# Finding Factors

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## Objective

SWBAT determine all the factors of a number less than or equal to 100

#### Big Idea

Every positive integer (except for 1) can be represented as the product of at least 2 factors.

7 minutes

Students work in pairs to complete the Think About It problem.  As they create their rectangles and multiplication sentences, I let students know that they don't need to display each rectangle twice.  What I express to them, without using numbers, is that a 1x24 rectangle is the same as a 24x1 rectangle - I don't want to give away one of the factor pairs!  I hold up a piece of paper and let students know that holding it horizontally or vertically doesn't change that it is one rectangle.

The most common student error here will be that students don't organize their work, and therefore forget one or more factor pair.

I have students share out the rectangles that they were able to create, and then ask students to raise a hand if they forgot a factor pair on their lists.  I frame this lesson by telling students they'll have the chance to practice their division fluency and learn a way to organize their work that will help them to capture all of the factor pairs for a given number.

## Intro to New Material

10 minutes

Students worked with factors and multiples in 4th grade.  This lesson is a review of previously learned materials, and it's designed to set students up for our work finding the greatest common factors of numbers and solving application problems involving GCFs.

The Intro to New Material section in this lesson is a short one.  The goal is to get kids using the academic vocabulary that goes along with this lesson, and then also have them work systematically using t-charts to organize their factor pairs.  Below are the steps that students will follow.

Steps for Finding Factors of  a Number

1)      Create a small T-Chart and write the number on top.

2)      On the left side of the chart start with the first factor, 1. On the right side of the chart across from the 1, next to the 1 write what number you multiply 1 by to get the original number, which will be that original number.

3)      Continuing in numerical order, try each number mentally (or by  dividing off to the side), and write each factor pair.

4) Stop when a factor pair repeats itself.

After we go through the process of creating a t-chart for the factor pairs of 36, I guide students through filling in the definitions at the bottom of the page.  These terms are not new for students.  I want my students to use these terms throughout the lessons in this unit.

Factor – one of two or more whole numbers that are multiplied to get a product.

Factor pair – two whole numbers that are multiplied to get a product.

Divisible – a number is divisible by another number when it can be divided by that number and the remainder is zero.

Relatively prime numbers - two numbers are relatively prime if they have no common factors other than 1.

Prime number – a number with only 2 factors, 1 and itself. Or, a number with one factor pair.

Composite number – a number with more than 2 factors. Or, a number with more than one factor pair.

## Partner Practice

20 minutes

Students work in pairs on the Partner Practice  problem set.  As students work, I circulate around the room and check in with each group.  I am looking for:

• Are students talking about their thinking with their partners?
• Are students identifying the prime numbers?
• Are students working systematically to find all of the factors?

• Explain you determined the factors.
• How did you begin looking for the factors?
• How did you know you found all of the factors?
• What numbers is ___ divisible by?  How do you know?  Is it divisible by ______?  How do you know it isn’t?

In the t-chart section, 3 of the 6 problems are prime numbers.  This was intentional, as I want students to encounter prime numbers and become fluent in recognizing them.  This will be important in our upcoming work, with prime factorization and GCF.

Before students work on the Check for Understanding, I use the partner practice problems to quickly review vocabulary.  I display one student's t-charts on the document camera.  I ask students which numbers had only 2 factors, and then ask students for the word for numbers that have only 2 factors.  I then ask for the term we use for numbers that have more than one factor.  Students then complete the CFU questions independently.

## Independent Practice

15 minutes

Students work on the Independent Practice problem set.

In this problem set, there is less of a reliance on t-charts.  Students will get more practice with finding all of the factors of numbers as we work with greatest common factors.  Students have the opportunity to justify their responses when answering the 'how do you know' questions.

Problem 16 gives students a preview of the work to come.  This problem may be difficult for kids, but it is a good chance to have them build perseverance in problem solving.

## Closing and Exit Ticket

8 minutes

After independent work time, we come together as a class to discuss Problem 17.  I ask if anyone has come up with what they think is the mystery number.  If there are responses, I record them on the write board and then ask students to test each of the responses against the clues with their partners.  After 2-3 minutes of work time (depending on the number of responses), I then have students share out their thinking about each possible answer.

If students do not have an answer for Problem 17, then we'll talk about each clue as a class.  I'll ask students what each clue tells us, and we'll come up with possible answers ourselves, until we get to 64.

Students then complete the Exit Ticket independently to close the lesson.