Factors Over the Rainbow
Students will be able to find all factor pairs for a given number.
Introduction (5 minutes)
- Write the number 12 on the board.
- Ask students to shout out factors (i.e. what numbers go into 12?)
- Write the factors around the number and prompt students as needed to get all factors (1,2,3,4,6,12).
- Explain, "These are all factors of 12. Factors are numbers we can multiply together to get another number. For example, 3 x 4 is 12, so 3 and 4 are both factors of 12."
- Write the definition of factor on the board (a whole number that divides exactly into another number).
Explicit Instruction/Teacher Modeling (10 minutes)
- Tell students, "You helped me find the factors of 12. But when we list factors, we have to be sure we don’t forget any. It can be easy to miss a factor if we just list them from memory."
- Explain, "One way to ensure that we have listed all the factors of a number is by finding factor pairs: a set of two numbers that, when multiplied together, result in a given product."
- Tell students that we can use factor rainbows as a way to list factor pairs and find all the factors of a number, in order from least to greatest.
- Make a rainbow with the factors of 12 (see resources for examples).
- Write another number, like 15, on the board.
- Remind students that 1 and the number itself are always factors of every number.
- Draw a factor rainbow for 15 starting with 1. Draw a big arch from 1 to 15, leaving room for other factors inside the first arch.
- Ask students, "Is 2 a factor? Is 3 a factor? What times 3 is 15?" Draw an arch from 3 to 5 to continue the rainbow.
- Tell students that when making a factor rainbow, they should keep counting up from 1 and adding factors until reaching a factor that is already listed (5 in this case). When they reach a factor that is already listed, the factor rainbow is complete.
- Guide students through an example of a square number, like 16, and demonstrate how to make a rainbow when a factor is used twice (i.e. write the factor once as the center number in the rainbow without an arch drawn above it).
- Optional Extension: Display factor rainbows for 12 and 16 side by side. Circle the common factors (1,2, and 4) and explain, "When two numbers have factors in common, we call these common factors. The term greatest common factor refers to the largest, or greatest, common factor between two or more numbers. In this case, 4 is the the greatest common factor."
Guided Practice/Interactive Modeling (15 minutes)
- Hand out the Factor Rainbows worksheet.
- Review the example problem and and complete the try it problem as a class.
- Have students complete the worksheet with their partner.
- Go over the worksheet with the class.
Independent Working Time (15 minutes)
- Have students count off one through four and assign each number a different problem (i.e. if you are a number one, make a factor rainbow for 48; other numbers could be 56, 60, or 72).
- Hand out a sheet of paper to each student.
- Instruct students to use markers to make a factor rainbow for their assigned number.
- When students are finished, invite a few students up to share so that each of the four assigned numbers is represented.
- Optional Extension: Have students pair up with a partner whose assigned number was different than their own. Instruct partners to find the greatest common factor for their two numbers.
- Provide a multiplication table for students to refer to as needed.
- Provide partially completed factor rainbows for students to finish.
- Allow students to use a calculator to find factors during independent practice.
- Have students make factor rainbows for larger numbers.
- Have students find the greatest common factor of two numbers (see resources).
Assessment (5 minutes)
- In small groups, give students a number and ask them to tell you the factors. Create a factor rainbow as a small group.
- As an alternative assessment: Hand out index cards to each student. Write a number, like 54, on the board and have students make a factor rainbow. Collect the cards and check for understanding.
Review and Closing (5 minutes)
- Ask and discuss, "Why are factors important? How can we use what we’ve learned about factors to help us in math?"