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Topic 1: Solving
one step equations
Name: ________________________
Per: ________________________
Date: ________________________
Class: Algebra 1A________________
Notes:
Integer Review:
Adding and subtracting
integers
A ________ number
a ________ number =
a ________ number
A ________ number
Example: 11 + 8 = _________
a _________ number =
Example: 6  8 = ________
a _________ number
So, two negatives make a _________ ____________.
And, two positives make a ________ _____________.
When adding, a positive and a negative _____________
and take the sign of the __________ ___________.
Examples:
A.
Simplify each
expression
C.
–3 – 9
B.
–4 + 8
6 – 13
D.
7+15
A double negative _____________ and becomes a __________
Examples:
Simplify each
expression
D.
– 5 + ( – 16)
E.
– 6 + (– 14)
F.
– 6.2 – (– 6.9)
G.
8.8 – (– 7.9)
Multiplying and dividing When multiplying or dividing,
integers
A positive # divided by a positive # = ______________________.
A positive # divided by a negative # = _____________________.
A negative # divided by a positive # = _____________________.
Note: The same is true
if you multiply.
A negative # divided by a negative # = ____________________.
When several terms are multiplying or dividing, an ________
number of negatives become __________________.
And an ______ number of negatives stays _________.
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Examples:
I.
– 6 (5)(– 7)
J.
(–24)/6
K.
6(–6)
L.
12
3
M.
4  9
N.
15
3
O.
(2)(7)
P.
18
2
Simplify each
expression
More Examples:
The goal of Algebra
The goal of solving equations is to get ________________
(you’re trying to get __ by __________).
Opposite Operation
We may have to move terms to get the variable by itself. We do this
by using the opposite operation.
So, if something is adding to the variable, we ____________.
If something is subtracting from the variable, we _______.
If something is multiplying to the variable, we _________.
If something is dividing from the variable, we _________.
Examples
Solve each equation for x. Be sure to show your work.
Q.
x + 4 = 11
R.
x + 2 = –5
S.
x – 12 = 10
T.
x – 6 = –1
V.
5x  15
U.
3x  18
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More Examples
Solve each equation for x. Show your work.
W.
x
6
12
X.
x
2
6
Y.
x
 4
4
Z.
x  4
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DAY 2: STAR skills: Key words or phrases that will be used throughout your math classes!
Deductive Reasoning
Reasoning _____________ from given statements to form a
_____________________.
If a student is on the cross country team, then the student can run a
Example
long way. What can we conclude?
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Example 2 If a student wears plaid, then the student must be Scottish.
Conclude: ___________________________________________
Example 3 If a number is a prime number, then the number can only be divided
by itself and one.
Conclude: ___________________________________________
Inductive vs. Deductive ___________ reasoning goes from _____________ examples to a
Reasoning
______________ conclusion. Many times inductive reasoning will
have examples that use ______________ (which are specific).
___________ reasoning goes from _____________ examples to a
______________ conclusion.
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Which statement is an
A) Bob concluded that 5  8  a positive number, because when a
example of deductive
positive number and a negative number are added together the
reasoning and which is
answer is always the sign of the larger number.
an example of
inductive reasoning?
B) Betsy concluded that two negative numbers always add to make
a larger negative by calculating the sums of several pairs of
negative numbers.
Q #1: Which of the following best describes deductive reasoning?
A using logic to draw conclusions based on accepted statements
B accepting the meaning of a term without definition
C defining mathematical terms to correspond with physical objects
D inferring a general truth by examining a number of specific examples
Q #3: Consider the arguments below.
I. Every multiple of 4 is even. 376 is a multiple of 4. Therefore, 376 is even.
II. A number can be written as a repeating decimal if it is rational. Pi cannot
be written as a repeating decimal. Therefore, pi is not rational.
Which one(s), if any, use deductive reasoning?
Conditional Statements If-then statements (like the _____________ _________)
These are often used in logical reasoning.
Hypothesis
The hypothesis of a conditional follows the _______
Circle the hypothesis: If x 2  4 , then x  2 or x  2 .
Conclusion
The conclusion of a conditional follows the ________.
Circle the conclusion: If you want to find the perimeter of a shape,
then you add up all the sides lengths.
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Examples What is the conclusion of the statement below? (circle it)
If x  5  9 , then x  5  5  9  5.
What is the conclusion of the statement below? (underline it)
If 2 r  c , then r 
Perimeter
c
2
To find the perimeter of any shape, ________ all the side lengths.
Example The lengths of the sides of a triangle are 6, 12, and 14 centimeters.
Find the perimeter of the triangle.
The lengths of the sides of a triangle are y, y+2, and 8 centimeters.
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If the perimeter is 48 centimeters, what is the value of y?
Reminder:
To find the perimeter of
any shape, you
____ _____ ________
A tougher perimeter
If the sides lengths of one triangle are x, 2x + 3, and 3x, and the
problem
sides of a second triangle are x, 2x – 1, and x + 4, what is the
Step 1: _________
difference between the perimeters of the two triangles?
your ______________.
Step 2: Difference
means ____________.
Step 3: Set up an
_______________.
Step 4: __________
the equation.
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Sometimes/always/
Statements made may be true sometimes, always, or never.
never
For example, if x  2  5 , then x  7 is _________________ true.
Or, two negatives make a positive is __________________ true
(explain here)
Counterexample
A counterexample is an example that _________________ an
argument.
For example, if I say that every odd number is prime, a
counterexample to that would be the odd number ___________.
Example The chart below shows an expression evaluated for one value of x.
Finish filling in the table for the three missing values.
x
x2  3
1
4
3
7
9
Reading this chart, it appears that for every positive value of x,
x 2  3 produces a number divisible by 4. Can you find a value of x
that would serve as a counterexample?
Now turn the page and fill out the summary.
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Summary: Below is a model used to show adding negative and positive
integers. Write the expression you think is being modeled and then simplify it.
_________________
_________________
What did
represent in your expression?
What did
represent in your expression?
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Topic 2: Solving
two step equations
(Day 3)
Name: ________________________
Per: ________________________
Date: ________________________
Class: Algebra 1A________________
Notes:
Variable
Usually ____, but any letter that represents the ______________
quantity.
Constants
The terms in an equation that don’t have a _______________.
Example Identify the variables and the constants in the following equation:
2x  3  x  7
Like terms
If two or more terms are the same variable, they are ______ _____.
If two or more terms are constants, they are _______ _________.
If like terms are on the same side of the ________ sign, they ______
their signs and ____________ together.
If terms are on the _______________ side of the ________ sign,
you must use _____________ _____________ to move the like
terms to the same side of the equal sign.
Example Identify the like terms in each equation(circle the variables and
underline the constants).
A) 2x  4  12
B) 3x  4x  13
C) 14  2x  4
D) 12  4x  2x
E) 4x  12  16
F) 6  2x  6
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A helpful tool: the
To determine whether you need to use _____________
equal sign
______________ before combining like terms, it can help to draw a
_____________ _________ through the equal sign. Do this on
examples A through F on the last page. When do like terms need to
move?
Solving two step
equations
Goal: Get all of the _________________ on one side, and
all of the _____________on the other side.
Then (usually) divide to get ___________ or ____________
Try this 1) 3x  2  14
2) 4x  4  8
Solve for x.
3) 2x  3x  11
More Practice 5. 5  7 x  2
4) 3  5x  18
6.
7  3x  2
8.
12  2  7x
What value of x is the
solution to the
equation?
7. 8  5  3x
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More Practice 9. 4x  20  4
10. 7  13  2x
What value of x is the
solution to the
equation?
11. 6  x  10
12. 14  2x  18
Next step: Moving two
Identify which example you need to use opposite operations before
terms
combining like terms.
A) x + 13 = – x + 21
Practice C) 4x + 14 = 6x – 38
B) 3y – 5y = 27 + 19
D) 5x + 14 = 3x + 6
What value of x is the
solution to the
equation?
E) x – 6 = 2x + 12
F) -2x + 8 = x - 7
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More practice G) 3m – 21 = 28 – 3m
H) 3x – 6 = x + 11
What value of x is the
solution to the
equation?
Moving the variable to
x = 5 is equivalent (means “the ______ ___” ) to ____ = ______
the right side and
Sometimes you may want or need to move your variable to the right
solving.
side of the equation to solve (usually either to keep your variables
___________ or because your constants are only on the _______
side of the equation). For example,
Case 1) –x + 12 = 2x – 6
Case 2) 3x + 52 = 7x
Try this Solve each of the two cases above by moving the variables to the
right side of the equation. In which case was it better to use the right
side for variables? Why?
When like terms are
Sometimes your like terms may be on the same side of the equation.
on the same side of
For example, 4n + 14 – 8n = 38. Explain in sentences the two ways
the equation
you would combine like terms in this equation.
Explain here:
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Examples: 1) 4x  3x  5  2x  25
2) 10x  4x  2x  32
What value of x is the
solution to the
equation?
3) 2x  5x  15  x  21
Try this 5) 12x  6x  7 x  10  16
4) 22  12x  7  7 x  2x
6) 6x  4x  8x  13  5
What value of x is the
solution to the
equation?
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What value of x is the
solution to the 7) 2x  x  6  18  9x
8) 14  2x  7  5x  2x
equation?
Summary: In the model below
the number 1. So 3x  2 looks like
represents a variable and
and 2x  4 looks like
represents
.
Explain how to solve
=
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Topic 3: Problem
solving strategies:
Drawing a picture
and using rate.
(Day 4)
Draw a _____________
Name: ________________________
Per: ________________________
Date: ________________________
Class: Algebra 1A________________
Notes:
To help ____________ a word problem, sometimes it is helpful to
draw a _________________. Depending on the problem, you
could _________ a _____________ of a ________, or a
________________, or a _________ or ___________________.
Then, you can ____________ the _____________ or __________
and try to set up an ______________ which you can solve.
A 60-foot-long rope is cut into 3 pieces. The first piece of rope is
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twice as long as the second piece of rope. The third piece of rope
is three times as long as the second piece of rope. What is the
length of the longest piece of rope?
Hint:
1) Draw a ___________
2) Label each
_________
3) Set up an
__________
Try again
A 24-inch-long piece of Fruit Rollups is divided among three
people. The first piece of Rollups is three times as long as the
second piece of Rollups. The third piece of Rollups is twice as
long as the second piece of rollups. What is the length of the
longest piece of Fruit Rollups?
A. 16 inches
B. 12 inches
C. 8 inches
D. 4 inches
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Another example
Three friends earned $120 by creating a wedding picture
montage. Bob spent twice as long as Amy working on the video,
while George spent three times as many hours working as Amy.
How much money should each person make?
Rate problems Distance is the _______________ of something
Rate is the _____________ of something
Time is the _____________ of time (usually in hours or minutes)
_______________ = __________ x __________ or __ = __ x __
Conceptual
If Ed drives 40 miles per hour, he travels ____ miles in 1 hour.
understanding
If Kay drives 50 miles per hour, she travels ____ miles in 2 hours.
If Bob drives 65 mph, he travels _____ miles in 2 hours.
If Vern drives 45 mph for 4 hours, he travels _____ miles total.
If Lily drives 35 mph for 3 hours, she travels _____ miles total.
Andy’s average driving speed for a 4-hour trip was 50 miles per
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hour. During the first 3 hours he drove 40 miles per hour. What
was his average speed for the last hour of his trip?
Hint: How many total
miles did Andy drive in
four hours?
How far did Andy drive in
the first three hours?
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Similar question
Miranda’s average driving speed for a 4-hour trip was 60 miles per
hour. During the first three hours she drove 65 miles per hour.
What was her average driving speed for the last hour of her trip?
A harder skill
Suppose Pat drove for five hours at an average speed of 50 miles
per hour. If Pat drove the first three hours at 40 miles per hour,
what was Pat’s average speed for the last two hours?
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Two airplanes left the same airport traveling in opposite directions.
similar to:
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If one airplane averages 300 miles per hour and the other airplane
averages 230 miles per hour, in how many hours will the distance
between the two planes be 1325 miles?
Do you add or subtract the speeds? ___________
What word determines that in the above problem? ____________
Set up the equation to solve this:
Now solve
(expect a decimal/fraction
answer)
Similar skill
What if two cars leave the same location at the same time and
travel in the same direction on a road. The first car travels at a
rate of 50 miles per hour, and the second car travels at a rate of
30 miles per hour. How much time will pass before the cars are
exactly 80 miles apart?
Do you add or subtract the speeds? ___________
What word determines that in the above problem? ____________
Set up the equation to solve this:
Now solve
Now turn the page and fill out the summary.
17
Summary: A grocer is stacking soup cans in a pyramid. The bottom row will
have 7 cans. Each row will have 1 can fewer than the row below it. How many
rows of cans will be stacked in the pyramid? How many cans will be stacked in
all?
Draw a picture, and then explain how you drew your picture and how it
helped you solve the problem.
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Topic 4: Problem
solving strategies:
Error Analysis.
Name: ________________________
Per: ________________________
Date: ________________________
Class: Algebra 1A________________
(Day 5)
Notes:
Correct vs. incorrect
When presented with an equation that is solved step by step, you
solutions
may be asked to identify whether there is an error in the solution.
To do this, it might be helpful to:
also known as
1) __________________________________________________
Error Analysis
_____________________________________________________
2) __________________________________________________
_____________________________________________________
Usually, ____________ the problem on ______ _____ next to the
solution _________ to identify _________ the _________ is
located.
STAR released
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Solve:
3 x  5  2 x  20
Step 1:
3x  15  2x  20
Step 2:
5x  15  20
Step 3:
5x  5
Step 4:
x 1
Which is the first incorrect step in the solution shown above?
19
Stan’s solution to an equation is shown below.
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Are all of the
steps correct?
Or is there an
error in one of
the steps?
Given: b  6  b  10  80
Step 1:
b  6b  10  80
Step 2:
7b  10  80
Step 3:
7b  80  10
Step 4:
7b  70
Step 5:
7b 70

7
7
Step 6:
b  10
What is true about Stan’s solution?
Similar STAR question
Four steps used to simplify the equation are shown below.
skills
4(2  4 x)  6  2(1  3x)
similar to:
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I
8  16x  4  6x
II
22x  4
III
8  16x  6  2  6x
IV
22x  4
Hint: Try solving this out
for yourself, showing
your work.
What is the correct order for these steps?
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Jeremy, Michael, Shanan, and Brenda each worked the same
math problem at the chalkboard. Each student’s work is shown
below. Their teacher said that while two of them had the correct
answer, only one of them had arrived at the correct conclusion
using correct steps.
Jeremy’s work
Shanan’s work
6  4  2x  8
6  4  2x  8
6  4  2x  8
6  4  2x  8
2  2x  8
2  2x  8
2x  6
2 x  10
x3
x  5
Michael’s work
Brenda’s work
6  4  2x  8
6  4  2x  8
6  4  2x  8
6  4  2x  8
2  2x  8
2  2x  8
2x  6
2 x  10
x3
x5
Which is a completely correct solution?
Now turn the page and fill out the summary.
21
Summary: Tanya solved the equation 3  6x   15 as follows.
3  6x   15
6x  3  15
6 x  18
x3
Myra discovered that Tanya’s solution was wrong. Explain how she knew that
Tanya’s solution was wrong.
22
Topic 5: The
Distributive Property
(Day 7)
Name: ________________________
Per: ________________________
Date: ________________________
Class: Algebra 1A________________
Notes:
The Distributive Property
When parentheses are present, this indicates that you need to
distribute the term immediately in front of the parentheses (so you
are _____________).
This looks like 2(x + 3) and becomes ___ + ___ .
In an equation, you should distribute BEFORE combining like
terms or performing opposite operations.
Why?
Why do we distribute before adding?
PEMDAS = we __________ before we _____ or ___________.
And, the parentheses show that we are multiplying the term
outside the parentheses to ______ term inside.
Examples as a class 1) 3 x  4  6
3) 15  5 x  1
2) 4  x  2  12
4) 4  x  4  16
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Distributing a negative
If the term in front of the parentheses is negative, then the signs
term
of the terms inside the parentheses must ____________.
Example: 2  x  4  6
Why?
3 x  2  9
A negative times a negative is a ______________________.
A negative times a positive is a ______________________.
Ex) -4(-2) =
Ex) -9(4) =
Ex) 3(-12) =
If there is not a number (shown) before of the parentheses, but
only a sign, then distribute the sign.
A positive sign or no sign means all signs _______ the _______.
A negative sign means all signs _____________.
  2x  4  8
Solving two step
14    x  2
First step: ___________________.
equations with distributing Then, solve normally.
Examples Solve for x.
A) 2  x  3  4 x  8
B) 3( x  3)  4( x  2)
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More Examples
Solve for x.
C) 4 x  12  2  x  4
D) 2  x  3  4  x  2
Distribute first, then
In equations with parentheses, and thus distributing, you should
combine like terms
_______________________ first, then ______________
_______ ____________.
What is the value of x in
E) 3 x  2  12  2 x  9
F) 4  x  5  24  2  x  6
the equations to the
right?
Equivalent
Equivalent means _____ ___________ ______.
Example: Is the equation 2  3x  6  18 equivalent to
6x  12  18 ?
________, by the __________________ Property.
The Distributive Property
When there are parentheses, and thus distributing, you should
of Multiplication over
always ___________________ before _________________
Addition
________ __________.
Example: What is the next step in the equation
4  3  2 x   6  31  3x  ?
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Example What equation is equivalent to 4 x  2  6 x  1  12 x ?
Simplify the left
side (only) here
Practice Solving 1) 3  2  x  4  1
 12x
 12x
2) 7  3 2 x  4  1
What is the value of x in
each of these equations?
3) 6   x  3  10
4) 10   3x  6   2
5) 6 x  2  x  4  16
6) 4  2 x  3  4  2 3x  6
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Summary: 4  x  3 means I have 4 groups of x  3 . We can show this with a model.
1st group
2nd group
3rd group
4th group
Explain how this model shows that 4  x  3  4 x  12 .
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Questions dropped from the notes
Testable Questions
The total cost of renting a truck for n days is given by the equation
c  100  50n . If the total cost was $350, for how many days was
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the truck rented?
Hint: For which variable do you substitute 350? ______
The cost to rent a construction backhoe is $500 per day plus $200
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per hour of use. What is the maximum number of hours the
backhoe can be used each day if the rental cost is not to exceed
Hint:
$1900 per day?
Would drawing a picture
be helpful here?
_______
So we should _____
___
___ _______________.
Solving equations with
Some equations have decimals in them, especially certain word
decimals
problems. To solve, follow the steps of solving a normal equation.
However, there are two ways to work with the decimals:
1) Keep the decimals in the equation until the end. Just
remember, when you divide by a decimal, you have to move the
decimal of the term on the __________ of the division sign.
Example:
.65 .26
2) Multiply by a power of 10 to clear the equation of decimals.
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Multiplying by 10 moves the decimal _____ time, multiplying by
100 moves the decimal _____ times, etc. Remember, each term
has to be multiplied (if there is already something multiplying into a
parentheses, be careful to multiply that out first, and then multiply
each term by the power of 10).
Bob will make punch that is 20% fruit juice by adding pure fruit
juice to a 2-liter mixture that is 10% fruit juice. How many liters of
pure fruit juice does he need to add?
similar to:
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To solve:
What percent is pure fruit juice? ___________
What variable can we use to represent how much fruit juice Bob is
adding? ____
How much fruit juice does Bob start with in the mixture? _______
How much fruit juice will Bob have after he adds more?
_____________
Set up an equation and solve
.20 (
A harder problem
similar to:
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+
) = ____ (
) + 2 (.10)
A pharmacist mixed some 10%-saline solution with some 15%saline solution to obtain 100 mL of a 12%-saline solution. How
much of the 10% saline solution did the pharmacist use in the
mixture?
To solve:
What are we trying
to solve for?
What should we call x? ________________________
Still using x, how much of the other saline solution did the
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pharmacist add if there is a total of 100 mL? _______________
Set up an equation and solve.
.10 (
) + .15 (
) = .12 (100)
Q #1: Which of the following best describes deductive reasoning?
A
using logic to draw conclusions based on accepted statements
B
accepting the meaning of a term without definition
C
defining mathematical terms to correspond with physical
objects
D
inferring a general truth by examining a number of specific
examples
Q #3: Consider the arguments below.
I. Every multiple of 4 is even. 376 is a multiple of 4. Therefore,
376 is even.
II. A number can be written as a repeating decimal if it is rational.
Pi cannot
be written as a repeating decimal. Therefore, pi is not
rational.
Which one(s), if any, use deductive reasoning?
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