# Ten optional supplements

Pick one to revisit an idea or stretch it. These are ordinary learning activities with visible feedback. They are not fresh assessments. Drawn materials are sufficient.

## S01 · Sort the comparisons

For a picture with 3 triangles and 2 circles, sort these into part:part, part:whole or incorrect-for-this-picture: 3:2 triangles:circles; 3:5 triangles:all; 2/5 fraction of all marks that are circles; 3/2 fraction of all marks that are triangles. Explain one placement.

**Check:** the first is part:part; second and third compare a part with the whole; the last is incorrect for that stated fraction. Triangle fraction is 3/5.

## S02 · Scale down as well as up

A continuous length mix has red:grey = 6:9. Show equivalent 2:3 and 4:6 mixes. For total 10 m, find each length. Explain why dividing both terms by three preserves the mix.

**Check:** five equal parts share 10 m, giving 4 m red and 6 m grey. Both terms change by the same nonzero scale factor.

## S03 · Three parts, one total

A paper count has dot:dash:ring = 1:3:2. Find every count for total 42. Decide which totals from 35, 36 and 37 can work exactly with whole marks.

**Check:** seven groups give 7, 21, 14. Six-part groups fit 36, yielding 6, 18, 12; 35 and 37 do not fit exact whole counts.

## S04 · Find the rate you actually need

A fictional table says 12 metres in 8 seconds. Write metres/second and seconds/metre. Choose which rate answers “How long for 3 metres?” and calculate the time under a constant-rate assumption.

**Check:** 3/2 m/s and 2/3 s/m; time 3 × 2/3 = 2 s. Multiplying 3 m by 3/2 m/s does not produce a time unit.

## S05 · Same unit price, different fee

A fictional offer sells one paper label at $0.25 with no fee. Another sells whole packs of 8 for $2 plus a $1 order fee. Need 16 labels. Compare the complete costs and describe a condition under which the comparison would change.

**Check:** singles cost $4; two packs cost $5 including one fee. Unit prices before the fee are both $0.25/label. Removing the order fee makes the stated costs equal.

## S06 · Prove a sign with an inverse

Use multiplication to check each quotient: −24 ÷ 6, −24 ÷ (−6), 24 ÷ (−6). Write the multiplication that returns the dividend.

**Check:** −4, 4, −4 respectively; 6 × (−4) = −24, (−6) × 4 = −24 and (−6) × (−4) = 24. The check must return the original dividend.

## S07 · Repair a fraction claim

A note says 1/4 + 2/3 = 3/7. Draw equal whole bars or use equivalent fractions to repair it. Explain why the corrected sum is greater than 2/3.

**Check:** 3/12 + 8/12 = 11/12. Adding a positive quarter to 2/3 makes a greater number. The denominator names part size; it is not a count to add here.

## S08 · Choose a useful cancellation

Calculate (8/15) × (25/12) exactly. Show a method that simplifies factors before multiplying. Explain why the same cancellation cannot be applied to (8 + 25)/(15 + 12).

**Check:** cancel 8/12 to 2/3 and 25/15 to 5/3, giving 10/9. The second expression is 33/27 = 11/9, a different value; sums are not products of those factors.

## S09 · A quotient and a remainder

Stock length is 1¾ m and a full section needs 1/3 m. Find the exact quotient, full sections and remaining length. Check that the used plus remaining length returns the stock.

**Check:** (7/4) ÷ (1/3) = 21/4 = 5¼ section-lengths. Five sections use 5/3 m; 1/12 m remains. 5/3 + 1/12 = 21/12 = 7/4 m.

## S10 · Test a conjecture of your own

On paper, choose a positive two-part ratio and compare totals 6, 12 and 18 in one named length unit. Predict what will remain the same before calculating. Test your prediction. Then change just one ratio component and describe what changed. If any selected total creates fractional counts, switch to a length model or explain why whole counts cannot meet it.

**Check:** a valid response uses the same positive scale factor on both components, labels units and distinguishes a part:part ratio from a fraction of the whole. For the optional example 1:2 in metres, totals 6, 12, 18 give (2, 4), (4, 8), (6, 12). The proportion of the first part is always 1/3. Your own different ratio may give different answers; the reasoning is the check.