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Factor Findings

  • Lesson
Number and Operations
Jamie Piecora
Farmingville, NY

In this lesson, students first create factor posters for a variety of different numbers that will be displayed in the classroom to be utilized as a resource throughout the school year. They make discoveries about factors using color tiles, represent their discoveries using graph paper, and display their information on poster board as find factors of an assigned number.

Begin the lesson by dividing the students into pairs. For this lesson it would work best to pair up students based on ability. Those students of similar abilities should be paired together. Give each pair of students 12 plastic color tiles. Instruct those students to arrange these 12 tiles into an array. Students will create one of the following:

3280 2x6 3280 3x4 3280 1x12 

Ask the students to identify how they would express these arrays in numbers. [6X2, 3X4, 12X1.]

This example will lead to a discovery of the commutative property. Some students will have 6X2 and others will have 2X6. This is a great opportunity to discuss the commutative property. Bring attention to the fact that the arrays still look the same they are just positioned differently. Both problems still only provide the factors 6 and 2.

Then write just the numbers on the board: 1,2,3,4,6,12.
Explain that these are the factors of 12 because they are the only numbers that can divide evenly with no remainders into that number. This is easily seen when you refer back to the arrays that they created. Demonstrate that they can check to see if they have all of their factors by creating a factor rainbow as seen below.

3280 rainbowA factor rainbow is a way of showing factor pairs in a list of all of the factors of a number. Factor rainbows are used to check whether a list of factors is correct. To create a factor rainbow, the student must list the factors in order from least to greatest. They can then draw an arch that links the factor pairs. For square numbers, there will be no connecting arch in the middle; therefore the student can put a square around that number.

Provide the students with another example of finding factors of a number using the color tiles and then creating a factor rainbow. Possible numbers to use would be 16 or 9. Lead the students through the process for this example. Then have the pair of students decide on a number that they would like to find factors of. You may limit this number to 50, depending on the amount of tiles you have available. When they have decided on their number they can get that many tiles from a bucket of tiles located somewhere in the classroom and use them to find the factors of that number. Instruct the students to create the factor rainbow in their notebooks. Check their factor rainbows to validate the students’ understanding before you introduce the main activity.

For the main activity, instruct the students that they are each going to get a number and it is up to them to find all of the factors for that number and create a poster that the entire class can use throughout the year. Give each pair of students a bag of a different amount of color tiles. (ex. 18, 20, 24, 36, 40, 56, 60) Give the bags with the lower amounts to the pairs of students who may be struggling. Instruct the students what to do with tiles using the following directions:

  1. Challenge the pairs of students to find as many different arrays as they can, using the plastic color tiles they have been given. They must use all of the tiles each time.
  2. Each time they find an array they can then represent it on graph paper.
  3. The students color one square on the graph paper for every one plastic color tile in their array. This will form an array on their graph paper.
  4. They will cut it out and glue it to their poster and label it with the corresponding factors.
  5. They continue this process until they believe that they have found all of the factors.
  6. The students then check their factors by creating a factor rainbow at the bottom of their poster.
  7. The students must also include a title on their poster. Their poster should look similar to the following:

3280 poster 

To conclude the lesson the students will display their posters on the wall and the class will have a gallery walk. During a gallery walk each student will walk around the room and look at everyone’s work as if they were in a gallery. They will each be given some post-it notes they can use to anonymously comment on any piece of work and place on the poster. They will also write down two facts they discover after reviewing all of the posters. [Just because a number is larger doesn’t mean that it has more factors than a smaller number; all even numbers have a factor of two; the number 16 is a perfect square; etc.]

  • Plastic color tiles
  • Graph paper
  • Crayons
  • Poster board
  • Scissors
  • Glue
  • Computers or tablets with internet connection (optional; for extension)

Assessment Options

  1. The students’ Factor poster be used to assess the student’s understanding and ability identify the factors of a given number using color tiles, graph paper and factor rainbows.
  2. Throughout the lesson, circulate, observe and question the students as they create their Factor poster. Use the questions from Questions for Students to assess students’ understanding.
  3. An individual assessment can also be done to assess a student’s understanding of the concept of using color tiles, graph paper and factor rainbows to find factors. Put the number 24 on the board and have the students find the factors using one of the three methods and record them on a half sheet of paper at the end of the lesson. If you are going to use the number 24 try not to give a group 24 tiles during the lesson.
  4. Give students completed factor rainbows. Ask them to create the corresponding arrays and the original number.


  1. As an extension to this lesson the students can compare posters in order to find the greatest common factors and the least common factors of the numbers on the posters they are comparing.
  2. Students can play the Factor Game on the computer or tablet.
    appicon Factor Game

Questions for Students 

1. What do you notice about the posters that have the number two listed as a factor?

[That the given number is even]

2. What would a factor poster of the number three look like?

[It would just have one set of factors, one and three.]

3. What is a number called when the only factors of that number are one and itself?


4. When is it useful to know what the factors of a number are?

[It's useful to know factors when you divide, when you are finding common denominators and simplifying fractions, when you are measuring and you have to add fractions of an inch with different denominators.]

Teacher Reflection 

  • Were the students given enough time to investigate and then create a poster?
  • Could some pairs of students have more than one given number and create 2 factor posters?
  • If I hang these posters around the room as a resource for the students, what other lessons could I use to reference these posters?
  • Was this lesson enough to get the students to understand the concept of factors or should I reintroduce the topic again?

Learning Objectives

Students will:

  • Identify the factors of a given number using color tiles, graph paper, and a factor rainbow.

NCTM Standards and Expectations

  • Develop fluency in adding, subtracting, multiplying, and dividing whole numbers.
  • Select appropriate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tools.

Common Core State Standards – Mathematics

Grade 4, Algebraic Thinking

  • CCSS.Math.Content.4.OA.B.4
    Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

Grade 4, Num & Ops Base Ten

  • CCSS.Math.Content.4.NBT.B.5
    Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP1
    Make sense of problems and persevere in solving them.
  • CCSS.Math.Practice.MP4
    Model with mathematics.
  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP7
    Look for and make use of structure.