Carol Cashion, teacher of math and other high school equivalency subjects at the New York City College of Technology in Brooklyn, and I co-led this meeting. In October, I observed Carol’s class when she introduced factors using blocks. I was interested to see how the approach opened up a tangible way of playing with factors and concepts such as greatest common factor. In our teachers’ circle, Carol explained her lesson plan and then we explored “prime factor stacks” as a problem-posing and problem-solving method. -Eric
Carol’s lesson:
Carol generously shared her full lesson in Google Slides.
Introduce prime numbers with “I’m going on a picnic”
Can you figure out the rule below?
How about this one?
And this one?
And finally…
Carol prompted us to describe in different ways the numbers that we could bring on the picnic:
- Only prime numbers
- No numbers with factors other than 1 and the number itself
- A number that can’t be divided by more than two numbers. Can only be divided by itself and 1.
- The only way to get that number is to multiply that number with 1.
- If you make a rectangular array, there is only one you can make.
- No numbers that have 3 or more factors.
Write definitions for FACTOR, MULTIPLE, and PRIME NUMBER
Carol described how she works with the class to define concepts that they had already explored in the picnic game. The idea is to ask students to define these concepts before introducing the vocabulary words multiple, factor, and prime number. Usually, math teachers introduce the vocabulary and give students definitions to memorize. Carol is suggesting that we can move in the opposite direction. Students can explore factors and multiples without knowing those specific words, then we can attach words to the definitions they create.
Here are some definitions Carol’s students wrote, with her help:
Explore Prime Factors
You can find Carol’s full lesson in Google Slides above.
Problem-Posing with Prime Factor Stacks:
I shared the following image and asked the group what they noticed and then what mathematical questions we might pose with this model?
Noticings:
- Both have a 2 at the bottom.
- One is taller.
- Left tower is 60. Right tower is 42.
- Greater numbers are stacked on smaller numbers.
- Both have 2 and 3.
- Don’t have same number of blocks.
- Taller tower is also a larger number.
- You can make more combinations with larger tower. More ways to put them together.
- Both have odd number on top.
- Shorter tower has larger number on top. Are there other short towers that multiply to a bigger number?
What problems can we pose with this model?
- Create these two prime factor towers: 2|2|3|5 and 2|3|7. What are some factors that both towers have? What is the largest factor both towers have?
- Create a prime factor tower for 360. Is there more than one way to do it?
- Is there a way to calculate the difference between these two numbers by looking at their prime factors rather than doing the full multiplication calculation?
- Can we create a higher number with smaller factors or do we need higher number factors?
- Does a number with a taller tower always have more combinations? Is there a consistent/set number of combinations for each height of a factor tower?
- Is there a way to know how many factors a number has by looking at the prime factors?
- How can we use prime factor towers to bridge students toward working in the other direction – to FIND the prime factors of a number?
- Can you build your birthday day (day of the month)? If so, find someone whose birthday day has common factors. If not, find someone else who can’t.
- If you want a greater product, is it easier to use more factors, or larger factors ?
We then went into breakout groups to explore some of these questions.
Amy, Maya, and Sarah explored this question: What are some other ways to represent prime factors?
After the meeting, Sarah made the “atomic model” of prime factors in Desmos: https://www.desmos.com/calculator/skwbfmd9ra
Mark explored these questions:
- Does a number with a taller tower always have more combinations?
- Is there a consistent/set number of combinations for each height of a factor tower?
- Is there a way to know how many factors a number has by looking at the prime factors?
Polypad file: https://polypad.amplify.com/p/DqgEsVeA8Wuhw
Carol also shared her thinking below on how we might explore least common multiples with the prime factors stacks. I’ve never seen least common multiples taught this way. We would love to hear from anyone who tries it. Carol is planning to try it with her class, but probably not until the topic comes around again next fall…
Exploring Least Common Multiples with Prime Factor Stacks, by Carol Cashion
1. Prior knowledge: use with students who have a grasp on the concept of multiples (touched on it on the picnic!) or have created a multiples table. Ask students to “build” prime factor stacks to represent 12 and 9:
2. Set the stacks aside for the moment and create a side by side multiples tables for 12 and 9 on paper or on the board. If helpful, refer back to the stacks while building the table. Looking at the table, what is the first multiple in the table that is shared by 12 and 9?
3. Build a prime factor stack to represent that number:
4. What do you see? As students describe the third stack, encourage the use of the term “factor” once it is used by a student. Encourage them to hold each original stack up next to the “shared multiple” stack as the class answers “what do you see?” “What do you wonder?”
5. Here are two more stacks. What numbers do they represent? Let’s make a multiples table for each number, find the first (least) multiple they share, then build it:
6. What do you see? How does the least common multiple stack compare to the original two stacks?
7. Here are two more stacks. Can we find the first multiple they have in common without making a table? How? Make a table if you need to. Build a prime factor stack to represent their least common multiple.
8. On your own or with a partner, build two stacks and then build a least common multiple stack on your own. Using the stack method, how would you explain to a classmate how to find the first/least shared/common multiple of two numbers represented by prime factor stacks?
