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Factors and Multiples
When I taught fifth grade, my school had a meeting with incoming fourth graders and their parents. Some of the most asked questions had to do with math and multiplication facts. I remember telling them that it’s not just knowing your times tables, what’s more important is being able to recognize factors and multiples of numbers – to be able to look at a number and know what can divide into it. Factoring is a skill that students use in fractions and division, but also problem solving, algebra and even quadratic equations when they move through more advanced math in middle school.
How to Introduce Factors and Multiples
I like to introduce factors as building blocks. Have you ever seen the LEGO Movie? I have used scenes from the movie to talk about factors. If you are a “Master Builder” then you can see the blocks that fit together to build the number. Factors are the blocks that multiply together to create the product.
I like to tell my students that if they work to become a Master Builder, they will “see” the factors of any number. I also like to talk about the Factor Factory. Factors are equal to or less than the number and they multiply together to make the product. Just like a factory Factors make products.
Students Often Confuse Factors and Multiples
Factors (factories) make products. And Multiples Make More, they multiply. There are an infinite amount of multiples, but a finite number of factors.
I like to play scenes from the LEGO movie, especially where the main character realizes he is a Master Builder. You can find the movie or links to these scenes. When students successfully find all the factors, we also celebrate by rocking out to Everything is Awesome (I usually just use the first 48 seconds of the song).
Set up for the Factors and Multiples Activity
 Bring in LEGOS, duplos, blocks or unifix/snap cubes
 Index cards or small pieces of paper
 Coins or tokens
 Graph paper
 Playing cards
 Hundreds Chart
 Factors and Multiples Worksheet for every student
 Video clip or lego instructions book
Put students in small groups or create stations around the room with the following materials:
Station 1: Playing Cards, Graph paper
Station 2 : Dice and Playing cards, coins/tokens
Station 3: Coin, hundred chart, index cards
Station 4: Dice and Legos/blocks
Launch the Factor Factory Activity
You can find a clip from The LEGO Movie to help you talk about what it means to be a “Master Builder”. Being a Master Builder is not just following the instructions, but “seeing” the pieces or parts that you need to build the product, as shown in the image below.
A goal of knowing the factors or factoring a number is seeing ALL of the factors that can be used to create that number. A Master Builder will “see” them all. A good way to show how to gather all of the factors is to build up the number with factor pairs.
I like to talk about “closing it up, or wrapping up the product”.
So a good way to show this is to tell the students to write the number they want to factor on the top. That’s the goal. That’s the product they’re building. Can they see all of the factors, like a “Master Builder” can see the correct blocks?
The Universal Factor
The best way to find all of the factors is to start with the universal factor – 1. 1 is a factor for every number. A great way to show this is to have 6 coins or cards. Ask students to put them into equal rows. 1 row of 6, 2 rows of 3, 3 rows of 2 and 6 rows of 1. Talk about how the factors are the arrays or arrangements that they can make with equal rows. With any number, you can always have 1 row of something. 1 is always a factor.
Help the students see the factor pairs by using their cards or coins, but also show them how they can factor by knowing their multiplication facts.
Show the students how to write the number they are factoring, on the top of that “factory” and underline it, like this:
Then, inside the factory, write the factor pairs. Always start with 1 x it’s factor pair. Then try 2 and write its pair, then 3, then 4, then 5, then 6! If they build all the way to the factory number then they will find all of the factors.
Students can also show these pairs in arrays with their coins or cards.
Similar to image below:
Ask the students if they notice any similarities between the arrays and their factories?
A 2×3 array is basically the same as a 3×2 array. So show how in the factory, as soon as a pair is repeated, the factory can be “closed” or wrapped up. When they write the factors as pairs, the students can find all of the factors faster. Do they need to count all the way to 6? No. They can find all of the factor pairs 1,6,2,3 by the time they get to 3, because that pair gets repeated.
The Difference Between Factors and Multiples
This activity is to practice finding all of the factors. But before we begin, I like to talk about the difference between factors and multiples. One way I do that is either call on a volunteer or ask the class to stand up and say the multiples of 6 out loud. I stand there, I even help them out if they need it, and I wait. I wait longer than you would think. I let them keep going, and going, and going. (If you really want to make a point, you could do a smaller number like multiples of 2 or 3, but I like to use 6 to show that it has several factors, but the multiplies never end). The point is to show that multiples make more, they get bigger, and there is no end. But factors, even though they involve multiplication, have a set amount that can make up the number. Once you have the class list multiples of 6 and find the factors of 6, they can see the difference between the two terms.
Check their understanding by trying to factor 15.
Use the Factory as well as cards or tokens/coins to find all of the factors together.
Remind the students that as soon as a pair is repeated, they have found them all. They are Master Builders when they’ve found all of the factors!
The Factor Factory uses the idea of building up from 1 and finding the factor pair. This is a great way to help students work on their times tables while strategically becoming a master builder of math. Once they have found the factors, they can prove it with arrays of cards/coins. There are only 4 ways to show 15: 1×15 or 15×1 and 3×5 or 5×3. 2 pairs.
In the activity, the students must use a number (or product) and build it with its factor pairs. They must find all of the factors and become the Master Builder. Students can show all of the arrays as well as write out the factors in the Factor Factory.
The Factor Factory Activity
A good way to practice factors is to make stations and provide lots of practice. Stations can help students find numbers, show or color arrays, and create factor factories, all of which will help them increase their fluency with finding factors.
Here is a list of stations that can help you get the students moving, generate numbers and find factors.
Station 1 : Playing Cards: Draw 23 cards, multiply the face value together and then find ALL factors of that product.
Show arrays on graph paper.
Station 2: Roll a dice and draw a card. Find the product. Then find ALL of the factors of that product.
Show arrays with coins or tokens.
Station 3: Flip a coin onto a Hundreds Chart or close your eyes and randomly choose a number. Find the factors of the number you land on. Make a factor factory with index cards showing the factors.
Station 4: Roll 2 dice – if you roll a 1, roll again. Find the product. Build arrays with lego blocks or blocks. Each brick can be 1. For example: 8 could be built with 8 blocks, but 2 rows of 4 or 4 rows of 2 or 1 row of 8 etc…
Reflecting on the Factor Activity
Use these questions to guide a discussion with students (Note: Written reflection questions are also included at the end of the Factors and Multiples Worksheet):
 Which number that you found has the most factors? The least?
 How did the arrays help show the factors? Explain.
 We know that every number has a factor of 1 and itself. What do you know about factors of even numbers?
 Can you find ALL of the factors of 30? List them.
 Can you find ALL of the multiples of 30?
Factors and Multiples Extensions
 Students can Make a Factor City for a certain number. – For example 100. On graph paper or with blocks, create buildings that each have 100 blocks/squares. However, each building has a different shape because they are built in factor pair arrays. Using lego bricks or blocks, students can make it 3D.
 Draw a sign or poster that advertises certain “Factories.” Draw and color some factories with all of the factors inside for each one. Be sure to choose numbers that are higher than 24.
FREE Factors and Multiples Worksheets and Resources
These are all PDF Files. They will open and print easily. The Student Edition Files are labeled SE and the Teacher Editions Files are labeled TE. Click the links below to download the different resources.
 43 Assignment SE – Factors and Multiples (PDF)
 43 Assignment TE – Factors and Multiples (Member Only)
 43 Bell Work SE – Factors and Multiples (PDF)
 43 Bell Work TE – Factors and Multiples (Member Only)
 43 Exit Quiz SE – Factors and Multiples (PDF)
 43 Exit Quiz TE – Factors and Multiples (Member Only)
 43 Guided Notes SE – Factors and Multiples (PDF)
 43 Guided Notes TE – Factors and Multiples (Member Only)
 43 Interactive Notebook – Factors and Multiples (PDF)
 43 Lesson Plan – Factors and Multiples (PDF)
 43 Online Activities – Factors and Multiples (PDF)
 43 Slideshow – Factors and Multiples (PDF)
Factors and Multiples Worksheets and Resources
To get the Editable versions of these files Join us inside the Math Teacher Coach Community! This is where we keep our full curriculum of 4th Grade Math Lessons and Activities.
 43 Assignment SE – Factors and Multiples (Member Only)
 43 Assignment TE – Factors and Multiples (Member Only)
 43 Bell Work SE – Factors and Multiples (Member Only)
 43 Bell Work TE – Factors and Multiples (Member Only)
 43 Exit Quiz SE – Factors and Multiples (Member Only)
 43 Exit Quiz TE – Factors and Multiples (Member Only)
 43 Guided Notes SE – Factors and Multiples (Member Only)
 43 Guided Notes TE – Factors and Multiples (Member Only)
 43 Interactive Notebook – Factors and Multiples (Member Only)
 43 Lesson Plan – Factors and Multiples (Member Only)
 43 Online Activities – Factors and Multiples (Member Only)
 43 Slideshow – Factors and Multiples (Member Only)
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Don’t Forget to Pin The Factor Factory Activity…
This lesson is from…
Unit 4 – MultiDigit Whole Number Division
 41 Division Using Area Models
 42 Whole Number Quotients and Remainders
 43 Factors and Multiples
 44 Prime and Composite Numbers
 45 Division of Multiples of 10, 100, and 1,000 by SingleDigit Numbers
 46 Three and FourDigit Division with Divisors of 2, 3, 4, and 5
 47 Division with a Zero in the Dividend or in the Quotient
 48 Division with Divisors of 6, 7, 8, and 9
 49 Solve OneStep Division Word Problems