math-spine.md §6 notes: Multiplicative compare is GRADE 4 ONLY — do NOT introduce in Gr3 (distinct from additive compare, more complex, K2 p.22). Keep word-problem work about interpretation, not formal order-of-operations notation (that waits for Gr6, K2 p.27). instrument_gap: true representations: N: move: read a multi-step real-world problem (buy, share, then have some left over), plan the ORDER of operations from the situation, and decide what the remainder MEANS in context (round up for buses, drop for full boxes, keep as leftover). check: 26 kids need vans that hold 4 each — how many vans? Why can't the answer have a remainder left un-rounded? down: math.multdiv.mult-div-multidigit C: move: 'explore factors/multiples and prime/composite physically: try to build every rectangle for a number of tiles — 12 tiles make 1x12, 2x6, 3x4 (composite, many rectangles); 7 tiles make ONLY 1x7 (prime, one rectangle).' check: make all the rectangles you can with 16 tiles — is 16 prime or composite? Try 13 tiles — what does that tell you? G: move: use an array / multiplication-table view to see factor pairs and multiples, and to justify multiplicative comparison (35 = 5x7 = '5 times as many as 7') as a scaled bar — distinct from additive 'more than'. check: 'on the table, list all factor pairs of 24. Then: ''5 times as many as 7'' — is that 12 or 35?' S: move: write the multi-step solution as a sequence of equations (interpreting each result), and write factor pairs / multiplicative-compare equations; keep it about interpretation, NOT formal order-of-operations notation yet. check: Maya has 3 boxes of 8 markers and gives away 5 — write the steps and the final number left. - id: math.fractions.equal-shares title: Partition shapes into equal shares (halves, thirds, fourths); fair shares strand: fractions band: - 6 - 8 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: partition a shape into 2/3/4 equal shares; name halves/thirds/fourths; recognize that equal shares need not be the same SHAPE (equal area)' L3: SM-2 spaced gate on mixed items across separate days requires: - math.geometry.compose-decompose-shapes helps: - math.measure.money-coins-bills - math.measure.tell-time-5min - thinking.reasoning.self-explanation resources: - type: dimensions ref: Dimensions Math 1B ch15 note: Fractions (halves, fourths) — first exposure record: true - type: packet ref: fractions-2026-07-12 note: 'parts 1-3: fair shares → the half → quarters & thirds' - type: singapore ref: PMUS 2B u4 Fractions (Halves and Quarters) - type: gumball ref: generator:fraction_identify sources: - CC 1.G.3 - CC 2.G.3 - research/math-spine.md §7 - research/math-spine.md §8 notes: '''Equal parts'' means equal MEASUREMENT, not equal shape (K4 p.7): 1/4 of a square can be 4 differently-shaped equal-area pieces. helps: clock-quarters ↔ coin-quarters ↔ fraction-quarters transfer (the ed-packet panel''s empirical finding, DESIGN §2). Seed Penelope L0 — packet built but NOT completed (evidence-inventory).' representations: C: move: Fold a paper strip (or paper circle) in half, then fold again for fourths; open it and trace the crease lines. Then hand her a rectangle of clay/dough and a plastic knife and have her cut a 'fair share' for 3 people — she keeps re-cutting until every piece would make each customer smile. Physically stack the pieces to prove they match. check: Fold this paper strip so two people get a fair share. Now show me the folds — are the two parts exactly the same amount? How do you know? down: math.geometry.compose-decompose-shapes G: move: 'Draw the packet''s pie and bar figures: a straight cut through the center of a pie / through the middle of a bar makes 2 equal parts; add a cut for fourths. Contrast an HONEST equal partition against a distractor where the cut is off-center — she circles which one is really cut into equal shares. Keep tilted pies OFF the ''is-this-equal'' items (a tilted slice looks unequal even when it isn''t).' check: Here are three cut pies. Circle every one that is cut into equal shares — and cross out the one that only looks fair. S: move: 'Attach the names to the pictures: two equal parts → ''halves'', three → ''thirds'', four → ''fourths/quarters''. Write the word under each partitioned figure so the picture and the name sit together. Not yet the a/b symbol — words first (that''s the next node).' check: This circle is cut into 4 equal parts. What is each part called — halves, thirds, or fourths? N: move: 'Run the bakery story from the packet: Shortchange Sam hands out unfair slices and it''s her job to catch him. ''Four kids are sharing one brownie — cut it so nobody complains.'' Use a real snack (a graham-cracker rectangle, an orange) and split it fairly at the table; the lived unfairness of a bad cut is the whole lesson.' check: You and your sister are splitting one cookie. Show me how to break it so it's a fair share — and tell me why the big-little way isn't fair. - id: math.fractions.unit-fraction-meaning title: Unit fraction 1/b; a/b as a copies of 1/b; specify the whole strand: fractions band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: interpret 1/b as one part when a whole is split into b equal parts; read 3/4 as three 1/4s; explain that a fraction is meaningless without a specified whole' L3: SM-2 spaced gate on mixed items across separate days requires: - math.fractions.equal-shares helps: - thinking.metacog.spot-the-trap resources: - type: dimensions ref: Dimensions Math 2B ch11 note: Fractions (unit fractions) record: true - type: packet ref: fractions-2026-07-12 note: 'parts 4-5: counting pieces (numerator) → fraction of a set' - type: singapore ref: PM2022 G3 ch7 §7A Unit Fractions / §7B More Fractions - type: beast ref: BA 3D ch10 Fractions - type: gumball ref: generator:fraction_identify sources: - CC 3.NF.1 - research/math-spine.md §7 notes: 'Unit fractions are THE building block (K4 p.7), like 1 for whole numbers. Specify-the-whole is first-class: the same shaded region is 2/3 or 1/3 depending which square is the whole. No need for ''proper''/''improper'' yet. Gr3 denominators limited to 2,3,4,6,8.' representations: C: move: 'Give her fraction tiles/circles: pull ONE 1/4 tile out of the four that tile the whole — that single piece is the unit fraction 1/4. Then physically lay 3 of the 1/4 tiles side by side to build 3/4, saying ''three one-fourths''. Swap the whole (a small circle vs a big circle) with the same tiles to show 1/4 depends on which whole you started from.' check: Grab one-fourth of this whole with the tiles. Now build three-fourths — how many of the same piece did you use? down: math.fractions.equal-shares G: move: Draw a bar split into b equal parts, shade exactly one part, and label it 1/b — the unit fraction. Then shade a copies of that one part and count them out loud ('1/4, 2/4, 3/4') so a/b reads as a copies of 1/b. Draw the SAME shaded region inside a small square and inside a big square to show it's 1/2 of one and 1/4 of the other — the whole must be named. check: This bar has 4 equal parts and 3 are shaded. Say it as counting one-fourths — how many one-fourths are shaded? S: move: 'Connect the picture to the notation: the bottom number (denominator) = how many equal parts the whole was cut into; the top (numerator) = how many you counted. Write 3/4 next to its shaded bar and read it ''three one-fourths'', not ''three out of four''. Stress that 3/4 is meaningless until you say ''3/4 OF WHAT''.' check: Write the fraction for three shaded parts when the whole is cut into four. What does the bottom 4 tell you? N: move: 'Chocolate-bar sharing: a bar scored into 6 squares — one square is 1/6 of the bar, and if you eat 4 squares you ate 4 of those one-sixths = 4/6. Then trick her: ''1/2 of my little candy bar vs 1/2 of the giant one — same amount?'' The ''of what'' question is the point.' check: I ate 3 squares of a chocolate bar that has 8 equal squares. What fraction of the bar did I eat — and one-eighth of WHAT? - id: math.fractions.fraction-number-line title: Fraction as a number on the number line; whole numbers as fractions strand: fractions band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: place a fraction on a number line (partition the unit interval 0-1 into b parts, iterate 1/b); express whole numbers as fractions (3 = 3/1, 4/4 = 1)' L3: SM-2 spaced gate on mixed items across separate days requires: - math.fractions.unit-fraction-meaning helps: - math.measure.length-ruler resources: - type: singapore ref: PM2022 G3 ch7 §7F Fractions on a Number Line / §7C Fractions Greater Than 1 - type: beast ref: BA 3D ch10 Fractions sources: - CC 3.NF.2 - CC 3.NF.3c - research/math-spine.md §7 notes: Number line = the unit interval (0→1) as the whole; 'an abstraction and generalization of their work with length measurement' (K4 p.2-3,8). helps FROM length-ruler — wire it or the map looks disconnected (spine §11.4). instrument_gap: true representations: C: move: 'Use a physical number line: a length of ribbon or a ruler-strip marked 0 and 1. Have her fold/step the unit interval into b equal jumps with a paper strip and mark the tick for 1/b, then keep iterating jumps (1/b, 2/b, 3/b...) walking her finger along. This is length measurement — she''s measuring how far from 0, not shading a region.' check: Here's the strip from 0 to 1. Make equal jumps to land the tick for one-third, then keep jumping — where does two-thirds land? down: Measure lengths on a ruler first (partition a unit interval into equal parts) G: move: Draw a number line 0→1, partition the unit interval into b equal segments, and locate a/b by iterating the unit fraction from 0 (NOT by counting tick marks). Extend past 1 to place 4/4 = 1 and to show whole numbers as fractions (3 = 3/1). Emphasize the whole is the DISTANCE 0→1, an abstraction of the length-bar model. check: Mark 3/4 on this 0-to-1 number line. Now mark where 4/4 is — what other name does that point have? S: move: 'Write whole numbers in fraction form and match them to points: 1 = 2/2 = 3/3 = 4/4 (all land on the ''1'' tick); 3 = 3/1. Pair each written fraction with the point you land on so the symbol names a NUMBER, not just a shaded piece.' check: Write 1 as a fraction with a bottom number of 4. Write 3 as a fraction. Which tick does each one sit on? - id: math.fractions.equivalent-compare-3 title: Recognize/generate simple equivalent fractions; compare (same denom / same numer) strand: fractions band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: recognize/generate simple equivalents (1/2 = 2/4, same point); compare same-denominator (more unit fractions = bigger) and same-numerator (bigger denom = smaller pieces) fractions of the SAME whole' L3: SM-2 spaced gate on mixed items across separate days requires: - math.fractions.fraction-number-line helps: [] resources: - type: dimensions ref: Dimensions Math 2B ch11 note: Fractions (compare/order) record: true - type: packet ref: fractions-2026-07-12 note: 'part 6: bigger-denominator = smaller piece' - type: singapore ref: PM2022 G3 ch7 §7D-7E Compare/Order & Equivalent Fractions - type: gumball ref: generator:fraction_compare sources: - CC 3.NF.3a - CC 3.NF.3b - CC 3.NF.3d - research/math-spine.md §7 notes: 'THE comparison misconception: 1/4 > 1/2 ''because a fourth of the big pizza is bigger than a half of the small pizza'' — comparisons valid only for the SAME whole (K4 p.9). Same-numerator (bigger bottom = smaller) is counterintuitive, its own item type. Gr3 = preliminary equivalence (recognize); Gr4 = the general rule.' representations: C: move: 'Lay fraction tiles on top of each other: place two 1/4 tiles end-to-end against one 1/2 tile — they cover the exact same length, so 2/4 = 1/2. For same-numerator comparison, set one 1/3 tile beside one 1/6 tile from the SAME whole and physically feel that the sixth piece is smaller — bigger bottom number, skinnier piece.' check: Cover one-half with the smaller tiles. Which tiles make a matching length, and how many did it take? So one-half equals how many fourths? down: math.fractions.fraction-number-line G: move: 'Draw two same-size bars stacked: split one into halves, one into fourths, shade 1/2 and 2/4 — the shaded lengths line up, so they''re equivalent (same point on the line). For comparison, draw same-numerator pairs (1/3 vs 1/4) of the SAME whole and shade one piece each: more cuts → skinnier piece. This is the packet''s Part 6 skinny-slice figure.' check: Here are 1/3 and 1/4 shaded on two same-size bars. Which shaded piece is bigger? Why does the bigger bottom number give the smaller piece? S: move: 'Write the equivalence and comparison statements the pictures show: 1/2 = 2/4; 3/4 > 2/4 (same bottom, more pieces = bigger); 1/3 > 1/4 (same top, bigger bottom = smaller). Have her write the >, <, or = between two fractions after reading it off the shaded figure — picture first, symbol second.' check: 'Fill in >, <, or =: 2/4 __ 3/4 , 1/3 __ 1/4 . Say the rule you used for each.' N: move: 'The pizza-whole trap from the node''s own misconception: ''1/4 of a giant pizza vs 1/2 of a tiny personal pizza — which is more food?'' Use two really different-sized paper pizzas to show that comparing fractions only works when the WHOLE is the same size; then compare fairly with two same-size pizzas.' check: A fourth of the big pizza looks bigger than a half of the little one. Is 1/4 really more than 1/2? What has to be true about the two pizzas before we can compare? - id: math.fractions.equivalence-compare-4 title: Explain fraction equivalence (n×a/n×b); compare unlike fractions via benchmarks strand: fractions band: - 8 - 10 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: explain a/b = (n×a)/(n×b) and generate equivalents; compare two fractions with unlike numerators AND denominators via common denominator or a benchmark (1/2)' L3: SM-2 spaced gate on mixed items across separate days requires: - math.fractions.equivalent-compare-3 helps: [] resources: - type: singapore ref: PM2022 G3 ch7 §7E Equivalent Fractions - type: singapore ref: PMUS 3B u6 Fractions (Equivalent Fractions) - type: beast ref: BA 3D ch10 Fractions sources: - CC 4.NF.1 - CC 4.NF.2 - research/math-spine.md §7 notes: 'Gr4 general equivalence rule + benchmark/common-denominator comparison, split from the Gr3 ''recognize'' rung (K4 p.8). Gr4 denominators: 2,3,4,5,6,8,10,12,100. BA-L3 already reaches 4.NF.1/4.NF.2 (§5.C).' instrument_gap: true representations: G: move: Draw a bar split into 3rds with 1/3 shaded; overlay a split that cuts each third into 4, turning it into 12ths with 4/12 shaded — same shaded length, showing 1/3 = 4/12 = (4×1)/(4×3). For unlike-fraction comparison, draw both fractions against a 1/2 benchmark bar to see which sits above/below one-half. check: Cut each third of this bar into 4 smaller parts. What does 1/3 become in twelfths? Which is closer to one-half, 3/8 or 5/8? down: math.fractions.equivalent-compare-3 S: move: 'Generalize to the symbolic rule a/b = (n×a)/(n×b) and use it to make a common denominator, then compare. Also teach the benchmark move symbolically: 3/8 < 1/2 because 3/8 < 4/8; 5/6 > 1/2 because 5/6 > 3/6. Write the multiplied form beside the original so the equivalence is explicit, not memorized.' check: Rewrite 2/3 and 3/4 with the same bottom number, then tell me which is bigger. Is 5/8 more or less than 1/2 — how do you know without a picture? - id: math.fractions.add-sub-mult-like title: Add/subtract like-denominator fractions & mixed numbers; multiply fraction × whole strand: fractions band: - 8 - 10 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: add/subtract fractions & mixed numbers with like denominators; multiply a fraction by a whole number (5/4 = 5 × 1/4); decompose a fraction into a sum' L3: SM-2 spaced gate on mixed items across separate days requires: - math.fractions.fraction-number-line - math.multdiv.mult-div-meaning - math.addsub.wp-result-unknown helps: [] resources: - type: beast ref: BA 3D ch10 Fractions - type: singapore ref: PM2022 G3 ch7 §7G Fractions of a Set sources: - CC 4.NF.3 - CC 4.NF.4 - research/math-spine.md §7 notes: Fraction add/sub reuses the SAME Add-To/Take-From/Put-Together situations as whole numbers (K4 p.4) — softly requires wp-result-unknown. Multiply fraction×whole builds on mult-as-equal-groups. a/b = a copies of 1/b is the load-bearing idea. instrument_gap: true representations: C: move: 'Fraction tiles again: to add 2/8 + 3/8, physically push together 2 of the 1/8 tiles and 3 of the 1/8 tiles — you now hold 5 of the same 1/8 piece = 5/8. The denominator is the NAME of the piece (it doesn''t change); you''re just counting pieces. For 5 × 1/4, lay out five 1/4 tiles and see 5/4 (past one whole).' check: Add 3/8 and 2/8 with the eighth-tiles — how many eighths do you have now? Does the bottom number change? down: math.fractions.fraction-number-line G: move: 'Bar/number-line model: shade 1/5, then shade 3 more fifths to show 1/5 + 3/5 = 4/5 as counting same-size pieces along one bar; for subtraction, cross off pieces. For fraction × whole, draw 5 copies of a 1/4 segment jumping along a number line to land on 5/4, connecting to repeated equal groups.' check: On this bar cut into fifths, shade 2/5 then 2/5 more. What's the total? Draw four jumps of 1/3 on a number line — where do you land? S: move: 'Write the operation on the notation, keeping the like denominator: 2/5 + 1/5 = 3/5 (add the tops, keep the bottom because the pieces are the same size); decompose 5/8 = 1/8+1/8+1/8+1/8+1/8; write 5 × 1/4 = 5/4. Connect each to the picture so the ''add the tops only'' rule is grounded, not a trick.' check: What is 4/6 − 1/6? Write 3/4 as a sum of one-fourths. What is 3 × 1/5 written as a single fraction? N: move: 'Same Add-To / Take-From stories she already knows, now with fractions: ''You ate 2/8 of the pizza, your brother ate 3/8 — how much is gone?'' ''The bottle had 5/6 of a liter, you poured out 2/6 — how much is left?'' The situation is identical to whole-number add/sub; only the unit is a fraction.' check: You had 7/8 of a chocolate bar and gave away 3/8. How much do you have left? What operation did you use? - id: math.fractions.decimal-notation title: Decimal notation for fractions (denominators 10/100); compare decimals strand: fractions band: - 8 - 10 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: express fractions with denominator 10/100 as decimals; write/read decimal notation; compare two decimals to hundredths' L3: SM-2 spaced gate on mixed items across separate days requires: - math.fractions.equivalence-compare-4 - math.placevalue.big-numbers-round helps: [] resources: - type: singapore ref: PMUS 3A u5 Money (Dollars and Cents) sources: - CC 4.NF.5 - CC 4.NF.6 - CC 4.NF.7 - research/math-spine.md §7 notes: Decimals as denominator-10/100 fractions; ties to money (dollars.cents). MA spends 8 topics here (6.19); we keep one node. Money-notation is the concrete anchor. instrument_gap: true representations: G: move: 'Use a 10×10 hundredths grid (and a 10-strip for tenths): shade 7 of 10 columns to show 7/10 = 0.7; shade 23 small squares to show 23/100 = 0.23. Line two shaded grids side by side to compare 0.3 vs 0.27 by area shaded — more shaded = bigger, which breaks the ''longer string of digits is bigger'' error.' check: Shade 0.4 on the tenths strip and 0.40 on the hundredths grid — same amount? Which is bigger, 0.3 or 0.27, and why? down: math.fractions.equivalence-compare-4 S: move: 'Connect fraction↔decimal notation by place value: 7/10 = 0.7 (tenths place), 45/100 = 0.45 (hundredths place); 3/10 = 30/100 = 0.30. Write the fraction directly above its decimal so the denominator (10 or 100) names the place. Compare decimals by lining up the decimal points and comparing tenths, then hundredths.' check: Write 6/10 and 9/100 as decimals. Which is greater, 0.6 or 0.06? Write 0.75 as a fraction over 100. N: move: 'Money is the concrete anchor: $0.75 is 75/100 of a dollar = 3 quarters; $0.5 = 50¢ = half a dollar = 5/10. Count real coins into dollar-and-cents amounts and write them both as money ($1.25) and as a mixed decimal, so the decimal point is the dollars-cents divider she already knows.' check: Three quarters is what fraction of a dollar, and how do you write it as a decimal? Write two dollars and five cents as a decimal number. - id: math.measure.compare-order-length title: Compare & order length; indirect comparison (transitivity) strand: measure band: - 5 - 7 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: describe/compare measurable attributes (longer/shorter, heavier); order 3 objects by length; compare two objects via a third (transitivity)' L3: SM-2 spaced gate on mixed items across separate days requires: [] helps: [] resources: - type: dimensions ref: Dimensions Math KA ch5 note: Compare Height, Length, Weight, and Capacity record: true - type: singapore ref: PM2022 G2 ch5 §5C Compare and Order Lengths - type: activity ref: order toys/sticks by length; compare via a string sources: - CC K.MD.1 - CC K.MD.2 - CC 1.MD.1 - research/math-spine.md §8 notes: 'Length develops in a FIXED order: direct comparison → transitivity/indirect → iterate units → ruler (K4 p.2-4). Do NOT skip transitivity.' instrument_gap: true representations: C: move: 'Lay two pencils side by side with their ends flush at one edge; the one that sticks out farther is longer. Then hand her three sticks and have her physically line them up shortest->longest, re-checking each pair by aligning ends. For transitivity, hide the pencils: cut a piece of string exactly as long as pencil A, then carry the STRING over to pencil B to decide which is longer without the two pencils ever touching.' check: Show me these three ribbons in order from shortest to longest. Which end do you line up first? down: count objects — she should already track 'more/fewer' before 'longer/shorter' G: move: 'Draw two bars starting from the same vertical baseline, one longer; ask which is longer just from the picture (no objects). Then draw three bars off a common line for ordering. Draw the transitivity case: bar A next to a string, string next to bar B, and reason A vs B through the string.' check: Here are three drawn bars starting at the same line. Circle the longest and put 1-2-3 under them from shortest to longest. S: move: 'Attach the words to the model: write ''A > B'' / ''A is longer than B'' beside the flush-ends comparison; write ''A > B and B > C, so A > C'' beside the string chain. The symbol just RECORDS what the objects showed.' check: The blue rod is longer than the red, and the red is longer than the green. Without seeing them, which is longer — blue or green? Write it with 'longer than'. N: move: 'Use a lived height/length situation: ''You''re taller than Cora, and Cora is taller than the teddy — so are you taller than the teddy?'' Or line up the family''s shoes by length. Narrate that measuring lets us compare things that can''t stand next to each other.' check: Dad's bike is longer than your bike, and your bike is longer than Cora's trike. Whose is longest? How do you know without lining them all up? - id: math.measure.length-ruler title: Measure length with iterated units & a ruler; two-unit inverse; length word problems strand: measure band: - 6 - 8 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: measure by laying same-size units end-to-end (no gaps); measure with a ruler (in/cm/ft/m) aligning the zero-point; measure the same object in two units (smaller unit → more units); solve length add/sub word problems; place whole numbers on a number-line diagram' L3: SM-2 spaced gate on mixed items across separate days requires: - math.measure.compare-order-length helps: [] resources: - type: dimensions ref: Dimensions Math 1B ch10 note: Length record: true - type: dimensions ref: Dimensions Math 2A ch4 note: Length - type: singapore ref: PM2022 G2 ch5 Length - type: singapore ref: PMUS 2A u3 Length - type: beast ref: BA 2C ch7 Measurement sources: - CC 1.MD.2 - CC 2.MD.1 - CC 2.MD.2 - CC 2.MD.3 - CC 2.MD.4 - CC 2.MD.5 - CC 2.MD.6 - research/math-spine.md §8 notes: 'Four named length understandings, each a misconception (K4 p.3): (1) unit iteration no gaps; (2) accumulation of distance (''eight'' = whole span); (3) zero-point alignment (the classic ''start measuring at 1'' error); (4) numerals count units-so-far. Inverse-unit relationship (smaller unit → more units) later powers same-numerator fraction comparison — helps into fractions.' instrument_gap: true representations: C: move: 'First iterate a UNIT: measure the table with same-size paperclips laid end-to-end, no gaps and no overlaps, and count them. Then hand her a real ruler and measure the same object — critically, START AT THE ZERO mark, not the ''1'', pointing out that the ruler''s numbers count units-so-far. Measure one object twice: once in cm, once in a bigger unit, and notice the smaller unit gives MORE units.' check: Measure this book with the ruler. Where does the ruler have to line up before you read the number? (Show me.) down: math.measure.compare-order-length G: move: 'Show a drawn ruler under an object with the classic trap: the object starts at the ''1'' mark. Ask what''s wrong, then redraw it aligned at zero. Use a number-line diagram: place whole numbers as tick marks and read a length as the distance from 0 to the end, so ''eight'' means the whole span, not just the last tick.' check: This drawing shows a crayon starting at the 1 on the ruler and ending at 7. The child said '7 cm'. Is that right? What's the real length? S: move: 'Write the measurement with its UNIT every time: ''14 cm'', ''3 in'', never a bare ''14''. Turn a two-part measure into a length number sentence for word problems: ''12 cm + 9 cm = 21 cm''. Name that the unit is part of the answer.' check: The ribbon is 8 cm and you cut off 3 cm. Write the number sentence and the answer WITH the unit. N: move: 'Measure real things she cares about: how tall she''s grown against the doorframe, how long her bed is, whether the couch fits a wall. Frame a word problem from a real cut: ''The tape is 15 cm; you use 6 cm for a bracelet — how much tape is left?''' check: Your bookshelf is 90 cm wide and the wall is 120 cm. Will it fit? How much room is left over? - id: math.measure.measure-fraction-inch-convert title: Measure to nearest 1/2, 1/4 inch (line plot); convert units within a system strand: measure band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: measure to nearest 1/2 and 1/4 inch and build a line plot; state relative unit sizes and convert within a system (larger → smaller unit)' L3: SM-2 spaced gate on mixed items across separate days requires: - math.measure.length-ruler - math.fractions.unit-fraction-meaning helps: [] resources: - type: singapore ref: PM2022 G3 ch8 Mass and Liquid Volume - type: beast ref: BA 3C ch9 Measurement sources: - CC 3.MD.4 - CC 4.MD.1 - CC 4.MD.2 - research/math-spine.md §8 notes: Fractional-inch measurement requires unit-fraction meaning (cross-strand requires). Conversion (larger→smaller) leans on multiplication. MA splits conversion per unit type × direction (7.20, 9 topics); one node here. instrument_gap: true representations: C: move: Measure real objects (a leaf, a crayon, her finger) to the nearest half then quarter inch using a ruler whose quarter-inch marks she can see and touch; say 'between 2 and 2 and a half — closer to the half'. Collect several measurements and stack physical unit cubes or sticky notes above each length value to physically build a line plot. check: Measure these four crayons to the nearest quarter inch and stack a cube for each one over its length. Which length has the tallest stack? down: math.measure.length-ruler G: move: 'Draw a ruler zoomed in so the 1/2 and 1/4 tick marks are labeled 1/4, 1/2, 3/4; have her mark where an object ends and name the fraction. Then draw the line plot: an X above each measured value on a number line, spacing the X''s evenly, and read ''how many measured 2 and 1/4 inches''.' check: On this line plot of pencil lengths, how many pencils were 3 and 1/2 inches? How many were longer than 3 inches? S: move: 'Write the fractional measure in notation (2 1/4 in) and connect it to unit conversion: 1 ft = 12 in, so 3 ft = 36 in — a LARGER unit converts to MORE of the smaller unit by multiplying. Build a small conversion table (ft->in, m->cm) from the model.' check: Write '2 and three-quarter inches' in numbers. And if a table is 4 ft long, how many inches is that? down: math.fractions.unit-fraction-meaning N: move: 'Measure everyone''s height to the quarter inch and make a family line plot; or measure a set of toy cars and plot their lengths, then answer ''what''s the most common length?''. For conversion, use a recipe/height context: ''You''re 4 feet tall — the doctor writes it in inches. How many?''' check: You measured 6 of your stuffed animals and made a line plot. What length shows up the most? How much longer is the biggest than the smallest? - id: math.measure.mass-liquid-volume title: 'Mass (grams/kilograms) & liquid volume (liters/milliliters): measure, estimate, one-step problems' strand: measure band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: measure & estimate mass in grams/kilograms and liquid volume in liters/milliliters using a scale / graduated container; read the scale; solve one-step +/−/×/÷ word problems involving masses or volumes given in the same units' L3: SM-2 spaced gate on mixed mass & volume items across separate days requires: - math.measure.length-ruler - math.addsub.add-sub-fluency-100 helps: [] resources: - type: dimensions ref: Dimensions Math 2A ch5 note: Weight (grams/kilograms) record: true - type: dimensions ref: Dimensions Math 2B ch13 note: Capacity (liquid volume) - type: singapore ref: PM2022 G3 ch8 Mass and Liquid Volume - type: beast ref: BA 3B ch8 Mass and Liquid Volume - type: beast ref: BA 3C ch9 Measurement - type: activity ref: weigh fruit on a kitchen scale; fill a measuring jug to a target mL sources: - CC 3.MD.2 - research/math-spine.md §8 - research/dimensions-math.md §2 - research/singapore-beast.md §5.D notes: 'MISSING-NODE ADD (adj #7). CC 3.MD.2: measure & estimate liquid volumes and masses of objects using g/kg/l; add/subtract/multiply/divide to solve ONE-STEP word problems involving masses or volumes given in the same units (excludes multiplicative comparison). Requires scaled-instrument measurement (length-ruler: reading a scale, unit iteration, zero-point) + add-sub-fluency-100 (one-step problem arithmetic). Slotted in the measure length/quantity cluster before Time. Resources verified: Dimensions 2A ch5 Weight + 2B ch13 Capacity (dimensions-math.md §2), Singapore PM2022 G3 ch8, Beast 3B ch8 / 3C ch9 (singapore-beast.md §5.D). No gumball generator exists for mass/volume -> instrument_gap (adj #6).' instrument_gap: true representations: C: move: Put fruit on a real kitchen scale and read the grams; feel that a bag of flour is about 1 kg. Fill a graduated measuring jug to a target line (e.g. 250 mL) and pour, reading the scale on the side at eye level. Compare 'which is heavier' by holding two objects, then confirm on the scale — the instrument settles the argument. check: Pour water into this jug until it reads 200 mL. Now add 100 mL more — read the new line. How much is in the jug? down: math.measure.length-ruler G: move: 'Draw a scale dial or a jug with tick marks and have her read the value where the needle/water sits — reinforcing zero-point and reading the scale (same skill as the ruler). Draw a mass bar or jug for a one-step problem: ''450 g of apples + 300 g of pears'' shown as two segments.' check: This drawn jug is filled to a line halfway between 400 and 600 mL. How much water is in it? S: move: 'Write quantities with correct units (g, kg, mL, L) and solve one-step word problems with a number sentence: ''2 L - 750 mL'' or ''3 bags x 250 g''. Keep units on the answer; note that mixing L and mL needs the same unit first.' check: A bottle holds 1 L and you drink 350 mL. Write the number sentence for how much is left. N: move: 'Cook or shop: weigh out 500 g of pasta, measure 1 L of milk for a recipe, or figure out how many 250 mL cups fill a 1 L jug at a party. Narrate that mass and volume are how the whole grocery world is measured and sold.' check: A recipe needs 750 g of flour. You've weighed out 500 g. How many more grams do you need? - id: math.measure.tell-time-hour-half title: Tell & write time to the hour and half-hour (analog + digital) strand: measure band: - 6 - 8 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: read and write time to the hour and half-hour on analog and digital clocks; track the hour hand''s position' L3: 'SM-2 spaced gate (via the clock review pack: 3+ spaced reps, 4+ lifetime correct) across separate days' requires: - math.counting.count-sequence-100 helps: [] resources: - type: dimensions ref: Dimensions Math KB ch13 note: Time (to the hour, early) record: true - type: dimensions ref: Dimensions Math 1B ch18 note: Time - type: packet ref: analog-clocks-2026-07-12 note: 'parts 1-4: clock face → o''clock → half-past → the sneaky hour hand' - type: gumball ref: generator:clock_read - type: gumball ref: generator:clock_pick sources: - CC 1.MD.3 - research/math-spine.md §8 - research/singapore-beast.md §5.D notes: Penelope FINISHED the analog-clocks packet 2026-07-12 end-to-end ('TOUGH for her') → seed L1. Clock review pack ACTIVATED 2026-07-13; first review 2026-07-14 → NO retention (L3) evidence yet, seed L1 not higher. NO Beast-L2 resource (BA defers Gr2 time to L3C ch9, §5.D) — packet + Singapore + gumball carry it. representations: C: move: Use a real GEARED teaching clock so the hour hand physically CREEPS as she turns the minute hand — set 3 o'clock, then walk the minute hand a full turn and watch the hour hand move from the 3 toward the 4, landing exactly ON the 3 at o'clock and HALFWAY between 3 and 4 at half-past. She reads it off the actual hands, not a picture. check: Turn the hands to 3 o'clock. Now make it half past 3 — where does the SHORT hand end up, and how do you know? down: math.counting.count-sequence-100 G: move: Use real-looking clock-face diagrams (the analog-clocks packet style, with a properly PROPORTIONAL short hand). At o'clock the hour hand points dead-on a number and the minute hand is straight up; at half-past the minute hand points down at 6 and the hour hand has crept halfway to the next number. Have her READ drawn faces and also DRAW the hands for a given time. check: Draw the hands for 'half past 2'. Where exactly does the short hand go — on the 2, or between 2 and 3? S: move: 'Connect the face to digital notation: o''clock reads ''3:00'', half past reads ''3:30''. Read the hour off the SHORT hand''s position and the minutes off the long hand. Practice writing both forms for the same face; note that ''3:30'' still belongs to the 3 o''clock hour even though the hour hand isn''t on the 3 anymore.' check: This clock shows the short hand between 4 and 5, long hand on the 6. Write the digital time. N: move: 'Anchor times to her real day: ''We eat dinner at 6 o''clock — set the clock. Bath is half past 7.'' Read the kitchen clock together at real moments. Time is the schedule she lives inside, so tie each reading to an event.' check: It's half past 8 and bedtime is 9 o'clock. Show me both times on the clock. Which comes first? - id: math.measure.tell-time-5min title: Tell & write time to the nearest 5 minutes; a.m./p.m. strand: measure band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: read/write time to the nearest 5 minutes (each number = 5); use quarter-past / quarter-to; distinguish a.m./p.m.' L3: SM-2 spaced gate across separate days requires: - math.measure.tell-time-hour-half - math.counting.skip-count-2-5-10 helps: - math.fractions.equal-shares - thinking.reasoning.deep-structure-transfer resources: - type: dimensions ref: Dimensions Math 1B ch18 note: Time (to 5 min) record: true - type: dimensions ref: Dimensions Math 2B ch12 note: Time - type: packet ref: analog-clocks-2026-07-12 note: 'parts 5-7: counting minutes (each number = 5) → any time → quarter past/to' - type: singapore ref: PM2022 G2 ch7 Time - type: gumball ref: generator:clock_read sources: - CC 2.MD.7 - research/math-spine.md §8 - research/singapore-beast.md §5.D notes: Needs skip-count-5 (each clock number = 5 min). helps clock-quarters ↔ fraction-quarters (quarter-past = 1/4 turn). BA defers to L3C ch9. representations: C: move: 'On the geared clock, skip-count by 5 out loud around the face while pointing at each number: the 1 is 5 past, the 2 is 10 past... so each NUMBER is worth 5 minutes for the long hand. Physically set the minute hand to ''3:45'' by walking it to the 9 while she counts 5-10-15...-45. Do quarter-past (long hand on 3) and quarter-to (long hand on 9) as landmark turns.' check: Count around the clock by fives with me. Now set the long hand to 20 minutes past — which number does it point to? down: math.measure.tell-time-hour-half G: move: Show a clock-face diagram with a faint minute-count ring (5,10,15... outside each number). Have her read times to the 5 minutes and mark quarter-past / quarter-to. Draw hands for '5 minutes to 4' — the trap is she must read the SHORT hand as still near the 3, almost at 4. check: This face has the long hand on the 9 and the short hand just before the 12. What time is it — and is it quarter-to something? S: move: 'Write times like 3:45, 8:20, and label a.m. vs p.m. from the day''s events. Connect ''quarter past'' = :15, ''half past'' = :30, ''quarter to'' = :45. Reinforce the leading zero: five past three is 3:05, not 3:5 — the notation always fills two minute digits.' check: Write 'twenty-five minutes past seven' as a digital time. Is it a.m. or p.m. if it's breakfast? N: move: 'Read real schedules: ''The show starts at 7:15 — set it.'' ''School is out at 3:05.'' Use a.m./p.m. against her routine (7:30 a.m. is getting-up, 7:30 p.m. is winding-down). Cross-reference: a quarter-past on the clock is the same ''quarter'' as a coin quarter and a fraction quarter — one-fourth of the turn.' check: The park closes at quarter to 6 in the evening. Write that time. Is it a.m. or p.m.? down: math.fractions.equal-shares - id: math.measure.elapsed-time title: Time to the minute; elapsed-time / time-interval word problems strand: measure band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: read time to the minute; find minutes between two times (count up, crossing the hour in hops); compute start + duration = end; handle a.m./p.m. spanning the day' L3: SM-2 spaced gate across separate days requires: - math.measure.tell-time-5min - math.addsub.add-sub-fluency-100 helps: - math.addsub.counting-on - thinking.reasoning.deep-structure-transfer resources: - type: dimensions ref: Dimensions Math 2B ch12 note: Time (intervals) record: true - type: packet ref: elapsed-time-2026-07-12 note: 'full packet: refresher → minutes between → crossing the hour → hours-then-minutes → start+minutes=end → AM/PM' - type: singapore ref: PM2022 G3 ch5 Time - type: gumball ref: generator:elapsed_time sources: - CC 3.MD.1 - research/math-spine.md §8 - research/singapore-beast.md §5.D notes: 'Elapsed time = clocks + arithmetic; uses count-up (helps counting-on: count-up-change ↔ elapsed-time is the ed-packet panel''s empirical transfer, DESIGN §2). Seed Penelope L0 — packet built but NOT completed (evidence-inventory).' representations: C: move: 'Use the geared clock to COUNT UP from the start time: set 2:40, then hop the minute hand forward in fives to the next o''clock (2:40 -> 3:00 is 20 min), then continue to the end time, adding the hops. Crossing the hour is a physical stop-and-restart at the 12. Also do it forward: set a start, walk the hands ahead by a duration to find the end.' check: Set the clock to 2:50. Now move it forward to 3:20 by hopping. How many minutes did you move? down: math.measure.tell-time-5min G: move: 'Draw an open NUMBER LINE (the elapsed-time packet method): mark the start time, hop to the next whole hour, then hop the remaining minutes to the end, and add the two hops. This makes ''crossing the hour'' visible as two clean jumps instead of borrowing across 60.' check: On the number line, jump from 4:45 to 6:10. Draw the hops and add them up. How long is that? S: move: 'Write the interval as arithmetic: start + duration = end, or end - start = duration, minding that an hour is 60 (not 100) minutes so you regroup at 60. Handle a.m./p.m. spans by noting the crossing of noon or midnight.' check: A movie starts at 3:50 p.m. and lasts 1 hour 30 minutes. Write the end time. Does it cross into a new hour? down: math.addsub.counting-on N: move: 'Use her real intervals: ''Swim practice is 4:15 to 5:00 — how long is that?'' ''We leave at 9:20 and it''s a 40-minute drive — when do we arrive?'' Say out loud that count-up change-making (money) and count-up elapsed time are the SAME move — hop to a friendly landmark, then finish.' check: Your show starts at 6:40 and it's 25 minutes long. What time does it end? Which way did you count — up or back? - id: math.measure.money-coins-bills title: Coins & bills; count mixed amounts; make change (count up); money word problems strand: measure band: - 6 - 8 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: identify coin values, skip-count one coin type, count a mixed pile biggest-first, recognize same amount from different coins, know the dollar = 100¢, and make change by counting up; solve $ and ¢ word problems' L3: SM-2 spaced gate on mixed items across separate days requires: - math.counting.skip-count-2-5-10 - math.addsub.add-sub-within-100 helps: - math.fractions.equal-shares - math.addsub.counting-on - thinking.reasoning.deep-structure-transfer resources: - type: dimensions ref: Dimensions Math KB ch14 note: Money (coins, early) record: true - type: dimensions ref: Dimensions Math 1B ch19 note: Money - type: dimensions ref: Dimensions Math 2B ch10 note: Money - type: packet ref: money-2026-07-12 note: 'full packet: meet the coins → skip-count → biggest-first → the dollar → making change (count up) → shop' - type: singapore ref: PM2022 G2 ch1 §1E Count Money - type: singapore ref: PMUS 2B u3 Money (U.S. Currency) - type: gumball ref: generator:money_count - type: gumball ref: generator:money_change sources: - CC 2.MD.8 - research/math-spine.md §8 - research/singapore-beast.md §5.D notes: 'Leans on skip-count (5s/10s/25s) + coin values. helps: coin-quarters ↔ clock-quarters ↔ fraction-quarters, and count-up-change ↔ elapsed-time (DESIGN §2 empirical transfers). BA defers Gr2 money to L3C ch9 (§5.D). Seed Penelope L0 — packet built but NOT completed.' representations: C: move: 'Sort REAL coins into piles by kind and name each value (penny 1, nickel 5, dime 10, quarter 25). Count a mixed pile BIGGEST-FIRST: start with quarters skip-counting by 25, then add dimes by 10, nickels by 5, pennies by 1. Trade physically: show that 5 pennies = 1 nickel, 2 dimes + 1 nickel = 1 quarter (same amount, different coins). Make change by COUNTING UP from the price to the money given, handing over coins as you go.' check: 'Here''s a real pile: 2 quarters, 1 dime, 3 pennies. Count it biggest coin first. What did you get?' down: math.counting.skip-count-2-5-10 G: move: Use true-to-size coin pictures (the money packet style) so she identifies by real diameter, not guesswork. Show a drawn mixed set and have her write the running total under each coin as she counts biggest-first. Draw a 'count-up change' line from 63 cents to $1.00 showing the hop coins. check: This drawn set is a quarter, a quarter, a nickel, and 2 pennies. Write the running total under each and circle the total. S: move: 'Write amounts in money notation: 25 cents, $1.25, and 100 cents = $1. Solve $ and cent word problems as number sentences, keeping the sign/point consistent. Connect the dollar to place value: cents are the hundredths of a dollar.' check: 'Write ''one dollar and thirty cents'' two ways: in cents and with a dollar sign. How many cents in a dollar?' down: math.addsub.add-sub-within-100 N: move: 'Play shop with real prices: she''s the cashier, you buy something for 68 cents with a dollar, she makes change by counting up (69, 70, 75, 100 — that''s 32 cents back). Beware the trap that MORE coins can be LESS money (6 pennies < 1 nickel? no — but 1 penny < 1 dime though it looks similar). Cross-reference: a coin quarter, a clock quarter, and a fraction quarter are all one-fourth.' check: You buy a snack for 55 cents and pay with a dollar. Count up to give me my change. How much do I get back? - id: math.measure.data-graphs title: Classify & count categories; picture graphs & bar graphs (single-unit) strand: measure band: - 5 - 8 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: classify/sort objects into ≤3 categories and count each; read & build a single-unit picture graph and bar graph; answer put-together / take-apart / compare (how-many-more) questions from the graph' L3: SM-2 spaced gate on mixed items across separate days requires: - math.counting.count-objects-cardinality helps: - math.addsub.wp-change-addend-compare - thinking.reasoning.compare-contrast resources: - type: dimensions ref: Dimensions Math 1B ch11 note: Comparing (picture graphs) record: true - type: dimensions ref: Dimensions Math 2B ch14 note: Graphs (picture & bar) - type: singapore ref: PM2022 G2 ch8 Data (Picture/Bar Graphs, Line Plots) - type: singapore ref: PMUS 2B u7 Graphs sources: - CC K.MD.3 - CC 1.MD.4 - CC 2.MD.9 - CC 2.MD.10 - research/math-spine.md §8 notes: 'NO Beast resource — BA ''does not include picture or bar graphs'' (2.MD.D.9/.10 Not Included, §5.D). Build/buy signal if BA is the daily core: Singapore + ed-packets carry it.' instrument_gap: true representations: C: move: 'Sort a real collection (buttons, toy animals, colored blocks) into up to 3 category piles and count each pile. Then BUILD a picture graph physically: line the objects up in neat rows on a grid so each column is one category, one object per cell — turning the piles into a graph she can see and touch.' check: Sort these blocks by color into piles and line them up in rows. Which color has the most? How many more than the fewest? down: math.counting.count-objects-cardinality G: move: Draw a single-unit picture graph (one icon = one thing) and a bar graph from the same data, with a clear category axis and count axis. Have her READ values off the bars and answer put-together (total), take-apart, and compare (how-many-MORE) questions directly from the picture. check: On this bar graph of favorite fruits, how many more kids chose apples than bananas? How many kids in all? S: move: 'Write the counts as a small tally/frequency table beside the graph, then as compare number sentences: ''apples 7, bananas 4, 7 - 4 = 3 more''. The graph is the picture; the number sentence records the comparison it shows.' check: The graph shows 6 cats and 9 dogs. Write the number sentence for 'how many more dogs than cats'. N: move: 'Graph something from her life: survey the family''s favorite ice-cream flavors, or count the cars of each color that pass the window for five minutes, then build the bar graph together and answer real questions (''which flavor wins?'').' check: You counted the colors of cars that drove by. Which color was most common, and how many more than the least common? down: math.addsub.counting-on - id: math.measure.scaled-graphs-lineplots title: Scaled picture & bar graphs; line plots (whole & fractional scale) strand: measure band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: read/build scaled picture & bar graphs and answer how-many-more/less; build a line plot with a whole-number scale and with a fraction (1/2,1/4,1/8) scale' L3: SM-2 spaced gate on mixed items across separate days requires: - math.measure.data-graphs - math.multdiv.mult-div-meaning helps: [] resources: - type: singapore ref: PM2022 G3 ch9 Data sources: - CC 3.MD.3 - CC 4.MD.4 - research/math-spine.md §8 notes: Scaled graphs need multiplicative reasoning (each symbol = n). Fractional line plot ties to fractional-inch measurement. NO Beast resource (BA 3.MD.3 'Not Included', §5.C) — Singapore/ed-packet only. instrument_gap: true representations: C: move: 'Build a SCALED picture graph where each icon stands for MORE than one (each sticker = 2 kids): physically place half as many stickers and reason ''this is worth 10, not 5''. For a line plot, measure real objects and place a physical marker above each value on a number line — then switch the axis to fractional units (1/2, 1/4, 1/8) and plot again.' check: In this graph each apple sticker means 5. There are 4 stickers in the row — how many apples is that? down: math.measure.data-graphs G: move: Read scaled bar/picture graphs where the axis counts by 2s, 5s, or 10s — she must multiply the bar's units by the scale, not just count squares. Build a line plot with a whole-number scale, then one with a fractional scale (1/4-inch marks) tying it to fractional-inch measurement. check: This bar reaches the 3rd mark and each mark is worth 5. What's the value? If the scale were 2 per mark, what would it be? S: move: 'Write the reading as multiplication: ''each symbol = 5, there are 6 symbols, 6 x 5 = 30''. Record line-plot answers with counts and fractions (''three items at 2 and 1/4 in''). The scale makes graph-reading a multiplicative act, not a count.' check: A picture graph uses one star = 4 books. A row has 7 stars. Write the multiplication and the total. down: math.multdiv.mult-div-meaning N: move: 'Make a scaled graph of real data too big to draw one-per: ''We read 40 books this month — draw it where each book icon = 10.'' Or plot the family''s shoe lengths to the quarter inch on a line plot and ask what''s most common and the range.' check: You want to graph 60 minutes of screen time where each icon is 15 minutes. How many icons do you draw? - id: math.measure.area-tiling-multiply title: Area as unit-square tiling; relate area to multiplication (length × width) strand: measure band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: find area by counting unit squares tiling a region (no gaps/overlaps); relate area to multiplication (area = length × width) and use distributive tiling to break a rectangle apart' L3: SM-2 spaced gate on mixed items across separate days requires: - math.geometry.array-rows-columns - math.multdiv.mult-div-meaning helps: [] resources: - type: singapore ref: PM2022 G3 ch6 §6A-6B Area / Area of Squares and Rectangles - type: beast ref: BA 3A ch3 Area and Perimeter - type: beast ref: BA 3D ch12 Area sources: - CC 3.MD.5 - CC 3.MD.6 - CC 3.MD.7 - research/math-spine.md §8 notes: Area relies on area-invariant, area-additive, unit-tiling-no-gaps (K4 p.4). Perceiving a region as tiled = spatial structuring (helps from array-rows-columns). Author area-tiling BEFORE any formula node. instrument_gap: true representations: C: move: TILE a real rectangle (a book cover, a placemat) with square tiles or pattern-block squares — no gaps, no overlaps — and count the squares to get the area. Then arrange the SAME tiles into equal rows and columns (an array) and see that rows x columns gives the same count faster. Physically split a big rectangle into two smaller tiled pieces to show area adds up. check: Cover this rectangle with square tiles, no gaps. How many tiles? Now — without counting one by one — how many rows and how many in each row? down: math.geometry.array-rows-columns G: move: 'Draw a rectangle on grid paper and shade the unit squares to find area by counting; then label the side lengths and show area = length x width matches the count. Draw a rectangle split into two parts (distributive tiling: a 4x7 as 4x5 + 4x2) to make the multiplication visible.' check: This drawn rectangle is 5 squares wide and 3 tall. Count the squares, then check with a multiplication. Do they match? S: move: 'Write area as a product with square units: ''area = length x width = 6 cm x 4 cm = 24 square cm''. Name that area is always in SQUARE units. Show the distributive break as 4 x (5+2) = 4x5 + 4x2.' check: A rectangle is 8 cm by 3 cm. Write the area with its unit. What does 'square cm' mean? down: math.multdiv.mult-div-meaning N: move: 'Measure a real area she cares about: how many square tiles cover the bathroom floor, or how much wrapping paper covers a box face, or the area of her bedroom to lay a rug. Area = ''how much surface it takes to cover''.' check: Your rug is 6 feet by 4 feet. How many square feet of floor does it cover? Show the multiplication. - id: math.measure.perimeter-formulas title: Perimeter of polygons; apply area & perimeter formulas for rectangles strand: measure band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: find perimeter of polygons; apply area & perimeter formulas for rectangles; compare same-perimeter/different-area (and vice versa)' L3: SM-2 spaced gate on mixed items across separate days requires: - math.measure.area-tiling-multiply - math.measure.length-ruler helps: [] resources: - type: singapore ref: PM2022 G3 ch6 §6C-6D Perimeter / Composite Figures - type: beast ref: BA 3A ch3 Area and Perimeter sources: - CC 3.MD.8 - CC 4.MD.3 - research/math-spine.md §8 notes: Area↔perimeter confusion is the classic error; 3.MD.8's same-perimeter/different-area comparison directly targets it (K4). Requires area-tiling first + ruler for measured sides. instrument_gap: true representations: C: move: Run a finger (or a piece of string) all the way AROUND the edge of a real shape and measure the string — that's the perimeter, the distance around. Contrast with area (covering the inside). Build two rectangles from the same loop of string that have the SAME perimeter but different area, so she feels the two are different measures. check: Wrap this string around the whole edge of the placemat and measure it. Is that the perimeter or the area? What's the difference? down: math.measure.length-ruler G: move: Draw a polygon with all side lengths labeled and have her ADD the sides for perimeter; draw a rectangle and derive both area (l x w) and perimeter (add all four, or 2l + 2w). Draw two rectangles with equal perimeter but different area side by side to target the classic area/perimeter mix-up. check: This drawn rectangle is 7 by 3. Find its perimeter AND its area. Which one is 'around' and which is 'inside'? S: move: 'Write the perimeter as a sum (5 + 3 + 5 + 3 = 16) and introduce the rectangle formulas P = 2l + 2w (or 2(l+w)) and A = l x w side by side, with UNITS (perimeter in cm, area in square cm). Stress: perimeter is a length, area is a square measure — the units tell them apart.' check: A rectangle is 9 m by 4 m. Write both formulas and compute perimeter and area — each with its unit. down: math.measure.area-tiling-multiply N: move: 'Perimeter is real fencing/trim: how much fence goes around the garden, how much ribbon around a picture frame, how much baseboard around a room. Area is the covering (sod for the garden, glass for the frame). Same shape, two different shopping questions.' check: You're putting a fence around a 10 m by 6 m garden. How many meters of fence? Is that perimeter or area? - id: math.measure.angle-measure title: Angle as turn (1° = 1/360); measure with a protractor; angles are additive strand: measure band: - 8 - 10 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: understand an angle as a turn (1° = 1/360 of a full turn); measure & draw angles with a protractor; find an unknown angle using additivity' L3: SM-2 spaced gate on mixed items across separate days requires: - math.geometry.lines-angles helps: - math.fractions.unit-fraction-meaning resources: - type: beast ref: BA 3A ch1 Shapes sources: - CC 4.MD.5 - CC 4.MD.6 - CC 4.MD.7 - research/math-spine.md §8 notes: '1° = 1/360 of a turn ties angle to unit fractions (helps). Additivity: total angle = sum of parts. MA 7.22 splits angle work fine; one node here.' instrument_gap: true representations: C: move: 'Make an angle with two rulers (or a clock''s hands) hinged at a point and physically OPEN it — the angle is the amount of TURN, not the length of the arms. Walk a full turn = 360 degrees, a quarter turn = 90. Use a real protractor: line the center on the vertex, the baseline on one arm at zero, and read where the other arm crosses. Cut a straight angle into two pieces and show the pieces ADD to 180.' check: Open these two rulers to a right angle. About how many degrees? Now open them a little more — did the angle get bigger even though the rulers are the same length? down: math.geometry.lines-angles G: move: Draw angles and a protractor overlay so she reads the degree where the arm crosses the scale (watch the two scales — read from the zero side). Show a full circle marked in degrees so 1 degree = 1/360 of the turn. Draw an angle split into two parts labeled, e.g. 30 and unknown inside a 90, to find the missing part by subtraction. check: 'Read this drawn angle on the protractor. Then: inside a right angle, one part is 35 degrees — what''s the other part?' S: move: 'Write angle measures in degrees and use ADDITIVITY as arithmetic: if a straight angle (180) is split into 110 and x, then x = 180 - 110 = 70. Connect 1 degree = 1/360 of a full turn to unit fractions of a whole.' check: Two angles sit together on a straight line. One is 125 degrees. Write the equation for the other and solve it. down: math.fractions.unit-fraction-meaning N: move: 'Find turns in the real world: how far the clock''s minute hand turns in 15 minutes (a quarter turn, 90 degrees), the angle a door opens, a skateboard ''180'' or ''360'' being a half and full turn. Angle is how much something rotates.' check: A skater does a '180'. How many degrees is that? What about a '360' — where do they end up facing? - id: math.geometry.name-shapes-visual title: Name 2-D/3-D shapes regardless of orientation/size; 2-D vs 3-D strand: geometry band: - 4 - 7 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: name 2-D and 3-D shapes correctly regardless of orientation, size, or non-prototypical variant (skinny/rotated); identify shapes as flat (2-D) vs solid (3-D)' L3: SM-2 spaced gate on mixed variants across separate days requires: [] helps: [] resources: - type: dimensions ref: Dimensions Math KA ch4 note: Shapes and Solids record: true - type: dimensions ref: Dimensions Math 1A ch8 note: Shapes - type: singapore ref: PM2022 G2 ch9 §9A 2-D and 3-D Shapes - type: activity ref: shape hunt with rotated/skinny/scaled variants sources: - CC K.G.1 - CC K.G.2 - CC K.G.3 - research/math-spine.md §9 notes: 'van Hiele Level 0 (Visual/syncretic). Concept-image error: kids reject a ''very skinny'' isosceles triangle and accept a difficult distractor (K5 p.6). REMEDY: vary examples widely, include non-prototypical variants. Orientation is a NON-defining attribute (K5 p.8).' instrument_gap: true representations: C: move: 'Sort a bin of real objects and blocks into 2-D-vs-3-D piles (a paper triangle vs a wooden pyramid, a circle vs a ball), then a shape hunt around the house naming what she finds. Crucially, hand her NON-prototypical variants — a very skinny/long triangle, a rotated square (diamond), an oversized circle — so she names by shape, not by the ''typical'' picture (the concept-image error: kids reject a skinny triangle).' check: Sort these into flat shapes and solid shapes. Now — is this skinny pointy one still a triangle? What makes it a triangle even though it's not the fat kind? G: move: Show shapes in many orientations and sizes on paper and have her name each; mix in a hard distractor (a shape that looks triangle-ish but isn't closed, or has a curved side). Rotate the same square to sit on a corner ('diamond') and ask if it's still a square — orientation is a NON-defining attribute. check: Name each of these shapes. Two of them are turned sideways — does turning a shape change what it's called? N: move: 'Point out shapes in the real world: ''that clock is a circle, the window is a rectangle, the roof is a triangle, the ball is a sphere, the cereal box is a rectangular prism (3-D).'' A walk or a picture book becomes a naming game — and note when a real shape is skinny or tilted so the everyday example fights the prototype.' check: Look around the room — find me something shaped like a circle and something shaped like a triangle. Is the tall skinny door a rectangle even though it's not square-ish? - id: math.geometry.compose-decompose-shapes title: Compare shapes by informal parts; build, compose & decompose shapes strand: geometry band: - 5 - 7 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: compare shapes by informal parts (sides, vertices, corners); model shapes from components (sticks/clay); compose simple shapes into a larger shape and decompose a composite; use a composite as a unit' L3: SM-2 spaced gate on mixed items across separate days requires: - math.geometry.name-shapes-visual helps: [] resources: - type: dimensions ref: Dimensions Math 1A ch8 note: Shapes (compose/decompose) record: true - type: singapore ref: PM2022 G2 ch9 §9B Partition 2-D Shapes - type: activity ref: 'pattern blocks / tangrams: compose and take apart' sources: - CC K.G.4 - CC K.G.5 - CC K.G.6 - CC 1.G.2 - research/math-spine.md §9 notes: Compose → decompose → spatial-structuring is a real prereq chain (K5 p.4-5). Feeds array-rows-columns (the multiplication/area hub) and equal-shares (fractions). instrument_gap: true representations: C: move: 'Pattern blocks and tangrams: put two triangles together to make a square or a rhombus; fill a hexagon outline with triangles and trapezoids; then take a composite apart. Build shapes from components too — toothpicks/sticks and clay balls for corners, so she feels sides and vertices as she counts them. Use a composite (two triangles = a square) as a single unit to make a bigger shape.' check: Make a bigger triangle out of these smaller triangles — how many did it take? Now take the square apart into two triangles. Can you build a rectangle the same way? down: math.geometry.name-shapes-visual G: move: 'Draw a composite figure and have her draw the cut-lines that break it into shapes she knows (a ''house'' pentagon = a square + a triangle); and go the other way — show puzzle pieces and have her sketch the whole they''d compose into. Compare two shapes by their informal parts: ''this one has more corners / more sides.''' check: Draw a line to split this house-shape into a square and a triangle. Which of these two shapes has more corners? N: move: 'Real composing/decomposing: cutting a sandwich into two triangles then rearranging them; building a fort or a block tower where a big wall is made of smaller blocks used as a unit; a quilt/tile floor where small shapes tile a bigger pattern. Name the parts and the whole as she builds.' check: You cut your square sandwich corner to corner — what two shapes did you make? If you push them back together, what shape do you get? - id: math.geometry.defining-attributes title: Defining vs non-defining attributes; name/draw shapes by attributes; ID quadrilaterals strand: geometry band: - 6 - 8 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: distinguish defining attributes (sides, closed, angles) from non-defining (color, size, orientation); name & draw a shape given attributes (# sides/angles); identify triangles, quadrilaterals, pentagons, hexagons, cubes' L3: SM-2 spaced gate on mixed items across separate days requires: - math.geometry.compose-decompose-shapes helps: [] resources: - type: dimensions ref: Dimensions Math 2B ch15 note: Shapes (attributes) record: true - type: beast ref: BA 3A ch1 Shapes - type: singapore ref: PM2022 G2 ch9 Shapes sources: - CC 1.G.1 - CC 2.G.1 - research/math-spine.md §9 notes: van Hiele Level 1 (Descriptive). Directly targets the orientation-as-non-defining misconception ('doesn't matter which way it's turned', K5 p.8). BA defers Gr2 geometry to L3A (§5.B) — BA 3A ch1 is a DEEP resource (triangles, quadrilaterals, polyominoes). instrument_gap: true representations: C: move: 'Sort-and-say with blocks: give her a mixed pile and a rule card (''4 straight sides, closed'') — she pulls every shape that fits, ignoring color and size. Use a geoboard/rubber bands or sticks to BUILD a shape to a spec (''make a shape with 3 sides and 3 corners''). Physically rotate a built triangle to prove turning it doesn''t change what it is (orientation is non-defining).' check: Pull out every shape that has exactly 4 straight sides — does the color matter? Does it matter which way it's turned? Build me a shape with 5 sides. down: math.geometry.compose-decompose-shapes G: move: 'Give a ''What makes it a ___?'' contrast: two triangles (one fat, one skinny, one rotated) all count because all have 3 straight sides + closed; a color/size difference does NOT. Then dictate attributes and have her DRAW the shape (''draw a closed shape with 4 sides''). Name triangles/quadrilaterals/pentagons/hexagons by counting sides on the diagram.' check: Circle every shape that is a triangle. Why do the skinny one AND the fat one both count? Draw a shape with exactly 6 sides — what's it called? S: move: 'Attach category names to side/angle counts: 3 sides = triangle, 4 = quadrilateral, 5 = pentagon, 6 = hexagon. Write the defining attributes (''closed, 4 straight sides'') as the shape''s ''rule'', and label the non-defining ones (color, size, orientation) as things that DON''T decide the name.' check: A shape has 4 straight sides and is closed. What family is it in? Is 'it's blue' a reason it's a quadrilateral? - id: math.geometry.array-rows-columns title: Partition a rectangle into equal rows & columns of unit squares; count them strand: geometry band: - 7 - 8 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: partition a rectangle into rows and columns of same-size squares and count them without double-counting or losing count; see a rectangle as rows-and-columns' L3: SM-2 spaced gate on mixed items across separate days requires: - math.geometry.compose-decompose-shapes helps: [] resources: - type: beast ref: BA 3A ch3 Area and Perimeter - type: activity ref: grid/graph-paper rectangle rows-and-columns count sources: - CC 2.G.2 - research/math-spine.md §9 notes: HUB NODE — spatial structuring feeds multiplication arrays (D), area (measure), and partition-equal-area (geometry/fractions). 'More learning than is sometimes assumed'; at low levels kids double-count or lose count (K5 p.4-5,11). BA defers 2.G.2 to L3A ch3 (§5.B). instrument_gap: true representations: C: move: Tile a rectangle with same-size square tiles/cubes with NO gaps or overlaps, filling it completely; then count the tiles by touching each once. Build it row by row so she sees equal rows and equal columns. Snap unit-cube rows together so the rectangle is literally rows of columns — the hands-on antidote to double-counting or losing count. check: Fill this rectangle with the square tiles — no gaps, no overlaps. How many tiles fit? Count them by rows so you don't count one twice. down: math.geometry.compose-decompose-shapes G: move: On graph/grid paper, draw a rectangle on the lines and have her mark the equal rows and columns of unit squares, then count them systematically (skip-count by the row length, or count row-by-row) rather than one-at-a-time. Highlight one row and one column so 'rows AND columns' is visible — this is the structure that becomes area and multiplication arrays. check: This grid rectangle is 3 rows of 4 squares. Count the squares without counting any twice — how many, and how could you count faster than by ones? N: move: 'Real arrays: a muffin tin (rows and columns of cups), an egg carton, floor tiles, a chocolate bar''s grid of squares, window panes. ''How many muffins fit — count the rows.'' The everyday grid makes rows×columns concrete before it''s a formula.' check: This muffin tin has 2 rows and 6 cups in each row. How many muffins fit? How did you count them? - id: math.geometry.partition-equal-area title: Partition shapes into equal areas; each part = unit fraction of the whole strand: geometry band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: partition a shape into equal areas and express each part as a unit fraction of the whole (e.g. one part of a rectangle split into 4 = 1/4 the area)' L3: SM-2 spaced gate on mixed items across separate days requires: - math.geometry.array-rows-columns - math.fractions.unit-fraction-meaning helps: [] resources: - type: packet ref: fractions-2026-07-12 note: reinforces area-as-fraction (part 1 fair shares) - type: beast ref: BA 3D ch10 Fractions sources: - CC 3.G.2 - research/math-spine.md §9 notes: 'Bridges geometry and fractions: area partitioning = the geometric model of unit fractions. BA folds this into L3D ch10 Fractions (§5.B, 2.G.A.3 → BA 3D).' representations: C: move: Fold a paper rectangle into equal parts (halves, then fourths) and shade one part — that shaded part is 1/4 of the whole area. Cut it apart and stack the pieces to prove all four are the same amount of paper (equal AREA, even if you fold them into differently-shaped pieces). Bridges the fraction fold-work to the geometric whole. check: Fold this rectangle into 4 equal parts and shade one. What fraction of the whole rectangle is shaded? Are all four parts the same amount of paper? down: math.geometry.array-rows-columns G: move: 'Draw a shape and partition it into equal-area parts (using the grid/array structure), shade one part, and label it as a unit fraction of the whole: a rectangle split into 4 equal parts → each is 1/4 the area. Show TWO different partitions of the same square into fourths (four strips vs four squares) — different shapes, same 1/4 area (the equal-area-not-equal-shape idea).' check: This rectangle is split into 4 equal parts. What fraction of its area is one part? Here's a different way to cut it into fourths — is each part still 1/4? S: move: 'Connect the picture to the unit-fraction symbol: an equal-area part of a shape split into b parts = 1/b of the whole; write 1/4 under the shaded region of a fourths-partition. Tie it back to the fraction strand''s notation (denominator = number of equal parts of the whole).' check: Write the fraction that names one part when a shape is cut into 4 equal-area parts. If it were cut into 6 equal parts, what fraction is one part? - id: math.geometry.quadrilateral-categories title: Shapes share attributes → larger categories (quadrilateral hierarchy, intro) strand: geometry band: - 7 - 9 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: recognize that shapes sharing attributes form larger categories (rhombus/rectangle/square are all quadrilaterals); sort shapes into categories by shared attributes' L3: SM-2 spaced gate on mixed items across separate days requires: - math.geometry.defining-attributes helps: [] resources: - type: beast ref: BA 3A ch1 Shapes sources: - CC 3.G.1 - research/math-spine.md §9 notes: 'Gr3 = ''shares attributes → larger category'' ONLY; keep the full Venn/hierarchy (all squares are rectangles) OUT of the Gr3-4 band — kids resist it via visual prototype (K5 p.13,16). TRAPEZOID definition conflict (K5 p.3): prefer the INCLUSIVE definition (''at least one pair of parallel sides'') per the progression''s own diagrams; flag if a resource uses exclusive — don''t mark ''a parallelogram is a trapezoid'' wrong. Flagged for adjudication.' instrument_gap: true representations: C: move: Physically sort a pile of cut-out quadrilaterals (squares, rectangles, rhombi, trapezoids, general 4-sided shapes) into a 'quadrilateral' hoop because they all share '4 straight sides' — then sub-sort by extra attributes (right angles, all sides equal). Handling the shapes and grouping by SHARED attributes builds the 'these belong to a bigger family' idea (Gr3 level, not the full hierarchy yet). check: Put all the shapes with 4 straight sides in this hoop. What do a square, a rectangle, and this slanted shape all have in common that makes them one family? down: math.geometry.defining-attributes G: move: Draw a set of varied four-sided shapes and have her circle 'all the quadrilaterals' (everything with 4 straight sides), showing that rhombus/rectangle/square all qualify by the shared attribute. Keep it to 'shares attributes → same category' — do NOT push the full 'all squares are rectangles' hierarchy yet (Gr3-4 kids resist it via the visual prototype). Use the INCLUSIVE trapezoid (at least one pair of parallel sides). check: Circle every shape here that is a quadrilateral. Is the square in the same family as the long rectangle? What attribute do they share? S: move: 'Name the shared-attribute categories in words: quadrilateral = any closed shape with 4 straight sides; a square, rectangle, and rhombus are all KINDS of quadrilateral because they share that attribute. Write the category above its members. (Hold the nested ''a square is also a rectangle'' statement out of scope at this band.)' check: What do we call the whole family of shapes with 4 straight sides? Name two different shapes that both belong to that family. - id: math.geometry.lines-angles title: Points, lines, rays, angles (right/acute/obtuse); parallel & perpendicular; symmetry strand: geometry band: - 8 - 10 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: draw/identify points, lines, rays, angles (right/acute/obtuse) and ⊥ / ∥ lines; classify 2-D figures by parallel/perpendicular lines & angle size (incl. right triangles); recognize & draw lines of symmetry' L3: SM-2 spaced gate on mixed items across separate days requires: - math.geometry.defining-attributes helps: [] resources: - type: singapore ref: PM2022 G3 ch10 §10A Angles and Shapes - type: singapore ref: PMUS 3B u8 Geometry (Angles, Right Angles) - type: beast ref: BA 3A ch1 Shapes sources: - CC 4.G.1 - CC 4.G.2 - CC 4.G.3 - research/math-spine.md §9 notes: van Hiele Level 1→2. Prereq for angle-measure (measure strand). Merges 4.G.1/4.G.2/4.G.3 (lines/angles, classification, symmetry) into one node — one MA-module-sized cluster. instrument_gap: true representations: C: move: Build angles with two sticks/pencils or a bendy straw hinged at a point — open them to a square corner (right angle), narrower (acute), wider (obtuse); check 'square corner' against the corner of an index card. Use straws or road-lines to lay out parallel (never meet) vs perpendicular (meet at a square corner) lines. Fold a paper shape to find its line of symmetry (the halves match when folded). check: Open the sticks to make a right angle — check it with the card corner. Now make an angle that's smaller than a right angle. Fold this shape so the two halves match exactly — where's the fold line? down: math.geometry.defining-attributes G: move: Draw and label points, lines, rays, and angles; sort drawn angles into right / acute / obtuse using a right-angle corner as the referent. Mark parallel lines (arrows) and perpendicular lines (the small square). Classify 2-D figures by their angles/parallel sides (e.g. a right triangle has one right angle). Draw the line(s) of symmetry on a figure. check: Label the right, acute, and obtuse angle in this row. Draw a line of symmetry on this butterfly shape. Which of these figures has a pair of parallel sides? S: move: 'Introduce the notation and vocabulary that names the pictures: point, line, ray, angle; right/acute/obtuse; parallel (∥) and perpendicular (⊥). Match each symbol/term to the drawn figure so the words are grounded (a ray has one endpoint and goes on forever; ⊥ means a square-corner crossing). Classify figures in words by angle and line properties.' check: Which symbol means 'perpendicular'? What do you call an angle bigger than a right angle? Is a right angle the same as, bigger than, or smaller than an acute angle? - id: math.placevalue.decimal-place-value-round title: Deeper decimal place value to thousandths; read/write/compare; round decimals strand: placevalue band: - 8 - 11 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: name decimal place value to thousandths (each place = 1/10 the place to its left), read/write decimals in numeral/word/expanded form, compare two decimals, and round a decimal to any place' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.fractions.decimal-notation - math.placevalue.big-numbers-round helps: [] resources: - type: aops ref: AoPS Prealgebra §6.2 Rounding record: true - type: singapore ref: PM2022 G4/G5 Decimals (place value to thousandths) sources: - CC 5.NBT.1 - CC 5.NBT.3 - CC 5.NBT.4 - research/prealgebra-extension.md §5 (D1) notes: 'Deeper decimal place value: 5.NBT.1 (each place = 1/10 the one to its left, 10× to its left), 5.NBT.3 (read/write/compare to thousandths), 5.NBT.4 (round decimals). Placed in placevalue (base-ten) not a decimals strand, per adjudication. Misconception: ''longer decimal = bigger'' (0.45 > 0.5 because 45 > 5) — align by place value, not digit count; treating the decimal point as a mirror line for symmetry of place names (there is no ''oneths'').' instrument_gap: true representations: G: move: shade thousandths on a base-ten flat/rod/unit grid; stack two decimals in a place-value chart aligned on the decimal point to compare check: which is greater, 0.4 or 0.38? show it on the grid S: move: write the decimal in expanded form (0.372 = 3×0.1 + 7×0.01 + 2×0.001); round by finding the nearest benchmark place check: round 4.276 to the nearest hundredth N: move: read a race time or a price to the thousandth of a second / cent and say which of two is smaller check: runner A is 12.08s, runner B is 12.1s — who won? down: math.fractions.decimal-notation - id: math.placevalue.decimal-four-ops title: Add, subtract, multiply & divide decimals; fluent decimal arithmetic strand: placevalue band: - 9 - 12 tier: L4 fluency_aim: metric: problems_per_min_written target: 9 sustain: 2 of 3 separate days, ≤2 errors/min source: research/fluency-aims.md §decimal-ops (UNGROUNDED — section not yet authored) gates: L2: '≥90% on 10 fresh items, untimed: add/subtract/multiply/divide decimals to hundredths using models, drawings & place-value strategies; explain where the decimal point goes' L3: SM-2 spaced gate on mixed four-op decimal items across separate days OR parent-observed spaced re-check after a gap of days (understanding gate — fires BEFORE the L4 rate gate) L4: 1-minute written timing on mixed multi-digit decimal +/−/×/÷ items via the standard algorithm; rate per fluency_aim (UNGROUNDED — pending research/fluency-aims.md §decimal-ops); ≤2 errors; 2 of 3 separate days requires: - math.placevalue.decimal-place-value-round - math.addsub.add-sub-standard-algorithm - math.fractions.multiply-divide-fractions helps: - math.ratio.percent-concept resources: - type: aops ref: AoPS Prealgebra §6.1 Arithmetic with Decimals record: true - type: aops ref: AoPS Prealgebra §6.4 Repeating Decimals - type: beast ref: BA-4 Decimals - type: beast ref: BA-5 Decimals sources: - CC 5.NBT.7 - CC 6.NS.3 - research/prealgebra-extension.md §3 - research/prealgebra-extension.md §5 (D2) notes: 'UNGROUNDED: fluency_aim target is a directional prior — no research/fluency-aims.md §decimal-ops section exists yet; needs grounding before use. THE CC-''fluently'' L4 of §3 (6.NS.3: fluently add/subtract/multiply/divide multi-digit decimals via the standard algorithm; 5.NBT.7 is the strategies/understanding rung). fluency_aim is UNGROUNDED: research/fluency-aims.md has no §decimal-ops rate aim yet — build-signal, NOT a guessed rate. Directional prior (to validate, NOT authored as fact): ~8–10 problems/min, multidigit-compound band. L4 gate fires only AFTER the L3 understanding gate. Requires standard-algorithm (÷ leans on that) + multiply-divide-fractions (decimal ÷ ties to fraction division). Misconception: ''count the total decimal places'' misapplied to addition (line up the points, not the right edges); dropping the point in multiplication; not extending the dividend with zeros in division.' instrument_gap: true representations: G: move: add/subtract on a place-value chart with the decimal points stacked; multiply via an area (box) model where tenths×tenths tile into hundredths check: why does 0.3 × 0.2 = 0.06, not 0.6? show the box S: move: line up the decimal points for +/−; for × ignore the point, multiply, then count total decimal places; for ÷ shift both points to make a whole-number divisor check: compute 4.2 × 0.05 and explain the point placement N: move: total a grocery receipt, split a bill evenly, or convert a per-item price — real money forces the point to matter check: $12.60 split among 4 people — how much each? down: math.placevalue.decimal-place-value-round - id: math.numbertheory.factors-multiples title: Factors, multiples, factor pairs; multiplicative structure to 100 strand: numbertheory band: - 8 - 11 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: list all factor pairs of a number ≤100, list multiples of a number, and tell a factor from a multiple in context' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.multdiv.mult-div-facts-memory-100 - math.multdiv.multistep-4ops-patterns helps: [] resources: - type: aops ref: AoPS Prealgebra §3.1 Multiples / §3.6 Divisors record: true - type: beast ref: BA-5 Factors & Multiples sources: - CC 4.OA.4 - CC 6.NS.4 - research/prealgebra-extension.md §4A (N1) notes: 'Consolidation/deepening node — the grade-4 4.OA.4 seed (factor pairs / prime-composite / multiples) already lives in multistep-4ops-patterns; this makes factor/multiple structure a first-class rung feeding divisibility, primes, GCF/LCM. Do NOT re-file the grade-4 seed. 6.NS.2 long-division fluency is an extension of the existing multidigit node, noted here, not a separate L4. Misconception: ''factor'' vs ''multiple'' confusion (a factor divides IN, a multiple is built OUT); forgetting 1 and n are always factors of n.' instrument_gap: true representations: C: move: build a number as a rectangle of tiles every way it factors (12 = 1×12, 2×6, 3×4); a prime has only the 1×n strip check: make every rectangle for 18 — what are its factors? G: move: draw a factor rainbow pairing factors from the outside in; skip-count on a number line to land the multiples check: draw the factor rainbow for 24 N: move: arrange 24 cookies onto identical plates with none left over — each arrangement is a factor pair check: how many equal plates can hold 24 cookies with none left over? - id: math.numbertheory.divisibility-rules title: Divisibility rules (2, 3, 4, 5, 6, 9, 10) and why they work strand: numbertheory band: - 9 - 12 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: apply divisibility rules for 2/3/4/5/6/9/10 to decide (without dividing) whether a number is divisible, and explain why the digit-sum rule works for 3 and 9' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.numbertheory.factors-multiples helps: - math.numbertheory.prime-factorization resources: - type: aops ref: AoPS Prealgebra §3.2 Divisibility Tests record: true - type: beast ref: BA-4 Factors (divisibility) sources: - research/prealgebra-extension.md §4A (N2) - AoPS Prealgebra §3.2 notes: 'AoPS-surfaced first-class node (per adjudication). Not a standalone CC standard — CC folds divisibility into 4.OA.4 / 6.NS.4. Cited to AoPS §3.2, NOT fabricated to CC. Misconception: applying the digit-sum rule (÷3, ÷9) to other divisors; treating ''ends in an even digit'' as sufficient for ÷4 (need the last TWO digits); assuming a ÷6 rule is just ''even'' (need ÷2 AND ÷3).' instrument_gap: true representations: S: move: test 2/5/10 by the last digit, 4 by the last two digits, 3/9 by the digit sum, 6 by ÷2 and ÷3 together check: is 3,564 divisible by 9? by 6? show the test N: move: decide if a bag of items can be split evenly among 3 (or 4, or 6) friends WITHOUT dividing — just by inspecting the number check: can 132 stickers be shared evenly among 6 kids? use a rule, not division - id: math.numbertheory.primes-composites title: Prime & composite numbers; the sieve; primes as building blocks strand: numbertheory band: - 9 - 12 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: classify a number ≤100 as prime or composite, use the Sieve of Eratosthenes to find primes, and explain why 1 is neither prime nor composite' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.numbertheory.divisibility-rules helps: [] resources: - type: aops ref: AoPS Prealgebra §3.3 Prime Numbers record: true - type: beast ref: BA-4 Factors - type: beast ref: BA-5 Factors & Multiples sources: - CC 4.OA.4 - research/prealgebra-extension.md §4A (N3) - AoPS Prealgebra §3.3 notes: 'The grade-4 prime/composite seed (≤100) is in multistep-4ops-patterns; this node OWNS it going forward + adds the sieve and ''why 1 is neither.'' Uses divisibility rules to test candidates quickly. Misconception: ''1 is prime'' (it is neither); ''all odd numbers are prime'' (9, 15, 21…); assuming primes must be small.' instrument_gap: true representations: C: move: try to build each number as a NON-strip rectangle of tiles — the ones that only make a 1×n strip are prime check: which of 7, 8, 9 can only be a single strip? those are prime G: move: 'run the Sieve of Eratosthenes on a hundreds chart: circle 2, cross its multiples, repeat; the survivors are prime' check: after crossing multiples of 2, 3, 5, 7 — why is everything left prime? N: move: primes are the 'atoms' of numbers — you cannot split a group of a prime size into equal smaller groups (except 1s) check: can 13 chairs be set in equal rows bigger than 1? why not? - id: math.numbertheory.prime-factorization title: Prime factorization; factor trees; exponent form of a factorization strand: numbertheory band: - 9 - 12 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: find the prime factorization of a number with a factor tree and write it in exponent form (72 = 2^3 × 3^2)' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.numbertheory.primes-composites helps: - math.algebra.write-evaluate-exponents - math.fractions.multiply-divide-fractions resources: - type: aops ref: AoPS Prealgebra §3.4 Prime Factorization record: true - type: beast ref: BA-4 Factors (factor trees) - type: beast ref: BA-5 Factors & Multiples sources: - research/prealgebra-extension.md §4A (N4) - AoPS Prealgebra §3.4 notes: 'Fundamental Theorem of Arithmetic, informally: every integer >1 has a UNIQUE prime factorization. Feeds GCF/LCM and fraction simplest-form. Exponent notation first appears here (helps into write-evaluate-exponents; also helped by it). No standalone CC code — cited to AoPS §3.4. Misconception: stopping the factor tree before all leaves are prime; thinking two different trees give different factorizations (uniqueness — different trees, same primes).' instrument_gap: true representations: G: move: grow a factor tree, splitting each composite branch until every leaf is prime; circle the prime leaves check: build the factor tree for 60 — what are the prime leaves? S: move: collect the prime leaves and write the product in exponent form, smallest prime first check: write 200 as a product of primes in exponent form - id: math.numbertheory.gcf-lcm title: Greatest common factor & least common multiple (via listing and via prime factorization) strand: numbertheory band: - 10 - 12 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: find the GCF of two numbers ≤100 and the LCM of two numbers ≤12 by listing AND from prime factorizations; use GCF to factor a sum (36 + 8 = 4(9 + 2))' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.numbertheory.prime-factorization helps: - math.fractions.multiply-divide-fractions - math.fractions.mixed-numbers - math.algebra.expressions-evaluate-write resources: - type: aops ref: AoPS Prealgebra §3.5 LCM / §3.7 GCD record: true - type: beast ref: BA-5 Factors & Multiples sources: - CC 6.NS.4 - research/prealgebra-extension.md §4A (N5) - AoPS Prealgebra §3.5 notes: '6.NS.4: GCF of two numbers ≤100, LCM of two ≤12, and the distributive-factoring of a sum (36 + 8 = 4(9 + 2)) — that last part helps into expressions. Feeds fraction simplest-form + adding unlike denominators. Misconception: swapping GCF↔LCM; ''GCF is always bigger than the numbers'' (it is ≤ both); computing LCM as the plain product (only correct when the numbers are coprime).' instrument_gap: true representations: G: move: list factors of each number, ring the shared ones, take the biggest (GCF); list multiples of each, ring the shared ones, take the smallest (LCM); OR use a Venn of the prime factors — overlap = GCF, whole diagram = LCM check: put the prime factors of 12 and 18 in a Venn — read off GCF and LCM N: move: GCF = biggest equal-size groups you can make from two piles with none left over; LCM = when two repeating events (buses every 12 min, every 18 min) next coincide check: two lights blink every 4s and every 6s — when do they blink together again? - id: math.fractions.multiply-divide-fractions title: Multiply fractions (× fraction, × whole); divide fractions; multiplication as scaling strand: fractions band: - 9 - 12 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: multiply fraction × fraction and fraction × whole (area meaning), divide a unit fraction by a whole and a whole by a unit fraction and (a/b)÷(c/d); predict whether × makes a number bigger or smaller (× by <1 shrinks); give the answer in simplest form' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.fractions.add-sub-mult-like - math.fractions.equivalence-compare-4 helps: - math.numbertheory.gcf-lcm resources: - type: aops ref: AoPS Prealgebra §4.2 Multiplying Fractions / §4.3 Dividing by a Fraction / §4.5 Simplest Form record: true - type: beast ref: BA-4 Fractions (multiplying, division by unit fractions) - type: beast ref: BA-5 Fractions sources: - CC 5.NF.4 - CC 5.NF.5 - CC 5.NF.6 - CC 5.NF.7 - CC 6.NS.1 - research/prealgebra-extension.md §4E (F1) notes: '5.NF.4–7: (a/b)×q, area with fractional sides, multiplication as scaling/resizing (×<1 shrinks, ×>1 grows), divide unit-fraction÷whole & whole÷unit-fraction; 6.NS.1 completes GENERAL fraction division ((a/b)÷(c/d) = ad/bc) with a visual justification. Simplest form leans on GCF (helps from gcf-lcm — kept as helps to avoid a numbertheory→fractions hard gate that would over-couple the strands). Misconception: ''multiplying always makes bigger'' (false for factors <1, 5.NF.5b targets it); inverting the WRONG fraction when dividing; ''flip and multiply'' with no meaning.' instrument_gap: true representations: G: move: multiply with an area model — shade 2/3 of a rectangle one way and 3/4 the other; the double-shaded overlap is the product; for division, ask 'how many 1/4s fit in 2?' check: show 2/3 × 3/4 on a grid — why is the answer 1/2? C: move: 'fold paper: fold in thirds, then fold that in fourths, unfold — count the double-folded region against the whole' check: fold to show 1/2 of 1/3 N: move: half of a half-pizza; how many quarter-cups fit in 3 cups of flour — real 'of' and 'how many fit' situations check: a recipe needs 3/4 cup, you make half — how much? down: math.fractions.add-sub-mult-like - id: math.fractions.mixed-numbers title: Mixed numbers ↔ improper fractions; operate with mixed numbers (all four operations, unlike denominators) strand: fractions band: - 9 - 12 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: convert mixed↔improper, and add/subtract/multiply/divide mixed numbers including UNLIKE denominators (via equivalent fractions)' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.fractions.multiply-divide-fractions helps: - math.numbertheory.gcf-lcm resources: - type: aops ref: AoPS Prealgebra §4.7 Adding and Subtracting Fractions / §4.8 Mixed Numbers record: true - type: beast ref: BA-4 Fractions - type: beast ref: BA-5 Fractions sources: - CC 4.NF.3c - CC 5.NF.1 - research/prealgebra-extension.md §4E (F2) notes: '5.NF.1: add/subtract fractions with UNLIKE denominators (incl. mixed) via equivalent fractions — the piece past the existing add-sub-mult-like (LIKE-denominators only). Uses GCF/LCM for the common denominator (helps). Misconception: adding whole parts and fraction parts separately then mishandling the regroup; converting 3¼ → 13/4 by ADDING (3+4+1) instead of (3×4)+1.' instrument_gap: true representations: G: move: draw the whole circles plus the fractional part; to add unlike denominators, re-slice both pictures to a common number of pieces (the LCM) first check: draw 1½ + ¾ — what common slice size makes both pictures match? S: move: convert mixed→improper before multiplying/dividing; for +/− find a common denominator, then regroup wholes check: compute 2⅓ − ⅚, showing the common denominator step N: move: combine board lengths, recipe amounts, or times given as whole-plus-part quantities check: two boards are 2¼ ft and 1½ ft — total length? - id: math.integers.negatives-number-line title: Negative numbers in context; the two-way number line; opposites & absolute value strand: integers band: - 9 - 12 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: place negatives on a two-way number line, name a number''s opposite, interpret negatives in context (temperature, credit/debt, elevation), and give the absolute value as distance from 0' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.placevalue.big-numbers-round helps: - math.measure.length-ruler resources: - type: aops ref: AoPS Prealgebra §1.4 Negation record: true - type: beast ref: BA-4 Integers sources: - CC 6.NS.5 - CC 6.NS.6a - CC 6.NS.6c - CC 6.NS.7 - research/prealgebra-extension.md §4B (I1) notes: '6.NS.5: pos/neg describe opposite directions (temp above/below 0, credits/debts). 6.NS.7c: absolute value = distance from 0. requires big-numbers-round (whole-number number line + place value); the number line = a two-way ruler (helps from length-ruler). Misconception: ''−8 > −3 because 8 > 3'' (ordering flips for negatives, 6.NS.7a); |−5| read mechanically as ''make it positive'' instead of ''distance''; confusing ''−'' as an operation vs a sign.' instrument_gap: true representations: C: move: use two-color counters (yellow = +1, red = −1); a yellow-red pair is a 'zero pair' worth 0 — build 3 and −3 as opposites check: show me −4 and its opposite with counters G: move: extend the number line left of 0; mark a number and its opposite as equal hops on either side; |n| is the length of the hop check: on the line, why are −5 and 5 the same distance from 0? N: move: read a thermometer below zero, a bank balance in the red, or floors below ground — negatives are directions from a chosen zero check: the temperature is −7°F and rises to −2°F — did it get warmer or colder? - id: math.integers.compare-order-rationals title: Compare & order rational numbers (negatives, fractions, decimals) on the number line strand: integers band: - 10 - 12 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: order a mixed set of negatives, fractions and decimals on a number line; read −3 > −7 as ''right of on the line''; interpret absolute value in a debt/balance context' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.integers.negatives-number-line - math.fractions.decimal-notation helps: [] resources: - type: aops ref: AoPS Prealgebra §1.4 Negation record: true sources: - CC 6.NS.7a - CC 6.NS.7b - CC 6.NS.7d - research/prealgebra-extension.md §4B (I2) notes: '6.NS.7a: interpret −3 > −7 as ''−3 is right of −7 on the line''; 6.NS.7c–d: absolute value in real-world debt contexts (a balance of −30 is a debt of 30; ''less than −30'' = a bigger debt). requires decimal-notation so a mixed negative-fraction / negative-decimal set can share a representation. Misconception: conflating ''farther from zero'' with ''greater''; comparing a negative fraction and a negative decimal without converting to a common form.' instrument_gap: true representations: G: move: plot the whole mixed set on one number line; leftmost = least; convert fractions/decimals to a common form to place them precisely check: order −1.5, −3/4, 0.2, −2 on the line N: move: rank bank balances or temperatures — a −$50 balance is 'less' (worse) than −$20 even though 50 > 20 check: 'who is worse off: someone at −$50 or someone at −$20? which number is smaller?' - id: math.integers.integer-add-subtract title: 'Add & subtract integers (and rationals): number-line hops, zero pairs, subtract = add the opposite' strand: integers band: - 10 - 13 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: add and subtract integers using number-line hops and zero pairs; rewrite subtraction as adding the opposite (p − q = p + (−q)); give the distance between two numbers as |difference|' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.integers.compare-order-rationals helps: - math.algebra.one-step-equations resources: - type: aops ref: AoPS Prealgebra §1.4 Negation / §1.5 Subtraction record: true - type: beast ref: BA-4 Integers (adding & subtracting) sources: - CC 7.NS.1 - research/prealgebra-extension.md §4B (I3) notes: '7.NS.1 (all of a–d): p+q as a distance |q| from p in the +/− direction; opposites sum to 0 (additive inverses); subtraction of rationals = adding the additive inverse (p − q = p + (−q)); distance between two numbers = |difference|. THE canonical error: ''−5 − 3 = −2'' (treating the magnitude as shrinking); ''−5 − (−3)'' sign-of-sign confusion; ''two minuses make a plus'' over-applied to addition.' instrument_gap: true representations: C: move: with two-color counters, adding is combining piles and removing zero pairs; subtracting is 'take away' — if you can't take away, add zero pairs first to make the removal possible check: model −5 − (−3) with counters — what do you take away? G: move: on the number line, start at p, then hop |q| RIGHT for +q or LEFT for −q; rewrite any subtraction as add-the-opposite before hopping check: walk −2 − 4 on the line by first rewriting it as −2 + (−4) N: move: track a temperature that rises and falls, or a balance with deposits and withdrawals — a withdrawal from a debt deepens it check: you owe $5, then spend $3 more — what's your balance? - id: math.integers.integer-four-ops title: Multiply & divide integers; fluent four-operation signed arithmetic strand: integers band: - 10 - 13 tier: L4 fluency_aim: metric: problems_per_min_oral target: 30 sustain: 2 of 3 separate days, ≤2 errors/min source: research/fluency-aims.md §integer-facts (UNGROUNDED — section not yet authored) gates: L2: '≥90% on 10 fresh items, untimed: multiply and divide integers with correct signs, explain (−1)(−1)=1 via the distributive property, and apply −(p/q) = (−p)/q = p/(−q)' L3: SM-2 spaced gate on mixed signed +/−/×/÷ items across separate days OR parent-observed spaced re-check after a gap of days (understanding gate — fires BEFORE the L4 rate gate) L4: oral 1-minute timing on mixed signed +/−/×/÷ single-digit items; rate per fluency_aim (UNGROUNDED — pending research/fluency-aims.md §integer-facts); ≤2 errors; 2 of 3 separate days requires: - math.integers.integer-add-subtract - math.multdiv.mult-div-facts-memory-100 helps: - math.algebra.two-step-equations resources: - type: aops ref: AoPS Prealgebra §1.3 Multiplication / §1.4 Negation record: true - type: beast ref: BA-4 Integers sources: - CC 7.NS.2 - CC 7.NS.3 - research/prealgebra-extension.md §3 - research/prealgebra-extension.md §4B (I4) notes: 'UNGROUNDED: fluency_aim target is a directional prior — no research/fluency-aims.md §integer-facts section exists yet; needs grounding before use. The pedagogically-motivated L4 of §3 (not CC-''fluently'' but the same automaticity logic — signed-arithmetic automaticity frees WM for equation-solving, DESIGN.md §1). 7.NS.2a: multiplication extends to rationals preserving distributivity → (−1)(−1)=1 and the sign rules; 7.NS.2b: integer division (nonzero divisor), −(p/q) = (−p)/q = p/(−q). fluency_aim is UNGROUNDED: research/fluency-aims.md has no §integer-facts rate yet — build-signal, NOT a guessed rate. Directional prior (to validate, NOT authored as fact): ~30–40/min oral (near the mult-facts band). L4 gate fires only AFTER the L3 understanding gate. Misconception: ''−3 × −4 = −12'' (sign rule half-applied); mishandling division signs; forgetting 0/n = 0 but n/0 is undefined.' instrument_gap: true representations: C: move: 'multiply as repeated groups of counters: 3 × (−2) is three groups of two red; negative × is ''remove that many groups'' — removing red counters leaves yellow, so −×− is +' check: why does −3 × −2 come out positive? model it S: move: chant the sign rule (same signs → +, different signs → −) then compute the magnitude; check division against multiplication check: compute −24 ÷ 4 and −24 ÷ −6 N: move: repeated debt (−$2 four times = −$8); 'undoing' repeated debt reverses the sign — the ledger interpretation of the sign rules check: losing $2 per day for 3 days changes your balance by how much? - id: math.algebra.write-evaluate-exponents title: 'Exponents: powers, squares & cubes; write/evaluate powers; light square-root intro' strand: algebra band: - 9 - 12 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: write and evaluate whole-number powers (2^3 = 8), recognize squares and cubes, and use that √ undoes squaring for perfect squares (√49 = 7)' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.multdiv.mult-div-facts-memory-100 helps: - math.numbertheory.prime-factorization - math.algebra.order-of-operations resources: - type: aops ref: AoPS Prealgebra §2.1 Squares / §2.2 Higher Exponents / §9.1 From Squares to Square Roots record: true - type: beast ref: BA-4 Exponents (perfect squares) - type: beast ref: BA-5 Exponents / Square Roots sources: - CC 6.EE.1 - CC 5.NBT.2 - research/prealgebra-extension.md §4C (A1) notes: '6.EE.1: write/evaluate numerical expressions with whole-number exponents; 5.NBT.2 (powers of 10). Squares/cubes/higher powers; 2^3 = 8; light intro that √ undoes squaring (perfect squares only, AoPS §9.1). Do NOT push zero/negative exponents (AoPS §2.3–2.4) or non-perfect-square roots (§9.2) — pre-algebra-DEEP, kept as stretch in notes, honoring ''not excessively far.'' Misconception: ''2^3 = 6'' (base×exponent); ''5^2 = 10''; conflating 2^3 and 3^2.' instrument_gap: true representations: C: move: build a square (n rows of n) and a cube (n×n×n) with tiles/blocks — the count IS n² and n³; a perfect square is the one that makes a full square check: build 5² with tiles — how many? why is it called 'squared'? S: move: expand a power as repeated multiplication (2^4 = 2×2×2×2) before evaluating; read √ as 'what squares to this?' check: evaluate 3^3 by writing it out; find √64 N: move: area of a square patch (side²), volume of a cube box (side³) — powers count repeated growth, not repeated addition check: a square garden is 6 ft on a side — how many square feet? - id: math.algebra.expressions-evaluate-write title: Variables; write & evaluate algebraic expressions; identify parts (term, coefficient, factor) strand: algebra band: - 10 - 13 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: write an expression from words (''subtract y from 5'' = 5 − y), read 3x as ''3 times x'', identify terms/coefficients/factors/sum/product, and evaluate an expression (incl. from a formula like V = s³) at given values, including signed values' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.algebra.write-evaluate-exponents - math.multdiv.multistep-4ops-patterns - math.integers.integer-add-subtract helps: - math.numbertheory.gcf-lcm resources: - type: aops ref: AoPS Prealgebra §5.1 Expressions record: true sources: - CC 6.EE.2 - CC 5.OA.2 - research/prealgebra-extension.md §4C (A2) notes: '6.EE.2 (a–c): write expressions with letters for numbers, identify sum/term/product/factor/quotient/coefficient, evaluate at values incl. from formulas; 5.OA.2 (write simple expressions). requires integer-add-subtract so evaluation at SIGNED values works. Also absorbs the light 6.EE.3/6.EE.4 equivalent-expression / distributive work, helped by gcf-lcm''s 36+8=4(9+2) factoring (per §8 folding note; a separate equivalent-expressions node is the most defensible 43rd node if this feels overloaded — flagged). Misconception: writing ''5 − y'' as ''y − 5'' (order); reading 3x as ''3 and x'' not ''3 times x''; treating the letter as a fixed unknown vs a varying quantity.' instrument_gap: true representations: C: move: with algebra tiles, an x-tile is an unknown length and a unit tile is 1; build 2x + 3 as two x-tiles and three units; 'evaluate at x=4' means swap each x-tile for a length-4 strip check: build 2x + 3 with tiles, then find its value when x = 4 S: move: translate word phrases to symbols term by term; name the coefficient, the variable, and the constant; substitute and simplify check: write 'seven more than triple a number' and evaluate at n = 2 N: move: a rule for a real quantity — cost = 3 × (number of tickets); the variable stands for the thing that can change check: if each ticket is $3, write the cost for t tickets, then the cost for 5 - id: math.algebra.order-of-operations title: Order of operations (grouping, exponents, ×÷, +−); GEMS/PEMDAS with nested grouping strand: algebra band: - 10 - 13 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: evaluate multi-operation numerical expressions with grouping symbols (parentheses, brackets, braces) and exponents in the conventional order, doing ×÷ left-to-right and +− left-to-right' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.algebra.write-evaluate-exponents helps: - math.algebra.expressions-evaluate-write resources: - type: aops ref: AoPS Prealgebra §5.1 (order of operations) record: true - type: beast ref: BA-4 Exponents (order of operations) sources: - CC 5.OA.1 - CC 6.EE.2c - research/prealgebra-extension.md §4C (A3) notes: '5.OA.1 introduces grouping symbols; 6.EE.2c folds exponents into the conventional order. Teach GEMS (Grouping, Exponents, Mult/div left-to-right, Add/sub left-to-right) to defuse the PEMDAS ''multiply-before-divide'' trap. Misconception: doing ALL multiplication before division (they are co-equal, left-to-right); assuming a − b + c = a − (b + c).' instrument_gap: true representations: G: move: box the innermost grouping first and work outward; underline exponents as the next layer; annotate each step so the order is visible check: evaluate 2 + 3 × (4 + 1)² — box what you do first S: move: apply GEMS as a checklist, resolving ×÷ strictly left-to-right and +− strictly left-to-right (not P-then-E-then... blindly) check: why is 8 ÷ 2 × 2 = 8, not 2? - id: math.algebra.one-step-equations title: Solve one-step equations (px = q, x + p = q) by inverse operations; a solution makes it true strand: algebra band: - 10 - 13 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: solve x + p = q and px = q for nonnegative rationals by applying the inverse operation to BOTH sides; check a candidate by substitution; explain the solution as the value that makes the equation true' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.algebra.expressions-evaluate-write - math.integers.integer-add-subtract - math.addsub.equal-sign-unknown helps: - math.multdiv.mult-div-word-problems resources: - type: aops ref: AoPS Prealgebra §5.2 Solving Linear Equations I record: true sources: - CC 6.EE.5 - CC 6.EE.6 - CC 6.EE.7 - research/prealgebra-extension.md §4C (A4) notes: '6.EE.5: solving = find which value makes the equation true (check by substitution); 6.EE.7: solve x + p = q and px = q for nonnegative rationals. THE load-bearing prereq is the equal-sign-unknown node (''='' as balance, not ''answer comes next'') — the K–4 spine flagged this misconception directly. Misconception: ''undoing'' with the SAME operation instead of the inverse; applying the inverse to only one side.' instrument_gap: true representations: C: move: 'the balance/pan-scale model: an equation is a level balance; whatever you remove or add, do to BOTH pans to keep it level; x + 3 = 7 → take 3 off each pan' check: model x + 3 = 7 on a balance — what keeps it level? S: move: isolate the variable by applying the inverse operation to both sides; substitute the answer back to verify check: solve 4x = 20 and check by substitution N: move: '''I had some, added 3, now have 7 — how many did I start with?'' — the unknown is the start, the inverse recovers it' check: a number times 5 is 35 — what's the number? down: math.addsub.equal-sign-unknown - id: math.algebra.two-step-equations title: Solve simple two-step equations (px + q = r, p(x + q) = r) — the FRONTIER equation node strand: algebra band: - 11 - 13 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: solve px + q = r and p(x + q) = r for rational p,q,r by undoing operations in the correct order; compare the algebraic solution to an arithmetic one' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.algebra.one-step-equations - math.integers.integer-four-ops helps: [] resources: - type: aops ref: AoPS Prealgebra §5.2–5.3 Solving Linear Equations I & II / §5.4 Word Problems record: true sources: - CC 7.EE.4a - CC 7.EE.3 - research/prealgebra-extension.md §4C (A5) notes: '7.EE.4a: solve px + q = r and p(x + q) = r (p,q,r rational) FLUENTLY; compare the algebraic to an arithmetic solution. THE TOP OF THE GRAPH — stops short of Algebra 1 (no variables-on-both-sides, no general linear, no systems). requires integer-four-ops (fluent signed arithmetic makes the two steps affordable). Misconception: undoing in the wrong order (add/sub before mult/div when isolating); distributing errors in p(x + q).' instrument_gap: true representations: C: move: 'balance model in two moves: first remove the constant q from both pans, THEN split the remaining pans into p equal parts (undo × last)' check: on a balance, solve 2x + 3 = 11 — which do you undo first, the +3 or the ×2? S: move: 'undo in reverse order of operations: subtract/add the constant, then divide/multiply; distribute first for p(x + q); verify by substitution' check: solve 3(x − 2) = 12 two ways (distribute vs divide first) N: move: '''a taxi costs $3 plus $2 a mile; the ride was $11 — how many miles?'' — the two steps peel off the flat fee then the per-unit rate' check: set up and solve the taxi problem down: math.algebra.one-step-equations - id: math.algebra.inequalities-intro title: 'Inequalities: write & graph x > c / x < c; solution SETS; simple two-step inequalities' strand: algebra band: - 11 - 13 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: write x > c / x < c for a constraint, recognize such inequalities have infinitely many solutions, graph the solution set on a number line (open/closed circle), and solve simple px + q > r' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.algebra.one-step-equations - math.integers.compare-order-rationals helps: [] resources: - type: aops ref: AoPS Prealgebra §5.5 Inequalities record: true sources: - CC 6.EE.8 - CC 7.EE.4b - research/prealgebra-extension.md §4C (A6) notes: '6.EE.8: x > c / x < c as a constraint, recognize INFINITELY many solutions, graph on a number line; 7.EE.4b extends to px + q > r. Keep it one-variable; the flip-on-multiply-by-a-negative subtlety is an edge item (noted), NOT a fluency requirement here. Misconception: treating an inequality solution as a single number; open vs closed circle on the graph; forgetting the sign flips when multiplying/dividing by a negative.' instrument_gap: true representations: G: move: 'graph the solution on a number line: open circle for > / < (not included), closed for ≥ / ≤, arrow toward all the values that work' check: graph x > 3 — why an open circle and an arrow, not a single dot? S: move: solve like an equation to find the boundary, then decide the direction; the solution is a SET, not one value check: solve 2x − 1 < 5 and describe every number that works N: move: '''you must be at least 48 inches to ride'' or ''spend under $20'' — real constraints that many values satisfy' check: if a ride needs height ≥ 48 in, graph who can ride - id: math.algebra.dependent-independent-vars title: 'Two related quantities: dependent vs independent variable; equations, tables & graphs of a relationship' strand: algebra band: - 11 - 13 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: for two quantities that change together, name the dependent and independent variable, write an equation (dependent in terms of independent, e.g. d = 65t), and move among equation, table, and graph' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.algebra.expressions-evaluate-write - math.geometry.coordinate-plane - math.ratio.rates-unit-rates helps: [] resources: - type: aops ref: AoPS Prealgebra §5.4 (word problems) record: true sources: - CC 6.EE.9 - CC 5.OA.3 - research/prealgebra-extension.md §4C (A7) notes: '6.EE.9: variables for two quantities that change together, write an equation (dependent in terms of independent), analyze with graphs & tables (d = 65t); 5.OA.3 (generate patterns, ordered pairs). DELIBERATELY STOPS SHORT of ''function'' — relationship-between-quantities, NOT function notation/domain/range (grade 8, OUT). The conceptual seed of functions without the machinery. requires coordinate-plane + rates-unit-rates (the y=kx proportional relationship it graphs). Misconception: swapping which variable depends; reading a table as unordered.' instrument_gap: true representations: G: move: fill a table of matched values, plot the ordered pairs on the coordinate plane, and read the pattern (straight line through origin for d = 65t) check: table and plot d = 65t for t = 0,1,2,3 — what shape appears? S: move: write the dependent variable in terms of the independent one; substitute independent values to generate the table check: for c = 3t, which variable depends on which? make the table N: move: distance depends on time at a fixed speed; total cost depends on how many you buy — one quantity drives the other check: at 65 mph, how far in 3 hours? which quantity is the independent one? down: math.ratio.rates-unit-rates - id: math.ratio.ratio-concept title: Ratios & ratio language; equivalent ratios; tables & tape/double-number-line diagrams strand: ratio band: - 10 - 13 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: describe a relationship with ratio language (2:1, ''for every 2 …''), build a table of equivalent ratios, and represent a ratio with a tape diagram or double number line' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.multdiv.mult-div-facts-memory-100 - math.fractions.equivalence-compare-4 helps: [] resources: - type: aops ref: AoPS Prealgebra §7.1 What is a Ratio / §7.2 Multi-way Ratios record: true - type: beast ref: BA-5 Ratios & Rates sources: - CC 6.RP.1 - CC 6.RP.3a - research/prealgebra-extension.md §4D (R1) notes: '6.RP.1: ratio as a multiplicative relationship (''2:1, for every 2 wings there was 1 beak''); 6.RP.3a: tables of equivalent ratios, tape diagrams, plot on the plane. requires equivalence-compare-4 (equivalent-ratio reasoning is equivalent-fraction reasoning). Misconception: treating a ratio ADDITIVELY (2:3 → 4:5 by adding 2, instead of ×2 → 4:6); part:part vs part:whole confusion.' instrument_gap: true representations: C: move: 'lay out physical groups in the ratio (2 red : 1 blue), then copy the group to keep the ratio while the counts grow' check: 'build 2 red : 1 blue, then double it — what''s the new ratio?' G: move: draw a tape diagram (2 boxes to 1 box) or a double number line with the two quantities scaling together in lockstep check: on a double number line, if 2:1 gives 6:?, what's the missing value? N: move: recipe ratios (2 cups flour per 1 cup sugar), mixing paint, 'for every' rates in a story check: a recipe is 2:1 flour to sugar — for 6 cups flour, how much sugar? - id: math.ratio.rates-unit-rates title: Unit rates; rate language; constant of proportionality; proportional vs non-proportional strand: ratio band: - 11 - 13 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: find a unit rate (incl. from a complex fraction, ½ mi per ¼ hr = 2 mph), use rate language, identify the constant of proportionality k in a table/graph/equation (y = kx), and test whether a relationship is proportional (straight line through the origin)' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.ratio.ratio-concept - math.fractions.multiply-divide-fractions helps: - math.geometry.coordinate-plane resources: - type: aops ref: AoPS Prealgebra §7.3 Proportions / §7.5 Speed / §7.6 Other Rates record: true - type: beast ref: BA-5 Ratios & Rates sources: - CC 6.RP.2 - CC 6.RP.3b - CC 7.RP.1 - CC 7.RP.2 - research/prealgebra-extension.md §4D (R2) notes: '6.RP.2 unit rate a/b; 7.RP.1 unit rates from complex fractions; 7.RP.2 constant of proportionality k in tables/graphs/equations y = kx, straight-line-through-origin test, meaning of (0,0) and (1,r). y = kx is PROPORTIONAL reasoning, NOT function/slope machinery — stays below the Algebra-1 line. requires coordinate-plane (to graph a proportional relationship) + multiply-divide-fractions (complex-fraction unit rates). Misconception: inverting the rate (mph vs hours-per-mile); calling a line NOT through the origin proportional.' instrument_gap: true representations: G: move: on a double number line or a plane, scale down to the '1' column to read the unit rate; a proportional relationship is a straight line through the origin check: if 3 lbs cost $6, scale to 1 lb — what's the unit price? S: move: divide to make the denominator 1; write the relationship as y = kx where k is the unit rate; check k is constant across the table check: find k for a table where y is 10,20,30 when x is 2,4,6 N: move: miles per hour, price per ounce, pages per minute — the 'per one' amount that lets you scale to any quantity check: a car goes 150 miles in 3 hours — what's the speed per hour? down: math.ratio.ratio-concept - id: math.ratio.proportions-solve title: Set up & solve proportions; scaling; unit conversion via ratio reasoning strand: ratio band: - 11 - 13 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: set up a proportion a/b = c/d with matching units, solve for the unknown by scaling (or cross-multiplication as a derived move), and convert measurement units by ratio reasoning' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.ratio.rates-unit-rates helps: [] resources: - type: aops ref: AoPS Prealgebra §7.3 Proportions / §7.4 Conversions record: true - type: beast ref: BA-5 Ratios & Rates (unit conversions) sources: - CC 6.RP.3d - CC 7.RP.2c - research/prealgebra-extension.md §4D (R3) notes: '6.RP.3d: ratio reasoning to convert measurement units; solve a/b = c/d for an unknown (cross-multiply as a DERIVED technique, not a first move). Misconception: cross-multiplying part:part with part:whole; setting up a proportion with mismatched units top/bottom.' instrument_gap: true representations: G: move: line up the two ratios in a table with units labeled; find the scale factor between rows/columns and apply it to the unknown check: 3 apples cost $2; scale the table to find the cost of 12 apples S: move: write a/b = c/d with like units aligned; scale up/down, or cross-multiply and solve the resulting one-step equation check: solve 4/6 = x/15 and say what you scaled by N: move: map scale, recipe scaling, unit conversion (miles↔km) — same ratio held across different sizes check: a map says 1 in = 5 mi; how far is 3.5 in? - id: math.ratio.percent-concept title: Percent as a rate per 100; percent↔fraction↔decimal; percent of a number; find the whole strand: ratio band: - 10 - 13 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: read a percent as a rate per 100, convert among percent/fraction/decimal, find a percent OF a number, and find the whole given a part and the percent' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.ratio.ratio-concept - math.fractions.decimal-notation - math.fractions.equivalence-compare-4 helps: [] resources: - type: aops ref: AoPS Prealgebra §8.1 What is a Percent / §8.2 Word Problems record: true - type: beast ref: BA-5 Percents sources: - CC 6.RP.3c - research/prealgebra-extension.md §4D (R4) notes: '6.RP.3c: percent = rate per 100 (30% = 30/100 of a quantity); find the whole given a part and the percent. The decimal↔fraction↔percent triangle completes here (½ = 0.5 = 50%). requires decimal-notation for the conversions. Misconception: ''of'' as addition not multiplication; ''50% always means half of the answer'' regardless of the whole; treating 5% as 5.' instrument_gap: true representations: G: move: shade a 10×10 grid (100 cells) — the shaded count IS the percent; the same grid reads as a fraction of 100 and a two-place decimal check: shade 25 of 100 cells — that's what percent, what fraction, what decimal? S: move: convert percent→decimal (÷100) to compute 'percent of' by multiplying; set up part = percent × whole and solve for whichever is unknown check: find 20% of 45; then 12 is 25% of what? N: move: a test score, a battery level, a sale sign — percent as 'how much out of the whole' check: you got 18 of 20 — what percent? - id: math.ratio.percent-change-applications title: Percent increase/decrease; discounts, tax, tip, markup, interest, percent error strand: ratio band: - 11 - 13 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: solve multistep percent problems — tax, tip, discount/markup, simple interest, percent increase/decrease, percent error — using the correct base' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.ratio.percent-concept - math.ratio.proportions-solve helps: [] resources: - type: aops ref: AoPS Prealgebra §8.3 Percent Increase and Decrease record: true - type: beast ref: BA-5 Percents (percent change) sources: - CC 7.RP.3 - research/prealgebra-extension.md §4D (R5) notes: '7.RP.3: multistep percent problems — simple interest, tax, markups/markdowns, gratuities/commissions, fees, percent increase/decrease, percent error. Misconception: computing % change off the WRONG base (new vs original); a 20% rise then 20% fall does NOT return to start; adding percentages taken of different wholes.' instrument_gap: true representations: G: move: draw a bar for the original (the 100% base), then extend it for an increase or cut it for a decrease; the change is measured against the ORIGINAL bar check: draw a $50 item marked up 20% — where does the tax/tip get measured from? S: move: identify the base, compute change = percent × base, then add (increase/tax/tip) or subtract (discount); final = base ± change check: a $40 shirt is 25% off then 8% tax — final price? N: move: shopping (sale + tax), tipping, interest on savings — money situations where the base is the thing you must pin down check: leave a 20% tip on a $30 meal — how much total? - id: math.ratio.decimals-fractions-percents-fluency title: Convert fluently among fractions, decimals, and percents (benchmark set) strand: ratio band: - 10 - 13 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: convert quickly among fraction, decimal, and percent for the benchmark set (½, ¼, ⅓, ⅕, ⅛, 1/10, 1/100)' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.ratio.percent-concept helps: [] resources: - type: aops ref: AoPS Prealgebra §6.3 Decimals and Fractions / §8.1 record: true - type: beast ref: BA-5 Percents sources: - CC 6.RP.3c - research/prealgebra-extension.md §4D (R6) notes: 'OPTIONAL consolidation node (§4D R6) — the benchmark FDP set (½, ¼, ⅓, ⅕, ⅛, 1/10, 1/100). Fold into percent-concept if trimming — it is not a distinct CC standard, it is a fluency-of-conversion rung. Left L3 (not L4): benchmark recall, not a rate-gated fundamental. Flagged for adjudication (keep vs fold). Misconception: memorizing conversions without the /100 meaning; misplacing the decimal (¾ = 0.75 = 75%, not 7.5%).' instrument_gap: true representations: G: move: 'build a triple-labeled benchmark strip / anchor chart: each landmark shows its fraction, decimal, and percent side by side' check: cover the chart — what are ⅛ as a decimal and a percent? S: move: fraction→decimal by dividing, decimal→percent by ×100; drill the benchmark set until recall is instant check: give ⅓ as a decimal and a percent - id: math.geometry.coordinate-plane title: 'Coordinate plane: plot ordered pairs (four quadrants); reflect across axes; distance along an axis' strand: geometry band: - 9 - 13 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: plot & name ordered pairs in all four quadrants, map a point''s quadrant from its signs, reflect a point across an axis (sign change), and find the distance between two points sharing a coordinate as |difference|' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.geometry.lines-angles - math.integers.negatives-number-line - math.placevalue.big-numbers-round helps: [] resources: - type: aops ref: AoPS Prealgebra §11.1 (measuring segments, part) record: true sources: - CC 5.G.1 - CC 5.G.2 - CC 6.NS.6b - CC 6.NS.6c - CC 6.NS.8 - CC 6.G.3 - research/prealgebra-extension.md §4F (G1) notes: 'The AoPS-surfaced coordinate node. Deliberately SPANS grade-5 (Q1 only, 5.G.1–2) and grade-6 (all four quadrants via negatives, 6.NS.6b; distance between points sharing a coordinate = |difference|, 6.NS.8). ONE node, harder items (negative quadrants, distance) gate L3 — do NOT split per-quadrant (granularity rule). requires the integers node (negatives-number-line) for quadrants II–IV — the load-bearing cross-strand edge — and lines-angles (axes = perpendicular lines). Misconception: (x,y) order reversed (''over then up''); sign→quadrant mapping; counting grid LINES vs SPACES for distance.' instrument_gap: true representations: C: move: 'walk a floor/tape grid: from the origin, step right/left for x then up/down for y; standing on a spot names its ordered pair' check: walk to (−3, 2) — which way for the −3, which for the 2? G: move: plot on grid paper; read a point's quadrant from the sign pattern of (x,y); reflect across an axis by flipping one coordinate's sign check: plot (4,−3) and its reflection across the x-axis — what changed? N: move: a map or game grid with an origin — coordinates locate a treasure, a chess-like square, a city on a plan check: the treasure is 3 east and 2 south of camp — write it as an ordered pair - id: math.measure.stats-center-spread title: Statistical questions; mean, median, mode, range; measures of center vs variability strand: measure band: - 10 - 13 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: recognize a statistical question (anticipates variability), compute mean/median/mode/range for a data set, and say whether a number describes CENTER or SPREAD' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.measure.scaled-graphs-lineplots - math.addsub.add-sub-standard-algorithm helps: [] resources: - type: aops ref: AoPS Prealgebra §13.1 Basic Statistics / §13.2 Limits of Basic Statistics record: true sources: - CC 6.SP.1 - CC 6.SP.2 - CC 6.SP.3 - CC 6.SP.5c - research/prealgebra-extension.md §4F (S1) notes: '6.SP.1–3: a statistical question anticipates variability; a data set has a distribution with center/spread/shape; a measure of center summarizes with one number, a measure of variation describes spread. Mean = ''fair-share/balance point''; median = ''middle value.'' AoPS §13.2 (when the mean misleads) is the thinking hook. requires add-sub-standard-algorithm for the mean''s arithmetic. BA omits basic stats — build/buy signal. Misconception: median = middle of the RANGE not the middle VALUE; ''average'' always = mean; assuming the mode is unique; forgetting to SORT before taking the median.' instrument_gap: true representations: C: move: make each data value a stack of cubes; even the stacks out to the same height — that height is the MEAN (the fair-share/balance-point meaning) check: with stacks of 2, 4, 6, even them out — what's the mean? G: move: line up a dot/line plot, then read the median as the middle dot and the range as end-to-end width check: on a sorted line plot of 7 values, which dot is the median? N: move: average test score, typical shoe size, spread of daily temperatures — one number to summarize 'what's typical' vs 'how varied' check: five friends' ages — is the mean or the median more 'typical' if one is much older? - id: math.measure.stats-displays title: 'Display & summarize data: dot plots, histograms, box plots; describe a distribution' strand: measure band: - 11 - 13 tier: L3 gates: L2: '≥90% on 10 fresh items, untimed: build & read dot plots, histograms, and box plots; report n, the attribute & units, a measure of center and of variability, and relate the choice of measure to the distribution''s shape' L3: SM-2 spaced gate on mixed items across separate days OR parent-observed spaced re-check after a gap of days requires: - math.measure.stats-center-spread helps: [] resources: - type: aops ref: AoPS Prealgebra §13.3 Tables, Graphs, and Charts record: true sources: - CC 6.SP.4 - CC 6.SP.5 - research/prealgebra-extension.md §4F (S2) notes: '6.SP.4: dot plots, histograms, box plots; 6.SP.5: report n, describe the attribute/units, give center + variability, relate the choice of measure to the shape. Bar/picture graphs already exist (data-graphs, scaled-graphs-lineplots) — this node is the NEW display types (histogram, box plot, dot plot) + describing shape. Misconception: histogram (binned continuous) vs bar graph (categorical) confusion; box-plot quartiles read as equal-COUNT not equal-WIDTH.' instrument_gap: true representations: G: move: turn the same data set into a dot plot, then a histogram (bin it), then a box plot (five-number summary) and compare what each shows about shape check: for one data set, what does the box plot show that the dot plot hides? N: move: 'describe a real distribution — heights in the class, wait times at a ride — in words: where''s the center, how spread, any outliers, what shape' check: look at a histogram of class heights — is it symmetric or skewed?