2. 2.7 Reversing Operations
Launch:
Think about different actions.
What are some examples of actions that can be reversed?
What are some examples of actions that can't be reversed?
What are some math examples that can be reversed?
What are some math examples that can't be reversed?
2
3. 2.7 Reversing Operations
To show that something can't be reversed, use a counterexample.
Counterexample:
1. Mary found the sum of the digits in a number to be 6. What number
was she using?
2. Robert squared a number and got 81. What was his number?
3. Another example???
3
4. 2.7 Reversing Operations
Backtracking can be used to reverse some mathematical operations.
Backtracking: a method of reversing operations
1. Use backtracking to find the value of x that produces 37 from the
expression 3x ‐ 14.
Step 1: Write the steps, in order, that explain how to get from x to
the output value of 37.
Step 2: Reverse each step and apply to the output value (37).
2. Use backtracking to find the value of a that produces ‐12 from the
expression x + 6 .
5
Step 1:
Step 2:
4
5. 2.7 Reversing Operations
Understanding Check:
1. Getting into a car involves these steps.
• Open the car door.
• Sit down.
• Close the car door.
• Buckle the seat belt.
Describe the steps for getting out of a car.
With your partners please complete the following problems. (7 mins.)
3. Write each algebraic expression as a list of operations. If the operation
is reversible, describe the operation that reverses it.
a. n + 13
b. b
‐2
c. 3(5m ‐ 12)
d. 15m ‐ 36
4. Hidecki says, "I take a number, multiply it by 12, and then subtract 9. My
final result is ‐5." What is Hidecki's starting number?
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6. 2.7 Reversing Operations
Backtracking using Input‐Output Tables:
Below is the input‐output table for y = x2, where the expressions has be
evaluated for x = ‐4, ‐3, ‐2, ‐1, 0, 1, 2, 3, and 4. Some values of x2 are missing.
1. Complete the table.
Input ﴾x﴿ Output ﴾y﴿
2. How can you tell from the table that
squaring a number isn't reversible? ‐4 16
‐3
3. Make a similar table for cubing (y = x3) for ‐2
x = ‐2, ‐1, 0, 1, and 2. Is cubing reversible?
‐1
Why or why not?
0
Input (x) Output (y)
1
‐2
‐1 2 4
0 3
1 4
2
4. Do the input‐output tables below represent a reversible operation?
Why or why not?
a. b.
Input (x) Output (y) Input (x) Output (y)
‐6 37 0 ‐5
‐3 10 2 ‐3
0 1 4 ‐1
3 10 6 1
6 37 8 3
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