# Open up problems

Using open problems can help students to recognise that a number can often be partitioned in different ways. For example, 24 is 24 ones, 2 twelves, 3 eights, 4 sixes, 6 fours, 8 threes, etc. The table below provides examples of open questions that involve multiplicative thinking.

 Multiplication Division Equal grouping: There are 6 tables in the classroom. Each table has the same number of chairs. How many chairs might there be altogether? A student could work on this problem using a drawing or using counters. Help the student to recognise that because the problem involves equal groups, they could also display it in an array. Discuss with the student what a reasonable upper limit might be. Equal grouping: I have 24 chairs, and I want to arrange them in groups of equal size. How many chairs could I put in each group? Give the student 24 counters. A student might begin to solve this problem using trial and error. Work with them to show them that if the counters can be arranged in an array, then they can use the array to see the number of groups and how many chairs would be in each group. Rotate the page to show the student that if there can be 8 groups of 3 chairs, there can also be 3 groups of 8 chairs. Rate: Misha’s mum went to the shop to buy some cards. Each card cost \$4. How many cards might she have bought if she spent less than \$20? Rate: I spent \$24 on cards. If each card cost the same amount, how many cards might I have bought? Comparison: Shona has 4 times as many toy cars as John. John has fewer than 10 cars. How many cars might Shona have? Comparison: I have 4 times as many toy cars as John. If I have fewer than 20 cars, how many cars might John have?

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