Because You Can!

I love to get into classrooms to do demo lessons or to co- teach. I have been in Canberra all last week and have had opportunities to try out the Show Bags problem (see the blog below) in various guises with students in Grades 1, 2, 3 and 4. The results were interesting and informative.

Students respond well to being challenged when they are in an environment that values risk taking and sharing of ideas rather than right answers and where students are able to invent their own approaches to a problem. These children have a ‘because you can’ attitude to maths that enables them to get going on a challenge rather than shy away from engaging with difficulties that the challenge presents.

Grade 1

For the Grade 1s I reduced the number of show bags to 4 and chose some different quantities for the items:

4 bouncey balls

6 balloons

3 dice

10 counters

5 sticks of chalk

We did not get to the Sting in the Tail part of the problem but reflected on several of the student strategies, how they worked, how some of them ‘saved brain space’ and how they were each related mathematically to each other.

Some students simply added the number of items initially not realising that they had to fill 4 bags each time. The range was as expected very wide with some students drawing pictures and counting all or counting on . Some combined pictures and chunking as shown below for the bouncey balls.

Note how the secret code makes the thinking visible. In this instance we see that a double followed by a count on was used for the solution to 4 lots of 4 bouncey balls. Using tallies was also used to find the number of bouncey balls:

Other strategies included using skip counting sequences (often by counting on each time), using known facts, (e.g., double 4 is 8, double it again is 16) and in a few instances 4 × 5 = 20 and so 4 × 10 = 40.

One student used the 2s to get to the 4s by doubling his answer to 2 × 4 for 4 × 4 and later used his answer to 4 × 10 to find 4 × 5 by halving. This is an early indicator that multiplicative thinking and proportional reasoning are beginning to naturally develop as part of being given the opportunity to grapple with a challenging situation.

Grade 2

The responses in the Grade 2s was similar but here the number of show bags was increased 7 and I added 25 paper clips to the list of items for the show bag.

One student looked at the 25s and said he knew three 25s was 75 so double it is 150 and 25 more is 175. The distributive property was used by several students who worked out 7 × 25 by first working out 7 × 20 and then 7 × 5 and adding the two products

Another student wasn’t sure what 7 × 3 dice would be, but quickly worked out 7 × 2 by doubling then simply added one more 7 for 7 × 3, knowing that that was a fast strategy.

These indicators of deep understanding are always exciting to see and, when students are invited to share them in their own terms with the class, are often contagious.

Grades 3 and 4

There were still some students who did not recognise that this was a multiplication problem initially and as teachers we need to ensure that students can recognise the appropriate operation for a given problem. Often this is experiential and students need many more experiences interpreting and possible writing their own problems.

Some students were still using counting, skip counting and repeated addition strategies. For these students it is important to help them make connections between repeated addition and multiplicative strategies. Often students do not make these connections easily. One student, given the 26 lots of 25 paper clips as a sting, knew that 4 × 25 was 100. The 26 was split into 2 + 24 because 24  = 6 × 4 which gives 24 × 25 = 6 × 4 × 25 = 6 × 100 = 600 and another 2 lots of 25 is 650.

It is only by offering challenging problems that we can get windows into what the students do know, what they can do as well as how they do it.

Hope you have tried the problems too. Simply adapt them to your students.

P.S. By the way we did need an extra sting. We had to quickly come up with realistic prices for each of the packets so that the Year 1 teacher would know how much it was all going to cost.