I was recently co-planning a lesson with a group of teachers for a grade 9 academic class. The class had worked a bit on writing expressions but the teacher wanted to introduce equations and solving them! We looked at a few different problems and then settled on this one:
We weren’t sure how kids would approach it… would kids just do trial and error? Would they start writing equations? How long would it take them?
We made the decision to take the part about finding 2 ways to solve the problem out. In hindsight, we should have left it in. We ended up asking most groups to show it another way anyway, and often that was the part that challenged the students and lead them to writing equations.
As students worked we monitored to see what strategies were happening. There was a great variety of strategies! Not many students did trial and error. Most did a type of “undoing” of the mathematics (there are 2 frogs and 2 lions, I know the frogs are each 7 so that is 14, 30-14=16, 16/2=8, therefore the lion is 8). Some were more formal about their work. We made a decision to put the problem on the board but not give students their own copy. This really encouraged the use of variables. Some pairs of students redrew the table but used f’s, l’s, b’s and h’s to represent the different animals. Some went further and wrote out expressions. One or two groups wrote an equation and solved it.
As we were monitoring the groups, we tried to organize a sequence that would make sense for students to present their strategies. Our learning goal (we had this in mind… we didn’t tell the students this at the beginning) was for students to see how the situation could be described by an equation, understand what an equation is and what it means to solve an equation, and start to understand the balance model for solving equations. With this in mind we found three groups to present to (hopefully) make these concepts clear to the class.
The first group that presented had used the undoing strategy that I mentioned earlier. This seemed to be a good one to start with because almost everyone in the class could understand what they were doing.
Next we had a group that originally had used the undoing strategy but then extended their work by writing the expressions that represented each row (2L+2F, etc.). Originally when I saw their work there was no equal sign, but by the time they got up to present they had added the equal sign and the total to each of their expressions. They hadn’t solved the equations, but they had them written there so that allowed a nice class discussion about why those equations described the situation, what the difference is between an equation and expression, and what an equal sign represents (one student yelled out “it means what is the answer” when I asked this question, but after some turn and talk time he changed his mind). We were also able to show how solving those equations related to the undoing method the first group had used, and introduced the balance model!
Finally we showed a totally different approach to the problem. This group had decided that the sum of the rows had to be equal to the sum of the columns, so found the total of column 1 by doing (28+30+20+16)-(19+20+30). This didn’t relate directly to the algebra we were discussing, but it was so cool and such a neat way of thinking about it we had to share!
It was great to try this problem as an introduction to solving equations. I think it’s a nice way to initiate some of those important conversations that need to happen for students to understand this concept.