Take a look at the interactive model below (and here). Most of the numbers in the array are shaded orange, but several are blue. What is special about these blue values? They are the factors of 32, the largest number in the array.
Try dragging the red point to change the dimensions of the array. You’ll see that the pattern of blue and orange changes, with the blue-shaded numbers indicating the factors of the largest number in the array.
If the goal of studying factors is to factor actual numbers, then this model is horrible—it does the work for you! But mastering factoring, while important, is not nearly as interesting as exploring the mathematical relationships between numbers and their factors. And that’s the benefit of this Sketchpad model: It makes the factoring itself easy and allows students to focus on the numerical and visual patterns of the blue and orange-shaded numbers.
Here is a partial list of questions and observations that students might make while exploring the interactive model above:
- 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 can 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 with us!