GET READY BEFORE THE BELL RINGS!

Take out homework and a pencil to prepare for the homework quiz!

Check the file folder for your class to pick up graded work Located by the bookshelf, blue crate. Look behind the tab for your class period and

take home anything with your name on it.

2.07 REVERSING OPERATIONS

Goals:•Reverse, or undo, a series of steps•Understand the relationships between opposite operations such as addition and subtraction or multiplication and division.•Use backtracking to solve a problem.

LAUNCH

Turn to page 120 in your textbook.Read through the list of reversible actions and operations.

Last week we investigated the examples given here for operations that cannot be reversed. Read about proving by counterexample.

MINDS IN ACTION

Pair up with another student and read Tom and Takashi’s conversation on page 121.

FOR DISCUSSION 1

What did Takashi do wrong?What number do you think Tom chose?

Takashi didn’t realize that 169 is the square of two numbers; –13

THINKING FURTHER

In general, the operation of “unsquaring,” or taking the square root of, an unknown number is not reversible since there are two numbers with the same output. Specifically, the second number is the opposite of the first.

For example, find the two numbers with a square of 121.11 and –11

EXAMPLEProblem: Find the value of x that solves the equation 3x – 14 = 37.Solution: Use backtracking to solve the equation. We read this equation as, “Three times some starting number, minus 14, is equal to 37.” The equation is like one of Spiro’s number tricks. Step 1: write the steps, in order, that show how to get from the input variable x to the output value 37.

EXAMPLEStep 2: make a list that reverses the order of the 1st list and shows how to undo each operation (reverse the machine).

37

Step 3: start with the output and go through the list of reverse steps to find the value of the input variable x.

37 + 14 = 51

51 ÷ 3 = 17

x = 17

DEVELOPING HABITS OF MINDSUMMARY: ESTABLISH A PROCESS

Step 1: make a list of steps, in order, that show how to get from the input variable to the output. Alternatively, you can build a machine diagram or a flowchart to show your steps.

Step 2: make a list that reverses the order of the 1st list and shows how to undo each operation (reverse the machine or flow chart).

Step 3: start with the output. Perform each step on the list of reverse steps (or go through the reverse machine) to find the value of the input variable.

FOR DISCUSSION 2

Derman solves an equation such as 3x – 14 = 37 by making a guess, checking it, and then making a better guess until he finds the solution. Compare Derman’s method with the backtracking method. What advantages does backtracking have? Are there any disadvantages to backtracking?

Sample answer: backtracking has the advantage of always leading to the correct solution for certain types of equations. sometimes, backtracking can involve an unnecessarily long series of steps. For example, 2 (10 – x) = 0 has an obvious solution—what is it? x = 10

CHECK YOUR UNDERSTANDING 1

Find a partner. Each person thinks of a number. Take your number and follow these steps (don’t show your partner!): Add 6 Divide by 4 Multiply by 8 Add 7 Multiply by 10

Exchange only your ending number with your partner, and find his/her starting number.

CHECK YOUR UNDERSTANDING 2

Write each algebraic expression as a statement of one or more operations. For each operation that is reversible, describe the reverse operation.

a) n + 13b) b/–2c) 3(5m – 12)d) 15m – 36

Check answers:a) “Add 13 ”; subtract 13.b) “Divide by –2 ”; multiply by –2 c) “Multiply by 5, subtract 12, and

then multiply by 3” ; divide by 3, add 12, and then divide by 5.

d) “Multiply by 15 and then subtract 36” ; add 36 and then divide by 15.

CHECK YOUR UNDERSTANDING 3Here is a table for the input and output of , where x has integer values from –4 to 4. Some values are missing from the table.Copy and complete the table.

2y x

CHECK YOUR UNDERSTANDING 3Your table should look like this:

ANSWER THESE QUESTIONS

1. From your table, how do you know that squaring is not a reversible operation?

2. Now add another column to your table for cubing the input, label the column “ Output, ”. Is this operation reversible?

3. Make tables for the outputs of Which powers produce reversible operations?

4. Why is the result positive? What powers of –3 produce negative numbers?

3x

4 5 6 7, , , and .x x x x

43

All the outputs except zero have more than one input.

Yes, because each output came from only one input.

All the odd powers are reversible.

Because when we multiply a negative number an even amount of times, the product is positive. All odd powers of –3 are negative.

CHECK YOUR TABLES

2.08 SOLVING EQUATIONS BY BACKTRACKING

Goals:•Understand the relationships between an equation and its solutions.•Use backtracking to solve a problem.•Understand basics of equations, including when equations are always true and when they are always false.

LAUNCH

Consider the following equations.x + 4 = 7, 2x + 2 = 2(x + 1), x = x + 1

• Is x = 3 a solution of any of the equations?

• Is x = 10 a solution?

•How many solutions does each equation have?

Yes; the first two equations.

Yes; the second equation.

The first equation has only one solution, the second equation is true for all values of x, and the third is never true.

DEFINITIONS

Equation An equation is a mathematical sentence stating that two quantities are equal.

This definition doesn’t state that an equation must be true; it only states that it is a complete thought about numbers. For example, the equation 3 + 4 = 7 is true, but 2 + 1 = 9 is an equation that’s false.

SolutionsThe values of the variables that make an equation true are solutions of the equation.

FOR DISCUSSION

1. How do you know that x + y = y + x is always true?

2. How do you know that is always true?

3. How do you know that x = x + 1 is always false?

Commutative Property of Addition.

By definition of squaring a number.

No number can equal more than itself.

EXAMPLE 1Why is the value 3 a solution to the equation x + 4 = 7?

The value 3 is a solution to the equation because it makes the equation true. Any other value makes the equation false, so 3 is the only solution.

You can use the term solution set for the collection of all solutions of an equation. The equation has the solution set {-3, 3}.

To find out if a number is a solution to an equation, just test it out. A variable such as x represents a number, so every time you see an x in an equation, replace it with the same number. If you get a true statement, that number is a solution.

When an equation is always false, it has no solutions. We can say the solution set is the empty set, or null set.

EXAMPLE 2

Is the number 7 a solution to the equation 3x – 28 = 46?Replace x with 7 to determine if the result is true.

≟ 46 21 – 28 ≟ 46 7 is not a solution to the equation above.

FOR DISCUSSION

4. Suppose you want to find the solution to 3x – 28 = 46 by guessing. How can you do it?

5. Can you solve this equation by backtracking?

6. Suppose you want to find both solutions to How can you do this?

EXAMPLE 3

Solve the equation to find the value of q. Suppose you divide the starting number q by 3 and then add 76. Backtrack by reversing each step in the opposite order. To find q, start with the ending number, 81, and follow these steps: Subtract 76 Multiply by 3The starting value of q is 15. After the 1st backtracking step, the remaining equation is . You find the value of q by multiplying.Verify the solution in the original equation.

FOR DISCUSSION

7. Solve the equation 3(a – 1) – 5 = 34.a = 14

8. Explain why backtracking helps you solve the equation above. Use the phrase “reversible operations.”You can use backtracking because all the steps involved in undoing the equation to get from a to 34 were reversible operations. All the steps we took above in solving the equation were reversible operations.

CHECK YOUR UNDERSTANDING

Turn to pages 128-129 in your book and work to complete the following problems:# 3, 5, 6, and 7