I'm Thinking of a Number. Can You Find It (Using Counting Boards)?

        

Supplies: Two identical counting boards, as shown above
                     About 20 two-sided (white and red) tokens
                     Four or five variable counters (such as pennies)

Warning: The boards we are using limit the numbers we can handle, as you will see below.


In this unit, some concepts that underlie algebra will be introduced. We will use variable counters with two boards to find numbers in such problems as

"I am thinking of a number t. When you add 3 to it, you get 7. What is my number?"
"I am thinking about a number p. Three times p equals 2 times p plus 5. What is my number?"

Here, we want to name the number with one letter. (In algebra, you use one letter as the name of a variable). On a counting board, a variable is a counter (or token) that can have any value that we need it to have. In different problems, it has different values.

We use the two boards to represent the two expressions on either side of an equals sign, and whatever we do to an expression on one board, we do to the expression on the other board.

Let's try it on the problems below, and then make up some more.

  1. I'm thinking of a number. When you add 7 to it, you get 10. What is the number?

Here's how it may look using the boards:


        
My number is the green triangle. My number plus 7 = 10.


        
I subtract 7 from both sides (both boards).


        
I regroup 10 as 5 + 5.


        
My number equals 5 minus 2.


        
I regroup 5 as 3 + 2.


        
2 minus 2 is zero, so my number equals 3.
Let's check.
I'm thinking of three. When you add seven to it, do you get 10? Yes!

  1. I'm thinking of a number. When you double it and add one, you get 9. What is the number?

Here's how it may look using the boards:


        
My number is the green triangle. My number doubled plus 1 = 9.


        
I subtract 1 from both sides (both boards).


        
I half both sides by moving all tokens down one row.


        
My number equals 4.
Let's check.
I'm thinking of four. When you double it and add one, do you get 9? Yes!

  1. When you subtract 5 from the number I am thinking of, you get 3. What is my number?
  2. I added 4 to my number and then subtracted 5. I got 2. What is my number?
  3. Twice my number plus 4 is 10. What is my number?
  4. I multiplied my number by 4 and got 24. What is my number?
  5. Three times my number plus 4 equals 13. What is my number?
  6. p + 6 equals 2p - 3. What is p?
  7. 5p + 4 = 6p - 6. What is p?
  8. 3p + 10 = 4. What is p?
  9. 8p - 4 = 10p - 4

Now let's just put configurations of tokens on the two boards and regroup as far as we can!


Webpage Maintained by Owen Ramsey
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