Odd Man Out

• ## Odd Man Out: Identifying Sums of Numbers as Odd or Even

Grade: PreK to 2nd,3rd to 5th
Periods: 1
Author: Brenda Lidestri

### Instructional Plan

For a brief overview of this lesson plan, please watch the following video.

Snap cubes are preferred for this activity. If none are available, provide 1" squares cut from cardstock. Instead of having students build "two towers" with snap cubes, have students build "two towers" with the squares—two columns of squares next to each other on a rigid piece of paper. Make sure each student also has a copy of each activity sheet.

Begin the lesson by introducing the term "odd man out" by presenting a fictional scenario in which the students’ physical education teacher will be providing dance lessons. For the lessons, each student will need a partner. Have students form 2 equal lines facing each other, with students directly across from each other. Tell the students that the person they are facing is their "partner." If there is one student left over, he/she is the "odd man out" or "odd person out" because he/she doesn't have a partner. If all the students have partners, tell the students you will be joining in. This makes you the "odd man out." Have students return to their seats. Ask students what they think "odd man out" means. Ask: "If there were 2 more students in the class, would there still be an 'odd man out'?" [Yes.]

After students return to their seats, tell them they will be looking at odd and even numbers today. Introduce "two towers" by showing students how to snap sets of two cubes together and snapping the pairs of cubes together to form a tower that is two cubes wide and some number of cubes tall. (If snap cubes aren't available, demonstrate with paper squares. You may choose to use a document camera to demonstrate the building of “towers”. Otherwise, you may do it on a desk or table where students can gather around.) Reinforce the concept of pairs as you distribute the snap cubes to students.

Tell students that they will be building towers based on a number. Each cube will be paired up with a "partner." Have students count off from 1 through 9, returning to 1 after the 9th student. Have each student create a "two tower" for the number he/she counted off.

Give students time to build their towers. Once everyone has finished, ask two students with even towers to bring them to the front of the classroom (or the center of the classroom, especially if paper squares are used).

Ask, "What do these towers have in common?" [Each cube has a partner; there is no odd man out.] Ask, "If we put these two towers together, do you think we'll get an odd number or an even number?" [We'll get an even number because every cube will have a partner.]

Demonstrate putting the towers together to make a new "two tower" to represent the sum. Record the information on the board by creating a chart, shown below, or use the document camera to display the Tower Power Activity Sheet.

(Example)

 Addends (even/odd)? + (even/odd)? Is the sum odd or even? 2 + 4 Even + Even 2 + 4 = 6; Even

Reinforce the meaning of odd and even numbers as you fill in the chart.

Have two more students with even numbers combine their towers in front of the class. Record the results in the chart. Ask, "Why has the sum been even both times we've added even numbers?" [When even numbers are added, they can always be divided into groups of 2. There is no odd man out in either tower we are adding.]

Now have two students who have "two towers" with an odd man out show their towers. Ask, "What do these towers have in common?" [Both have an odd man out. They both have a cube without a partner.]

Have the two students put their towers together and extend the table with their results:

(Example)

 Addends (even/odd)? + (even/odd)? Is the sum odd or even? 3 + 5 Odd + Odd 3 + 5 = 8; Even

Repeat the process with two more students with "odd man out" towers and record the results in the table. Ask, "Why has the sum been even both times we've added odd numbers?" [When odd numbers are added, the two "odd men out" become paired. There is no odd man out in the new tower.]

Next, have one student with an even numbered tower and one student with an odd numbered tower show their towers. Ask, "If we put these two towers together, do you think we'll get an odd number or an even number?" [We'll get an odd number because the odd man out still won't have a partner.]

Distribute the Tower Power Activity Sheet and have students put together their own towers. Make sure students show the sum and record the information in the chart:

(Example)

 Addends (even/odd)? + (even/odd)? Is the sum odd or even? 4 + 5 Even + Odd 4 + 5 = 9; Odd

Once students have about half of their charts filled out, have students that combined an even number and an odd number tower to raise their hands. Ask each student to state their addends and whether the sum was odd or even. You may wish to write these on the board to have students comprehend that an even + odd = odd. Have students also record the results in the activity sheet. Ask, "Why has the sum been odd when we've added an even and an odd number?" [The even number can be arranged in into groups of 2 with no cube left over, but when you add the odd number to it, there's one cube left over.]

Ask students what they learned about adding even and odd numbers. [Adding 2 evens gives an even sum; adding 2 odds gives an even sum; adding one even and one odd gives an odd sum.] Students should first record this on their Tower Power Activity Sheet, and then, share their findings with the class.

Distribute the Odd Man Out Activity Sheet and have students complete it.  After students complete the activity sheets, discuss the results as a class. Look for additional patterns, or "sub-patterns." For example, the order of the numbers does not matter when adding an even number and an odd number. Addition is commutative. So two conjectures could be made:

1. Odd plus even is always odd.
2. Even plus odd is always odd.

Have students share any additional patterns they find.

### Ideas for Differentiation

• Students who need a challenge can be asked to determine whether or not the following sums will be odd or even:
• Even + Even + Even [= Even.]
• Even + Even + Odd [= Odd.]
• Even + Odd + Odd [= Even.]
• Odd + Odd + Odd [= Odd.]
• Encourage students to explain their thinking.
• For struggling students, the activity sheet can be modified so that all or some of the numbers are single-digit.

### Assessments and Extensions

Assessment Options

1. The activity sheets, Tower Power and Odd Man Out, can be used to check for student comprehension.
2. Have students fill in each blank with "odd" or "even" and then complete the statements:
The sum of two even numbers is always _______ because… [even.]
The sum of an even number and an odd number is always _______ because… [odd.]
The sum of two odd numbers is always _________ because… [even.]
3. Give students a pair of number cubes. Have them roll it to find the sums of different combinations of numbers (i.e., 3 + 7, 4 + 8, 9 + 6, 3 + 7, 4 + 8, 9 + 6, etc..). Have students use drawings to show whether the sum would be odd or even. If no dice are available, simply give students a combination of numbers.

Extensions

1. Have students use the "two towers" to model subtracting even and odd numbers to determine whether or not the differences are even or odd.
2. Ask students what happens when any number is doubled. Will the result be even or odd and why? [The sum will be even.] If 2 × n (or 2n) represents an even number, how would you represent an odd number? [Sample answer: 2n+ 1.]
3. Extend the concept to multiplication. Have students work in groups of 3. In each group, each student creates a "two tower" to show the same even one-digit number. They should then write an addition sentence to represent the sum of the towers. Then, they can write a multiplication sentence to represent the same total. (For example, if there are 3 students and they each used a "two tower" to show the number 4, (each student would have a two tower two cubes high) their addition sentence would be 4+4+4=12, and their multiplication sentence would be 3 × 4=12.). Then repeat with all three students using the same odd number. Have each group share their findings, recording on the board whether or not the factors were even or odd and whether the product was even or odd.
4. If you roll two number cubes each with faces numbered one through six, which is more likely to occur: a sum that is odd or a product that is odd? [For the sum to be odd, one cube must be even and the other odd. This occurs 18 times out of the possible 36 combinations. The probability of an odd sum is 1/2. For the product to be off, both number cubes must be odd. This occurs 9 times out of the possible 36 combinations. The probability of an odd product is 1/4. It is more likely to get an odd sum than an odd product.]

### Questions and Reflections

Questions for Students

Refer to the instruction plan.

Teacher Reflection

• What were your observations regarding student use of the model to represent even and odd numbers?
• How did the visual model help students generalize about sums of even and odd numbers?
• How did you challenge high achievers?

### Objectives and Standards

Learning Objectives

Students will:

• Create models of numbers.
• Determine whether numbers are odd or even.
• Determine whether sums of numbers are odd or even.
• Recognize patterns of the sums of odd and even numbers.
Common Core State Standards – Mathematics

Pre K to 2nd

• Kindergarten
• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP7
Look for and make use of structure.
• CCSS.Math.Practice.MP8
Look for and express regularity in repeated reasoning.

Pre K to 2nd

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP7
Look for and make use of structure.
• CCSS.Math.Practice.MP8
Look for and express regularity in repeated reasoning.

Pre K to 2nd

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP7
Look for and make use of structure.
• CCSS.Math.Practice.MP8
Look for and express regularity in repeated reasoning.

3rd to 5th

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP7
Look for and make use of structure.
• CCSS.Math.Practice.MP8
Look for and express regularity in repeated reasoning.

3rd to 5th

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP7
Look for and make use of structure.
• CCSS.Math.Practice.MP8
Look for and express regularity in repeated reasoning.

3rd to 5th

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP7
Look for and make use of structure.
• CCSS.Math.Practice.MP8
Look for and express regularity in repeated reasoning.
Common Core State Standards – Practice
• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP7
Look for and make use of structure.
• CCSS.Math.Practice.MP8
Look for and express regularity in repeated reasoning.