Download Section 2

Section 2.7 Reversing Operations (p. 120)
Goal: Understand the relationships between opposite operations.
Vocabulary building:
Operation – add, subtract, multiply, or divide
Reverse – to go backwards, to go back where you came from
Backtracking – reversing operations
Inverse operations – operations which are the opposite of each other; ex. Add and
subtract
Mathematical symbols or formulas:
/
a fraction bar means to divide
(x)(3) two parenthesis next to each other tells you to multiply
3(4) a number next to a parenthesis tells you to multiply
Main ideas:
1. Think about things you do which can be done in reverse (undone) such as if you open a
window, you can close it again. If you change the speed on the fan, it can be put back to
where it was.
2. Some things are irreversible such as baking cookies. Once the ingredients have been
mixed, you can’t take them apart again.
3. Reversing in math is working with inverse operations. If you add 3 and 5, you get 8. If
you take 8 and subtract 5, you get 3.
4. In this section we will work with “backtracking”. We will take an equation and go
through the operations in reverse order to get back to the beginning. Look at this example
from p. 121:
Talk through the problem out loud: three times x minus fourteen equals thirty-seven.
Notice how these two steps are the opposite of what happens in step 1.
5. On p. 122 take notes over the “rules” or steps to backtracking.
6. Practice with the idea of backtracking:
a. n + 3
Read it as n plus three.
The reverse is subtract three.
b. 2n – 4
Read it as two times n minus 4.
The reverse is add four,
divide by 2.
c. -2(4m + 6) Read it as four times m plus 6 all times negative two.
The reverse is divide by negative two, subtract six, and divide by 4.
Section 2.8 Solving Equations by Backtracking (p. 126)
Goal: Understand the relationship between an equation and its solutions.
Vocabulary building:
Equation – mathematical sentence that states two quantities are equal
Ex. 2x + 3 = 6
Solutions – answers; values of the variable that make an equation true
Solution set – The number or set of numbers that make an equation true
Empty set – when there is no solution
Null set – (same as empty set)
Mathematical symbols or formulas:
{ }
represent “empty set” or “null set”
{1, 2, 3} when numbers are inside, they would be the solution set
0
another way to represent the “empty set” or “null set”
Main ideas:
1. An equation says two things are equal, but it does not have to be a true statement. See
p. 126 for examples.
Example of true:
Example of not true:
2. Look at the example at the top of p. 128. Notice they checked their answer when they
got done to make sure they had backtracked correctly.
INVESTIGATION 2C (p. 134)
This part of the chapter will teach you about linear equations and how to solve them.
Section 2.9 Getting Started (p. 135)
Goal: Evaluate and simplify expressions using the basic rules.
Vocabulary building:
Mathematical symbols and formulas:
Main ideas:
This section brings us back to number tricks. This time however, we need to see if we can
use backtracking to find the original number. Try some of the questions on p. 135.
Section 2.10 When Backtracking Doesn’t Work (p. 138)
Goal: Evaluate and simplify expressions using the basic rules
Vocabulary building:
Mathematical symbols and formulas:
Main ideas:
1. Sometimes backtracking doesn’t work. For example, in the equation 3t + 12 = 5t + 6
there are variables on both sides of the =. We can’t backtrack this because we don’t know
the result.
2. We can solve the equation by using a number line. (from p. 138)
Notice t jumps by the same amount top and bottom. The 12 was randomly drawn as a
jump. The 6 has to meet the 12 because the red has to equal the blue.
If you think about getting rid of the t’s top and bottom, you’d lose the 3 on the left and
have this left over:
which is 2t + 6 = 12. Now you could use
backtracking to find the answer.
3. Now try this one which comes from p. 139:
Notice again that w always takes up the same amount of space each time it jumps. The
red and the blue do finish at the same place on the right.
Again, if you start at the left and look top and bottom, you can cancel out some w’s. You
are left with:
Now you have 3w + 8 = 23, and you can use
backtracking to solve.
(If you understand the number line, you can skip the next part of the notes. If the number
line confuses you, you might try this method instead.)
4. Since an equation has an = sign, it is like the balance you might use in science class.
You can solve an equation by thinking about balancing. Let’s go back to the original
problem of 3t + 12 = 5t + 6. Instead of using the number line, you could use symbols and
create a “balance”.
(Notice the = sign.)
You could now think about removing t’s from the left and right sides.
Now you have 12 = 2t + 6. You could either
backtrack or start removing the red triangles from both sides.
Now you have 6 = 2t, and it’s very easy to backtrack to
find out t = 3.
5. Practice this one using the number line method:
6. Practice this equation using the balance method:
Section 2.11 Basic Moves for Solving Equations (p. 143)
Goal: Solve equations using the basic moves.
Vocabulary building:
Mathematical symbols and formulas:
Main ideas:
1. Read the conversation between Tony and Sasha on p. 143.
2. How is Sasha’s method similar to the number line and balance we practiced with last
class period? (If you have trouble answering, try solving it using the number line or
balance and then compare.)
3. Look at the basic moves of solving equations on p. 144. Write the basic moves into
your notes.
4. Look at the example on p. 145. Notice how the equation is being solved kind of by
backtracking (using opposites) and kind of like you did with the number line or balance
by taking away from both sides.
5. Finally, notice how every example has shown how to check your answer to see if it’s
correct. This is very important.
Section 2.12 Solutions of Linear Equations (p. 148)
Goal: Understand equations can have multiple solutions or no solutions.
Vocabulary building:
Theorem – something which has been proven true
1
2
3
Reciprocal – flipped fraction; ex. 2 and are reciprocals; and are reciprocals
2
3
2
Linear equation – an equation which only has the power of 1
Ex. 3x + 2 = 5 but NOT 3 x 2  5 because the x is to the 2nd power.
Mathematical symbols or formulas:
Main ideas:
1. Write down the theorems from p. 148.
2. What word do you see in “linear”? A linear equation represents a line. It only goes to
the first power. (Remember that x is the same as x1 , and we usually don’t use an
exponent.)
2
m  6  15
Examples: 3x + 2 = 5
-4x -6 = 12
3
3. Linear equations can get a solution in one of three ways:
No solution – when you solve the equation, you might end up with something
like 2 = 4 which is a false statement. (Remember your answer
can be given as the empty set or the null set.)
One solution – This is what we’ve been working with. Your answer might
be n = 3.
Many solutions – When you solve the equation, you might end up
with -4 = -4 which is a true statement.
4. Read the top of p. 149 to understand “no solutions”.
5. Try writing examples for the bottom of p. 149. (You will run into these on your quiz.)
Section 2.13 Focus on the Distributive Property (p. 153)
Goal: Solve an equation involving many variations of the distributive property.
Vocabulary building:
Mathematical symbols and formulas:
Main ideas:
1. Review how to use the distributive property. Whatever is outside the ( ) is multiplied
by everything inside the ( ). Example: 3(x + 4) = 3x + 12