# Ten daily mathematics lessons · Days 11–20

Every day is a **25-minute core block**. On Days 11–14 and 16–19, use 2-minute recall, 5-minute explicit model, 6-minute guided test, 7-minute learner route, 3-minute exit and 2-minute evidence note. Days 15/20 use a new 12-minute check inside the same total. [Worked models](MATERIALS.md), [A/B/C learner cards](STUDENT-CARDS.md), [held-out checks](STUDENT-CHECKS.md), [teacher key](teacher/TEACHER-KEY.md) and [optional extensions](DAILY-EXTRAS.md) are separate. A child can reason with objects, tactile place cards, speech, AAC or writing; record what was prompted. Codes below are **partial** national links, not a year-level verdict.

## Day 11 · Estimate the size before finding a product

**Success:** choose a benchmark for a larger-number product, calculate and explain why the exact result is plausible. **Prepare:** first model in [MATERIALS](MATERIALS.md), blank estimate mat. **Codes:** AC9M5N06, AC9M5N08.

1. **Recall · 2 min.** Ask if `50×20` is a useful scale for `48×19`. **Key:** yes, about 1,000; it is not the exact seat count.
2. **Model · 5 min.** On the invented 48-by-19 stand, show `48×(20−1)=960−48=912`. Both factors were rounded up for 1,000, so an answer above 1,000 would deserve a recheck. Say the original unit is *drawn seats*, not attendance.
3. **Guide · 6 min.** Compare `246×4` with `250×4=1,000`. Partition `246×4=(200+40+6)×4=800+160+24=984`. Ask which estimate is intentionally a little high and why.
4. **Choose · 7 min.** Offer one [Day 11 A/B/C card](STUDENT-CARDS.md). Ask for an estimate **before** exact multiplication, a visible method and one sentence comparing them.
5. **Exit · 3 min.** “Would 9,120 be reasonable for 48×19?” **Key:** no, about ten times a useful benchmark. Repair a misplaced zero with the bound.
6. **Record · 2 min.** Note estimate, exact value and explanation separately. A child's reasonable estimate can be sound even if their exact arithmetic needs reteaching.

## Day 12 · Partition a larger number by one digit

**Success:** multiply hundreds, tens and ones by a single digit without losing a zero place. **Prepare:** 324×6 model, three place columns. **Codes:** AC9M5N06.

1. **Recall · 2 min.** Ask for `3×6` and `30×6`, then why they differ. **Key:** 18 and 180; the second has tens.
2. **Model · 5 min.** Build `324×6=300×6+20×6+4×6=1,800+120+24=1,944`. Circle the sum, not a concatenation of partial answers.
3. **Guide · 6 min.** Test `407×8 = 400×8 + 0×8 + 7×8 = 3,200+56=3,256`. Ask whether `0` in the tens place can be silently turned into a 7 tens. **Key:** no.
4. **Choose · 7 min.** Day 12 A/B/C learner card. Permit place-value blocks, a typed table, vertical algorithm or spoken partitions; require each part to retain its value.
5. **Exit · 3 min.** “Which is a useful check for `324×6`: near 1,800 or near 18,000?” **Key:** near 1,800. If 18,000, rebuild the hundreds group.
6. **Record · 2 min.** Mark arithmetic versus place-value error; a correct answer without inspectable strategy needs a question, not an automatic failure.

## Day 13 · Twenty groups and three groups

**Success:** split a two-digit multiplier into tens and ones, then combine partial products. **Prepare:** [partial-products mat](print/partial-products-mat.pdf) or its text/tactile build. **Codes:** AC9M5N06.

1. **Recall · 2 min.** Ask what `×20` means. **Key:** twenty groups/two tens of the number, not merely `×2`.
2. **Model · 5 min.** Write `128×23=128×20+128×3=2,560+384=2,944`. Put 2,560 under the *twenty groups* row and 384 under *three groups*; align the sum by place. Compare with a loose `130×20=2,600`, noting why the exact can be higher.
3. **Guide · 6 min.** Calculate `64×36 = 64×30 + 64×6 = 1,920+384=2,304`. Ask child to audit the false partial `64×30=192` and restore the tens value.
4. **Choose · 7 min.** Day 13 A/B/C route: two-digit multiplication in a new fictional domain. Learner selects mat, expanded notation, grid or another efficient correct method and labels the tens product.
5. **Exit · 3 min.** “Would 294 be plausible for `128×23`?” **Key:** no; 128×20 alone is 2,560. Ask for a lower bound.
6. **Record · 2 min.** Save the two partial products and the child's check, not only the final numeral.

## Day 14 · Choose a strategy that helps

**Success:** use compensation, partitioning or a standard algorithm deliberately and explain a bound. **Prepare:** 247×18 model. **Codes:** AC9M5N06, AC9M5N08.

1. **Recall · 2 min.** Ask why 18 might be rewritten as 20−2. **Key:** 20 is easy to multiply by; subtract two extra groups.
2. **Model · 5 min.** Show `247×18=247×20−247×2=4,940−494=4,446`. Check against `250×18=4,500`; exact is 54 below because each of 18 groups has 3 fewer.
3. **Guide · 6 min.** Compare `125×32=(125×(4×8))` with `125×(30+2)=3,750+250=4,000`. A learner may use 125×8=1,000 then ×4=4,000. Ask why either route is efficient here, and why `125×30` is 3,750 rather than 375.
4. **Choose · 7 min.** Day 14 A/B/C route with two possible strategies; child selects and defends one. Correct alternatives count if exact and inspectable.
5. **Exit · 3 min.** “Is 44,460 a reasonable result for 247×18?” **Key:** no; scale near 4,500, about ten times smaller. Revisit place alignment if needed.
6. **Record · 2 min.** Note choice, correctness and reasonableness separately; do not insist on one prescribed algorithm.

## Day 15 · Fresh check A: product and bound

**Success:** transfer estimate, two-digit multiplication and explanation to an unseen count. **Prepare:** reveal [Check A](STUDENT-CHECKS.md) today only; keep [worked key](teacher/TEACHER-KEY.md) private. **Codes:** AC9M5N06, AC9M5N08.

1. **Recall · 2 min.** Ask from memory what makes an estimate useful. **Key:** nearby, justified and suited to the decision; do not preview the new digits.
2. **Directions · 3 min.** Explain response modes. Children may use paper mat or tactile place cards; if a calculator is allowed, record its role. Do not solve a parallel problem with check numbers.
3. **Access · 3 min.** Read the new fictional card once neutrally or provide a large-print/text copy. Check understanding of *each* and *groups* without suggesting an operation result.
4. **Fresh response · 12 min.** Learner estimates, calculates and checks Check A independently. A child needing more processing time finishes in a separately recorded continuation; time pressure is not the target.
5. **Reflect · 3 min.** Ask which part of their exact result agrees with their benchmark; collect before discussing a key.
6. **Record · 2 min.** Mark independent/prompted and method; secure named work under school policy. Offer after-check [Day 15 routes](STUDENT-CARDS.md) only **after collection**.

## Day 16 · A price needs an exact decision

**Success:** use multiplication and estimation in an invented budget, then state money left and the decision limit. **Prepare:** model $16 per kit, 28 kits, $470 available. **Codes:** AC9M5N06, AC9M5N08, AC9M5N09.

1. **Recall · 2 min.** Ask whether “about $450” is enough to say exactly how much remains from $470. **Key:** no.
2. **Model · 5 min.** State the complete fictional problem. Compute `28×$16 = 28×(10+6) = $280+$168=$448`; `$470−$448=$22`. Rough `30×$15=$450` checks scale, but $450 does not replace exact payment. No actual shop price or family budget is claimed.
3. **Guide · 6 min.** Try $23 per item for 17 items with a $400 fictional limit: `23×17=23×(10+7)=230+161=391`; $9 remains. Ask if $39.10 could be right for 17 items near $20 each. **Key:** no, scale near $340–$400.
4. **Choose · 7 min.** Day 16 A/B/C route. Learner writes/points to `price × count → total → compare with given limit → money left`, including the dollar unit. No discussion of real household spending.
5. **Exit · 3 min.** “If a rough estimate is close to the limit, what next?” **Key:** exact calculation and constraint check.
6. **Record · 2 min.** Note if child chose the operation and interpreted the $ difference; arithmetic alone is not the whole modelling target.

## Day 17 · Check with the inverse, then a second route

**Success:** use multiplication/division as inverse facts and a bound to inspect a product. **Prepare:** 468×7 model. **Codes:** AC9M5A01, AC9M5N06, AC9M5N08.

1. **Recall · 2 min.** From `6×7=42`, ask for a related division fact. **Key:** 42÷7=6 or 42÷6=7.
2. **Model · 5 min.** Compute `468×7=400×7+60×7+8×7=2,800+420+56=3,276`. Divide back `3,276÷7=468`; compare with `470×7=3,290`. Distinguish inverse and estimation checks.
3. **Guide · 6 min.** Let class audit `324×8=2,592`. Ask what division statement would undo it and what rough product `300×8=2,400` predicts. If a child says 259.2, use place-value scale.
4. **Choose · 7 min.** Day 17 A/B/C product plus inverse fact, using a fact family, manipulatives or selected equation cards. No child must perform long division as the only evidence; a valid inverse multiplication check is accepted.
5. **Exit · 3 min.** “Why is 32,760 implausible for 468×7?” **Key:** near 3,290, about one zero too many.
6. **Record · 2 min.** Note whether the child understands which number is group size/count; copied reverse digits alone are not an inverse explanation.

## Day 18 · Find the missing group count

**Success:** solve a multiplication equation with an unknown, then check by substitution. **Prepare:** `24×?=720` card. **Codes:** AC9M5A02, AC9M5A01.

1. **Recall · 2 min.** Ask what 24×10 and 24×3 are. **Key:** 240 and 72.
2. **Model · 5 min.** `24×?=720`: use inverse `720÷24=30` or note 24×3=72 then tenfold to 720. Substitute: `24×30=720`. Name `?` as number of equal groups in this model.
3. **Guide · 6 min.** Solve `?×16=672`: `16×40=640`, `16×2=32`, total 672, so `?=42`. Check with `672÷16=42`. Correct a plausible `?=4.2` by scale.
4. **Choose · 7 min.** Day 18 A/B/C unknown-value route; child may use related facts, an area model, division or trial and refinement. Require a substitution check.
5. **Exit · 3 min.** “Would 3 or 30 fit 24×?=720?” **Key:** 30; 24×3=72. Ask why the zero changes the total.
6. **Record · 2 min.** Note strategy and check rather than speed or preferred symbol format.

## Day 19 · Equal totals can hide different constraints

**Success:** formulate and compare two multiplicative plans, then choose the one that meets a stated limit. **Prepare:** 24×36 versus 32×27 paper-bundle model. **Codes:** AC9M5N09, AC9M5N06, AC9M5N08.

1. **Recall · 2 min.** Ask if equal totals guarantee equal practicality. **Key:** no; another constraint can differ.
2. **Model · 5 min.** Both invented arrangements hold `864` cards: `24×36=864` and `32×27=864`. If the fictional workspace allows **at most 30 bundles**, only 27 bundles fits. Do not invent a price, environmental impact or real policy.
3. **Guide · 6 min.** Compare `18×42=756` with `27×28=756` under at most 30 bundles. The 28-bundle arrangement fits; 42 does not. Let children state the multiplication equation and constraint separately.
4. **Choose · 7 min.** Day 19 A/B/C route. Require two products, constraint test and a recommendation limited to the supplied facts; a learner may challenge missing information.
5. **Exit · 3 min.** “Can we name the cheapest option from these numbers?” **Key:** no, no prices were given. If a child asserts a cost, ask which source supplies it.
6. **Record · 2 min.** Note product, constraint and limit to conclusion as separate evidence. A correct total with unsupported recommendation needs feedback.

## Day 20 · Fresh check B: multiply, check and decide

**Success:** transfer a two-digit product, benchmark, inverse and context decision to a new unseen arrangement. **Prepare:** reveal [Check B](STUDENT-CHECKS.md) only now; staff [key](teacher/TEACHER-KEY.md) remains private. **Codes:** AC9M5N06, AC9M5N08, AC9M5N09, AC9M5A01.

1. **Recall · 2 min.** Ask what a complete recommendation needs: product, bound and stated constraint. Do not preview card values.
2. **Directions · 3 min.** Explain accessible response routes and that every quantity is fictional; provide no worked parallel solution using test digits.
3. **Access · 3 min.** Read the card once neutrally or supply the text/tactile version; record any adult paraphrase.
4. **Fresh response · 12 min.** Learner calculates a new product, gives a useful estimate, checks inversely and judges a stated package limit. Finish later with a recorded continuation if access/time requires it.
5. **Reflect · 3 min.** Ask what evidence in the card supported the recommendation. Collect initial and revised work before feedback.
6. **Record · 2 min.** Mark calculation, estimation, inverse and interpretation separately; no whole-year grade follows from this check. After collection, optional [Day 20 routes](STUDENT-CARDS.md) are available.

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