Lesson 21

Odd and Even Numbers

These materials, when encountered before Algebra 1, Unit 7, Lesson 21 support success in that lesson.

21.1: Math Talk: Evens and Odds (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for combining even and odd numbers. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to justify whether the results of combining even and odd numbers in different ways are even or odd.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Student Facing

Evaluate mentally.

$$64+88$$

$$65+89$$

$$14 \boldcdot 5$$

$$14\boldcdot 4$$

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate $$\underline{\hspace{.5in}}$$’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to $$\underline{\hspace{.5in}}$$’s strategy?”
• “Do you agree or disagree? Why?”

21.2: Always Even, Never Odd (10 minutes)

Activity

In this activity, students examine some properties of even and odd numbers. In the associated Algebra 1 lesson, students prove some concepts about rational and irrational values. The work of this activity supports students with concrete examples before the abstract proof of the next activity and the work with rational and irrational numbers in the Algebra 1 lesson.

Student Facing

Here are some statements about the sums and products of numbers. For each statement:

• decide whether it is always true, true for some numbers but not others, or never true
• use examples to explain your reasoning
1. Sums:
1. The sum of 2 even numbers is even.
2. The sum of an even number and an odd number is odd.
3. The sum of 2 odd numbers is odd.
2. Products:
1. The product of 2 even numbers is even.
2. The product of an even number and an odd number is odd.
3. The product of 2 odd numbers is odd.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The purpose of the discussion is to show that using some examples can help understand a statement, but does not necessarily prove it for all values. Select students to share their solutions and examples. Ask students,

• “We said that the sum of 2 even numbers is always even, but nobody gave an example using numbers greater than 10,000. How can you know that it is always true if you don’t check all the possibilities?” (It is okay for students to struggle to answer this at this point. Sample response: We must use properties of numbers that are even or odd to show that it is always true, since we cannot check all possible values.)
• “If I claim that adding 2 integers is always another integer, how would you begin thinking about the statement to say it is true or false?” (I might start by thinking about specific examples. I would try a few easy examples, and if they work, I would try to get creative by using large numbers, negative numbers, or trying to find other ways to interpret the statement more creatively.)

21.3: Even + Odd = Odd (25 minutes)

Activity

The goal of this activity is to introduce students to a proof by contradiction before they examine a similar proof in the associated Algebra 1 lesson. Students follow a proof by contradiction for why the sum of an even and an odd number is always odd. Throughout the process, students answer questions explaining some of the steps.

Launch

Arrange students in groups of 3–4.

Student Facing

How do we know that the sum of an even number and an odd number must be odd? Examine this proof and answer the questions throughout.

Let $$a$$ represent an even number, $$b$$ represent an odd number, and $$s$$ represent the sum $$a + b$$.

1. What does it mean for a number to be even? Odd?

Assume that $$s$$ is even, then we will look for a reason the original statement cannot be true. Since $$a$$ and $$s$$ are even, we can write them as 2 times an integer. Let $$a = 2k$$ and $$s = 2m$$ for some integers $$k$$ and $$m$$.

2. Can this always be done? To convince yourself, write 4 different even numbers. What is the value for $$k$$ for each of your numbers when you set them equal to $$2k$$?

Then we know that $$a + b = s$$ and $$2k + b = 2m$$.

Divide both sides by 2 to get that $$k + \frac{b}{2} = m$$.

Rewrite the equation to get $$\frac{b}{2} = m - k$$.

Since $$m$$ and $$k$$ are integers, then $$\frac{b}{2}$$ must be an integer as well.

3. Is the difference of 2 integers always an integer? Select 4 pairs of integers and subtract them to convince yourself that their difference is always an integer.

4. What does the equation $$\frac{b}{2} = m - k$$ tell us about $$\frac{b}{2}$$? What does that mean about $$b$$?

5. Look back at the original description of $$b$$. What is wrong with what we have discovered?

The logic for everything in the proof works, so the only thing that could’ve gone wrong was our assumption that $$s$$ is even. Therefore, $$s$$ must be odd.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The purpose of the discussion is to understand how a proof by contradiction works. Select groups to share their responses and reasoning.

Tell students that this type of proof is called a proof by contradiction. In a proof by contradiction, we prove something is true by wondering what would happen if it were not true. These proofs begin by assuming that the claim is not true, then following that assumption through a series of logical steps to arrive at something that either contradicts itself or cannot be true. Since each step follows logically from the assumption, the assumption must be false. If the assumption is false, then the original claim must be true because that is the only other option.

• “What is the contradiction that meant the assumption must be wrong?” (The logical steps led to saying that $$b$$ is even, but the original assumption was that $$b$$ is an odd number. This is a contradiction.)