Teaching factors is so underrated! When kids understand relationships between factors and multiples, patterns of mathematics unfold before their eyes. Let’s talk about how to tackle Common Core State Standard 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.*

This standard offers no simple solution. Kids (and teachers) need to work hard to accomplish it. Factors must be considered conceptually and numerically. Arrays, patterns, multiplication facts (yes – multiplication facts), tables, factor trees, ladders, games, and more move students toward the final goal. Here’s how my class tackled this standard.

#### Awakening Conceptual Awareness

Fourth graders have been building understanding of factors for years. In early grades, skip counting and number towers paved the way. As a starting point, my class constructed some arrays. We began with arrays for multiples of a small number: 2, 3, 4, or 5. Each member of a group drew one set. They labeled the area (multiple) and sides (factors).

After discussing their findings, each student looked for additional arrays for some of the multiples they’d drawn. As I walked around, I heard comments like, “This number doesn’t have any more arrays,” and “Hmm, all of these multiples of four can have sides that are two too.”

#### Exploring Related Numbers

For more practice, I gave each pair of students a slip of paper with two related numbers. They generated all possible arrays for each number, cut them out, and pasted them on posters. One group came to me, puzzled that they could only find one array for 17. After going through all of the possibilities with me once more, they decided that 17 only had one factor pair. I was tickled when I saw they had crossed off the “s” on their poster title. “Arrays for 17” now read “Array for 17.”

This activity took way longer than I anticipated, but the apparent thinking and discussion of numbers was powerful. It was well worth it!

#### Looking at Prime, Composite, and Square Numbers

The next day, we hung all of the posters on our chalkboard. After the terms prime, composite, and square were introduced, students walked through the posters. They found three prime, three composite, and three square numbers and listed their factors. With just three days of work with arrays, my students were well on their way to conceptual understanding of factors and multiples.

#### Learning Multiplication Facts

Even when using a conceptual approach, multiplication facts are important. For kids who know their multiplication facts, numeric patterns are ready and waiting. The inverse is true too. When kids conceptualize multiplicative patterning, they get better at their multiplication facts. The two go hand in hand.

My incoming fourth graders had been practicing their multiplication facts for a year or so, but many of them had not reached mastery. Those with mastery needed work on fluency. We began our quest when school began. Our plan included these steps:

- Identify which fact groups a student knew (usually twos, threes, fours, fives, and sometimes nines).
- Start working with the lowest unknown group.
- Skip count that group and repeat orally over and over.
- Move onto Quick Flash II for flash card practice on that group. Master and move onto the next group.
- Play Math Facts Baseball twice each week.
- Increase speed with Fact Navigator.

Although a few students are still working on it, more than 90% of our class mastered their facts in the first four weeks of school. In my experience, fluency with multiplication facts is critical to mastering 4.OA.B.4.

#### Playing Games

Factor Captor is my all-time favorite for teaching factors! It’s now available to play online as well. Kids can play against the computer or other kids. This challenging game forces kids to think strategically *and *builds their familiarity with factors.

#### Investigating Prime Factorization

In the end, a number’s factors stem from its prime factors. This is the key to finding all factors for a number. My students built factor trees and used the ladder method. Click here to download these freebies.

#### Considering Divisibility Rules

As numbers get larger, it becomes more difficult to determine its factors. Learning divisibility rules can really help! If you’re up for a fun challenge, ask kids to generate these rules themselves.

#### Teaching Factors with Cards

I decided to create factor cards for each number between 1 and 100. Each of my 25 students received four cards. They were asked to find all factor pairs for the number on each card. After all of the arrays, multiplication fact practice, games, divisibility rules, and prime factorization, they still struggled.

I asked students with numbers from 1 to 20 to present their factors. Their classmates put thumbs up if they thought all of the factors were listed; thumbs down if they weren’t. Next, kids with difficult numbers (like 98) were asked to present. We discussed how seeing the factors ** in **factors helps us find all factors. In this example, students can see 7 x 7 in 49.

This was a turning point for my students. They began to pull it all together. I can’t emphasize the importance of this step when teaching factors.

#### Analyzing Factors

I created a large Venn diagram using hula hoops. After finishing the factor cards, students wrote their numbers on it. They could easily see that 1 was the only square number that was not composite and that primes and composites form distinct sets with no overlapping. (Sorry, my students’ writing was too faint on our class diagram, so I’m showing you the answer key instead.)

#### Time to Practice

Even after the unit on teaching factors is over, kids need practice. My students will practice with two activities: Number of the Day and Identifying Factors (and Non-Factors).

**Number of the Day** – I just finished this set of 20 Number of the Day cards. My students will find the prime factorization of a number and use it to find all factors. They’ll determine if the number is even, odd, prime, composite, and/or square. Then they’ll name some multiples of the number. Finally, we’ll will discuss. In four weeks, this little warm-up activity will generate some good discussion and move closer to true mastery.

**Identifying Factors (and Non-Factors) for a Given Number** – We’ll also use some Finding Factors from Common Core Sheets. These sheets provide wonderful practice and test prep. The multiple-choice worksheets ask kids to find a number that is (or is not) a factor of a given number.

#### Listing Factors

This is the ultimate goal. But listing factors for whole numbers from 1 to 100, which the standard requires, is no easy task. When kids can fill in factor sheets, your job is done.

Enjoy teaching factors!