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Students discover the Identity Property by continuing their exploration of Circles of Evaluation.

Lesson Goals

Students will be able to…​

  • Recognize that multiplying and dividing by one does not affect the value of an expression.

  • Recognize that adding or subtracting zero does not affect the value of an expression.

  • Demonstrate their understanding of the Identity Property with and without variables via Circles of Evaluation, numeric expressions, and words.

Student-facing Lesson Goals

  • Let’s explore the Identity Property and Circles of Evaluation!

Materials

Supplemental Materials

🔗The Identity Property

Overview

Students determine if pairs of Circles of Evaluation represent equivalent expressions. The Circles of Evaluation make visible the structural changes that occur when we apply the Identity Property.

Launch

Build on prior knowledge by reminding students of the category names they created in Equivalence, and draw their attention to any category names that connect to the Identity Property. For example, a category like “multiplying by one” or "adding zero" is a strong hint that students have an intuition for the Identity Property!)

Complete Discover the Identity Property.

  • What did you Notice about the three Circles of Evaluation in Question 1? What do you Wonder?

  • Possible responses: All three Circles include 12 + 4. All three Circles evaluate to 16.

  • What did you observe about the Identity Property for division? Subtraction?

  • Possible responses: I can multiply or divide by 1 and get an equivalent result. I can add or subtract 0 and get an equivalent result. I need to remember that Commutativity does not apply for division and subtraction!

Multiplying or dividing an expression by 1 does not change its value; similarly, adding or subtracting 0 results in the original value.

Confirm that students understand the main idea about the Identity Property! They will apply this knowledge throughout the lesson.

Investigate

Sometimes, we will encounter the Identity Property in sneakier forms!

(+ 27 (​- 8 8))

How does the Circle of Evaluation above illustrate the Identity Property?

  • In the Circle, zero (or 8 - 8) is being added to 27!

Now, consider this equation, which is really a pair of equivalent fractions:

$$\displaystyle \frac{3}{4} = \frac{15}{20}$$

When we find equivalent fractions, we multiply by one! In the example above, 3 /4 was multiplied by 5 /5 to get 15 /20.

As students explain why various Circles of Evaluation do not belong with the others, encourage them to practice using vocabulary that they have encountered already.

Synthesize

  • For which operations does the Identity Property hold? Why?

  • Did you encounter any instances where the Identity Property was used in combination with another property?

🔗The Identity Property and Variables

Overview

Students use their knowledge of the Identity Property to determine if expressions with variables are equivalent or not.

Launch

We can represent the Identity Property with variables, in the same way that we could represent the Associative Property and the Commutative Property with variables. Here’s an example:

(+ 3 q)

=

(​* (+ 3 q) 1)

3 + q

=

(3 + q) × 1

Or like this:

(+ 3 q)

=

(+ 0 (+ 3 q))

3 + q

=

0 + (3 + q)

The Identity Property will hold no matter what values we substitute in for q or for t. Although computation can help us test equivalence, there’s no way to test every possible value. The Identity Property lets us prove that these expressions are equivalent even with variables.

Investigate

As they did with the Commutative Property and the Associative Property, students now make the transition from numeric values to variables.

If students would like, they may choose values to represent the variables. Early finishers can substitute in numbers of their choosing to confirm that their analyses of the Circles of Evaluation are correct.

Synthesize

  • Did you use Computation to check your work? Or do you prefer thinking about properties and equivalence?

  • There is a version of the Identity Property for each of the four basic operations - addition, subtraction, multiplication, and division. But the Commutative and Associative Properties only apply for addition and multiplication. Why is the Identity Property different from these other properties?

🔗Programming Exploration: Identity Property

Overview

Students explore WeScheme functions that take in an image and produce an image identical to the original.

Launch

You’ve discovered that multiplying or dividing an expression by 1 does not change its value. Similarly, adding or subtracting 0 results in the original value. Each of these four applications of the Identity Property is represented in the four Circles of Evaluation below.

(​* m 1)

(/ m 1)

(+ m 0)

(​- m 0)

Each of these four Circles evaluates to m which means the Identity Property requires an operation that does… nothing!

We can also represent these four applications of the Identity Property with the Circles of Evaluation below, with some made-up functions that take in just ONE input. The functions' describe what they do!

If students struggle with this idea, revisiting the idea of a "function machine" can be helpful!

(multiply-by-one m)

(divide-by-one m)

(add-zero m)

(subtract-zero m)

No matter what value we use for m, we will get that value back.

Select four students - one to act out each of the four functions above (multiply-by-one, divide-by-one, add-zero, and subtract-zero). Make it clear to the class what each function’s name is. Emphasize that each function expects a Number, and will produce Number.

Just as with any acting career, this one will begin with a rehearsal. When I say, "multiply-by-one 24", you say, "24" Let’s try it.

  • Teacher: multiply-by-one…​ 24

  • Student: 24!

  • Teacher: divide-by-one…​ 366

  • Student: 366

  • Teacher: add-zero…​ 4

  • Student: 4

  • Teacher: subtract-zero…​ 16

  • Student: 16

That was a great rehearsal. You’re ready for the stage! Now it’s the class' turn to give you cues! Who’s got an input for our one of our actors?

Go around the room soliciting expressions from students until it’s clear that everyone could run this script in their sleep.

Thank the fantastic actors who brilliantly played the roles of multiply-by-one, divide-by-one, add-zero, and subtract-zero!

Let’s make a list of other function names that follow the Identity Property. In other words, if we give the actor playing that function any number, the function will return that same number.

If students struggle, you can offer some suggestions to get them started. For instance, a + (8 - 8) or (24 - 23) /times h.

Investigate

We just talked about four different functions. Each one consumed a number and produced a number. But what about functions that consume images? Can the Identity Property apply to those?

  • an image of a dogWhat functions do you know that will transform this image of a dog?

  • Sample responses: scale, rotate, overlay, etc…​

  • What is something we can do to this dog (right) that will transform it…​ but still result in the exact same image?

Solicit student responses and record them on the board. If students do not volunteer answers, consider sharing one or two of the following possible responses: flip it vertically twice; flip it horizontally twice; rotate it 360 degrees clockwise; rotate it 360 counter-clockwise; scale it by 1; slide it some distance and then return it to its original position.

Students will be working in WeScheme soon - but we recommend keeping things unplugged for now! You might even print up an image of the dog - or any other image you choose - to display on the board at the front of the room. Invite students to the board to demonstrate their ideas by manipulating it.

  • Open Identity Property Starter File and click “Run.”

  • Type dog into the Interactions Area to see what the defined image looks like.

  • How many different ways you can transform dog and still get the same image back!

  • Spend a few minutes to see how many “do-nothing” transformations you can make.

  • Up for a challenge? Try applying multiple “do-nothing” functions to the dog image.

  • What “do-nothing” transformations did you come up with? Let’s share.

As students share, record their responses on the board by drawing the Circles of Evaluation that represent their ideas. Some possible responses are below.

  • Complete The Identity Property and Images, where you will explore scale, rotate, flip-vertical, and flip-horizontal further.

  • As you work through this activity, be sure to make predictions about the code before testing it out!

Synthesize

  • What did you discover? For what values did scale and rotate produce identical images of the dog?

  • In your own words, describe how functions in WeScheme helped you understand the Identity Property.

These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, 1738598, 2031479, and 1501927). CCbadge Bootstrap by the Bootstrap Community is licensed under a Creative Commons 4.0 Unported License. This license does not grant permission to run training or professional development. Offering training or professional development with materials substantially derived from Bootstrap must be approved in writing by a Bootstrap Director. Permissions beyond the scope of this license, such as to run training, may be available by contacting contact@BootstrapWorld.org.

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