Back to: Module: Helping Students Do Math

# Purpose

The purpose of this activity is to learn how to help students more effectively use calculators to support their math learning. Remember: Math problems can consist of calculation steps (for which calculators are quite useful) and thinking steps (for which calculators are of limited value). Helping students understand this distinction is an important basis for calculator use.

# Procedures

- Identify a student, either one you work with or one you know through your family or friends.

- Ask about the student’s willingness to work on two puzzles using a calculator. If the student owns a calculator already, use that one; if not, any four-function basic calculator or scientific calculator that you can bring or borrow will work fine.

- Tell the student that the first puzzle is called “The Broken Calculator.” The student must pretend that the calculator is mostly broken with only six keys still working: 3, 4, subtraction sign, multiplication sign, equal sign, and clear button. Using only the six available keys, the student must figure out how to produce as many of the numbers 1–10 as possible. For example, typing 3 × 4 – 3 – 4 would result in “5.” The “3” and the “4” can be produced simply by pressing the respective button. Let the student figure out as many of the rest as possible before trying to help.

- Tell the student that the second puzzle is called “The Evil Calculator.” With this calculator, not only is one button broken, but also when you press it, it calculates a different number. For example, if the “3” is the evil button, it might calculate an “8” every time you press it. If you typed in “3 + 3”, the calculator screen would actually calculate “8 + 8” and show 16 as the answer. Based on the following results produced by the evil calculator, can the student figure out which button is broken and which number it’s calculating instead? The student can use his or her calculator as much as desired. The answer can be found at the very end of this activity (you may want to try the puzzle out yourself before looking at the answer!).

16 – 6 = 10

6 + 6 = 4

4 – 3 = 1

5 + 9 = 14

6 + 4 = 6

If the student doesn’t know where to start, try giving the following hints, in order. Notice that the hints are questions—this is a warm-up for Unit 6 on asking questions!

Hint #1: Which equations are correct and which are not?

Hint #2: Is it possible for exactly one digit to be wrong and the overall equation to be correct?

Hint #3: Can you come up with an equation where two digits are wrong and the overall equation is correct? (For example, if “3” produces an “8”, then 8 – 8 = 0 is still correct, even though two digits are wrong.)

Thank the student for participating.

## TAKE NOTES

In what ways was the calculator helpful for working on the problems?

For what parts of the problems was the calculator not useful?

Now think of problems you encounter in your life that involve math.

How could calculators be useful in solving any of these types of problems?

In what ways would the calculators not be useful?

## Answer to the “Broken Calculator Puzzle”

One possible answer set to “The Broken Calculator”

1 = 4 – 3

2 = 4 × 4 – 4 – 4 – 3 – 3

3 = 3

4 = 4

5 = 4 × 3 – 4 – 3

6 = 3 × 3 – 3

7 = 4 × 4 – 3 – 3 – 3

8 = 4 × 3 – 4

9 = 3 × 3

## Solution to “The Evil Calculator”

Every time the 6 button is pressed, the evil calculator calculates a 2. (Notice that the “6” in the answer of the final equation really is a 6, because it came from a calculation rather than from the 6 button being pressed.)