Grade 8 - Newark Public Schools · Grade 8 Similarity and Congruency 8.G.1 - 4 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS

Grade 8 Similarity and Congruency

8.G.1 - 4

2012 COMMON CORE STATE STANDARDS ALIGNED MODULES

THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS

2012 COMMON CORE STATE STANDARDS ALIGNED MODULES

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Understand congruency and similarity using physical

models, transparencies, or geometry software.

8.G.1-4 Similarity and Congruency

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Goal: In this module, students will transform shapes using translations,

reflections, rotations, and dilations. Students will use their understanding

of transformations to identify and create sequences to obtain similar

figures.

Prerequisites:

1. Graphing on a coordinate grid.

2. Identifying the number of lines of

symmetry in a given shape

Essential Question(s):

1.Which transformations preserve

order?

2.What other words can be used for

rotation, reflection, translation,

and dilation?

Embedded Mathematical Practice(s) MP.1 Make sense of problems and persevere in

solving them.

MP.2 Reason abstractly and quantitatively

MP.3 Construct viable arguments

MP.6 Attend to Precision

Lesson 1

8.G.1a-1c & 2: Verify experimentally the properties of

translations.

Lesson 2

Reflections and Rotations

8.G.1a-1c & 2: Verify experimentally the

properties of reflections and rotations.

Lesson 3

Dilations

8.G.3: Describe the effect of dilations on two-

dimensional figures using coordinates.

Lesson 4

Sequence of

Transformations

8.G.4 Understand that two-dimensional figure

is similar to another I the second can be obtained

from the first by a sequence of transformations.

Lesson 5

Golden Problem

8.G.1-4 Understand congruency

and similarity using physical

models, transparencies, or

geometry software.

Lesson Structure Introductory Task

Guided Practice

Collaborative work

Journal Questions

Skill Building

Homework

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Introductory Task Guided Practice Collaborative Work Homework Assessment

8.G.1: Verify experimentally the properties of rotations, reflections, and translations:

8. G.1a: Lines are taken to lines, and line segments to line segments of the same length

8 .G.1b: Angles are taken to angles of the same measure

8.G.1c: Parallel lines are taken to parallel lines

8. G.2: Understand that a two-dimensional figure is congruent to another if the second can be

obtained from the first by a sequence of rotations, reflections, and translations; given two

congruent figures describe a sequence that exhibits the similarity between them.

8. G.3: Describe the effect of dilations, translations, rotations, and reflections on two-

dimensional figures using coordinates

8.G.4: Understand that a two-dimensional figure is similar to another if the second can be

obtained from the first by a sequence of rotations, reflections, translations, and dilations;

given two similar two dimensional figures, describe a sequence that exhibits the similarity

between them.

Prerequisite Competencies 1. Graphing in the coordinate plane 2. Identifying the number of lines of symmetry in a given shape

Lesson 1:Introductory Task – Translation

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Focus Question(s) 1) How can you find a general rule to describe a translation?

2) When a figure is translated what stays the same and what changes?

Introductory Task

Write a rule to describe the translation above

Explain how you arrived at your rule

Define what a translation is

Think of two real life situations that model translations.

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Lesson 1:Guided Practice

1. Triangle JKL has vertices J(0,2), K(3,4), and L(5,1). Translate triangle JKL 4 units

to the left and 5 units up.

a. Draw and label the vertices of the original triangle and its image

b. Using arrow notation write a rule to describe the translations

c. How do the two triangles compare?

2. Draw the image of triangle ABC after a rotation of

about the origin.

3. A rectangle has its vertices at M(1,1), N(6,1), O(6,5)

and P(1,5). The rectangle is translated to left 4 units

and down three units.

a. Graph the rectangle MNOP and M’N’O’P?

b. What are the coordinates of M’N’O’P?

c. Use arrow notations to write a rule that describes the translation of

M’N’O’P’ to MNOP.

4. The points T(0,0), U(-3,0),V(-3,5) and W (2,3) form a quadrilateral. Name the

coordinates of the image of quadrilateral T’U’V’W’ after a rotation of How

do the two quadrilaterals compare?

5. What is the angle of rotation of the figure below? Explain how you found your

answer.

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Journal Question(s)

1. Suppose you translate a point to the left 1 unit and up 3 units. Describe what you

would do to the coordinates of the original point to find the coordinates of the image

Lesson 1:Collaborative Work

Collaborative Work

1. The chessboard below shows for possible moves for the white knight.

Write a rule to describe each move as a translation, using the knight’s

original position as the origin.

2. What is the angle of rotation, in degrees, that maps A to A’ in the photo of the ceiling fan

on the right?

3. Draw the image of triangle ABD after a rotation of the

given numbers of degrees about the origin. Explain how you

arrived at your answers.

4. A square has rotational symmetry because it can be rotated 180 degrees so that its image

matches the original. Your friend says the angle of rotation is . What is wrong with this statement?

5. Write a rule to describe the translation of G(-5,3) to G’(-1,-2).

6. Match each rule with the correct translation.

a. (x,y) (x – 6, y + 2) i) P(4,-1) P’(3,-6)

b. (x,y)(x+3, y) ii) Q (3,0) Q’ (-3, 2)

c. (x,y) (x-1, y-5) iii) R (-2,4) R’ (1,4)

How could you check your answers?

Challenge:

Graph the equation y=1/2x. Translate the line right 2 units and up four units

a) b)

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1) Parallelogram PQRS is shown on the coordinate plane on the right.

a. What are the coordinates of the image of the new image after parallelogram PQRS is translated 6 units to the left and 8 units down? Graph and label the new image on the coordinate plane.

b. Write a rule to describe the translation.

2) Find the coordinates of the vertices of each figure after the given transformation.

a. rotation of about the origin; E(2,-2), J(1,2), R(3,3), S(5,2)

b. translation: 7 units right and 1 unit down; J(-3,1), F(-2,3), N(-2,0)

c. rotation clockwise about the origin; B(-2,0), C(-4,3), Z(-3,4), X(-1,4)

d. translation: 6 units left and 3 units up; S(-3,3), C(-1,4), W(-2,-1)

3) Rotate the figure the given number of degrees about the origin. List the coordinates of the

rotated figure.


Lesson 1: Homework

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Lesson 2:Introductory Task – Reflecting Lines

Introductory Task

Line segments AB and CD have the same length. Describe a sequence of reflections that exhibits

congruence between them.

Commentary:

Students' first experience with transformations is likely to be with specific shapes like triangles, quadrilaterals, circles, and figures with symmetry. Exhibiting a sequence of transformations that

shows that two generic line segments of the same length are congruent is a good way for students to begin thinking about transformations in greater generality

Focus Question(s)

How does the line of reflection affect the reflection of a shape?

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Guided Practice

1. Triangle JKL is shown on the coordinate plane on

the right

a. Triangle JKL is reflected over the y-axis. On

your grid, draw triangle J’K’L’. Be sure to

label the vertices

b. Triangle J’K’L’ is rotated 90 degrees

clockwise about the origin. Draw triangle

J”K”L”, Be sure to label the vertices

c. Suppose the vertices of J”K”L” are reflected

over the y-axis and the reflected over the x-

axis. Do the vertices of the resulting triangle

have the same coordinates as the vertices of

triangle JKL? Show and explain how you got

your answer.

2. Quadrilateral BCDE has vertices of B (-4,4) , C(-1,5), D(0,2), E(-2,1)

a. Graph quadrilateral BCDE and its image after it is reflected over the line through

(1,3) and (1,0).

b. Name the coordinates of the vertices B’C’D’E’

3. Triangle ABC has vertices A(-2,5), B(-2,2), and C (-5,2). If triangle ABC is reflected across

the line y=x, what are the new coordinates?

Lesson 2: Guided Practice

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1. The coordinates of the end points of and its image are given below

1.

W What single transformation was used to map to ? Explain your answer.

2. A shape was moved from Position A to Position B, as shown below.

Describe how the shape was moved from Position A to Position B.

3. The vertices of triangle PQR are P(2,1), Q(6,1) and R (6,3).

Graph the image of triangle PQR after its translation using the rule (x,y) (x+3, y-6) and its

rotation 90 degrees about the origin.

4. Triangle ABC have vertices at A(-3, -4), B (-2,-3) and C(-4.-1).

Graph the image of the triangle after the transformations sequence.

Reflection across the line y=1

Rotation 90 degrees about the origin

Does the order affect the final image? Explain your answer

Journal Question(s)

1) A polygon is reflected in the x-axis and then reflected in the y-axis. Explain how you can use

rotations to obtain the same result as this composition of transformations. Draw an example.

2) When you reflect a figure in a line, you can visualize reflecting the entire plane and taking the

figure along for the ride. Are any points in the plane unmoved by a reflection? That is, are there

any fixed points? Explain.551)

S (2,4) S’ (-2,-4)

T (-1,1) T’ (1,-1)

Lesson 2: Collaborative Work

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Homework 1. Point P(6, 7) and point Q(6, 4) are plotted on the coordinate grid to the

right. Point P is rotated 180° clockwise about point Q. What are the coordinates of the image of point P after this rotation? Graph on the coordinate grid. Explain how you found for your answer.

2.

a. On your grid, draw the image of triangle LMN after it is translated 4 units to the left. Label the image PQR. List the coordinates for points P, Q, and R.

b. On your grid, draw the image of triangle LMN after it is translated 6 units up and 3 units to the right. Label the image TUV. List the coordinates for points T, U, and V.

c. On your grid, draw the image of triangle LMN after it is reflected over the x-axis. Label the image XYZ. List the coordinates for points X, Y, and Z.

3. Graph the reflection and identify the new coordinates formed by the polygons using the given

line:

4. Graph triangle RST with vertices R(-1,3), S(4,-2), and T(2,-5). Graph the image and identify

the new coordinates formed by rotating the triangle about the origin:

a) b) c)

Lesson 2: Homework

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Lesson 3:Introductory Task – Dialations

Introductory Task

Focus Question(s) How does scale factor affect whether the dilation will be an enlargement

or a reduction?

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Guided Practice

1. Find the image of triangle DEF with vertices D(-2,2), E(1,-1) and F(-2,-1) after a dilation

with center D and a scale factor of 2

2. Find the coordinates of the image of quadrilateral KLMN

after a dilation with a scale factor of 3/2.

3. Figure TRSV shows the outline of a park. A city planner dilates

the figure to show the area of the park that can be used for

concerts.

Find the scale factor. Is it an enlargement or a reduction?

How does the area of Figure TRSV compare to T’R’S’V’

4. You are reducing a digital photo that is 2 in. high and 3 in. wide. If the reduced photo is

in.

high, what is its width? Write your answer as a number in simple form.

Make a plan: Draw and label the original photo and the reduced photo next to each other.

Label the missing width w.

5. A window on a computer screen is 1 ½ in. high and 2 in wide. After you click the “size

reduction” button, the window is reduced to

in high. And

in. wide. What is the

scale factor?

6. The blue figure is a dilation of the original figure. What is the scale factor?

Lesson 3:Guided Practice

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Collaborative Work

1. A picture frame has an opening that is 12 in by 15 in. If a matting is placed inside the frame

to create an opening that is 7 ½ in by 9 3/8 in., what is the scale factor of the reduction?

2. Graph the coordinates of rectangle HIJK with vertices H(1,2), I(1,7), J(14,7), and K(14,2).

Label the vertices.

a. Find the image of rectangle HIJK after dilation with a scale factor of 0.5 and center

H. Label the image LMNO.

b. Describe the relationship between the perimeters of rectangle HIJK and of rectangle

LMNO. Write a ratio to compare the perimeters

c. Describe the relationship between the areas of HIJK and LMNO. Write a ratio to

compare the areas.

d. What conclusions can you make about the ratio of the areas with a scale factor

of 0.5?

3. Draw triangle ABC with the following vertices: A(0,0), B(5,4), and C(6,1). Find the image

of triangle ABC after dilation with a scale factor of 2.5.

a. What are the coordinates of the image of triangle ABC

b. How do the areas of the triangle ABC and its image compare?

4. Jorge is enlarging a digital photo is 4 in. high by 6 in. wide. The enlarged photo is 18 ½ in.

high.

a. What is its width?

b. What is the scale factor?

5. Find the image of triangle ABC at the right after a dilation with the given center and scale

factor

a. Center B, scale factor of 3

b. Center A, scale factor of ½

6. Triangle EFG has three angles of 60 degrees and three sides that measure 60 cm each.

What scale factor should you use to dilate to triangle A’B’C’ with side lengths of 21 cm.

Journal: Suppose you know the coordinates of the vertices of a triangle. Explain how you would find the coordinates of the vertices of its image after a dilation with a scale factor

of r.


Lesson 3:CollaborativeWork

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.

1. Using the graph below, identify and label which transformation is a reflection, rotation, translation,

and dilation of triangle ABC. Explain how you know.

2. Graph the coordinates of the quadrilateral ABCD. Find the coordinates of its image A’B’C’D’

after a dilation with the given scale factor

a) A(2,-2), B(3,2), C(-3,2), D(-2,-2); b) A (6,3), B(0,6),C (-6,2), D(-6,-5);

scale factor: 2 scale factor ½

3. A triangle has coordinates A(-2,-2), B(4,-2) and C(1,1). Graph its image A’B’C’ after

dilation with scale factor 3/2.

a) Give the coordinates of A’B’C’,

b) What is the ratio of the areas of the figures A’B’C’ and ABC.

4. Find the coordinates of the image of ABCD with vertices A(0,0), B(0,3), C(3,3), and D(3,0)

after a dilation with a scale factor of 4/3

5. The figure to the right, triangle PQR shows the outline of a

playing field. A city planner dilates the design to show the area

available for community youth to play sports. Find the scale

factor. Is it an enlargement or a reduction?

6. A rectangle is dilated with a scale factor or 0.6. Is the image a reduction or an

enlargement? Explain.


Lesson 3:Homework

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Lesson 4:Introductory Task –

Introductory Task

On the coordinate plane, draw a quadrilateral with vertices at Q (1,2), R(4,2), S(1,-1), and

T(-2,-1).

a) What shape is figure QRST?

b) Draw the reflection of the figure over the y-axis. Is the image similar to the

original figure?

c) Dilate the reflected image by a factor of 2 using the origin as the center of dilation.

Compare the dilated image with the reflected image and other figure. Which

figures are similar? Explain your reasoning.

Focus Question(s)

How do you identify a similarity transformation in the coordinate plane

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Guided Practice

1. Given the coordinates A (1,1) B(5, 1) C ( 5, 4) and D (1,4)

a. Connect the points and record the area and perimeter of the figure

b. Reflect quadrilateral ABCD over the y-axis and record the new coordinate points

c. Translate the new quadrilateral 3 units to right and 4 units up. Record the new

coordinates.

d. Triple the lengths of each side, give the new coordinates. What is the new area and

perimeter of the larger quadrilateral? Is it similar, by definition to the smaller

quadrilateral? 2. Graph a quadrilateral with vertices at A(-1,1), B(1,1),C(3.-1), and D(1,-1)

a. Reflect it over the x-axis and record the new coordinate points

b. Dilate the new quadrilateral by a factor of 3 using the origin as the center of dilation

c. Which of the figures are similar? Explain your reasoning?

3. Coordinates of the vertices of a pre-image and image figure are given. Describe the

transformations that move the first figure onto the second.

a. O(0,0), B(0,3), C(2,0); O(O,O), D(O,-6), E(4,0)

b. A(-6,0), B(0,6),C(9,0),D (0,-15); W(O,-2),X(-2,0), Y(0,3), Z(5,0)

4. The coordinates of the vertices of a triangle ABC and its image triangle DEF are given. Are

there a series of transformations that move triangle ABC onto triangle DEF.

A(1,1), B(22,2), C(2,2); D(2,22) E(24,24), F(4,24)

5. Rectangle PQRS is transformed to rectangle P’Q’R’S’ as shown on the graph below:

a. Describe a sequence of transformations to map

rectangle PQRS to rectangle P’Q’R’S’

b. Identify all congruent lines and angles

c. Can you perform a sequence of translations,

reflections, rotations, or dilations that are not

congruent to the first? Explain

Lesson 4: Guided Practice

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1. Given points A (1, 2), B (1, 4), and C (4, 4):

a. Connect the points and record the area and perimeter of the figure.

b. Translate triangle ABC 6 units to the left and record the new coordinate points.

c. Reflect the new triangle over the y-axis and record the new coordinate points.

d. Rotate the new triangle about the origin 90 degrees clockwise and record the new

coordinates.

e. Double the lengths of each side, give the new coordinates. What is the new area and

perimeter of the larger triangle? Is it similar, by definition, to the smaller triangle?

2. A fractal called the Sierpinksi triangle can be imagined by thinking of an infinite sequence

of stages, the first of which are shown below.

Calculate the side lengths of the light blue

triangles in Stages 1,2, and 3. Then describe the

3 dilatations you can apply to any stage to

generate the next stage

3. Use transformations to explain why pentagons

EFGHD and RSTUV are not similar

4. Draw triangle EFG with vertices at E(-2,3), F(0,4) and G(0,0) and triangle JKL with

vertices at J(3,-2), K(4,-4) and L(0,-4).

a. Describe a sequence of transformations to obtain triangle JKL from triangle EFG

b. Describe a different sequence of transformations that you used in part a. to obtain

triangle JKL from triangle EFG

c. Do the transformations you chose affect the congruence of line segments or

angles? Explain.

Journal Question(s)

1. In a coordinate plane, does the order of transformations affect similarity? Explain.

Lesson 4:Collaborative Work

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Homework

1. The vertices of triangle ABC are A (2, 4), B (6, 8), and C (11, 3). Triangle ABC is translated 5

units down into triangle A'B'C' and reflected in the y−axis into triangle A''B''C''.

What are the coordinates of the vertices of triangle A''B''C''?

2. Starting with the smallest square, describe the

transformations that were used to make the drawing on the

left.

3. Using the diagram on the right answer the following question:

a) Describe the sequence of transformations that maps

trapezoid WXYZ to trapezoid W”X”Y”Z”

b) Find the length of . Which other line segments are

congruent to ?

c) Does a sequence of transformations maintain

congruence? Use the lengths of line segments to justify

your answer.

d) Dose a sequence of transformations maintain

congruency of angles?

4. Verify that the two figures in the graph are similar by

describing a composition of transformations, involving a

dilation then a translation, that maps triangle DEF to

triangle D’E’F’ .

5. Coordinates of the vertices of a pre-image and image figure

are given. Describe the transformations that move the first

figure onto the second.

a. O(0,0), B(0,8), Q(6,0); O(O,O), R(4,0), E(0,-3)

b. J(-4,0), K(0,4),L(6,0); L(6,0),M(-0,6), N(-9,0)

Lesson 4: Homework

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Lesson 5:Golden Problem

Golden Problem – 1

Samantha and Mario were both given the coordinates: P(-4, 4), Q(-5, 4), R(-6, 2) and S(-2, 2) for parallelogram PQRS. They were

told to translate the parallelogram: (x, y)(x + 6, y + 1) and then reflect it over the x –axis, and label the new parallelogram: P’Q’R’S’. Samantha said her coordinates are: P’ (5, -5), Q’(1, -5), R’(0, -3) and S’(4, -3). Mario said his coordinates are: P’(5, -5), Q’(1, -5), R’(2, -3) and S’(6, -3). Who has the correct coordinates for parallelogram PQRS? Explain what mistake(s) the student made who had the incorrect coordinates.

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Lesson 5:Golden Problem

Golden Problem – 2

Given: ABC, A(4, 8), B (6, 14), C (10, 12)

a) Reflect ABC across the x-axis. Label that reflection

A’B’C’. What are the coordinates of A’, B’, C’?

b) Use a dilation of A’B’C’ with a scale factor of

to find

A”B”C”. What are the coordinates of A”, B”, C”?

c) Find the coordinates of the vertices of A’”B’”C’” by

translating A”B”C”: (x, y)(x + 4, y – 5), then reflect it over the y-axis.

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Golden Problem I Rubric: 3-Point Response The student states that the student with the correct coordinates is Samantha

The student correct identifies Mario’s error.

2-Point Response

The student shows correct work and a genuine knowledge and understanding of the transformation, but

a slight error was made on the graph

OR

The student chooses Samantha as the student with the correct coordinates, but is unable to exactly

identify Mario’s error.

1-Point Response

The student chooses Mario as the student with the correct coordinates and attempts to show an error

made by Samantha.

0-Point Response

The student shows no attempt at completing the graph.

AND

The student makes no attempt to identify an error made by one of the students in the problem.

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Golden Problem II Rubric: 3-Point Response The student shows a genuine knowledge and understanding of the transformations: reflection, dilation, and

reflection – although a slight error may have been made on the graph.

2-Point Response

A working knowledge and understanding of the transformations is evident, although a major mistake has

been made on the graph resulting in stating wrong coordinated

OR

The student commits a significant error but basically provides the correct coordinates on the graph.

1-Point Response

The student demonstrates very little knowledge of transformation.

OR

All the coordinates on the graph are incorrect.

0-Point Response

The student shows no understanding of any of the transformations.

OR

The student shows no comprehension of the Cartesian Coordinate System.

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Systems of linear equations can also have one solution, infinitely many solutions or

no solutions. Students will discover these cases as they graph systems of linear

equations and solve them algebraically.

A system of linear equations whose graphs meet at one point (intersecting lines) has

only one solution, the ordered pair representing the point of intersection. A system

of linear equations whose graphs do not meet (parallel lines) has no solutions and the

slopes of these lines are the same. A system of linear equations whose graphs are

coincident (the same line) has infinitely many solutions, the set of ordered pairs

representing all the points on the line.

By making connections between algebraic and graphical solutions and the context of

the system of linear equations, students are able to make sense of their solutions.

Students need opportunities to work with equations and context that include whole

number and/or decimals/fractions.

Examples:

• Find x and y using elimination and then using substitution.

3x + 4y = 7

-2x + 8y = 10

• Plant A and Plant B are on different watering schedules. This affects their rate of

growth. Compare the growth of the two plants to determine when their heights will

be the same.

Let W= number of weeks

Let H= height of the plant after W weeks

Plant A Plant B

W H W H

0 4 (0,4) 0 2 (0,2)

1 6 (1,6) 1 6 (1,6)

2 8 (2,8) 2 10 (2,10)

3 10 (3,10) 3 14 (3,14)

*Grade 8 Required Fluency:

Solve simple 2 x 2 systems by inspection

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Given each set of coordinates, graph their corresponding lines.

Solution:

• Write an equation that represent the growth rate of Plant A and Plant B.

Solution:

Plant A: H = 2W + 4

Plant B: H = 4W + 2

• At which week will the plants have the same height?

Solution:

The plants have the same height after one week.

Plant A: H = 2W + 4 Plant B: H = 4W + 2

Plant A: H = 2(1) + 4 Plant B: H = 4(1) + 2

Plant A: H = 6 Plant B: H = 6

After one week, the height of Plant A and Plant B are both 6 inches.

*Fluent in the Standards means “fast and accurate”. It might also help to think of fluency as meaning the

same thing as when we say, that somebody is fluent in foreign language; when you’re fluent, you flow. Fluent

isn’t halting, stumbling,or reversing oneself. Assessing fluency requires attending to issues of time (and even

perhaps rhythm, which could beachieved with technology).

Source: http://www.sde.idaho.gov/site/common/mathCore/docs/mathStandards/MathGr8.pdf

http://www.sde.idaho.gov/site/common/mathCore/docs/mathStandards/MathGr8.pdf

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Grade 8 Fluency Problems

1. Tanisha and Rachel had lunch at the mall. Tanisha ordered three slices of pizza and two colas. Rachel

ordered two slices of pizza and three colas. Tanisha's bill was $6.00, and Rachel's bill was $5.25. What

was the price of one slice of pizza? What was the price of one cola?

2. When Tony received his weekly allowance, he decided to purchase candy bars for all his friends. Tony

bought three Milk Chocolate bars and four Creamy Nougat bars, which cost a total of $4.25 without tax.

Then he realized this candy would not be enough for all his friends, so he returned to the store and bought

an additional six Milk Chocolate bars and four Creamy Nougat bars, which cost a total of $6.50 without

tax. How much did each type of candy bar cost?

3. Sal keeps quarters, nickels, and dimes in his change jar. He has a total of 52 coins. He has three more

quarters than dimes and five fewer nickels than dimes. How many dimes does Sal have?

4. Ramón rented a sprayer and a generator on his first job. He used each piece of equipment for 6 hours at a

total cost of $90. On his second job, he used the sprayer for 4 hours and the generator for 8 hours at a total

cost of $100. What was the hourly cost of each piece of equipment?

5. At a concert, $720 was collected for hot dogs, hamburgers, and soft drinks. All three items sold for $1.00

each. Twice as many hot dogs were sold as hamburgers. Three times as many soft drinks were sold as

hamburgers. Find the number of soft drinks sold .

6. Solve the linear system:





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Grade 8 - Newark Public Schools · Grade 8 Similarity and Congruency 8.G.1 - 4 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS