I’ve been working with K-5 teachers on how to use tape diagrams to solve problems. During one of these workshops, a first grader peeked in and asked, “Teachers have to go to school too?” It got quite a laugh, but it’s true. We need time to sit with our peers and try new ways to solve problems. I also think it reminds us of what it feels like to be a student and see things through a different lens.
One of the problems we worked on was:
Ms. Placa spends a total of $42 for 3 sweaters. Each sweater costs the same amount. How much does each sweater cost?
Most of the groups solved this by using long division.
But let’s say you have students who don’t remember the long division algorithm (most likely because it never it held any meaning to them). If they have been working with a tape diagram or bar models, they can solve it using that.
First they can draw a bar and split it into three equal boxes to represent the 3 sweaters.
Then they can try guessing in order to determine what goes in the box. You want to work with them on using efficient strategies for guess and check. For example, 10′s are easy to count by, so they can start with 10s.
They can see that they still need 12 more. At this point, some students may know that they can divide 12 by 3 and it will give them 4. If not, they can count by 2′s and then by 2′s again.
When they add what is in each box, they get to $14 for each sweater.
The tape diagrams may seem strange to us at first if we haven’t used them before. We need to learn how to use them just like students need to learn how to use them in a meaningful way. However, once they do, the bar models begin to make sense to them. Later, the models can be used to build to the algorithm so that the algorithm will have meaning for them.
Research shows that students can do well when they use diagrams to solve problems. However, the diagram alone does not hold magical powers. While simply putting a tape diagram alongside a word problem aided higher ability middle school students, it did not seem to aid lower level students who received no instruction on how to use the diagram. As with any model or representation, students need guidance and opportunities to foster their understanding of it.
Want to know more? Check out:
Beckmann, S. (2004). Solving algebra and other story problems with simple diagrams: A method demonstrated in grade 4–6 texts used in Singapore. The Mathematics Educator, 14(1), 42-46.
Booth, J. L., & Koedinger, K. R. (2012). Are diagrams always helpful tools? Developmental and individual differences in the effect of presentation format on student problem solving. British Journal of Educational Psychology, 82(3), 492-511.
Yancey, A. V., Thompson, C. S., & Yancey, J. S. (1989). Children must learn to draw diagrams. The Arithmetic Teacher, 36(7), 15-19.