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Copyright © 3P Learning – These resources have been created in partnership with Dr. Marian Small.
For more information visit
www.mathletics.comQuestions to facilitate the learning
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What is the least possible answer? How do you know it is least?
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What is the greatest possible answer? How do you know?
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What did you notice about how far apart the possible answers are?
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Why do you think that happened?
Scaffolding the learning
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Why would you not even bother trying 60?
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How do you know that the number cannot possibly be an even number?
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What might be a good first try? Why that number?
What’s the point of this task?
In this problem, students deal with division with remainders, even if they don’t initially realise that’s what
they’re doing. In fact, they are looking for a number that leaves a remainder of 1 when divided by 3 (a
number 1 more than a multiple of 3) that happens to leave a remainder of 3 when divided by 4 (a number
also 3 more (or 1 less) than a multiple of 4). It turns out that there are an infinite number of answers: 7, 19,
31, 43, … continuing with numbers that are 12 apart. The numbers are 12 apart because using an extra 12
counters makes perfect groups of 3 as well as perfect groups of 4, so the remainder is not affected.
Theoretically, students with minimal skills in multiplication could still attack the problem by simply creating
their groups and counting the counters they used. But students with multiplication/division skills are likely
to use those skills to be more efficient.
When students are asked for a number greater than 50, they are likely to work a little harder than if
numbers like 7 and 19 are allowed.
Extending the learning
Students who are interested might be asked how they could change the rules for the problem so that
possible answers could be these: 8, 43, 78, 113, …
Leftovers
Number