# Introducing (or re-introducing) solving equations!

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.

I was recently working with a grade 6 teacher and his class. They had been working on probability and we gave them a fairly standard question to consider:

• What is the probability of rolling a 5 on a typical 6 sided die?
• If you rolled the die 30 times, how many times would you expect a 5 to come up?

Every pair in the class answered the first part of the question correctly but many of them struggled with the second question. What I found fascinating was that many of them believed the answer to the second question was 15/30. It wasn’t until I sat and had a conversation with one of the groups that I realized why. They said “well, there is a 50/50 chance of getting a 5 so that’s why its 15, it can either be a 5 or not be a 5”.

I never anticipated this misunderstanding (perhaps because I’ve never taught grade 6) and wondered how many of my senior students have been confused about this but couldn’t express it as well as these young girls. Is there confusion about 50/50 because it is a term that is used in everyday language and isn’t always used with the same precision that we use in math?

After having a conversation with the other teachers that were working in the room that day with us, it became evident that there were quite a few students dealing with this same misconception and we decided we needed to stop and have a class discussion about 50/50. I am going to do my best to recount the dialogue because I found it fascinating, but it was over a week ago now so there will be some paraphrasing and I can’t quite remember all the names (they aren’t my students).

We first asked the students to talk to the person next to them about “What does 50/50 mean?” Once they had a chance to chat with a partner we had them share their ideas with the whole class. I wrote down what they said on the board.

Student A: it means half and half
Me: Can you explain that a bit more?
Student A: well, 50 represents 50 out of 100 which is 50% which is the same as half
Aiden: Its like if I was walking down the street with my friend and we go past a store and we might go in, or we might not – so there is a 50/50 chance that we will go in
(I wondered if I should question this a bit more – is it really 50/50, does it depend on what kind of store it is…..but I left it for now)
RJ: When I get home from school, there is a 50/50 chance that I will get to watch TV.
Me: What do you do if you don’t watch TV?
RJ: Chores
Me: So, over the course of 4 days, how many days would you get to watch TV?
RJ: (thinks for a bit) about 1 or 2
Owen: If you draw a target and cut it in half, you would put chores on one side and tv on the other.
Me: (I draw the target as the student describes)
Student B: It has to be equal percentages
Me: (I draw another target with chores as 3/4 of it and TV as 1/4) So is this 50/50?
Student B: no – that is 75/25 because chores is 75% of the circle and TV is only 25%
RJ: smiles and says yes!

We move on to the next question: “Is the probability of rolling a 5 on a regular die 50/50?” and give them time to talk with a partner about it.

Student C: “No, because there are 6 different numbers you can get and the 5 is only one of them”
Tiffany: “It would only be 50/50 if there were three 5’s on the die”
Me: “What do the rest of you think about what Tiffany just said” (several nods) So, are you saying that if the die had numbers 1, 2, 3 and then 5, 5, 5, that it would be 50/50?
Tiffany nods yes but hands are frantically going up as I say it.
Syndan: I disagree with Tiffany. I think it would have to be three of one number and three of another number.
Me: So the die might have the numbers 3, 3, 3, and 5, 5, 5?
Tiffany: Yeah – I disagree with myself too. I think its what Syndan said.
Me: Can someone explain why?
Tiffany: Because for it to be 50/50, half needs to be one number and half needs to be another number.
Student D: (Comes up and draws a target next to the other ones) If you drew a target for a regular die there would be six pieces and the 5 is only one of them so its not 50/50.

At that point our time was up for the day and I walked away amazed at the level of their discussion, willingness to put out their theories and challenge one another. The entire conversation fascinated me and exhausted me because the things they were saying I had not anticipated. I didn’t expect them to be so clever, especially about changing the die.

I’m so glad we decided to stop our plan and take time for the conversation!