Choose one that interests you. Paper and pencil are enough; counters can be drawn. All numbers and prices below are made-up paper situations. There is nothing to buy, prepare, upload or report to a teacher. A family member can ask a question, point to a diagram or record the learner's explanation. Leave the decision with the learner.
H01 · A tiny pattern
Draw a group with 1 star and 4 dots, then draw three groups. Ask each other, “What is the star-to-dot ratio? What fraction of all the marks are stars?” Circle one whole group before you answer.
A way to check: the ratio is 1:4 and the star fraction is 1/5. Three groups have 3 stars and 12 dots. Talk about why 1:4 does not mean stars are one-quarter of all marks.
H02 · A fair split on paper
Draw 24 counters to be split into pale and dark groups in the ratio 1:2. Find a way to show the split without colouring every counter. Try a total of 25 and ask whether the exact ratio can still use whole counters.
A way to check: three equal parts give 8 pale and 16 dark for 24. For 25 the part size is 25/3, so an exact whole-counter split is impossible. You can choose a different total; you do not need to round the counts.
H03 · A longer border
On a diagram, a border has plain:striped length = 2:5 and total length 3½ m. Find the two lengths. Explain whether fractional lengths are allowed in this paper model.
A way to check: seven parts share 7/2 m, so one part is 1/2 m. The lengths are 1 m and 2½ m. Their sum is 3½ m and their ratio remains 2:5.
H04 · Say the units aloud
Imagine a drawing model making 16 cm of line in 4 minutes at a constant rate. Say its rate aloud, with both units. Predict the line length for 7 minutes. Ask what would change if it paused for the first minute of that seven-minute period.
A way to check: 4 cm/min gives 28 cm for 7 productive minutes. With one minute spent paused and 6 productive minutes, the same productive rate gives 24 cm. The meaning of the time matters.
H05 · Which paper offer?
You need at least 11 pretend tickets. Offer A is whole packs of 4 for $1.20; B is whole packs of 6 for $1.68. There are no fees and all tickets are usable. Compare the sufficient orders, including spare tickets. Which would you choose?
A way to check: A needs 3 packs: 12 tickets, $3.60, one spare. B needs 2 packs: 12 tickets, $3.36, one spare. B costs $0.24 less under these conditions. Its unit price is $0.28/ticket; A's is $0.30/ticket.
H06 · A score story
A fictional score begins at −2, rises by 6 and then falls by 9. Draw or describe the two changes. Write an expression for the final score. Try reversing the order of the two signed changes and compare the final value with the path.
A way to check: −2 + 6 − 9 = −5. Reordering the additions +6 and −9 keeps the final sum −5 but changes the intermediate score. It does not give permission to rearrange any division or subtraction expression.
H07 · Find the missing piece
A paper length is 1½ m. One section is 5/8 m. What length belongs to the remaining section? Show the whole and both parts in a sketch, then check by adding.
A way to check: 12/8 − 5/8 = 7/8 m. The check is 5/8 + 7/8 = 12/8 = 1½ m. Keep the same whole and unit in every step.
H08 · Enough for complete sections?
A diagram has 2 m of usable stock. Each complete section needs 3/10 m. Find how many complete sections fit and what length remains. Explain why a calculator quotient alone does not answer the complete-section question.
A way to check: 2 ÷ (3/10) = 20/3 = 6⅔ section-lengths. Six full sections use 9/5 m; 1/5 m remains. Seven would need 21/10 m, more than the stock.
H09 · Same output, different rate
Imagine two constant drawing models. One draws 1/2 m/min for 4 minutes; the other draws 4/5 m/min for 2½ minutes. Calculate each output. Ask whether equal output means equal speed.
A way to check: both outputs are 2 m. The rates differ and the durations differ. Say the metre and minute units when explaining the calculation.
H10 · Choose a useful question
A fictional paper diagram needs four sections of 3/4 m and two of 1/2 m. Stock comes in whole 1 m units at $1.25 each, with no fee. Joins are allowed and every length is usable. Budget is $6. Ask a useful question about the order, calculate and explain your decision. Then ask one next question if these were real materials.
A way to check: total length 3 + 1 = 4 m; buy 4 units for $5, leaving $1 of the budget and no unused length under this model. A useful next question is whether joins or cutting loss would matter in the actual use. No real purchase is invited.