The Additive Inverse Property

Students discover the Additive Inverse Property, which tells us that adding a number and its opposite always produces zero.

We’ve learned that the Commutative Property and Associative Property apply to addition…​ but not division. What if there was a way to rewrite subtraction as addition? Then we could apply the Commutative and Associative Properties to subtraction expressions, too! Let’s explore this idea.

I’m going to draw a two-column table on the board. I’m labelling the left-hand side "Starting Value," and I’m labelling the right-hand side "Additive Inverse".

I’m going to write a number in the left-hand column. You are going to tell me what value I should add to that number, to get a sum of zero. I’ll record your response in the right-hand column.

@table{

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Allow a variety of students to share. Record responses on the table.

These number pairs all represent opposites.

Every number has an additive inverse.

Ensure that students are comfortable with this concept before moving onto the next activity.

Students rewrite addition expressions as subtraction, and subtraction expressions as addition by applying their knowledge of the Additive Inverse Property.

Now that we understand what the additive inverse is, we are ready to think about how we can put it to use…​ perhaps it can make some computations simpler?

During this exploration, you should have discovered that:

One powerful advantage emerges when we write subtraction as addition: we now can apply the Commutative and Associative Properties to a much broader set of expressions!

Now, students are ready to continue their exploration of subtraction as the inverse of addition. Note: To complete this worksheet, students do not need to be fluent at integer addition and subtraction. We just want students thinking about when expressions are equivalent based on what they have learned about the additive inverse.

Have students share strategies for determining equivalence. What are the different ways that they thought about the Additive Inverse Property?

Students examine whether rigid adherance to the "left-to-right" rule is needed when adding and subtracting.

Did you work from left to right to arrive your result? This solving strategy can be represented by the Circle of Evaluation, below.

Is it essential to solve from left to right?

Ask if anyone opted to subtract before adding. If so, invite them to share their method and then invite other students to weigh in.

Evaluate the Circle of Evaluation below. Is it equivalent to the previous Circle of Evaluation?

We’ve learned that the Associative Property applies for expressions with only addition…​ not addition and subtraction. Many of us have also learned that when an expression includes addition and subtraction, we must work from left to right. So… what’s going on!? It appears that we get the same result regardless of how we simplify this expression.

Does subtracting first work every time? Can we rearrange the groupings of any expression with both addition and subtraction? Let’s investigate.

Encourage students to think deeply about why this algorithm works – and if you’d like, invite them to consider and discuss why students all across the country are typically taught just one algorithm when, typically, there are an abundance to choose from!

Let’s put our new knowledge to use!

Project the problems below one at a time, and invite students to solve using mental math.

Students learn about examples in Pyret, and use their new knowledge to think about addition and subtraction as inverse operations.

Here are some common Noticings. Is there anything on this list that you didn’t notice?

Lines and lines of code can be difficult to read! Sometimes programmers want to write down their thinking, or leave notes to help others understand what the code is doing. That’s what comments are for: a hash at the start of a line tells the computer that the rest of the line is a comment instead of code. To make comments stand out, they are colored dark orange.

The comments in lines 2, 14, and 26 are used to break up the code into sections that make the starter file easier for users to interpret.

Sometimes a comment isn’t enough. A programmer might want to write down their thoughts so that the computer can test their thinking. These are called examples.

Take a look at the first examples block (lines 4-12). We start by typing examples:, then writing one or more examples of how we want some code to work before closing with with end. Notice that all of the examples are indented slightly, grouping them together between the bolded words. When we click "Run", Pyret will test each of our examples, and report back which ones are correct and which ones are not.

Debrief with students to ensure that they are looking at the messages that appear in Pyret. This activity not only provides practice thinking about the additive inverse; it also gives students exposure to tests - bits of code used to verify that code is working as we would expect. Examples and tests are widely used in programming! We explore examples in greater depth in Functions: Contracts, Examples & Definitions.

For the remainder of the activity, we will examine examples blocks. The first one includes only examples that pass. The second one has some errors! And the third one includes just one examples…​ you will be responsible for providing additional examples!

Optional: If you would like to offer students additional practice with the additive inverse in Pyret, two optional activities include Are They Identical? (Additive Inverse) and Writing Equivalent Code.