Decomposing Numbers

Explicit Instruction/Teacher modeling

• Introduce the key term decompose and explain that it means to break apart. Provide an example of things that decompose in different contexts. (e.g., In our every day lives, food will decompose, or break apart, over time and it is no longer good to eat. In science, a tree's leaves will decompose into dirt.) Ask students to think of any additional things they know that decompose.
• Explain that in math, we decompose numbers to break them apart and make them smaller and easier to work with. Tell students you have a certain number of cookies, and you'd like to break them up into different amounts. Write the number 10 on the board, and underneath it write two equations: ____ + ____ = 10 and ____ + ____ + ____ = 10.
• Model thinking aloud about how you could break apart the number 10 and create different equations. (e.g., Say, "I could decompose 10 by adding five and five. Or I could decompose 10 by adding two and three and five.") Point out that 10 is a friendly number to work with, so decomposing 10 would not always be necessary. The reason for decomposing is to make a problem simpler with friendlier numbers.
• Provide an additional example of a number that you'd like to decompose. For example, share that you have a multiplication problem (e.g., You need to figure out how many seats there are in the theater if there are 13 rows of five seats. You know the expression is 13 x 5) and you'd like to break one factor, which is any one of two or more numbers that are multiplied together to give a product. Tell students that you'd like to break 13 into easier numbers since math facts with 13 aren't as easy as others. Use manipulatives while you think aloud by sorting 13 blocks or counters to model decomposing the number.

Guided Practice

• Instruct students to take out their whiteboard and whiteboard marker. Distribute enough manipulatives to each student so they can visually decompose the numbers in the following practice problems.
• Put students into groups of 3-4 students and ask them to decompose the number 9. Challenge them to show at least two different ways to decompose the number. Discuss as a class by calling on nonvolunteers to share one way to decompose the number. Then, follow the same process with the number 17.
• Write a multiplication expression on the board (e.g., 8 x 4) and provide a real-world example to give students context for the numbers. Then, model decomposing one of the numbers. Think aloud about how you'd like to decompose the number eight, which is the larger factor, because it will make the problem easier for you. Explain that when you decompose one of the factors, you multiply each of the parts by the other factor in the expression, and then you add them together. For example, the new expression will be (2 x 4) + (6 x 4). Share that you will not be fully solving the multiplication expressions, but instead focus on how to decompose and set up the new problem.
• Have small groups use their whiteboards, markers, and manipulatives to decompose one of the factors in the following problems without solving it:
  • The tray has six rows of five donuts each. How many donuts are on the tray? The expression is 6 x 5.
  • The farm has nine baskets of three apples each. How many apples are in the baskets? The expression is 9 x 3.
• Discuss each of the problems as a class and ask questions such as, "How are the two strategies or methods the same? How are they different?" and "Why did you choose to decompose this factor?" Provide sentence frames for students to use as they participate in discussions.

Group work time

• Introduce the Number Talk activity that students will do by explaining the following steps:
  • 1 - Independent Think Time: Tell students that they will see a multiplication expression displayed on the board and they will be given 1-2 minutes to decompose one of the factors and create a new multiplication expression without paper or talking.
  • 2 - Whole Class Share Time: Explain that this portion of the activity will be when students share their method or strategy they used to decompose one factor and create the new multiplication expression.
  • 3 - Display Ideas: Share that you will create a visual display for each of their methods while they share their strategies, but that they also have the option to create their own visual displays to show.
  • 4 - Questions: Explain that you will ask questions to get students thinking and talking about the different methods, so this will be a time for discussion.
• Begin the Number Talk by sharing a real-life scenario, such as "My kitchen was really messy, so I organized the cabinets with my dishes. I made seven stacks of four plates. How many plates did I organize?"
• Display the following multiplication expression: 7 x 4. Follow the steps that you shared with the students. Ask questions such as "How are the two strategies or methods the same? How are they different?" and "Why did you choose to decompose this factor?" and "How can you represent that factor in a different way?" and "Do you agree/disagree with their answer? Why?" Support learners by displaying sentence stems that they can use while answering the discussion questions.