# Number Bonds – Critical Connections and a Student Introduction

I was asked to teach a demonstration lesson for a handful of teachers last November. It was a class of first-graders, and they had already completed addition and subtraction within 10, making 10, and they were ready to start adding within 20. The school had recently adopted a Singapore Mathematics program, and one of the topics their teachers wanted to focus on was introducing Number Bonds. In this post, we'll take a deep dive into how to introduce Number Bonds, and ensure your students are off to a good start when you're through.

### What Are Number Bonds?

Number bonds are one of the cornerstone strategies in a Singapore Mathematics classroom. They are a pictorial representation of ‘part-part-whole” relationships between three numbers. Number bonds are created using at least three circles that contain quantities. The parts are connected to the whole they make with lines. Below is an illustration from the think! Mathematics 1A textbook.

At the early stages of learning, number bonds consist of 3 numbers, and they correspond to a related family of number facts. The number bonds shown above corresponds to the fact family, 2 + 3 = 5, 3 + 2 = 5, 5 – 3 = 2, 5 – 2 = 3. These fact families are taught at a later stage.

Number bonds also illustrate other relationships between the parts of a number and the whole, such as the inverse relationship between addition and subtraction. Inverse operations are ‘reverse operations' or ‘opposite operations.' Addition and subtraction are inverse operations, and multiplication and division are inverse operations. A picture is worth a thousand words here:

Inverse relationships are a critical concept for students to understand when they start their work in solving algebraic equations and inequalities. If students do not know why balancing equations works, they will be more prone to both conceptual and procedural errors.

### History and Origins of Number Bonds

“Number bonds” is not another trendy, but soon to be extinct, term in mathematics education. Its origins can be traced back to the 1920s, and gained in popularity since it's introduction to the Singapore Mathematics curriculum in the early 1970s. Since then the term has found its way into countless curricula, standards, and programs around the world.

### How to Teach Number Bonds

Number bonds are typically introduced in kindergarten or the 1st grade, and this is true in the think! Mathematics series, as well as other Singapore Mathematics programs. Before I offer a couple of suggestions to introduce number bonds to your students, let's set some ground rules. We want to ensure we use the Concrete – Pictorial – Abstract approach and so that our lesson is as effective as it can be:

- Always introduce number bonds by letting students explore with some type of concrete manipulative. Allow them the time to play and make the connections they need to make.
- Do not teach number bonds and equations simultaneously. Remember that equations are an abstract concept for first graders and should be introduced as a separate topic after they completely understand number bonds.
- Provide students with a blank number bond and a recording sheet. This will reduce the extraneous cognitive load created when they are asked to create their materials.

The following is my preferred approach to introducing number bonds, and the one I've seen teachers have the most success with. It is taken from the think! Mathematics 1A textbook. You can download the entire lesson from our think! Mathematics page and feel free to use it in your classroom if you wish.

### Lesson Plan – Introduction to Number Bonds

**CCSS: K.OA.A.1, 1.OA.C.6**

**OBJECTIVE:**

To be able to form number bonds of a number up to 10

Materials:

5 Counters per student

1 Plate sheet per student

1 number bond mat per student

**PROGRESSION:**

1. Introduction – Distribute the material to the students and provide them with the Anchor Task. Tell the students that the counters they have represent cupcakes and the two circles that say “part” represent plates. They are going to explore the different ways they can place the cupcakes on the two plates.

2. Direction – Instruct the students to talk to each other about the different ways they can find out. Allow the students 5-10 minutes to talk and use the counters to find various ways to create the two groups.

3. Discussion – Have the students come together for discussion and share the various ways they were able to create the two groups. Help them to see that five counters/cupcakes on one plate is also an option if it doesn't occur to them.

4. Formalization – Draw a number bond on the board and write the number 5 in “whole circle.” Explain to the students that they can place 2 cupcakes on one plate, and three on the second plate. Write 2 and 3 in the respective circles while moving the counters into circles. Be sure to emphasize the key understanding by saying:

“This is called a number bond.”

“5 cupcakes can be split into 2 cupcakes on one plate and 3 cupcakes on the other plate.”

“We say that ‘2 and 3 make 5.'”

“The number bond is made up of two parts and one whole.”

“The two smaller numbers are the ‘parts' of the bigger number, which we call the ‘whole.'”

“Combining two numbers together will form another bigger number.”

Complete your board and leave posted while you ask students to get out their Journals.

5. Journal Writing – Have the students write in their journals to help them reflect on their learning and build a deeper understanding and recall of the concept of number bonds. Some possible prompts could be:

Prompt 1 – Pick a favorite number and make as many number bonds as you can in your journals

Prompt 2 – Make as many number bonds as you can for the number 10

I prefer prompt 2 as it sets the stage for “making 10.”

6. Guided Practice – Have students complete scaffolded number bonds that are represented with linking cubes. Before moving them to independent practice ensure that the students have a correct understanding and are using the correct vocabulary. This is a good time to ask the students if it makes a difference in which direction we write the number bonds in. Once students have completed making some number bonds with support, it may be a good time for a quick “Learning Check.” Students who demonstrate a solid understanding can be allowed to continue to independent practice in their workbooks, while those students who are still struggling can form a small group for a mini-lesson.

### Supporting Struggling and Advanced Learners

We find that when students struggle with number bonds, it is usually either language or the concept of parts and a whole. You can help these students with mini-lessons while others are working independently. You might give these students a strip of paper and tell them that you are giving them a “whole strip of paper,” then ask them to “split” or “break” the whole into two parts. Demonstrate this and then ask them:

Do I still have the same amount of paper I started with? (Join the two parts if necessary) Yes!

What has changed about the strip of paper that I started with? You tore it into two pieces.

Correct, we tore the WHOLE strip of paper into two PARTS.

Sometimes students struggle with translating the number of physical objects into the numerals in a number bond. A little push by asking them to count out objects and choose a numeral can help. Use linking cubes for this.

Give the students five linking cubes and ask them to count out two cubes and put them on one ‘part' of the number bonds mat. Ask them to count how many they have left and place them on the other ‘part.' Next, lay out numeral cards 0-5 and ask your students to choose the correct numeral card for each ‘part.' Count along with them again if you need to. Once they have associated the correct amounts and numeral cards you can ask them to count the total number of cubes they have in the ‘parts' and choose the correct numeral card for the ‘whole.'

### Making Number Bonds Practice Fun

Number bonds and the understanding of the ‘part-part-whole' relationships they represent are so important that we strongly suggest giving students ample time to practice, explore, and create with activities centered around number bonds. You can ask students to make their own “Number Bond Tool” which is some type of stick, string or chain with counters attached to it. Students should be able to slide or move the counters to make number bonds or use it to help them solve number bonds if they struggle with the concept. The image below links to a website with a few good projects.

Don't spend valuable class time having students create a single number bond from construction paper, plates, etc. All you'll accomplish is the memorization of a single number bond.

The Biggest Mistake Teachers Make When Teaching Number Bonds

This topic deserves its very own subheading. The biggest mistake we see teachers make when they introduce number bonds is teaching equations in parallel, or too soon after, the introduction to number bonds. Within the concept of number bonds, equations are abstract. Because of their abstract nature, they should only be introduced once students have completely mastered the idea through the concrete and pictorial representations of number bonds. We wouldn't want to have the ghost of Jerome Bruner (father of C-P-A approach) haunting us, now would we?

Remember, one of the most critical parts of our job is knowing when our students have learned what we've taught them. Number bonds are a cornerstone concept in a child's development of number sense and their future success in mathematics. Make sure you teach them well and be sure your kids have learned them well!

Don't forget that you can download this entire think! Mathematics unit from our think! Mathematics page.

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