Experience First

We break from our usual EFFL format today to do a strategy card sort. The idea of substitution is familiar to students from Algebra 2 and was revisited in Lesson 6.1. We wanted students to get a feel for the power of substitution and the multiple ways it can be applied to a problem. There are two ways to run the opening activity.

 

Option 1: Print the substitution problems on printer paper or cardstock and cut them into individual problems. Make a set for each pair of students. Give students instructions to sort the cards into groups based on how they would solve the problem (without actually doing the problem). You can give as much scaffolding as you would like. You can specify how many classifications they should make (anywhere from 2-4, based on our categorizations) and you can even provide the categories if you find that students need this additional structure. We’ve provided some ideas for categories in the document, but feel free to delete this portion before handing it to students!

 

Option 2: To get students moving around and talking with more people, I made slides with the substitution problems on them.  I instructed students to stand up, put their hand-up, and then find a partner. They were not allowed to have any paper or pencil out.  I would project the problem and have pairs only discuss what strategy they would use to solve, without actually solving it (yet!). Discussions were lively and rich with mathematical ideas and strategy. They would then switch partners and discuss a different system.

 

After students have done the strategy sort or discussion, have students actually work on solving the problems, either individually, in pairs, or in their small groups.

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Formalize Later

The big idea we want students to walk away with is that there is significant strategy involved in choosing when and how to make a certain substitution. We don’t always solve both equations for y, and not all word problems are about finding where the output is the same. We love throwing in problems where they can substitute in entire expressions like in problems 6 and 10.

 

Two problems that may on the surface seem very similar (like questions 2 and 8) have vastly different interpretations. Have students articulate the differing strategies they would use to solve these problems taking note of the fact that the two equations in question 2 both represent cost of a gym whereas in question 8 one equation represents quantity and the other represents price. The idea of “breaking even” is relevant to question 2 but not to question 8.

 

When debriefing the strategy sort, ask for various solution paths. An easy set of questions to use is “Who would do this problem in the same way?” and “Who would do this problem differently?” and then have students explain their thinking. Compare the validity and efficiency of various strategies. Emphasize that there are often multiple approaches to take but highlight some that may be particularly “clever.” 

 

Question 9 is a Calculus question! Students may need some prompting to realize that the quantity of sand is not changing at the instant when the rate that sand is being pumped onto the beach is equal to the rate at which sand is removed.