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Lessons

    Aligns to:

    (Also available in WeScheme)

    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
    Glossary
    Associative Property

    When adding three numbers or multiplying three numbers, it does not matter whether you start with the first pair or the last. The same is true when either adding or multiplying four numbers, five numbers, etc.

    Commutative Property

    For any expression involving only addition or only multiplication, changing the order of the numbers will not change the result.

    expression

    a computation written in the rules of some language (such as arithmetic, code, or a Circle of Evaluation)

    Identity Property

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

    variable

    a name or symbol that stands for some value or expression, often a value or expression that changes

    🔗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 that the class created, and then draw students’ attention to any category names that allude to or suggest some familiarity with the Identity Property. (Any mention of “multiplying by one” or "adding zero" is a strong hint that students’ have clued into the Identity Property!)

    • Now, turn to Discover the Identity Property. Look at the first problem. Translate the verbal expressions into Circles of Evaluation. What do you Notice? What do you Wonder?

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

    • Study the counter-examples in Q2. Explain why each Circle of Evaluation does not represent the Identity Property.

    • Complete the page by filling in the blanks in Q3 so that the value of the Circles of Evaluation does not change.

    • 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!

    Confirm that students understand the following main idea about the Identity Property:

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

    Students will apply this knowledge throughout the lesson.

    Investigate

    Explain to students that they will sometimes encounter the Identity Property in more sneaky forms.

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

    (+ 27 (​- 8 8))

    In the Circle above, 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?

      • To apply the Identity Property, we add or subtract zero, or we multiply or divide by one.

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

      • Yes! The Identity Property can be used in combination with the other properties we’ve discussed to create equivalent Circles.

    🔗The Identity Property and Variables 25 minutes

    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?

      • Student responses will vary.

    • There is a version of the Identity Property for each of the four operations - addition, subtraction, multiplication, and division. This is not the case for the Commutative Property or the Associative Property. Why is this so? How is the Identity Property different from these other properties?

      • We do not actually change the structure of the original Circle of Evaluation when we apply the Identity Property - we simply nest it inside of another Circle, a Circle which represents adding/subtracting zero or multiplying/dividing by 1. When we applied the Commutative Property and Associative Property, we fundamentally altered the structure of the Circles of Evaluation.

    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.