## Arithmetic Properties, Identities, and Inverses Through Number Talks

Like usual, the first few weeks of school came and went in a busy rush. I intended to write at least weekly, but few will find it surprising that this is my first post since the summer. Hoping to make it weekly (or better) from here on out!

I started the year in my Pre-Algebra course with Dan Meyer's Bucky the Badger activity. It did not go particularly well -- mostly because I needed to recalibrate my teaching after five year of teaching only fifth grade. Still, the activity gave me a chance to see how my students think and what they were ready for.

Next, the plan was to review arithmetic properties, identities and inverses while also providing more formal definitions for the students. Algebra is full of so many manipulations and calculations that often feel random or unjustified. I want my students' introduction to Algebra to be based in reason, not faith.

I know they have heard about these properties before -- particularly the commutative and distributive properties -- but I did not want to simply present them in a lecture. So I put together some number-talk-style questions where students were likely to use the properties as part of their mental calculations.

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First up was the commutative property. This expression was quite easy as expected and helped the students build confidence in their thinking. I collected all of the different strategies they used, and then we investigated the one that demonstrated the commutative property.

I posed the question: "Order of operations tells us that we have to complete addition and subtraction operations from left to right. Why is it okay to switch the places of the 59 and the 7 then?"

The students were quick to argue that it is "okay" or "allowed" because it never changes the result. This allowed me to opine about how that is a key part of Algebra: we use patterns to manipulate variables and numbers in ways that look different but do not change the result.^{[11]}I'll talk with them more about equivalency later when we start combining like terms.

Next, we investigated which operations allowed you to move operands around without changing the result:

We repeated this process for the associative and distributive properties (though I did a bit more lecturing for the distributive property). Ultimately, I had the students summarize these properties in their own words. These definitions became a handout^{[12]}Shared below to which we added numerical examples and algebraic generalizations (ex: 3+4 = 4+3 and *a+b=b+a*).

In case you are curious, below are all of the expressions I prompted the students with. Check the footnotes to see my goal for each one.

\(\) ^{[13]}This one was not focused on a specific property. Instead I wanted to give the students an easy one to build their confidence and to show them that I really was going to give them lots of wait time and ask to hear all of the different strategies they used.

\(\) ^{[14]}**Commutative Property: **That 7 is just dying to get combined with that 23.

\(\) ^{[15]}**Associative Property: **It is way easier to add that 28 and 12 first.

\(\) ^{[16]}This is the first multiplication question I gave them, so I wanted one that had a lot of different strategies. Some students used 25 as a friendly number, others did the traditional algorithm in their head, and others used the distributive property after decomposing 12 into (10+2).

\(\) ^{[17]}Another easier one with multiple accessible strategies.

\(\) ^{[18]}**Distributive Property: **After talking about decomposing numbers and then distributing in the previous two questions, the kids were sufficiently primed to handle this question.

That night, for homework, I had the students use GeoGebra to investigate the additive & multiplicative identities and the additive & multiplicative inverses. I wrote about this applet previous, but here it is in case you missed that post:

Without naming the properties, I challenged the students to accomplish each of the following:

**For Addition:**

- Get the sum to 0
- Get the sum equal to one addend

**For Multiplication:**

- Get the product equal to 1
- Get the product equal to one factor

The next day, I loaded the applet and we tried to recreate what they had done at home. One interesting thing that happened was that some students had found a single solution (ex: 0 = 0 + 0 for the first bullet) while others noticed that there were infinite way to achieve a goal. This led to some great discussion.

Finally, we filled in our reference sheet that I had printed prior to class. Here is one with the student-written definitions included before we wrote in examples.

**Assessment**

I assessed the students' understanding at the end of the week with a small quiz. During the quiz I let them use their reference sheet because I was more concerned about them being able to reason about the properties than keep all those esoteric names straight. The quiz had three sections:

**Part 1:** Match each property, identity, and inverse with a numerical equation. I included two non-examples that they had to consider also.^{[19]}1) Zero Product Property: 3 x 0 = 0 and 2) Decomposition: 89 = (80+9)

**Part 2:** Short answer: "We say that addition and multiplication have the **commutative property**, but subtraction, division, and exponentiation do not. Why is that the case? **Be sure to support your answer with an example or two."**

**Part 3: **I gave 4 examples of a students' thinking while trying to do mental calculations. For each, I asked for the property, identity, or inverse the student was making use of. Those questions are below:

Directions: Jim-Bob^{[20]}Duh, not his/her real name is doing mental calculations in math class. For each of the following problems, read his thinking and say which property, inverse or identity he is making use of.

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Jim-Bob’s thinking: I notice that the 3 will combine with the 7 in 17 to make a friendly number. So first I will add 17 + 3 to make the calculation easier. What property, identity or inverse is Jim-Bob making use of?

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Jim-Bob’s thinking: I don’t think I can do this calculation in my head, so I am going to break it down into an easier problem. I will decompose 12 into (10 + 2) and then multiply 17 x 10 and 17 x 2. Then I can add these products together to get the answer to the original problem. What property, identity or inverse is Jim-Bob making use of?

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Jim-Bob’s thinking: Trying to keep track of 199 in my head is tricky, so I am going to add 1 to it so it is 200. Then I just need to remember to subtract 1 at then end so that I really only added 0 to this expression. What property, identity or inverse is Jim-Bob making use of?

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Jim-Bob’s thinking: I notice that the 67 and the 13 are easy to add, so I will add those two numbers first. Then I will add 99 to their sum. What property, identity or inverse is Jim-Bob making use of?

**Progress**

I have not finished grading these yet, but a quick peek at the completed assessments looked pretty good. If I am feeling particularly un-busy, I will come back here and add some notes.