Grade 3 · Count groups of a chosen size · 548 of 750
Count groups of a chosen size: objects available: 69; objects required in each group: 9
At a classroom activity, 69 counters are packed in groups of exactly 9. Find the number of complete groups and the remainder, explain the calculation, and check a related question.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
At a classroom activity, 69 counters are packed in groups of exactly 9. First find the number of complete groups and the remainder. Then answer: Can every object be packed into complete groups of 9, with none left over?
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A starting point
Division by group size asks how many complete groups fit. A candidate factor divides a whole number exactly only when no objects remain outside the complete groups.
Work through the reasoning
Step 1
Identify the quantities and the task
At a classroom activity, 69 counters are packed in groups of exactly 9. The first task is to find the number of complete groups and the remainder. Division by group size asks how many complete groups fit. A candidate factor divides a whole number exactly only when no objects remain outside the complete groups.
Step 2
Calculate the original result
69 = 7 × 9 + 6. 7 full groups fit and 6 remain. Therefore 9 is not a factor of 69.
Step 3
Complete the follow-up calculation
Can every object be packed into complete groups of 9, with none left over? No. 7 groups use 63, leaving 6.
The answer
69 = 7 × 9 + 6. 7 full groups fit and 6 remain. Therefore 9 is not a factor of 69. Follow-up: No. 7 groups use 63, leaving 6.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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From experiment to screen.
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The mathematical idea
At a classroom activity, 69 counters are packed in groups of exactly 9. Division by group size asks how many complete groups fit. A candidate factor divides a whole number exactly only when no objects remain outside the complete groups. 7 full groups fit and 6 remain. Therefore 9 is not a factor of 69.
69 = 7 × 9 + 6
03 · Reflect and transfer
Explain what changes and why.
Watch the group colors separate successive groups of the specified size. A final incomplete group is a remainder, not another complete package. Explain which quantity changes and which relationship stays valid. Use the starting values again before checking your written answer.
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.