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.
Teaching Factors & Multiples
Our favorite fourth grade teacher, Ms. Sneed, sat at a table with her grade level team. Their topic for the week was CCSS 4.OA.B.4. “This standard offers no simple solution,” she said. “Kids (and teachers) need to work hard to accomplish it. Factors must be considered conceptually and numerically. Arrays, patterns, multiplication facts, tables, factor trees, ladders, games, and more move students toward the final goal. Would you like to hear how my class tackled this standard last year?”
The other teachers nodded. “Please,” said Mr. Frank. “I think you had better luck with it than the rest of us.”
Awakening Conceptual Awareness
“As you know,” Ms. Sneed continued, “kids in fourth grade math 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).” She passed around some sample papers.
“Hey,” said Mr. Frank, “this looks like the activity we do for area.”
“Yep,” responded Ms. Sneed. “There’s a definite correlation. Anyway, 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.’ Drawing the arrays really made them think.”
Exploring Related Numbers
Ms. Sneed grabbed a piece of purple construction paper and held it up. “You’re gonna love this,” she said. “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.'”
Ms. Sneed grinned. “They were so serious, but I had to laugh a little. Just a word of warning: the activity for teaching factors took longer than I thought. But it was well worth it.”
Looking at Prime, Composite, and Square Numbers
Next, Ms. Sneed passed around a picture. “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.”
“I like the fact that the kids could move around and share their work,” said Mr. Frank.
“Yes, this project really engaged them,” responded Ms. Sneed. “Even better, with just three days of work with arrays, my students were well on their way to conceptual understanding of factors and multiples.”
Extending Conceptual Understanding
Teaching Factors with Games
Ms. Sneed held up a copy of Factor Captor. “As you already know, this 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. When I tell them to play Factor Captor, they think it’s fun and games. But secretly, I know powerful learning is going on.”
“My kids even like playing this during indoor recess,” Mr. Frank added.
Investigating Prime Factorization
Next, Ms. Sneed handed out two pages: Factor Trees and Ladder Method. Then she continued. “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.”
“This method of teaching factors is really deep,” said Mrs. Price. “By my calculation, you’ve already talked about four days playing around with factors. Why don’t you just have them list them?”
“Sure, you could do that,” replied Ms. Sneed. “But in my experience, kids fail when simply asked to list. The scaffolded plans I’m showing you work to carefully build understanding so they can list them in the end.”
Mr. Frank chimed in, “I agree with Ms. Sneed. My kids weren’t successful with this last year, and I believe it hurt them with understanding of math concepts throughout the year. For example, when I asked them to simplify fractions, they struggled because they didn’t understand factors and multiples.”
Ms. Price shrugged and motioned for Ms. Sneed to continue.
Teaching Factors with Cards
“Next I used 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.” A few teachers let out deep sighs. They felt the struggle too.
“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.”
Mrs. Price smiled feebly. She was beginning to buy in to all of this factor practice.
“Then 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.”
Time to Practice
“Yay! The end!” said Ms. Price.
“Well, not quite,” Ms. Sneed responded. “Even after the unit on teaching factors is over, kids need practice. With this set of 20 Number of the Day cards, my students found the prime factorization of a number and used it to find all factors. They determined if the number was even, odd, prime, composite, and/or square. Then they named some multiples of the number. Finally, we discussed. In four weeks, this little warm-up activity generated some good discussion and moved them closer to true mastery.”
“Four weeks?!” Ms. Price hollered. “Are you kidding? How will we have time to teach anything else?”
“Relax,” replied Ms. Sneed. “These Number of the Day pages are simply warm-up exercises. They take very little time each day but deepen kids’ understanding. I continued with the next unit – but began class each day with these. Additionally, I postponed the test on factors until after they’d had this extra practice.”
Everyone looked relieved. Ms. Sneed continued, glancing at Mrs. Price nervously. “We also used some Finding Factors from Common Core Sheets. These sheets provided 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. Again, they’re short activities…”
“It’s okay,” said Mrs. Price. “Maybe I overreacted.”
Ms. Sneed took a deep breath. “Finally, my kids were ready to list factors of numbers from one through one hundred. This was the ultimate goal. But listing factors for whole numbers from 1 to 100, which the standard requires, is no easy task. When kids can simply list factors, your job is done.”
“Thank goodness,” said Mrs. Price.
“No,” Mr. Frank chimed in. “Thank Ms. Sneed.” He smiled. “She’s given us a clear path to mastery of this standard. It will take a while for us to get there, but in the end, it’s worth it.”
Ms. Sneed fell into her chair and took a sip of water. She looked gratefully at Mr. Frank and smiled her famous little smile.
Enjoy Teaching Factors
Teaching factors helps kids conceptualize relationships. Furthermore, working with factors and multiples prepares them for higher math (like fractions). As Ms. Sneed said, spending time on them is well worth it.