Math - Schoolwires
54
For families who need academic support, please call 504-349-8999 Monday-Thursday • 8:00 am–8:00 pm Friday • 8:00 am–4:00 pm Available for families who have questions about either the online learning resources or printed learning packets. 8 th Grade Math #JPSchoolsLove S h o w u s y o u r Book 1
Transcript of Math - Schoolwires
For families who need academic support, please call 504-349-8999
Monday-Thursday • 8:00 am–8:00 pmFriday • 8:00 am–4:00 pm
Available for families who have questions about either the online learning resources or printed learning packets.
8th GradeMath
#JPSchoolsLove
Show us your
Book 1
6th-8th GRADE DAILY ROUTINE
TTiimmee AAccttiivviittyy EExxaammpplleess
66--88 8:00a Wake-Up and
Prepare for the Day
• Get dressed, brush teeth, eat breakfast
9:00a Morning Exercise
• Exercises o Walking o Jumping Jacks o Push-Ups o Sit-Ups o Running in place o High Knees o Kick Backs o Sports
NNOOTTEE:: Always stretch before and after physical activity
10:00a Academic Time: Reading Skills
• Online: o Plato (ELA)
• Packet o Reading (one lesson a day)
11:00a Play Time Outside (if weather permits) 12:00p Lunch and Break
• Eat lunch and take a break • Video game or TV time • Rest
2:00p Academic Time: Math Skills
• Online: o Plato (Math)
• Packet o Math (one lesson a day)
3:00p Academic Learning/Creative Time
• Puzzles • Flash Cards • Board Games • Crafts • Bake or Cook (with adult)
4:00p Academic Time: Reading for Fun
• Independent reading o Talk with others about the book
5:00p Academic Time: Science and Social Studies
• Online o Plato (Science and Social Studies)
Para familias que necesitan apoyo académico, por favor llamar al 504-349-8999 De lunes a jueves • 8:00 am – 8: 00 pm Viernes • 8:00 am – 4: 00 pm Disponible para familias que tienen preguntas ya sea sobre los recursos de aprendizaje en línea o los paquetes de aprendizaje impresos.
Tiempo Actividad Detalles 8:00a Despierta y Prepárate para el día • Vístete, cepíllate los dientes, desayuna
9:00a Ejercicio Mañanero
NOTE: Siempre hay que estirarse antes y después de cualquier actividad física.
• Ejercicios o Caminar o Saltos de tijeras o Lagartijas o Abdominales o Correr en el mismo lugar o Rodillas altas o Patadas hacia atrás o Deportes
10:00a Tiempo Académico:
Habilidades de Lectura • En Línea:
o Plato (ELA) • Paquete:
o Leer(una lección al día) 11:00a Tiempo para jugar Afuera(si el clima lo permite) 12:00p Almuerzo y Descanso • Almorzar y tomar un descanso
• Este es tiempo para jugar videos y ver televisión
• Descansar 2:00p Tiempo Académico:
Habilidades de Matemáticas • En Línea:
o Plato (Matemática) • Paquete
o Matemática (una lección al día)
3:00p Aprendizaje Académico/Tiempo Creativo • Rompecabezas • Tarjetas Flash • Juegos de Mesa • Artesanías • Hornear o Cocinar( con un adulto)
4:00p Tiempo Académico: Leyendo por Diversión
• Lectura Independiente o Habla con otros acerca de
lo que leíste 5:00p Tiempo Académico:
Ciencias y Estudios Sociales • En Línea
o PLato(Ciencia y Estudios Sociales)
8.G Same Size, Same Shape?
Task
a. For each pair of figures, decide whether these figures are the same size and same
shape. Explain your reasoning.
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Illustrative Mathematics
1
b. What does it mean for two figures to be the same size and same shape?
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Illustrative Mathematics
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©Curriculum Associates, LLC Copying is not permitted. 213Lesson 20 Transformations and Similarity
Name:
Combine Dilations and Other Transformations
Study the example problem showing how to combine a dilation with other transformations. Then solve problems 1–6.
1 Suppose the scale factor of the dilation in the example
was 2 instead of 1 ·· 2 , but the dilation was still centered
about O and nABC was still rotated 180° about O . What
would the coordinates of the vertices of nHJK be?
H( ) J( ) K( )
2 Explain how a dilation is different from a translation, a reflection, or a rotation .
Example
In the diagram, nABC is similar to nHJK . A sequence of transformations was used to transform nABC to nHJK .
Describe the change in the coordinates .
A(2, 4) was transformed to H(21, 22) .
B(6, 22) was transformed to J(23, 1) .
C(2, 22) was transformed to K(21, 1) .
Each x-coordinate has the opposite sign and was multiplied by 1 ·· 2 .
Each y-coordinate has the opposite sign and was multiplied by 1 ·· 2 .
nABC was dilated about center O with a scale factor of 1 ·· 2 and rotated
180° about O.
Lesson 20
Ox
y
A
BCH
J K
Vocabularydilation a
transformation in which
the original figure and
the image are similar .
scale factor in a
dilation, the ratio of the
lengths of corresponding
sides of the figure and its
image .
center the center of a
dilation is the point that
is transformed onto itself
by the dilation .
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©Curriculum Associates, LLC Copying is not permitted.214 Lesson 20 Transformations and Similarity
Solve.
3 The coordinates of the vertices of Polygon RSTV are
R(2, 4), S(6, 4), T(6, 0), and V(2, 0) . The Polygon is dilated with
scale factor of 3 ·· 2 and center (0, 0) . Explain how you can find
the coordinates of the vertices of Polygon R9S9T9V9 from
the coordinates of the vertices of the Polygon RSTV .
4 Triangle PQR is shown at the right .
a. Reflect nPQR across the y-axis and then dilate it about center O with a scale factor of 2 . Sketch the final image .
b. Compare the coordinates of the corresponding vertices of the final image and nPQR.
5 In the diagram at the right, Polygon A is similar to Polygon W . What sequence of transformations transformed Polygon A to Polygon W?
6 Tracy dilates a figure with a scale factor of 3 ·· 4 and center O
and then dilates the image with a scale factor of 2 and center O . Carrie says that she can get the same final image using just one dilation . Is she correct? If so, how can she do that? If not, why not?
x
y
P Q
R O
x
y
A
W
O
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©Curriculum Associates, LLC Copying is not permitted. 215Lesson 20 Transformations and Similarity
Name:
1 Polygon ABCD is shown on the coordinate plane . Sketch the image after it is rotated 90° clockwise about O and then dilated with scale factor 2 and center O.
Transformations and Similarity
Solve the problems.
Make sure you rotate the polygon clockwise.
Lesson 20
x
y
A B
CD
O
2 The coordinates of nDEF are D(24, 4), E(2, 4), and F(0, 2) .
The triangle is dilated with scale factor 1 ·· 2 and center O .
What are the coordinates of the vertices of the image
of nDEF?
A (2, 22), (21, 22), (0, 21)
B (28, 8), (4, 8), (0, 4)
C (22, 2), (1, 2), (0, 1)
D (4, 24), (4, 2), (2, 0)
Sue chose A as the correct answer . How did she get that answer?
How do you use the scale factor to find the coordinates of the image?
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©Curriculum Associates, LLC Copying is not permitted.216 Lesson 20 Transformations and Similarity
Solve.
3 Tell whether each statement is True or False .
a. A dilation image is always congruent to the original figure . u True u False
b. A rotation image is always congruent to the original figure . u True u False
c. A reflection image is never congruent to the original figure . u True u False
d. A translation image is always congruent to the original figure . u True u False
4 Polygon LMNP was transformed to Polygon WXYZ .
Part A Describe a sequence of transformations that maps Polygon LMNP to Polygon WXYZ .
Part B Find the perimeters of Polygon WXYZ and Polygon LMNP . Then write the ratio of the perimeter of Polygon WXYZ to the perimeter of Polygon LMNP. How does this ratio compare to the scale factor you found in Part A?
What types of transformations keep the size and shape of the original figure?
What type of transformation can change the size of a figure?
x
y
Z
WL M
NP
Y
X
O22 2
22
3
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©Curriculum Associates, LLC Copying is not permitted. 19Lesson 2 Square Roots and Cube Roots
Name: Lesson 2
1 The formula for the surface area of a cube is S 5 6x2, where x is the length of one side . Find the length of the side of a cube with a surface area of 150 square inches .
A 5 inches C 30 inches
B 25 inches D 900 inches
Jack chose C as the correct answer . How did he get that answer?
Square Roots and Cube Roots
Solve the problems.
3 The formula for the volume of a square pyramid is V 5 (b2h) 4 3, where b is the length of one side of the square base and h is the height of the pyramid . Find the length of a side of the base of a square pyramid that has a height of 3 inches and a volume of 25 cubic inches .
Show your work.
Solution:
2 Choose Yes or No to tell whether the number is a perfect cube .
a. 27 u Yes u No
b. 100 u Yes u No
c. 125 u Yes u No
d. 1,000 u Yes u No
How do you find the value of x2?
Remember that a perfect cube is the product of three equal factors.
What is 3 ÷ 3?
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©Curriculum Associates, LLC Copying is not permitted.20 Lesson 2 Square Roots and Cube Roots
Solve.
4 The area of a square is a perfect square between 100 and 250 square centimeters . Which could be the area of the square? Select all that apply .
A 102 square centimeters
B 121 square centimeters
C 125 square centimeters
D 144 square centimeters
E 225 square centimeters
F 240 square centimeters
5 The base of a cube is shown . The area of the base is
1 ·· 4 ft2 . What is the volume of the cube?
Show your work.
Solution:
What is the square root of the numerator of 1 ·· 4
? What is the square root of the denominator?
Remember that a perfect square is the product of two equal factors.
x
x
15
16
17
18
19
20
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©Curriculum Associates, LLC Copying is not permitted. 9Lesson 1 Properties of Integer Exponents
Name: Lesson 1
1 Look at the equations below . Tell whether each equation is True or False .
a . 35 ? 375 335 u True u False
b . (63 ? 33)2 5 186 u True u False
c . 726 ? 1 ·· 74 5 1 ··· 710
u True u False
d . 44 ? 42 5 46 u True u False
e . 1324 ···· 134 5 130
u True u False
f . (23 ? 83)0 5 169 u True u False
Simplify Expressions with Exponents
Solve the problems.
3 Complete the table .
Expression 104 ? 1022 54 ? 74 (27 ? 47)3
Simplified Expression
2 Tyler simplified the expression 54 ? 529 . All of his work except his answer is shown below .
54 ? 529 5 541(29)
5 525
5 ?
Which expression is the correct answer for Tyler’s work?
A 55
B 1 ·· 55
C 1 ··· 525
D 5
You may have to apply more than one rule when working with exponents.
Remember what you know about negative exponents.
Are the bases equal? Are the exponents equal?
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©Curriculum Associates, LLC Copying is not permitted.10 Lesson 1 Properties of Integer Exponents
6 Which expression is equivalent to (34 ? 54)23?
A 1 ··· 155
B 15248
C 1 ···· 1512
D 155
Tania chose B as the correct answer . How did she get that answer?
Solve.
4 Simplify: 3221 ···· 326 . Write your answer with a
positive exponent .
Show your work.
Solution:
5 Write 96 as a power with a base of 3 .
What are the factors of 9?
Remember the order of operations. Simplify the expression within the parentheses first.
The expression is a quotient of powers.
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©Curriculum Associates, LLC Copying is not permitted. 133Lesson 12 Understand the Slope-Intercept Equation for a Line
Name: Understand the Slope-Intercept Equation for a Line
Lesson 12
Prerequisite: How can you represent and interpret proportional relationships?
Study the example problem showing how to represent and interpret a proportional relationship. Then solve problems 1–5.
1 How you can use the table to find the unit rate in the example problem?
2 What is the constant of proportionality in the example? What is the slope of the graph? What do they represent in the context of this problem? How do the constant of proportionality and slope relate to the unit rate?
Example
The table shows the costs for 2, 3, 4, and 5 vegetable seed packets . What is the unit rate?
Use the data to make a graph . Find the cost of 1 seed packet .
The unit rate is the cost in dollars for 1 packet . The graph shows that the unit rate is 1 .50 .
Cost
($)
Number of Packets
Vegetable Seeds
2 4 6 81 3 5 7Ox
y
2
1
4
6
3
5
7
8
(1, 1.50)
Vegetable Seeds
Number of Packets 2 3 4 5
Cost ($) 3 .00 4 .50 6 .00 7 .50
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©Curriculum Associates, LLC Copying is not permitted.134 Lesson 12 Understand the Slope-Intercept Equation for a Line
Solve.
3 The table shows how many words Julian can type if he types at a steady rate . Use the information in the table to make a graph . Find the slope of the graph and explain what it means in this situation .
Typing Rate
Number of Minutes 2 4 6 8
Number of Words 80 160 240 320
4 The price for movie tickets at Town Theater is shown in the graph . The price of 5 movie tickets at Center Theater is $3 .75 greater than the price of 5 movie tickets at Town Theater . What is the price per ticket at each theater?
5 A hardware store buys 300 feet of nylon rope . The store sells the rope by the inch . A customer can purchase 40 inches of the rope for $1 .60 . The store sells all of the rope and makes a profit of $54 . How much did the store pay for the rope in dollars per inch?
Show your work.
Solution:
Num
ber o
f Wor
ds
Number of Minutes
Typing Rate
2 4 6 81 3 5 7Ox
y
100
50
200
300
150
250
350
400
Pric
e ($
)
Number of Tickets
Town Theater Tickets
2 4 6 81 3 5 7Ox
y
12
6
24
36
18
30
42
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©Curriculum Associates, LLC Copying is not permitted. 135Lesson 12 Understand the Slope-Intercept Equation for a Line
Name: Lesson 12
Writing a Linear Equation in Slope-Intercept Form
Study the example problem showing how to write an equation in slope-intercept form. Then solve problems 1–6.
1 How is the equation y 5 2x similar to the equation y 5 2x 1 3 in the example problem? How is it different?
2 Graph y 5 2x in the diagram above . Compare the graphs . Does either graph represent a proportional relationship? Explain .
3 What value of b makes y 5 2x 1 b the same as y 5 2x? What does that value mean?
Example
Write an equation for the line shown in the diagram .
Find the slope of the line .
m 5 y2 2 y1 ······ x2 2 x1
5 7 2 3 ····· 2 2 0
m 5 4 ·· 2 , or 2
The line passes through (0, 3), so the y-intercept is 3 . Use the slope-intercept form y 5 mx 1 b to write an equation .
y 5 mx 1 b y 5 2x 1 3 Substitute 2 for m and 3 for b .
An equation for the line is y 5 2x 1 3 .
x
y
1
12122 2 3 4 5 6
3
4
5
6
7
8
O
Use the slope formula . Substitute 7 for y2, 3 for y1, 2 for x2, and 0 for x1 .
Simplify . The slope is 2 .
Vocabularyslope the ratio of the vertical
change (rise) to the horizontal
change (run) between any
two points on a line .
y-intercept the y-coordinate
of the point where a graph
intersects the y-axis .
The y-intercept is 3 .
x
y
1
O 121 2 3 4 6 721
2
4
5
6
7
(0, 3)
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©Curriculum Associates, LLC Copying is not permitted.136 Lesson 12 Understand the Slope-Intercept Equation for a Line
Solve.
4 Andy uses the table below to write a linear equation .
x 21 0 1 2
y 2 4 6 8
He says he can write an equation of the form y 5 mx for the given values . Is he correct? Explain your reasoning .
5 Look at these equations . Write each equation in slope-intercept form . Are the equations the same or different? Explain .
y 1 1 5 2x – 3 2x – 3 5 y 1 1 2y 1 2 5 4x – 6
6 Explain how you can write an equation for a line with
slope 1 ·· 2 that crosses the y-axis at the point (0, –1) .
Graph the line for your equation .
1
2 3 4 5
22
2122232425
23
24
25
2
3
4
5
x
y
O
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8.EE Proportional relationships,
lines, and linear equations
Task
Lines and have the same slope. The equation of line is . Line passes
through the point .
What is the equation of line ?
8.EE Proportional relationships, lines, and linear equations Typeset May 4, 2016 at 23:08:08. Licensed by Illustrative Mathematics under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .
L M L 4y = x M
(0, −5)
M
1
Illustrative Mathematics
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©Curriculum Associates, LLC Copying is not permitted. 143Lesson 13 Solve Linear Equations with Rational Coefficients
Name:
1 Check the solution to the example problem by replacing n in the original equation with 22 and evaluating both sides . What true statement do you get?
2 Suppose that you first want to eliminate the fraction in the example equation . What would your first step be? Is 22 still the solution when you start by eliminating the fraction first? Explain .
3 Trey solved the equation 1 ·· 4 (8x 1 16) 5 4x, as shown at
the right . Describe the error that he made . Then solve the problem .
1 ·· 4 (8x 1 16) 5 4x
2x 1 16 5 4x
16 ·· 2 5 2x ·· 2
8 5 x
Example
Solve the equation: 4n 5 1 ·· 2 (2n 2 12) .
4n 5 1 ·· 2 (2n 2 12)
4n 5 n 2 6 Step 1: Use the distributive property .
4n 2 n 5 n 2 6 2 n Step 2: Subtract n from both sides .
3n 5 26 Step 3: Simplify .
3n ··· 3 5 26 ··· 3 Step 4: Divide both sides by 3 .
n 5 22 Step 5: Simplify .
Lesson 13
Solve Equations with Rational Coefficients
Study the example showing how to solve an equation with rational coefficients. Then solve problems 1–6.
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©Curriculum Associates, LLC Copying is not permitted.144 Lesson 13 Solve Linear Equations with Rational Coefficients
4 Describe the first step you would use to solve the equation 20 5 7y 1 2 2 y . Is that the only possible first step?
5 Solve the equation in two different ways: 6p 5 0 .6(5p 1 15) .
Show your work.
Solution:
6 The two rectangles shown below have the same perimeter . Write and solve an equation to find the value of x . Then find the measures of the length and width of Rectangle B . All measurements are in inches .
4x 2 35
7
Rectangle A Rectangle B
x
Equation:
x 5
Length of Rectangle B:
Width of Rectangle B:
Solve.
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©Curriculum Associates, LLC Copying is not permitted. 145Lesson 13 Solve Linear Equations with Rational Coefficients
Name:
Solve Linear Equations with Rational Coefficients
Solve the problems.
3 In the equation below, for what value of c does x 5 4?
1 ·· 2 (2x 1 4) 5 3x 2 c
A 26 C 3
B 23 D 6
Jenn chose C as the correct answer . How did she get that answer?
2 Solve the equation for x: 3x 2 5 5 1 ·· 2 x 1 2x .
Show your work.
Solution:
1 Claire wants to solve the equation 2 1 ·· 4 (x 2 1) 5 2 ·· 3 x 1 2 .
Which step would not be an appropriate first step for Claire to take to solve for x?
A Multiply both sides by 24 .
B Use the distributive property to distribute 2 1 ·· 4 .
C Add 1 to both sides .
D Multiply both sides by 3 ·· 2 .
How can you check your answer?
What operations can you use to simplify both sides of the equation?
Lesson 13
What techniques can you use to simplify the equation?
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©Curriculum Associates, LLC Copying is not permitted.146 Lesson 13 Solve Linear Equations with Rational Coefficients
Solve.
4 Choose Yes or No to tell whether the equation has the given solution .
a. 2x 1 4 5 3x 2 2; x 5 6 u Yes u No
b. 1 ·· 4 x 1 3 5 3 ·· 4 x 1 1; x 5 8 u Yes u No
c. 3x 2 5 5 0 .5x; x 5 2 u Yes u No
d. 2 ·· 3 (3x 1 6) 5 3x 2 4; x 5 8 u Yes u No
5 The width of this rectangle is 1 ·· 3 of the length . Find the length and the width of the rectangle .
(2y 1 6) in.
(y 2 4) in.
Show your work.
Solution:
How can you use substitution to solve this problem?
What equation can you write to solve the problem?
32
©Curriculum Associates, LLC Copying is not permitted. 151Lesson 14 Solutions of Linear Equations
Name: Lesson 14
Determining the Number of Solutions of an Equation
Study the example showing how to identify the number of solutions an equation has. Then solve problems 1–7.
1 Suppose the right side of the equation in the example problem is 2x 2 3 1 8 . How many solutions would the equation have? Explain .
2 Suppose the right side of the equation in the example problem is 3x 2 3 1 6 . How many solutions would the equation have? Explain .
3 Look at the model at the right . Does it represent an equation that has one solution, no solutions, or infinitely many solutions? Explain how you know .
Example
How many solutions does the equation 2(x 1 2) 1 1 5 2x 2 3 1 6 have?
Simplify the equation .
2(x 1 2) 1 1 5 2x 2 3 1 6
2x 1 4 1 1 5 2x 1 3
2x 1 5 5 2x 1 3
The variable terms on each side of the simplified equation are the same but the constants are different, so the equation has no solution .
x x x
Number of Solutions
• An equation has infinitely many solutions when you simplify and the variable terms and constants are the same on each side, as in 2x 1 5 5 2x 1 5 or 4 5 4 .
• An equation has no solution when you simplify and the variable terms on each side are the same but the constants are different, as in 2x 1 5 5 2x 1 3 or 4 5 2 .
• An equation has one solution when the variable terms on each side are different, as in 3x 1 1 5 2x 2 5 .
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©Curriculum Associates, LLC Copying is not permitted.152 Lesson 14 Solutions of Linear Equations
Solve.
4 Consider the equation cx 2 d 5 2x 1 4 .
a. Replace c in the equation with 2 . For what value of d would the equation have infinitely many solutions? Explain .
b. Replace d in the equation with 2 . For what value of c would the equation have no solution? Explain .
5 Evelyn says that the equation 3(x 2 3) 1 5 5 3x 1 1 1 4 has infinitely many solutions because the variable terms on each side are the same . Do you agree with Evelyn? Explain why or why not .
6 Explain why the equation 5(2x 1 1) 2 2 5 6x 1 5 has only one solution . Then find the solution .
7 Write an equation that has one solution, an equation that has no solution, and an equation that has infinitely many solutions . Each equation should have one variable term on each side and a total of four terms .
One solution:
No solution:
Infinitely many solutions:
34
8.EE The Sign of Solutions
Task
Without solving them, say whether these equations have a positive solution, a negative
solution, a zero solution, or no solution.
a.
b.
c.
d.
e.
8.EE The Sign of Solutions Typeset May 4, 2016 at 22:18:42. Licensed by Illustrative Mathematics under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .
3x = 5
5z + 7 = 3
7 − 5w = 3
4a = 9a
y = y + 1
1
Illustrative Mathematics
35
©Curriculum Associates, LLC Copying is not permitted. 159Lesson 15 Understand Systems of Equations
Name: Lesson 15
1 Look at the slopes and the number of solutions for the first two systems of equations in the example . What do the systems have in common?
2 Use the coordinate grid showing a graph of the third system of equations in the example . Why does the system have no solution?
Example
You can compare the slopes and y-intercepts of a system of equations to predict how many solutions the system has .
System Slopes y-intercepts Number of Solutions
y 5 2x 2 1 y 5 x 1 1
2 and 1 Different
21 and 1Different
One solution
y 5 4x 1 3y 5 2x 1 3
4 and 2 Different
3 and 3 Same
One solution
y 5 3x 2 2y 5 3x 1 3
3 and 3 Same
22 and 3Different
No solution
y 5 2x 2 3y 5 2x 2 3
2 and 2Same
23 and 23Same
Infinitely many solutions
Determining the Number of Solutions of a System of Equations
Study the example showing how to determine the number of solutions for a system of equations. Then solve problems 1–6.
Vocabularysystem of linear equations a set of two
or more linear equations
that share the same
variables .
y 5 3x 2 2
x 2 y 5 1
1
2
21 3 4 5
22
232425
25
24
23
3
4
5
x
y
O22 2121
36
©Curriculum Associates, LLC Copying is not permitted.160 Lesson 15 Understand Systems of Equations
Solve.
Use these equations to solve problems 3–5.
Equation 1: y 5 2x 1 3 Equation 2: y 5 2x 2 3 Equation 3: 2y 5 4x 1 6
3 Form a system of equations with Equations 1 and 2 . Without graphing, explain how you can tell how many solutions the system has .
4 Form a system of equations with Equations 1 and 3 . Without graphing, explain how you can tell how many solutions the system has .
5 Tonya says that a system of equations formed by Equations 2 and 3 will have the same number of solutions as a system formed by Equations 1 and 2 . Is she correct? Use your answers to problems 3 and 4 to help you explain your reasoning .
6 The system of equations shown below has no solution . Change one number in one of the equations so that the system has one solution . Graph your new system on the coordinate grid to support your answer .
y 5 2x 2 1 y 5 2x 1 1
21
1
21 3 421
22
23
24
222324
3
2
4
x
y
O
37
38
39
40
41
42
©Curriculum Associates, LLC Copying is not permitted. 67Lesson 6 Understand Functions
Name: Lesson 6
1 Can you represent either of the functions in the example problem with an equation? Explain .
2 Suppose you reverse the inputs and outputs in Table B . Would the relationship be a function? Explain .
3 The table shows the number of concert tickets sold by five students . Is the relationship a function? Explain .
Student (input) 1 2 3 4 5
Tickets (output) 12 18 12 22 16
Example
Describe the relationship shown in each table . Is the relationship a function? Explain .
The input identifies the hours, and the output gives the cost for those hours . The relationship is a function because there is only one output for each input .
The input identifies the week and the output gives the growth for each week . The relationship is a function because there is only one output for each input .
Identify Functions
Study the example problem showing how to determine whether a relationship is a function. Then solve problems 1–7.
Table AHours (input) 1 2 3 4 5
Cost (output) $3 $6 $9 $12 $15
Table BWeek (input) 1 2 3 4 5
Plant Growth in Inches (output)
4 3 .25 2 2 1 .75
Vocabularyfunction a rule that
produces exactly one
output for each input .
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©Curriculum Associates, LLC Copying is not permitted.68 Lesson 6 Understand Functions
Solve.
Use the following situation to solve problems 4–5.
The table shows the number of calories in different numbers of servings of blueberries .
Servings (input) 1 2 3 4 5
Calories (output) 21 42 63 84 105
4 On the blank graph to the right, add a title and then label and number the axes . Then plot the ordered pairs on the graph .
5 Explain whether the relationship is a function . Can you represent the data with an equation? If so, write the equation .
6 Substitute values into the equation y 5 x 2 3 to complete the table . Then state whether the equation represents a function . Explain your reasoning .
7 Complete the table to show a relationship that is a function that you haven’t used yet . Be sure that you can represent your function with an equation .
x (input) 1 2 3 4 5
y (output)
Describe the relationship between the input and output values of your function . Then represent your function with an equation .
x
y
x (input) 22 21 0 1 2
y (output)
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©Curriculum Associates, LLC Copying is not permitted. 75Lesson 7 Compare Functions
Name: Lesson 7
1 What do the rates of change in the example represent?
2 What does it mean in the context of the example that Alyssa’s rate of change is greater than Sarah’s?
3 Write ordered pairs for the initial values of each function in the example . Tell what the initial values represent .
Example
Compare the rates of change for these two functions . Which function has a greater rate of change?
Tota
l Sav
ings
($)
Sarah’s Savings
Weeks2 4 6 8 91 3 5 7O
x
y
8
4
16
24
28
32
36
40
12
20
Tota
l Sav
ings
($)
Alyssa’s Savings
Weeks2 4 6 8 91 3 5 7O
x
y
8
4
16
24
28
32
36
40
12
20
Alyssa’s rate of change is greater than Sarah’s .
vertical change ·············· horizontal change 5 8 ·· 1 5 8 vertical change ·············· horizontal change 5 6 ·· 1 5 6
Vocabularyrate of change the rate
at which one quantity
increases or decreases
with respect to a change
in the other quantity . It is
the ratio of the vertical
change to the horizontal
change on a graph .
initial value the
starting value of a
function .
Interpret and Compare Rates of Change
Study the example problem showing how to compare rates of change. Then solve problems 1–5.
45
©Curriculum Associates, LLC Copying is not permitted.76 Lesson 7 Compare Functions
Solve.
4 The table shows the weight gain of a kitten over a 5-week period . The graph shows the weight gain of a second kitten over the same period . Compare the rates of change for these two functions .
Kitten AWeek Weight (oz)
0 3
1 7
2 11
3 15
4 19
5 23
5 Sonya sells bracelets once a month at a flea market . The table shows her profits for a 5-month period .
SonyaMonth 1 2 3 4 5
Total Profit ($) 30 60 90 120 150
a. Kirsten sells bracelets once a month at a different flea market . The rate of change for her profits is $10 per month . Complete the table and the graph to show her total profits .
Kirsten
Month 1 2 3 4 5
Total Profit ($) 10
b. Sonya says that her profit is increasing 4 times as fast as Kirsten’s profit . Do you agree? Explain .
Wei
ght (
oz)
Kitten B
Weeks2 4 6 8 91 3 5 7O
x
y
4
2
8
12
14
16
18
6
10
Tota
l Pro
�t ($
)
Kirsten
Month2 4 6 8 91 3 5 7O
x
y
10
5
20
30
35
40
45
50
15
25
46
©Curriculum Associates, LLC Copying is not permitted. 77Lesson 7 Compare Functions
Name: Lesson 7
1 What do the initial values mean in the context of the example problem?
2 Do the functions in the example show positive or negative rates of change? Explain .
3 Write an equation for each function, where x is the number of months and y is the amount owed .
Mr . Allen’s plan:
Mr . Jessup’s plan:
Example
Mr . Allen bought a new computer . His monthly payment plan is shown in the table .
Month 0 1 2 3 4 5 6 7
Amount Mr. Allen Owes ($)
560 480 400 320 240 160 80 0
Mr . Jessup buys a new computer for $400 . He makes monthly payments of $40 until the computer is paid for . Compare the initial values and rates of change of each function .
You can graph both functions to show that the amount Mr . Allen owes starts at $560 and decreases $80 per month . The amount that Mr . Jessup owes starts at $400 and decreases $40 each month .
Mr . Allen’s initial value is $160 more than Mr . Jessup’s . Mr . Allen’s rate of change is greater than Mr . Jessup’s rate of change .
Am
ount
Ow
ed ($
)
Payment Plans
Months2 4 6 8 9 101 3 5
Mr. Allen
7Ox
y
160
80
320
480
560
640
720
240
400
Mr. Jessup
Compare Negative and Positive Rates of Change
Study the example problem showing how to compare two functions. Then solve problems 1–6.
47
©Curriculum Associates, LLC Copying is not permitted.78 Lesson 7 Compare Functions
Solve.
4 Below are two companies’ rates to rent a bicycle . How much does it cost per hour to rent a bicycle at Company A? What is the cost to rent a bicycle for 6 hours from each company?
Company A: c 5 5h 1 4, where c 5 total cost (in dollars) and h 5 number of hours
Company B: $6 per hour per bicycle
5 Roy wants to buy a new television for $300 . Two stores offer different payment options . Compare the initial values and rates of change .
Show your work.
Solution:
6 Most plumbing companies charge a fee to come to your house plus a charge per hour of work . The fees and charges for two plumbing companies are shown .
Write an equation for each company, where c 5 total cost (in dollars) and h 5 number of hours . Explain what the initial values and rates of change mean in this context .
Company A:
Company B:
Store A Payment PlanMonth 0 1 2 3 4 5 6
Amount Owed ($) 300 250 200 150 100 50 0
Store B Payment PlanPay $100 at the time of purchase . Pay $50 per month until the television is paid for .
Company AFee: $50Charge per hour: $40
Company BFee: $25Charge per hour: $50
48
8.F Introduction to Linear
Functions
Task
a. Decide which of the following points are on the graph of the function :
.
Find 3 more points on the graph of the function.
b. Find several points that are on the graph of the function .
Plot the points in the coordinate plane. Is this a linear function?
Support your conclusion.
c. Graph both functions and list as many differences between the two functions as you
can.
8.F Introduction to Linear Functions Typeset May 4, 2016 at 22:34:27. Licensed by Illustrative Mathematics under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .
y = 2x + 1
(0, 1), (2, 5), ( , 2), (2, −1), (−1, −1), (0.5, 1)1
2
y = 2 + 1x2
1
Illustrative Mathematics
49
