King Philip Middle School
18 King Street, Norfolk, Massachusetts 02056
Assistant Superintendent
Phone: (508) 541-7324
Middle School Principal
Fax: (508) 541-3467
Assistant Principal
Mrs. Nancy Fischer
Website: www.kingphilip.org
Dr. Susan Gilson
2016 Summer Math Packet
Incoming 8th Grade Extended
Dear KPMS Students and Parents,
Research studies have shown that during an extended summer vacation, children can lose an average of 2.6
months worth of knowledge. The KPMS mathematics teachers want all students to be as successful as possible
in middle school math. This math packet has been designed so that students will maintain and review their
math skills during the summer months. Please monitor the progress of your child so that work is completed on
a weekly basis, not the day before school begins in September. All students are required to complete this
packet. This packet will count for a quiz grade (50 points) in the 1st quarter of the school year. To receive credit
for this packet, all of the problems must be completed and ALL SUPPORTING WORK MUST BE SHOWN! No
calculators are allowed. During the first week of school, your child’s math teacher will review and collect all
packets. Please note that the quality of effort is more important to us than getting all the correct answers.
The review packet is set up in two sections. The front section is a review of each concept. There are rules and
examples of each concept. Under each concept, there are a few examples for you to try and check your answers
at the end of the first section to make sure that you know what you are doing. The second section is the
summer math work. It is broken down into weekly assignments. The set-up is the same each week. For
example, the first problem each week is always the Equations problem. Be sure to check back to the first
section if you are having trouble.
This packet will be collected the first week of school and will help prepare your child for 8th grade.
Have a great summer!!!
KPMS Middle School Teachers
Some great resources: www.khanacademy.com www.learnzillion.com
www.coolmath.com
www.aaamath.com
www.mathchimp.com
www.mathisfun.com
KPMS – INCOMING 8TH GRADE EXTENDED SUMMER MATH
PACKET
www.mathgoodies.com
www.ixl.com
1
Review: Solving Linear Equations
Linear equations do not always have one solution. An IDENTITY is an equation that
is true for all values of the variable. Some linear equations have no solution.
Steps for Solving Linear Equations
1. Distribute and combine like terms, if possible, on each side of the equation.
2. Isolate the variable on one side of the equation by adding or subtracting to
combine like terms across the equal sign.
3. Divide or multiply both sides of the equation by the coefficient of the variable
to solve for “x”.
Example:
1
4
(12𝑥 + 16) = 10 − 3(𝑥 − 2)
3x + 4 = 10 – 3x + 6
3x + 4 = 16 – 3x
3x + 3x + 4 = 16 – 3x + 3x
6x + 4 = 16
6x + 4 – 4 = 16 – 4
6x = 12
6𝑥
12
= 6
6
x=2
Use distributive property
Simplify each side of the equation
Add 3x to both sides
Simplify
Subtract 4 from both sides
Simplify
Divide both sides by 6
Simplify
Review: Evaluate Variable Expressions
To evaluate an algebraic expression, substitute a number for each variable. Then use the
order of operations to find the value of the number.
Examples – Evaluate the expression for the given value of the variable(s).
1. 3x2 – 5x when x = -2
3(-2)2 – 5(-2)
3(4) – (-10)
12 + 10
22
KPMS – INCOMING 8TH GRADE EXTENDED SUMMER MATH
PACKET
2.
𝑎2 −4
𝑏
when a = -2 and b = -3
(−2)2 −4
−3
=
4−4
−3
=
0
−3
=0
2
Review: Slope of a Line
The slope of a nonvertical line is the ratio of vertical change
(rise) to horizontal change (run) between any two points on
the line. If a line in the coordinate plane passes through
points (x1, y1) and (x2, y2), then the slope m is
𝑚=
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛
=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 (∆𝑦)
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥(∆𝑥)
=
𝑦2 −𝑦1
𝑥2 −𝑥1
Examples – Find the slope of the line.
1.
𝑚=
𝑦2 −𝑦1
2.
𝑥2 −𝑥1
(3, 5) and (1, 1)
5−1
𝑦2 −𝑦1
𝑚=
𝑥2 −𝑥1
(1, 5) and (8, 1)
4
5−1
4
𝑚 = 3−1 = 2
𝑚 = 1−8 = −7
𝑚=2
𝑚 = −7
4
Find the slope of the line passing through the points.
3. (0, -10) and (-4, 0)
𝑚=
𝑚=
𝑦2 −𝑦1
𝑚=
𝑥2 −𝑥1
0−−10
−4—0
5
𝑚 = −2
4. (3,6 and 3,0)
10
= −4
𝑦2 −𝑦1
𝑦2 −𝑦1
𝑚=
𝑥2 −𝑥1
6−0
5. (4, 1) and (6, 1)
6
𝑥2 −𝑥1
1−1
0
𝑚 = 3−3 = 0
𝑚 = 6−4 = 2
𝑚 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
𝑚=0
Review: Slope-Intercept Form (function form)
The x-intercept of a line is the x-coordinate of the point where the
line crosses the x-axis. It occurs when y = 0.
The y-intercept of a line is the y-coordinate of the point where the
line crosses the y-axis. It occurs when x = 0.
KPMS – INCOMING 8TH GRADE EXTENDED SUMMER MATH
PACKET
3
A linear equation written in the form y = mx + b is in slope-intercept form (function form). The slope
of a line is m, and the y-intercept of the line is b.
y = mx + b
slope
y-intercept
If necessary rewrite in function form, then identify the slope and the y-intercept
of the graph of each linear equation.
1
1. y = -3x – 8
2. 𝑦 − 4 = 3 𝑥
1
𝑦 = 3𝑥 + 4
Add 4 to each side
1
m = -3; b = -8
𝑚= , 𝑏=4
3
Review: Graphing Equations
Example: Graph an equation using the slope (m) and
y-intercept (b)
1
𝑚 = 2; 𝑏 = 3
Plot the y-intercept (0, 3)
Use the slope to find your second point.
1
𝑚𝑒𝑎𝑛𝑠 𝑢𝑝 𝑜𝑛𝑒, 𝑡𝑤𝑜 𝑡𝑜 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡
2
Draw a line through your two points
Example: Graph a horizontal and vertical line.
Horizontal line: y = 3
The graph of the equation is a horizontal line 3
units above x-axis.
Vertical line: x = -2
The graph of the equation is a vertical line 2
units to the left of the y-axis.
KPMS – INCOMING 8TH GRADE EXTENDED SUMMER MATH
PACKET
4
Week of July 3rd – July 9th
Name: ________________________
PLEASE DO NOT USE A CALCULATOR AND REMEMBER TO SHOW ALL WORK!
Equations
Evaluate
Slope
Solve:
When a = 2 b = -3 and c = 4
Find the slope of the line
4 + ab
(0,0) and (5,2)
2(3x + 1) – x = 17
Slope Intercept Form y = mx + b
Graphing
Rewrite the equation in function form
Graph using the given slope and intercept
2x + y = 10
m=¾
b=1
Identify the slope and y-intercept
m=
y-intercept =
KPMS – INCOMING 8TH GRADE EXTENDED SUMMER MATH
PACKET
5
Week of July 10th – July 16th
Name: ________________________
PLEASE DO NOT USE A CALCULATOR AND REMEMBER TO SHOW ALL WORK!
Equations
Evaluate
Slope
When a = 2 b = -3 and c = 4
Find the slope of the line
5–b+c
(3, 3) and (-6, -4)
6 – (-5x) = 5x - 3
Slope Intercept Form y = mx + b
Graphing
Rewrite the equation in function form
Graph using the given slope and intercept
4x + 2y = 6
m=-3
2
b=4
Identify the slope and y-intercept
m=
y-intercept =
KPMS – INCOMING 8TH GRADE EXTENDED SUMMER MATH
PACKET
6
Week of July 17th – July 23rd
Name: ________________________
PLEASE DO NOT USE A CALCULATOR AND REMEMBER TO SHOW ALL WORK!
Equations
Evaluate
Slope
When a = 2 b = -3 and c = 4
Find the slope of the line
-10a + bc
(-6, -1) and (-6, 4)
24 – 6x = 6(4 – x )
Slope Intercept Form y = mx + b
Graphing
Rewrite the equation in function form
Graph using the equation
12x + 4y = 12
y = -2x + 5
Identify the slope and y-intercept
m=
y-intercept =
KPMS – INCOMING 8TH GRADE EXTENDED SUMMER MATH
PACKET
7
Week of July 24th – July 30th
Name: ________________________
PLEASE DO NOT USE A CALCULATOR AND REMEMBER TO SHOW ALL WORK!
Equations
Evaluate
Slope
When a = 2 b = -3 and c = 4
Find the slope of the line
3a2 + c
(0, -6) and (8,0)
12x – 7 = -3x + 8
Slope Intercept Form y = mx + b
Graphing
Rewrite the equation in function form
Graph using the equation
-2x + 4y = 8
y = ½x + 2
Identify the slope and y-intercept
m=
y-intercept =
KPMS – INCOMING 8TH GRADE EXTENDED SUMMER MATH
PACKET
8
Week of July 31st – August 6th
Name: ________________________
PLEASE DO NOT USE A CALCULATOR AND REMEMBER TO SHOW ALL WORK!
Equations
Evaluate
Slope
When a = 2 b = -3 and c = 4
Find the slope of the line
8 – b2
(1, -3) and (-4, -5)
9(b - 4) – 7b = 5(3b – 2)
Slope Intercept Form y = mx + b
Graphing
Rewrite the equation in function form
Rewrite the equation in function form to graph.
x + 6y = 12
3x + 4y = 12
Identify the slope and y-intercept
m=
y-intercept =
KPMS – INCOMING 8TH GRADE EXTENDED SUMMER MATH
PACKET
9
Week of August 7th – August 13th
Name: ________________________
PLEASE DO NOT USE A CALCULATOR AND REMEMBER TO SHOW ALL WORK!
Equations
Evaluate
Slope
When a = 2 b = -3 and c = 4
Find the slope of the line
8a – 4(-5a – 2) = 12a
abc -3a
(1,-2) and (-7,6)
Slope Intercept Form y = mx + b
Graphing
Rewrite the equation in function form
Rewrite the equation in function form to graph.
x + 6y = 12
x + 3y = 6
Identify the slope and y-intercept
m=
y-intercept =
KPMS – INCOMING 8TH GRADE EXTENDED SUMMER MATH
PACKET
10
Week of August 14th – August 20th
Name: ________________________
PLEASE DO NOT USE A CALCULATOR AND REMEMBER TO SHOW ALL WORK!
Equations
Evaluate
Slope
When a = 2 b = -3 and c = 4
Find the slope of the line
-(8n – 2) = 3 + 10(1 – 3n)
(5a)2 + bc
(2,2) and (-3,5)
Slope Intercept Form y = mx + b
Graphing
Rewrite the equation in function form
Graph the equation.
-2x + 3y = 9
x=5
Identify the slope and y-intercept
m=
y-intercept =
KPMS – INCOMING 8TH GRADE EXTENDED SUMMER MATH
PACKET
11
Week of August 21st – August 27th
Name: ________________________
PLEASE DO NOT USE A CALCULATOR AND REMEMBER TO SHOW ALL WORK!
Equations
4
−9
Evaluate
Slope
When a = 2 b = -3 and c = 4
Find the slope of the line
5𝑐+𝑏−𝑎
(0,-4) and (5, -4)
(2x – 4) = 48
1
𝑎
2
Slope Intercept Form y = mx + b
Graphing
Rewrite the equation in function form
Graph the equation
5x – 4y = 22
y = -4
Identify the slope and y-intercept
m=
y-intercept =
KPMS – INCOMING 8TH GRADE EXTENDED SUMMER MATH
PACKET
12
KPMS – INCOMING 8TH GRADE EXTENDED SUMMER MATH
PACKET
13