Little Brains, Big Concepts: Part-Whole Relationships

This post is part of the Little Brains, Big Concepts series. Check out the previous posts: Introduction, Number, Counting Strategies.

I find it difficult to describe the power and awe-inspiring nature of part-whole relationships. So, this haiku will have to do:

Knitted into the
Fabric of the universe
Building blocks of math

Part-whole relationships extend across disciplines and grade levels. Even college level math concepts are rooted in part-whole relationships. In order to hold the weight of future math learning, children’s understanding of part-whole relationships must be strong and deep.

The Big Concept

The concept of part-whole relationships is fascinating in its simplicity: You can break a whole into parts, and you can combine parts to make a whole. You will also hear this referred to as decomposition and composition.

Children experience part-whole relationships in the world around them before they experience them with numbers. Take a look at how you can apply this concept to a pan of brownies:

When applied to numbers, part-whole relationships might look like this:

You can also view the image through the lens of composition: you can combine each pair of groups to create the whole of 7 ducks.

While the amount of ducks in each part changes, the amount of the whole remains the same. A group of four and a group of three are equal to seven, just like a group of two and a group of five are equal to seven.

Take a moment to appreciate the added complexity this brings to children’s development of number. Their brains have worked hard to develop an understanding of what seven means. Now, they have to adjust that understanding so that seven can be thought of both as a whole and made of parts.

This emphasizes the flexible nature of math: numbers and shapes can be decomposed and composed in a variety of ways. This is an essential understanding that leads to powerful mental math and number sense strategies. Additionally, it lays the foundation for future skills across ages and grade levels, from fractions to geometry to factoring quadratic functions.

Experiences that Build Understanding

Play with Blocks

Putting blocks together to create a structure is all about combining parts to make a whole. While your child plays, say things like, “Each block is part of the wall!” This encourages your child to continue exploring math ideas on their own while also setting the foundation for important vocabulary words.

Explore Breaking Apart Groups of Objects

Have your child explore different ways to separate a group of objects. You can use snack crackers, paper clips, gummy bears, Legos – whatever your child is interested in!

• Give them a set of 5-10 objects and count the total together.
• Have your child separate the set into two groups.
• Count the amount in each group.
• Put the group back together and count the total. What do you notice?
• Repeat, having your child form different groups each time.

Extend the activity:

• Have your child move items from one group to the other group. Ask, How many are there in each group now? How many are there in all? What do you notice?

Tip: Narrate the process so that your child hears important vocabulary: “A group of 5 and a group of 3 are equal to 8. What other ways can we make 8?”

Dot Cards

Dot cards are cards that have between 1-10 black dots arranged in a variety of ways. Below are two example sets of dot cards for numbers 1-5. (Read more about dot cards here.)

To play, lay out one set of dot cards. Then ask, “Can you find two cards that make 5?” Then ask again, “Can you find two more cards that make 5?”

Extend the activity:

• Use multiple sets of dot cards. This will require your child to think about different representations of the same number.
• Add dot cards for numbers 6-10. 7 is especially fun since there are so many ways to make it!
• Challenge your child to find three cards that make a number.

Finger Counting Games

Finger games are great because you always have your fingers with you! You can play at the laundromat, at the park, in the car, you name it.

There are many different games you can play with your fingers, but these two specifically build an understanding of part-whole relationships:

Same or Different?

1. Use two hands to show your child a number. Ask, “How many fingers do you see?”
2. Then, use two hands to show the same number a different way. Ask, “Are these numbers the same or different?“

Can you show me?

1. Use two hands to show your child a number.
2. Ask, Can you show me the same number in a different way with your fingers?

Unifix Cubes

Unifix cubes are great for physically composing and decomposing. Create a tower of Unifix cubes using two colors and count the total number of cubes with your child. Then, break apart the tower into the two colors. Ask, “How many green cubes are there? How many yellow cubes are there?”. You can extend the activity by asking, “Can you make the tower in a different way?”

Find Part-Whole Relationships in the Real World

Once you start to think about part-whole relationships, you’ll see them everywhere! Be sure you point these relationships out to your child to show them that math is all around them.

• Cooking: When you chop vegetables, you’re decomposing a whole (the vegetable) into parts. Say, We cut the carrot into parts. How many parts are there?
• Money: The relationships between coins are great for discussing part-whole relationships. Ask, How many ways can you create a quarter with nickels? What about with nickels and dimes? How many ways can you create a dollar?
• Food: Many of our everyday foods involve splitting a whole into parts. Each time you eat a slice of pizza, you’re eating a part of the whole. Say, This slice of pizza is part of the whole pizza. How many slices are there?

Take Action

• Look for part-whole relationships in your everyday life. The more you see them, the more you’ll appreciate, or dare I say fall in love with, the concept.
• Choose one of the above activities to implement with your child this week.
• Look forward to next week’s big concept post about making ten!

References

Feikes, David., Schwingendorf, Keith. and Gregg, Jeff. (2018) Children’s Mathematical Learning. Retrieved from this website.

Dougherty, et al. (2010). Developing Essential Understanding of Number and Numeration: Pre-K-Grade 2. National Council of Teachers of Mathematics.