Building Arrays
As an introduction to factoring, Mr. Lazur pairs his students up to work with small, square color tiles, usually available in most mathematical manipulative kits. He gives each pair one or two whole numbers to work with and instructs them to count out that number of tiles. Mr. Lazur gives larger numbers to the partnerships he thinks are stronger, but he does not give anyone a number higher than 30. Mr. Lazur then instructs the partnerships to build as many arrays as they can from that number of tiles. An array is a rectangular shape composed of smaller shapes.
As his students build arrays, Mr. Lazur teaches them to write down the dimensions of the arrays they are coming up with. Nancy and Lucas are working with the number 24. They build many arrays and write a list of dimensions including 8 x 3, 4 x 6, and even 1 x 24. Quinn and Maya are working with the number 13, and despite a lot of trying, they can only come up with one set of dimensions: 13 x 1.
Mr. Lazur supplies each pair with graph paper so that they can sketch the relevant arrays for their numbers and write down their dimensions. After his students have worked with arrays, Mr. Lazur explains that what they have done is find factor pairs for different numbers. He explains the definition of the word 'factor' and asks his students to discuss what they've noticed.
Vocabulary
As they talk, he teaches them other relevant vocabulary for understanding factoring:
- Products are the numbers we get when we multiply factors, so factor x factor = product.
- Prime numbers have only two factors: 1 and the number itself.
- Composite numbers have more than two factors.
- Square numbers can be factored into a number times itself.
Factor Rainbows
Mr. Lazur teaches his students to make factor rainbows to break numbers into their factors. To make a factor rainbow, students write 1 on one side of the page and the number they're working with on the other side, drawing an arch in the shape of a rainbow to connect them. Students then work up to see what other factors their target number has. If 2 works, they write that down, then write its pair on the right side and draw a second arch. Students continue until they have found all of the factors.
Factor rainbows help Mr. Lazur's students make some important observations and ask conceptual questions. For example, they notice:
- Even numbers always have 2 as a factor.
- If a number is a factor of another number, it will also be a factor of any numbers the second number is a factor of. For example, since 2 is a factor of 8, 2 is also a factor of 16, 24, and all other multiples of 8.
- Except for the number itself, or the target number, there will never be a factor larger than half of the target number. For instance, except for 24 itself, no factors of 24 can be larger than 12.
Factor Trees
When he feels that they have a solid grasp of the basics of factoring, Mr. Lazur teaches his students to make factor trees. These help his students get to the prime factorization of a number, breaking the number down into only prime factors that compose it.
To make a factor tree, students write the target number at the top of a page. They then write one factor pair below that number. If either of the resulting factors is composite, they write a factor pair for that number below it. They continue working until they're left with only prime numbers.
For example, 24 can be broken into 3 x 8. 3 is prime, but 8 can be broken into 2 x 4. Then, 4 can be broken into 2 x 2. So the prime factorization of 24 is 2 x 2 x 2 x 3. Mr. Lazur teaches more advanced students how to write this in exponential form.
Lesson Summary
Factoring is an important mathematical strategy that brings up important concepts and vocabulary, such as products, prime and composite numbers, and square numbers. To factor a number means to break it down into the numbers that can be multiplied in order to reach it. Building arrays, making factor rainbows, and drawing factor trees are solid strategies for helping students learn how to factor.
