Today, I was able to pop into a 3rd grade classroom and have some fun with a true or false equation routine! This routine has become one of my favorites, not only for the discussion during the activity, but more for the journals after the talk. I haven’t figured out quite how to use them with the students, but it gives me such great insight into their understandings that I would love to think about a way to have students reflect on them in a meaningful way. I keep asking myself, *what conjectures or generalizations could stem from this work*?

I started with 4 x 3 = 3 + 3 + 3 + 3 to get students thinking about the meaning of multiplication and how we can solve for a product using repeated addition. I followed 6 x 4 = 8 + 8 + 4 to see how students talked about the 8’s on the right side. They could explain why it was false by either solving both sides or reasoning about the 8’s as two 4’s in some way.

My final problem was the one below, 8 x5 = 2 x 5 + 2 x 5 + 20. I chose this one because I wanted students to see an equation with multiplication on both sides. Up to this point, I structured them to be multiplication on one side and addition on the other. There was a lot of solving both sides – I think because of the ease of using 5’s – but, as the discussion continued the students made some really interesting connections about why the numbers were changing in a particular way. I really focused on asking them, “Where do you see the 8 and 5 in your response?” to encourage them to think relationally about the two sides.

I ended the talk with 8 x 6 = and asked the students to go back to their journals and finish that equation to make it true.

Some students knew it was equal to 48 right away and started writing equations that were equal to 48. For this student I probably would ask about the relationship between each of the new equations and 8 x 6.

There are so many interesting things in the rest of them, that I am not sure what exactly to ask student to look at more deeply.

In all of them, I see…

- Commutative property
- Multiplication as groups of a certain number
- Distributive property
- Doubling and halving & Tripling and thirding

The student below shared this one with the class during the whole class discussion:

8 x 6 = 7 x 10 – 3 x 10 + 2 x 4

From her explanation, she could explain how both sides were 48, but when I asked her how it related to 8 x 6, her wheels started spinning. You can see she played all around her paper trying to make connections between the two. That is the type of thinking I want to engage all of the students in, but based on their own personal journal writing – but what is the right prompt? “Where is one side in the other?” or “How are they related?” <—that one feels like it will lead to a lot of “They are both 48” so I need a follow up.

I actually left the room thinking about how I would explain how they two sides were related – in particular looking for either 8 groups of 6 or 6 groups of 8 on the right side. I found it was easier for me to find six 8’s, but now want to go back and find eight 6’s for fun. I can see how this could be so fun for students as well, but there is a lot of things going on here so I wonder how to structure that activity for them? Would love thoughts/feedback in the comments!

Christy PettisSo many interesting things in the students’ work here! Something I’ve found helpful is to talk about “near relations” and “far relations” when we look at examples. Some examples have fairly obvious, direct mappings between them. The examples you chose like 4×3=3+3+3+3 are good examples of this. An example like 6×4=8+8+4 is a little farther away but making connections between the two expressions is still pretty straightforward. They really look like “sisters.” The example the student produced though, that’s definitely a far relation. You have to really alter one or both expressions in some significant ways to see how they map to one another. While it can be a great exercise to play with expressions to see how they are related, being able to find “sister” or “cousin” expressions where the connections can be seen more readily is a very useful skill. Those more closely related expressions often help us see and make use of structure. Giving students some language for the type of related expressions they are making can help them see this as more than just “find two expressions that are equal to the same number”–and also help them name why some expressions can be easily related while others seem to take a whole page and still feel unsatisfyingly far apart.

If you do want to make connections between those two expressions that your student made, though, I think that thinking of the 8×6 as 8×5 + 8 might be helpful. It will give you 8 groups of 5 and those 5s will map more readily to the 10s on the right hand side, and the remaining 8 will connect to the 2×4 (it’s still pretty complicated though!).

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mathmindsblogPost authorThank you Christy! That is such a great way of thinking about how the expression relate to one another in terms of near and far. It was definitely a far relation and one I am sure she got from playing around to find an expression equal to 48 – regardless of the starting expression. Which don’t get me wrong, I think that was wonderful in and of itself and I love to see students playing around with it to see if they can find a connection, but probably not really helpful thing for all students to be doing.

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