Grade 2: Conceptual Understanding of Even & Odd
2.OA.3 Determine whether a group of objects (up to 20) has an even or odd number of members, e.g. by pairing objects or counting by 2's; write an equation to express an even number as a sum of two equal addends.
Students enjoy learning about even and odd numbers. It is a fun topic. Let's take a look at a few activities that can support students' understanding of even and odd numbers.Counters
Using linker cubes or counters allows students to physically pair the individual pieces. Let's say we have a group of 9 counters and we want to determine if the number is even or odd.
We can start by pairing the counters. There are 4 groups of 2 and 1 counter that does not have a pair. Think about how the visual can support the learning. Some visuals may support components of the topic better than others. Think about your lesson objective, the related standards and the sequence of the learning activities. Also consider possible misunderstandings that might arise. As you make choices about the sequence of activities, questions asked, and models used, keep in mind the ultimate learning goal and where you are in getting your students to that point. It is journey.
If all the counters are red, paired in 2's and one is not paired, students will be able to see the pairs and that one member does not have a partner. Turing the one counter to yellow highlights the fact that all of the red counters are paired and one counter does not have a pair - it is yellow. Students could flipped the counters over to the opposite color as the pair up the counters, which will leave the one that does not have a partner a different color. This is an instructional choice we will need to make.
Working in 2's shows that a number is even, or that the number is not even - it is odd. In the picture above, the counters are paired and somewhat scattered. If you are working on the connection between counting by 2's (starting at 0) and even numbers, this representation can be supportive of making the bridge from skip counting to even numbers. Students can write a number sentence 2 + 2 + 2 + 2 + 1 = 9.
However, it is also important that students have an understanding around even numbers as equal addends. Scattered pairs do not emphasize the understanding that 8 can be written as 4 + 4 = 8. So, let's take a look at a different way to arrange and represent even and odd numbers with counters and linker cubes.
If the counters we arranged in 2 columns (or 2 rows), pairs can be made as you create the array. If one counter does not have a partner, then the number is odd. Again, I flipped the one non-partnered member to yellow to emphasize that it was not paired. This can be done as counters are paired. Start with all counters yellow, as you make a pair flip to red and place in the array. An odd number will have one counter left as yellow.
Similarly, you can pair the counters,of different colors using the array representation. Once all counters have been paired - one red with one yellow - either all counters will have a partner (even) or one counter will not have a partner (you will have one more of one color).
The same idea can be used with Linker Cubes as well.
Linker Cubes
Lesson Idea - Check out the Math Coaching Consortium website, hosted by West Contra Costa Unified School District - www.wccusd.net/math. There are tons of lessons and resources on this website. The lesson for Even and Odd: A Conceptual Understanding can be found under Lessons - Grade 2.Good teaching always starts with the learning objective. There are several things happening in this lesson plan. You may not want to do all parts in one lesson. This may be something you revisit over a few days or over the course of the year as you see students are ready to add to their understanding around even and odd numbers. However, many students are ready to begin this discussion early in the school year.
Lesson Objective: Students will be able to determine if a number is even or odd by pairing linker cubes. Students will be able to write an equation for an even number using equal addends.
In the lesson, you start as a whole class and model how students are going to explore numbers by paring counters. After modeling a few numbers, 1, 2, 3, 4, 5, & 6... begin looking at patterns.
What patterns do you notice?
Define even numbers and odd numbers. Begin creating a class anchor chart that displays the information about numbers, the representation, equations, and whether the number is even or odd. Here is a sample of an in-progress anchor chart I need in a recent professional development workshop.
After creating the models, look at each number and determine if it is even or odd. Students should discover that every other number is even; every other number is odd. Let them discuss and discover these patterns. Then help students connect that mathematics to the patterns that they notice. Give them the vocabulary and structures that they need.
Writing the equations can be a tricky part to this activity. Writing equations for even numbers as equal addends will be easier than writing equations for odd numbers (writing equations for odd numbers is not a part of the second grade standard). This might be a piece of the lesson that is revisited or left out to use as a separate activity.
The second part of the lesson has students working on a number, pairing up members of the number using either linker cubes or counters. Students will need to tell if their number even or odd and why.
At the end of the lesson, students present their number to the class. Continue adding to the class anchor chart and making connections among the numbers, the equations, and whether the number is even or odd.
Evens as Equal Addends:
By displaying the linker cubes in two towers, the pairs are next to each other, one cube from the left tower with one cube from the right tower. This allows the equation to match the visual.
For example, if I want to write the equation for 10, we can see the the left tower has 5 cubes and the right tower has 5 cubes, therefore, 5 + 5 = 10.This works well for all even numbers.
Contrast this with with showing 5 groups of 2's. Although this shows that 10 is even, it leads students to write 2 + 2 + 2 + 2 + 2 = 10. Again going back to our purpose, if our lesson wants students to write an equation for an even number with two equal addends, then showing 2 columns (or rows) leads students to write the equation 5 + 5 = 10.
Now, your students might be curious about writing equations for odd numbers. There are a variety of ways to write equations for numbers. If I was doing a lesson on breaking a part number and recomposing numbers, then we would explore a variety of equations. But since we are on the topic of even and odd, let's look at how the equal addends extends to odds.
Continuing from the even number 10, we know that 11 is one more than 10. We can think 11 = 10 + 1. Then using the equal addends for 10,
we can write it as 5 + 5 + 1= 10.
Sentence Frames
To support students in their understanding and in their presentations, sentence frames are a great tool. Here is a sentence frame that I used.
The lesson includes an exit ticket that incorporates the sentence frame as well.
Go check out: www.wccusd.net/math
Go check out: www.wccusd.net/math
Go to Lessons, then Grade 2 ... look for the lesson titled Even and Odd: Conceptual Understanding
Tips for using Manipulatives:
- Allow students time to explore with manipulatives before requiring them to do the math activity.
- Use a placemat or construction paper - all materials should stay on the mat.
- Be realistic about your procedures & rules
- Model your procedures
- Follow-through with your rules
