Introduction to Substitution

This is a task from 50 Problem-solving Lessons Grades 1-6 by Marilyn Burns Math Solutions Press 2003, which helps introduce the idea of substitution in a less algebraic way. There is a tug of war competition between 4 acrobats and 5 grandmas and they are tied.

In round 2 a dog is tied when pulling against 2 grandmas and an acrobat

In round 3 we do not know who will win. We have 3 grandmas and a dog against 4 acrobats. We need figure out, and justify who will win.

Students were split up into small groups at the boards. They needed to figure out who would win round 3, and justify it.

some did beautiful work with sentences, explaining their thinking and logic.

others started to make equations and expressions using variables

Some got quite wound up in equations, but their solution was not easy to navigate without a map!

Others decided to make all of the relationships in terms of one variable, and use that to solve. They related 4m=5w so m=1.5w, and did the same for a dog d=w+w+m turns into d=3.25w.

There was a unique approach that I had never seen before. This group went from the equation 4A=5G and decided to place a value for A and G by relating it to an equation of 100=100 (since 100 can be divided by 4 and 5), so then A would be 25 and G would be 20

It was impressive to see the skills students were bringing to grade 10, and how they were working hard on communicating their thinking. The typical approach to solve this problem would be to substitute 2 grandmas and an acrobat in the place of a dog, and then to compare from that point. We can physically understand substituting 2 grandmas and an acrobat instead of a dog because of their same pulling force. This can help build the understanding of substituting equivalent expressions as a way to reduce the number of variables in use in a problem.