Mathematics · ten 25-minute… | Year 9 Maths · T1 W3–4 · Lessons

Use the original model cards, clean learner routes, optional extras and no-purchase home ideas, A4/text/tactile aids and public teacher worked key. Give fresh Check A/B as clean prompts on Days 15/20; the key is also public, so adapt a case locally if prior access matters. This continues the opening fortnight's attention to coordinates, rates, domains and model limits; Week 3 studies numerical integer exponents, then Week 4 variable bases and exponents. Every lesson is 2 + 5 + 6 + 6 + 4 + 2 = 25 minutes. Allow a calculator for arithmetic access while asking for the structure, condition and unit. Read-aloud is listening-supported access, not independent print reading. Count speech, AAC, tactile cards, typing and directed scribing as the learner's mathematics when their choices remain theirs; record any mathematical hint. All contexts and numbers are classroom fiction.

Week 3 Day 11 Product of powers counts combinations

Code: AC9M9A01. Goal: derive same-base product law from repeated independent choices and reject an invalid sum shortcut. Prepare: Card A, law and non-law mat, two sets of yes/no paper tokens.

Retrieve · 2 min. Ask what 2^3 counts and what the exponent 3 tells us. State base 2 and three independent binary slots, not “2×3”.
Model · 5 min. Arrange a three-slot and four-slot string. Joining them makes seven slots, so 2^3×2^4=2^(3+4)=128. Multiply 8×16 as a check. Name model strings, not real player behaviour.
Guided · 6 min. Try 3^2×3^3=3^5=243 by writing two and three factors of 3. Compare 3^2+3^3=9+27=36; the product law does not cross a plus sign.
Choose · 6 min. Learners take 11 A/B/C: token sequence, factor explanation or wrong-post repair. Each must state why the bases match and give one check.
Probe · 4 min. Ask whether 2^3×3^4 permits simply adding exponents. It has different bases, so evaluate or reorganise only with a justified property; no same-base shortcut.
Exit · 2 min. “The product works because ; the plus version .” If learner adds 8+16 to count pairs, make a two-row pairing grid before new values.

Week 3 Day 12 Quotient, zero exponent and the unit of the answer

Code: AC9M9A01. Goal: derive same-base quotient and zero-exponent rules from equal-sized groups. Prepare: Card B, quotient ladder, paper cells/group labels.

Retrieve · 2 min. Expand 3^2 and 3^5 as repeated factors. Ask what a division 243÷9 counts in Card B.
Model · 5 min. Cancel two non-zero factors of 3 from 3^5/3^2 to obtain 3^3=27 groups of 9 cells. State the model count and quotient unit.
Guided · 6 min. 5^3/5^3=1 exactly, so 5^(3−3)=5^0=1. This explanation requires non-zero 5; 0^0 is outside this rule. Then 2^6/2^2=2^4=16 with a factor check.
Choose · 6 min. Learners use 12 A/B/C: physical grouping, error analysis or verbal ratio. Require numerator/denominator and why subtraction applies.
Probe · 4 min. Ask whether 3^5/2^2 permits exponent subtraction. Different bases: no. A calculator may give a number but not justify a false rule.
Exit · 2 min. “For a non-zero same base, a^m/a^n=___; if exponents match, ___.” Attend to group units if the rule is correct but meaning is not.

Week 3 Day 13 A negative exponent is a reciprocal

Code: AC9M9A01. Goal: use quotient reasoning to rewrite a negative integer exponent as a positive reciprocal and maintain units. Prepare: Card C, quotient ladder, one paper metre strip drawing.

Retrieve · 2 min. Use 10^2/10^4=10^(2−4) and ask what ordinary fraction this quotient represents.
Model · 5 min. Cancel factors: 100/10000=1/100=10^−2. Scale a drawn 1 m strip by this dimensionless factor to 0.01 m = 1 cm. Mark centimetre as a unit, not part of the exponent law itself.
Guided · 6 min. 2^−3=1/2^3=1/8; 5^−1=1/5. Ask why a negative exponent does not make the result negative. For a negative base, (-2)^−3=−1/8; parentheses make the base negative.
Choose · 6 min. Take 13 A/B/C: reciprocal tiles, scale annotation or critique 10^−2=−100. Each includes the non-zero-base condition.
Probe · 4 min. Compare (-2)^4=16 with −2^4=−16. Show the brackets, not just a calculator key sequence. Do not apply 0^−1.
Exit · 2 min. Rewrite 4^−2 as a fraction and explain the base condition: 1/16, base non-zero.

Week 3 Day 14 A power of a power is repeated groups

Code: AC9M9A01. Goal: distinguish (a^m)^n=a^(mn) from a product of two powers, using brackets and a model. Prepare: Card D, law boundary mat.

Retrieve · 2 min. Name how many choices 2^3 represents in the fictional stage notebook.
Model · 5 min. Two independent eight-choice cards give (2^3)^2=2^3×2^3=2^6=64. Multiplying exponents is a shorthand for repeating the whole power twice.
Guided · 6 min. Compare 2^3×2^2=2^5=32, a different second stage. Then (3^2)^3=3^6=729; expand as three groups of two factors to check.
Choose · 6 min. Take 14 A/B/C: bracket tree, contrasting examples or caption editor. Each states exactly which expression matches which fictional cue structure.
Probe · 4 min. Contrast (2^3)^2 with 2^(3+2). Ask where the repeated group lies in each. Keep (-2)^4 brackets distinct from −2^4.
Exit · 2 min. Write one sentence explaining why a power of a power multiplies exponents; show a small expanded example.

Week 3 Day 15 Fresh numerical exponent check

Code: AC9M9A01. Goal: independently choose exponent laws for an new numerical source and explain one invalid claim. Prepare: fresh Check A and public teacher worked key. No practice with its values.

Frame · 2 min. State that the new card is a fresh source. Learners can use a blank law mat, calculator and their usual access method, with mathematical hints noted.
Read · 5 min. Give the source exactly; an adult read-aloud is logged as such. Do not identify the useful law.
First work · 6 min. Learners mark bases, operation signs and any denominator before simplifying. Preserve first response.
Respond · 6 min. Choose 15 A/B/C for the same product, quotient, reciprocal and claim-boundary thinking in the learner's own words.
Audit · 4 min. Ask students to compare one result with expanded factors or substitution without telling them which result to change. Keep revisions visible.
Collect · 2 min. Use criterion next moves; a small check cannot decide Year 9 attainment.

Week 4 Day 16 Variable bases and independent slots

Code: AC9M9A01. Goal: extend product and quotient laws to a variable base and distinguish the product's domain from the quotient's. Prepare: Card E, variable-domain aid.

Retrieve · 2 min. Replace each abstract slot with x choices. Ask what three independent slots count when x is a positive whole number.
Model · 5 min. Join three and two slots: x^3×x^2=x^5. At x2, 8×4=32=2^5. The product identity also makes sense at x0 with these positive exponents, though the count context has no strings.
Guided · 6 min. Divide x^5/x^2=x^3 by cancelling non-zero x factors. At x2, 32/4=8. For x0 the original quotient is 0/0, undefined; write x≠0 before cancellation.
Choose · 6 min. Use 16 A/B/C: slot diagram, algebra audit or domain comparison. Each gives a substitution check and explicit quotient restriction.
Probe · 4 min. Ask if x^3+y^2 may combine exponents. No: different bases and addition. Keep “cannot use this rule” separate from “cannot evaluate given values”.
Exit · 2 min. State one valid product identity and one domain condition for a variable quotient.

Week 4 Day 17 Negative variable exponents and signs

Code: AC9M9A01. Goal: rewrite a negative variable exponent as a reciprocal and test positive/negative non-zero substitutions. Prepare: Card F, quotient ladder.

Retrieve · 2 min. Ask whether u^2/u^5 divides by u when u0. Name the restriction first.
Model · 5 min. For u≠0, cancel: u^2/u^5=u^(−3)=1/u^3. At u2, result 1/8. Keep it a unitless ratio, not a count or length.
Guided · 6 min. At u−2, u^3=−8, so reciprocal −1/8. At u0, original denominator 0^5=0, so no value. Contrast (-u)^2 and −u^2 at u2.
Choose · 6 min. Take 17 A/B/C: cancellation ladder, error critique or signed substitution. Each includes u≠0 and fraction/negative-exponent forms.
Probe · 4 min. Challenge “negative exponent means negative result” with u2 and u−2. The sign comes from the base and parity, not the negative exponent by itself.
Exit · 2 min. Rewrite v^−2 as 1/v^2 with v≠0; say what fails at v0.

Week 4 Day 18 A variable appears inside the exponent

Code: AC9M9A01. Goal: add algebraic exponents for equal bases and test a whole-number context. Prepare: Card G, variable-power map.

Retrieve · 2 min. Read 2^(n+2) as a power with n+2 in the exponent; do not treat it as 2^n+2.
Model · 5 min. Join independent stages: 2^n×2^(n+2)=2^(n+n+2)=2^(2n+2). In Card G, n is a non-negative whole stage number.
Guided · 6 min. At n2, 2^2×2^4=4×16=64=2^6; at n0, 1×4=4=2^2. Both checks match the formula but do not replace the exponent-law argument.
Choose · 6 min. Use 18 A/B/C: stage table, algebra-line repair or spoken/tactile path account. Each writes the bracketed exponent and one domain statement.
Probe · 4 min. Compare 2^(2n+2) with 2^(n^2+2) at n3: exponents 8 and 11, not equal. Ask where the extra n×n came from; none is in the product law.
Exit · 2 min. Simplify 3^m×3^(m+1)=3^(2m+1) and state that m is a whole number in a slot-count model.

Week 4 Day 19 Choose a law for the design, not the other way round

Code: AC9M9A01. Goal: select the correct exponent structure for an original poster grid and explain a nearby non-example. Prepare: Card H, law-boundary and variable-power aids.

Retrieve · 2 min. Ask whether the design has two rows or 2^r rows. The source, not the symbol alone, decides.
Model · 5 min. A 2^r-by-2^r grid contains (2^r)^2=2^(2r) icons. At r3, 8×8=64. A different two-row design would contain 2×2^r=2^(r+1); label the changed condition.
Guided · 6 min. Check r2: first design 4×4=16, second 2×4=8. Discuss why a model icon count cannot guarantee legible printing, physical space or reader access.
Choose · 6 min. Take 19 A/B/C: compare grid sketches, create a different exponent model, or diagnose a misleading caption. Each names the count unit, r domain and what is unmodelled.
Audit · 4 min. Peer/teacher asks whether the learner's drawing represents the same rule they wrote. If not, revise either the words or expression, keeping first work.
Exit · 2 min. “I chose ___ because the source repeats ___; I would not claim ___ from the count alone.”

Week 4 Day 20 Fresh variable-exponent transfer check

Code: AC9M9A01. Goal: use the integer exponent laws in an new variable source with honest domains and one model limit. Prepare: fresh Check B and separate worked key.

Frame · 2 min. State that this is a fresh card. Blank aid, calculator or usual accessible representation may be used; keep mathematical prompting visible.
Read · 5 min. Give new source exactly. Record exact adult reading versus independent print. Do not model its values first.
First work · 6 min. Learners mark base, exponent expression, division and any zero-base exclusion. Keep unprompted work.
Respond · 6 min. Choose 20 A/B/C for the same algebra, substitution, reciprocal and source-reach criteria; preserve their wording.
Audit · 4 min. Ask for one substitution or expanded-factor check, then a second line distinguishing model count from a real-world promise. Record any change.
Collect · 2 min. Use the criterion-specific staff key for subsequent teaching, not a total score as Year 9 certification.