Pan Balance - Numbers

In understanding equality, one of the first things students must realize is that equality is a relationship, not an operation. Many students view "=" as "find the answer" (Principles and Standards, p. 39), so that "3 + 4 = ?" means "Find what 3 + 4 is," and "3 + 4 = 7" means that when you add 3 and 4, you get 7. To students with this operator-view of equality, a sentence like "11 = 4 + 7" makes no sense.

The number balance can be used to verify that mathematical sentences are equivalent, such as 3 × (5 – 2) = 15 – 6.

Click in either the red or blue pan, and enter an expression with numbers (such as 11 + 3 or 7 × 2). The keypad below the pans can be used like calculator buttons to enter an expression. Alternatively, the computer keyboard can be used, with  + for addition, – for subtraction, * for multiplication, and / for division.

• If the expressions in the red and blue pans are equivalent, the scale will balance. The corresponding equation will appear in the Balanced Equations area.
• If the expressions are not equivalent, the scale will not balance. The expression with the greater value will appear to be heavier.

Press the Reset Balance button to remove the expressions from the pans. Press the Reset Table button to remove all equations from the table.

Put 4 into the left pan by typing in the red space. Then, put 9 into the right pan by typing in the blue space. What happens? Why?

Reset the balance. Put 12 into the red pan. Then, put a sum (two numbers added together) onto the right pan that will balance with 12.

• Write another sum that balances with 12. How many different sums can you find? Make a list of the different sums you find.

• Can you add up three numbers that will balance with 12? Four numbers?

• What is the greatest number of numbers that you can use to balance with 12? What is the smallest number of numbers?

• Reset the balance. Put 12 into the red pan, and put a sum into the blue pan that will balance with 12. Now, put 3 + 9 into the red pan, and put something into the blue pan that will make it balance. Then find something else that will balance with 3 + 9. How many different answers can you find?