Special Numbers

This title may be slightly misleading because all numbers are special but it’s more about encouraging kids to adopt a number and really make it their own! We all know how important number sense is. If kids don’t have good number sense, then they don’t know that they’ve made mistakes in calculation and cannot then correct those mistakes. I think, on a more profound level, it also means that their ability to use numbers effectively in real life is considerably diminished. And that’s not what we want!

So, I started doing this project years ago and I didn’t invent it, although I have added to it a lot. The goal is to really become one with your number. Learn all about it! Live with it! Sleep with it! Feel the number in your core. What does it mean to be the number ___? It might seem a little ridiculous but I believe that if kids can really, really learn all about a special number, then they can begin to generalize those same ideas and concepts to other numbers, thus expanding and deepening their number sense. And, to give credit where it is due, the original idea came from a textbook called Prime Time, from the Connected Mathematics Project 2.

Here’s how I do it now in my sixth grade classes:

Step One: Students pick a number between 1 and 100.  First they make a big poster with their number in “bubble” style in the center. Then they color and decorate their number. We’ve gotten really into this step, using different mediums like pastels and watercolors. Kids have made really elaborate and beautiful designs. To encourage math-i-ness, you can show them pictures from Escher and mosaics, but I’ve found that they are pretty awesome at coming up with their own gorgeous patterns and designs.

Step Two: Then, we begin the math exploration with odds and evens to answer the question: is my number odd or even? To explore this idea we usually cut out grid paper arrays for a wide variety of numbers and see what we notice. Odd numbers, of course, will always have an extra square “sticking out”. At the same time we notice that even numbers are always divisible by two. It’s also interesting to look at sum rules: if I add two even numbers, the answer will always be….? if I add two odd numbers, the answer will always be….? if I add an even and an odd, the answer will always be….? To explore even further, you can play a simple dice game (just to warn you – it’s a set up and the game is unfair!). It goes like this: two players; one player gets a point if the sum is odd, the other player gets a point if the sum is even; take turns rolling two dice and adding the numbers together; keep track of points on a simple tally sheet. Play a designated number of turns and compare results in the class. This game really reinforces the odd and even concept. Can the kids connect this back to the arrays?

Now it’s time to add to Special Number Posters: My number is odd/even and I know this because____________________. It’s really interesting to see how kids will explain odd or even in their special number. Great opportunities for mathematical thinking, discussion and writing!

Note for working with younger grades: I think you could do both Step 1 and Step 2 with kids all the way down to kindergarten. For younger kids, pick smaller numbers. For younger kids, take more time to explore odd and even. Use manipulatives to create odd and even arrays. Use kids on the playground to create odd and even arrays. Play the dice game as a whole class (so no one cries when they lose!) and keep the discussion more open-ended. Remember the goal is to adopt a special number and really build a solid number sense as you move into the activities. Younger kids might just need more scaffolding to generalize ideas from one number to another.

Step Three: Products! (For younger students, try sums instead!) For this step of the project we create product lists using the special number times one through twelve. Then students are encouraged to explore this product list and look for patterns. Working with a partner and looking at each other’s lists can be helpful and often we will use a couple numbers that haven’t been claimed as practice together on the white board. Any kind of pattern is acceptable: odds, evens, repeated digits, additive patterns, multiplicative patterns….

Students add this list to the poster and then write about the patterns in the product list. It might look something like this: These are the products of my special number and the numbers one through twelve. I notice _______________ and __________________patterns in the products of my special number.

Step Four: Factors! Before we make a list of factors for our special number, we review factor trees, usually again with unclaimed special numbers. And this, of course, is where we get into the idea of Prime and Composite Numbers. Once a factor list has been created, then it’s easy for students to identify prime (no factors except one and itself) and composite (more factors than just one and itself). For younger kids, it would be cool to explore additive pairs. For example, if my number is 5, then the pairs that add up to 5 are 0+5, 1+4,2+3.

Students add the factor list to their poster. For older students they can also write about whether their number is prime or composite: My number is prime because __________________. Here, again, is another opportunity for mathematical thinking, discussion and writing as students share with each other and the whole class.

At this point, the special number posters could be finished and hung up for display. Students have explored some great ideas about numbers, they’ve spent a lot of time with their special number and, hopefully, they have a beautiful, full poster to share.

But there’s more, if you want to keep going…

Square numbers: is my special number a square number? What is a square number? Why is it called “square”? We usually go back to the arrays we made in the odds and evens exploration and look at the differences between arrays for numbers like 12 and 20 as compared to numbers like 16 and 25. Just asking the question, “what do you notice?” leads to great discussions. Often we will use calculators to explore the first twenty or so square numbers. If you have the capability, you can also explore square roots. Why does the symbol look like a division sign but different? Why is it called a “root”?

Perfect numbers: is my special number a perfect number? What is a perfect number? Perfect numbers are so cool! A perfect number is one in which the factors of the number (not including itself) add up to that number. 6 is the first perfect number because its factors: 1,2,3 add up to six: 1+2+3 = 6. What is the next perfect number? Can you find it? And the next one? Using calculators and working in teams is a great way to rise to this challenge. As the class works together to find the perfect numbers, they get great exposure to lots of numbers and they get to discuss the idea of “perfect”. It’s fun to go online and explore a bit about the origin of the idea of perfect numbers.

What else? Let your imagination take this project wherever you want to go and beyond! I’ve never been disappointed and I’ve always been amazed at what my students have created and learned.

Have fun with math every day!