OK, I admit it: I have factors on the brain.
First, I wrote about When Factoring Gets Personal. I followed that with a post describing what happens When Factors Put on Their Dancing Shoes. So what’s next—When Factors Apply for a Home Equity Loan? Thankfully no, but I do have an update to share.
Last week I had an idea for a new Dynamic Number Sketchpad model that’s easy to describe. A rectangular array of numbers from 1 to n appears on screen. Those numbers that are factors of n are shaded blue. The remaining numbers are shaded orange. By dragging a point in the upper-right corner of the array, you can change its dimensions.
With this concept in mind, I created the sketch. Here are two pictures of the array showing the factors of 24 and the factors of 27. Feel free to download the sketch.
What are the mathematical and pedagogical benefits of using this model?
If the goal of learning about factors is to recite the factors of any given number, then this model is horrible—it does the work for you! But I think the payoff from factors lies in analyzing the patterns that emerge when viewing the factors of lots of numbers. And that’s the benefit of this Sketchpad model. It makes the factoring itself easy and allows students to focus on what is interesting in the visual patterns of the blue and orange-shaded numbers.
Here is a partial list of questions and observations that I imagine a group of students might make while exploring the Sketchpad model:
- The number 1 is always shaded blue.
- The number in the bottom-right corner of the array is always shaded blue.
- Our array has 11 rows, and I see that the number 11 is shaded blue. In general, if we have n rows in our grid, then the number n will be shaded blue.
- What arrays have just two of their numbers shaded blue?
- There are only two ways to display the factors of prime numbers in the array—either as a single row of circles or a single column of circles.
- If the array has at least 2 columns and 2 rows, then the number it represents isn’t prime.
- If the number of rows and columns in the array are equal and prime, the array will contain exactly three numbers shaded blue.
- We found a way to create arrays with exactly four numbers shaded blue. Drag the red point to form a single row of numbers. Make sure the largest number in the row is prime. Then drag the red point straight up to create a prime number of columns. That does the trick.
- When our array contains an even number of rows, the rows in the upper half of the array are filled entirely with orange circles. Only the number we’re factoring is shaded blue. Why is that?
- We can pair every number that is shaded blue with a partner. For example, the factors of 18 are 1, 2, 3, 6, 9, and 18. Let’s pair 1 and 18, 2 and 9, and 3 and 6 together. In each pair, the product of the numbers is 18.
- What happens when we pair the factors of 25? Its factors are 1, 5, and 25. We can pair 1 and 25 together, but can we pair 5 with itself?
- In most of our arrays, there are an even number of circles shaded blue. But in some cases, the number of blue circles is odd. Is there a way to predict whether there will be an even or odd number of blue circles?
- We dragged the red point so that the numbers from 1-20 all appeared in a single row. We wanted to find other ways to display those 20 circles in the array. The numbers in blue—1, 2, 4, 5, 10, and 20—gave us a big hint. Since 2 is a factor of 20, we can make a 2 x 10 array. Similarly, we make a 4 x 5, a 5 x 4, and a 10 x 2 array.
- If the number of columns is even, the number 2 is always shaded blue. If the number of columns is odd, the number 2 alternates between orange and blue as I drag the red point up to add more rows.
- We created a game. We scrolled our sketch window so that you can only see the bottom row of circles. Your challenge is to make an educated guess about the total number of circles in the array.
What other questions and discoveries about factors can be made with the Sketchpad factor array? Share your ideas!