Math 8 Summer Study Guide

4.MD.5 Recognize angles as geometric shapes that are formed whenever two rays

share a common endpoint, and understand concepts of angle measurement:

a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc

between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be

used to measure angles.

b. An angle that turns through n one-degree angles is said to have an angle

measure of n degrees.

4.G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

4.G.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified

size. Recognize right triangles as a category, and identify right triangles.

4.G.3 Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching

parts. Identify line-symmetric figures and draw lines of symmetry.

5.NBT.A.2 Explain patterns in the number of zeros of the product when

multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use

whole-number exponents to denote powers of 10.

5.MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.

b. A solid figure which can be packed without gaps or overlaps using n

unit cubes is said to have a volume of n cubic units.

5.MD.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same

as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-

number products as volumes, e.g., to represent the associative property of multiplication.

b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge

lengths in the context of solving real world and mathematical problems.

6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is

3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”1

6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams,

double number line diagrams, or equations.

a. Make tables of equivalent ratios relating quantities with whole-number

measurements, find missing values in the tables, and plot the pairs of values

on the coordinate plane. Use tables to compare ratios.

1 Expectations for unit rates in this grade are limited to non-complex fractions.

b. Solve unit rate problems including those involving unit pricing and constant

speed. For example, if it took 7 hours to mow 4 lawns, then at that rate,

how many lawns could be mowed in 35 hours? At what rate were lawns

being mowed?

6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.

6.NS.C.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to

represent points on the line and in the plane with negative number coordinates.

a. Recognize opposite signs of numbers as indicating locations on opposite

sides of 0 on the number line; recognize that the opposite of the opposite of

a number is the number itself, e.g., –(–3) = 3, and that 0 is its own

opposite.

b. Understand signs of numbers in ordered pairs as indicating locations in

quadrants of the coordinate plane; recognize that when two ordered pairs

differ only by signs, the locations of the points are related by reflections

across one or both axes.

a. Find and position integers and other rational numbers on a horizontal or

vertical number line diagram; find and position pairs of integers and other

rational numbers on a coordinate plane.

2. A local park’s programs committee is raising money by holding mountain bike races on a course through the park. During each race, a computer tracks the competitors’ locations on the course using GPS tracking. The table shows how far each competitor is from a check point.

Number Competitor Name Distance to Check Point

223 Florence mile before

231 Mary

mile past

240 Rebecca mile before

249 Lita

mile past

255 Nancy

mile before

3. Mr. Kindle invested some money in the stock market. He tracks his gains and losses using a computer

program. Mr. Kindle receives a daily email that updates him on all his transactions from the previous day. This morning, his email read as follows:

Good morning, Mr. Kindle, Yesterday’s investment activity included a loss of , a gain of , and another gain of . Log in now to see your current balance.

a. Write an integer to represent each gain and loss.

SSSAAAMMMPPPLLLEEE AAANNNSSSWWWEEERRR———AAAnnnssswwweeerrrsss wwwiiilll lll vvvaaarrryyy

6.EE.A.1 Write and evaluate numeric expressions involving whole-number exponents. 6.EE.A.2 Write, read, and evaluate expressions in which letters stand for

numbers

a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract

from ” as .

b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression as a product of two factors; view as both a single entity and a sum of

two terms.

c. Evaluate expressions at specific values of their variables. Include

expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in

the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas and

to find the volume and surface area of a cube with sides of length

.

7.RP.2 Recognize and represent proportional relationships between quantities.

a. Decide whether two quantities are in a proportional relationship, e.g., by

testing for equivalent ratios in a table or graphing on a coordinate plane and

observing whether the graph is a straight line through the origin.

b. Identify the constant of proportionality (unit rate) in tables, graphs,

equations, diagrams, and verbal descriptions of proportional relationships.

c. Represent proportional relationships by equations. For example, if total

cost, t, is proportional to the number, n, of items purchased at a constant

price, p, the relationship between the total cost and the number of items can

be expressed at t = pn.

d. Explain what a point (x,y) on the graph of a proportional relationship means

in terms of the situation, with special attention to the points (0,0) and (1,r),

where r is the unit rate.

1. When a song is sold by an online music store, the store takes some of the money and the singer gets the

rest. The graph below shows how much money a pop singer makes given the total amount of money brought in by one popular online music store from sales of the song.

2. Over the break, your uncle and aunt ask you to help them cement the foundation of their newly purchased

land and give you a top-view blueprint of the area and proposed layout. A small legend on the corner states that 4 inches of the length corresponds to an actual length of 52 feet.

7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities

and commissions, fees, percent increase and decrease, percent error.

You may use a calculator for this part of the assessment. Show your work to receive full credit.

4. Tierra, Cameron, and Justice wrote equations to calculate the amount of money in a savings account after

one year with

interest paid annually on a balance of dollars. Let represent the total amount of

money saved. Tiara’s Equation: Cameron’s Equation: Justice’s Equation:

5. A printing company is enlarging the image on a postcard to make a greeting card. The enlargement of the

postcard’s rectangular image is done using a scale factor of . Be sure to show all other related math work. a. Represent a scale factor of as a fraction and decimal.

b. The postcard’s dimensions are inches by inches. What are the dimensions of the greeting card?

c. If the printing company makes a poster by enlarging the postcard image, and the poster’s dimensions are inches by inches, represent the scale factor as a percent.

d. Write an equation, in terms of the scale factor, that shows the relationship between the areas of the

postcard and poster. Explain your equation.

e. Suppose the printing company wanted to start with the greeting card’s image and reduce it to create

the postcard’s image. What scale factor would they use? Represent this scale factor as a percent.

f. In math class, students had to create a scale drawing that was smaller than the postcard image. Azra

used a scale factor of to create the smaller image. She stated the dimensions of her smaller

image as:

inches by inches. Azra’s math teacher did not give her full credit for her answer. Why?

Explain Azra’s error, and write the answer correctly.

You will now complete without the use of a calculator. 6. A MP3 player is marked up by and then marked down by . What is the final price? Explain

your answer.

7. The water level in a swimming pool increased from feet to feet. What is the percent increase in the

water level rounded to the nearest tenth of a percent? Show your work.

8. A -gallon mixture contains acid. A -gallon mixture contains acid. What percent acid is

obtained by putting the two mixtures together? Show your work.

9. In Mr. Johnson’s third and fourth period classes, of the students scored a or higher on a quiz. Let

be the total number of students in Mr. Johnson’s classes. Answer the following questions, and show your work to support your answers. a. If students scored a or higher, write an equation involving that relates the number of

students who scored a or higher to the total number of students in Mr. Johnson’s third and fourth period classes.

b. Solve your equation in part (a) to find how many students are in Mr. Johnson’s third and fourth period

classes.

c. Of the students who scored below , of them are girls. How many boys scored below ?

7.NS.A.2 Apply and extend previous understandings of multiplication and

division and of fractions to multiply and divide rational numbers.

a. Understand that multiplication is extended from fractions to rational

numbers by requiring that operations continue to satisfy the properties of

operations, particularly the distributive property, leading to products such as

(–1)( –1) = 1 and the rules for multiplying signed numbers. Interpret

products of rational numbers by describing real‐world contexts.

b. Understand that integers can be divided, provided that the divisor is not

zero, and every quotient of integers (with non‐zero divisor) is a rational

number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret

quotients of rational numbers by describing real‐world contexts.

c. Apply properties of operations as strategies to multiply and divide rational

numbers.

d. Convert a rational number to a decimal using long division; know that the

decimal form of a rational number terminates in 0s or eventually repeats.

1. Every month, Ms. Thomas pays her car loan through automatic payments (withdrawals) from her savings account. She pays the same amount on her car loan each month. At the end of the year, her savings account balance changed by from payments made on her car loan.

a. What is the change in Ms. Thomas’ savings account balance each month due to her car payment?

b. Describe the total change to Ms. Thomas’ savings account balance after making six monthly payments on her car loan. Model your answer using a number sentence.

2. Michael’s father bought him a -foot board to cut into shelves for his bedroom. Michael plans to cut the

board into equal size lengths for his shelves.

a. The saw blade that Michael will use to cut the board will change the length of the board by inches for each cut. How will this affect the total length of the board?

b. After making his cuts, what will the exact length of each shelf be?

3. The water level in Ricky Lake changes at an average of

inch every years.

a. Based on the rate above, how much will the water level change after one year? Show your calculations

and model your answer on the vertical number line, using as the original water level.

b. How much would the water level change over a -year period?

c. When written in decimal form, is your answer to part (b) a repeating decimal or a terminating decimal?

Justify your answer using long division.

7.EE.A.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

7.EE.B.4 Use variables to represent quantities in a real-world or mathematical

problem, and construct simple equations and inequalities to solve problems by

reasoning about the quantities.

a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution,

identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its

width? b. Solve word problems leading to inequalities of the form px + q > r or px + q

< r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a

salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you

need to make, and describe the solutions.

1. Use the following expression below to answer parts (a) and (b).

a. Write an equivalent expression in standard form and collect like terms.

b. Express the answer from part (a) as an equivalent expression in factored form.

2. Use the following information to solve the problems below.

a. The largest side of a triangle is six more units than the smallest side. The third side is twice the smallest side. If the perimeter of the triangle is units, write and solve an equation to find the lengths of all three sides of the triangle.

b. The length of a rectangle is inches long, and the width is

inches. If the area is

square

inches, write and solve an equation to find the length of the rectangle.

3. Ben wants to have his birthday at the bowling alley with a few of his friends, but he can spend no more than . The bowling alley charges a flat fee of for a private party and per person for shoe rentals and unlimited bowling. a. Write an inequality that represents the total cost of Ben’s birthday for people given his budget.

b. How many people can Ben pay for (including himself) while staying within the limitations of his budget?

c. Graph the solution of the inequality from part (a).

4. Jenny invited Gianna to go watch a movie with her family. The movie theater charges one rate for 3D admission and a different rate for regular admission. Jenny and Gianna decided to watch the newest movie in 3D. Jenny’s mother, father, and grandfather accompanied Jenny’s little brother to the regular admission movie. a. Write an expression for the total cost of the tickets. Define the variables.

b. The cost of the 3D ticket was double the cost of the regular admission ticket. Write an equation to represent the relationship between the two types of tickets.

c. The family purchased refreshments and spent a total of . If the total amount of money spent on

tickets and refreshments were , use an equation to find the cost of one regular admission ticket.

5. Gloria says the two expressions

and are equivalent. Is she correct? Explain

how you know.

7.G.A.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale

drawing at a different scale.

1. A printing company is enlarging the image on a postcard to make a greeting card. The enlargement of the postcard’s rectangular image is done using a scale factor of . Be sure to show all other related math work.

a. The postcard’s dimensions are inches by inches. What are the dimensions of the greeting card?

b. If the printing company makes a poster by enlarging the postcard image, and the poster’s dimensions are inches by inches, represent the scale factor as a percent.

c. Write an equation, in terms of the scale factor, that shows the relationship between the areas of the

postcard and poster. Explain your equation.

d. Suppose the printing company wanted to start with the greeting card’s image and reduce it to create

the postcard’s image. What scale factor would they use? Represent this scale factor as a percent.

e. In math class, students had to create a scale drawing that was smaller than the postcard image. Azra

used a scale factor of to create the smaller image. She stated the dimensions of her smaller

image as:

inches by inches. Azra’s math teacher did not give her full credit for her answer. Why?

Explain Azra’s error, and write the answer correctly.

7.G.A.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from

three measures of angles or sides, noticing when the conditions determine a

unique triangle, more than one triangle, or no triangle.

1. Use tools to construct a triangle based on the following given conditions. a. If possible, use your tools to construct a triangle with angle measurements , , and and

leave evidence of your construction. If it is not possible, explain why.

Solutions will vary. An example of a correctly constructed

triangle is shown here.

b. Is it possible to construct two different triangles that have the same angle measurements? If it is,

construct examples that demonstrate this condition, and label all angle and length measurements. If it is not possible, explain why not.

Solutions will vary. 2. In each of the following problems two triangles are given. For each: (1) state if there are sufficient or

insufficient conditions to show the triangles are identical, and (2) explain your reasoning.

a.

b.

c.

The triangles are identical by the two

sides and included angle condition.

△DEF ↔ △GIH.

There is insufficient evidence to

determine that the triangles identical.

In △DEF, the marked side is between

the marked angles but in △ABC, the

marked side is not between the

marked angles.

The triangles are identical by the two angles

and corresponding side condition. The

marked side is between the given angles.

△ABC ↔ △YXZ.

d.

3. Use tools to draw rectangle with cm and cm. Label all vertices and measurements.

4. Draw a right triangle according to the following conditions, and label the provided information. If it is not possible to draw the triangle according to the conditions, explain why not. Include a description of the kind of figure the current measurements allow. Provide a change to the conditions that makes the drawing feasible.

a. Construct a right triangle △ so that cm, cm, and cm; the measure of angle

is .

7.G.B.4 Know the formulas for the area and circumference of a circle and solve problems; give an informal derivation of the relationship between the

circumference and area of a circle.

1. A new park was designed to contain two circular gardens. Garden A has a diameter of m, and the garden B has a diameter of m. a. If the Gardner wants to outline the gardens in edging, how many meters will be needed to outline the

smaller garden? (Write in terms of .)

The triangles are not identical. In

△ABC, the marked side is opposite ∠B.

In △WXY, the marked side is opposite

∠W. ∠B and ∠W are not necessarily

equal in measure.

b. How much more fencing will be needed for the larger garden than the smaller one? (Write in terms of

.)

c. The Gardner wishes to put down weed block fabric on the two gardens before the plants are planted in

the ground. How much fabric will be needed to cover the area of both gardens? (Write in terms of .)

2. A play court on the school playground is shaped like a square joined by a semi-circle. The perimeter around the entire play court is ft., and ft. of the total perimeter comes from the semi-circle.

a. What is the radius of the semi-circle?

b. The school wants to cover the play court with sports court flooring. Using for how many square feet of flooring does the school need to purchase to cover the play court?

7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and use them to solve simple equations for

an unknown angle in a figure.

1. In each problem, set up and solve an equation for the unknown angles.

a. Four lines meet at a point. Find the measures of and .

b. Two lines meet at the vertex of two rays. Find the measures of and .

c. Two lines meet at a point that is the vertex of two rays. Find the measures of and .

d. Three rays have a common vertex on a line. Find the

measures of and .

n + 62 = 90

n + 62 – 62 = 90 – 62

n = 28°

m + 62 + (28) + 27 =

180

m + 117 – 117 =

180 – 117

m =

63°

m + 52 = 90

m + 52 – 52 = 90 –

52

m = 38°

40 + 52 + (38) + n =

180

130 + n =

180

130 – 130 + n =

180 – 130

n =

50°

50 + 90 + n = 180

140 + n = 180

140 – 140 + n = 180 –

140

n = 40°

m + 50 = 90

m + 50 – 50 = 90 –

50

m = 40°

n=90°, vertical angles

25 + (90) + 40 + m = 180

155 + m = 180

155 – 155 + m = 180 – 155

m = 25°

2. The measures of two complementary angles have a ratio of . Set up and solve an equation to

determine the measurements of the two angles.

3. The measure of the supplement of an angle is less than the measure of the angle. Set up and solve an

equation to determine the measurements of the angle and its supplement.

4. Three angles are at a point. The ratio of two of the angles is , and the remaining angle is more than the larger of the first two angles. Set up and solve an equation to determine the measures of all three angles.

2x + 3x + (3x + 32) = 360

8x + 32 = 360

8x + 32 – 32 = 360 – 32

8x = 328

(

)8x = (

)328

x = 41

Measure of Angle 1 = 2(41) = 82°

Measure of Angle 2 = 3(41) = 123°

Measure of Angle 3 = 3(41) + 32 = 155°

Let y be the number of degrees in the angle.

y + (y – 12) = 180

2y – 12 = 180

2y – 12 + 12 = 180 + 12

2y = 192

(

)2y = (

)192

y = 96

The measure of y = 96°; the measure of the supplement of y: (96) – 12 = 84°.

3x + 7x = 90

10x = 90

(

)10x = (

)90

x = 9

Measure of Angle 1 = 3(9) = 27°

Measure of Angle 2 = 7(9) = 63°

5. Draw a right triangle according to the following conditions, and label the provided information. If it is not

possible to draw the triangle according to the conditions, explain why not. Include a description of the kind of figure the current measurements allow. Provide a change to the conditions that makes the drawing feasible.

a. Construct triangle △ so that cm, cm, and cm; the measure of angle is

.

1. The three lines shown in the diagram below intersect at the same point. The measures of some of the

angles in degrees are given as , (

) , , .

a. Write and solve an equation that can be used to find the value of .

b. Write and solve an equation that can be used to find the value of .

It is not possible to draw this triangle because the lengths of the two

shorter sides do not sum to be greater than the longest side. In this

situation the total lengths of DE and EF are less than the length of FD;

there is no way to arrange DE and EF so that they meet. If they do not

meet, there is no arrangement of three non-collinear vertices of a triangle;

therefore, a triangle cannot be formed.

One possible change so that △ can be drawn: increase the length of

to .

2. Marcus drew two adjacent angles.

a. If ∠ has a measure one-third of ∠ , then what is the degree measurement of ∠ ?

b. If ∠ degrees, then what is the value of ?

3. The dimensions of an above-ground, rectangular pool are feet long, feet wide and feet deep.

a. How much water is needed to fill the pool?

b. If there are gallons in cubic foot, how many gallons are needed to fill the pool?

c. Assume there was a hole in the pool, and gallons of water leaked from the pool. How many feet did the water level drop?

d. After the leak was repaired, it was necessary to resurface (lay a thin layer of concrete to protect) the

sides of the pool. Calculate the area to be covered to complete the job.

4. Gary is learning about mosaics in Art class. His teacher passes out small square tiles and encourages the students to cut up the tiles in various angles. Gary’s first cut tile looks like this:

a. Write an equation relating ∠ with ∠ .

b. Solve for

3

c. What is the measure of ∠ ?

d. What is the measure of ∠ ?

7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles,

quadrilaterals, polygons, cubes, and right prisms.

1. In the following diagram, the length of one side of the smaller shaded square is

the length of square . What percent of square is shaded? Provide all evidence of your calculations.

2. The side of square , cm, is also the radius of circle . What

is the area of the entire shaded region? Provide all evidence of your calculations.

= (π)(2)2 = 4π; =

(4π) = 3π

cm2

= 4 cm2

= 4 + 3π cm2 or 13.4 cm2

Let be the length of the side of the smaller shaded

square. Then AD = 3x; the length of the side of the larger

shaded square is 3x – x = 2x.

AreaABCD= (3x)2 = 9x2

AreaLarge Shaded= (2x)2 = 4x2

AreaSmall Shaded = (x)2 = x2

AreaShaded = 4x2 + x2 = 5x2

Percent AreaShaded =

(100%) = 55

%

3. For his latest design, a jeweler hollows out crystal cube beads (like the one in

the diagram) through which to thread the chain of a necklace. If the edge of the crystal cube is mm, and the edge of the square cut is mm, what is the volume of one bead? Provide all evidence of your calculations.

4. John and Joyce are sharing a piece of cake with the dimensions shown in the diagram. John is about to cut the cake at the mark indicated by the dotted lines. Joyce says this cut will make one of the pieces three times as big as the other. Is she right? Justify your response.

5. A tank measures ft. in length, ft. in width, and ft. in height. It is filled with water to a height of ft.

A typical brick measures a length of in., a width of in., and a height of in. How many whole bricks can be added before the tank overflows? Provide all evidence of your calculations.

Volume in tank not occupied by water:

V = (4)(3)(0.5) = 6 ft3

VolumeBrick = (9)(4.5)(3) = 121.5 in3

Conversion (in3 to ft3): (121.5 in3)(

)=0.0703125 ft3

Number of bricks that fit in the volume not occupied by water: (

) =

85

Number of whole bricks that fit without causing overflow: 85

VolumeTrapezoidal Prism =

(5+2.5)(6)(10) = 225 cm3

VolumeTriangular Prism =

(2.5)(6)(10) = 75 cm3

Joyce is right, the current cut would give 225 cm3 of cake

for the trapezoidal prism piece, and 75 cm3 of cake for

the triangular prism piece, making the larger piece,

=

3 times the size of the smaller piece.

VolumeLarge Cube = (10)3 = 1,000 mm3

VolumeHollow = (10)(6)(6) = 360 mm3

VolumeBead = 1,000 – 360 = 640 mm3

10 mm

6 mm

6. Five, three-inch cubes and two triangular prisms have been glued together to form the composite three-dimensional figure shown in the diagram. Find the surface area of the figure, including the base. Provide all evidence of your calculations.

19 square surfaces: 19(32) = 171 in2

4 triangular surfaces: (4)(

)(3)(4) = 24 in2

3×5 rectangular surface: (3)(5) = 15 in2

3×4 rectangular surface: (3)(4) = 12 in2

6×5 rectangular surface: (6)(5) = 30 in2

6×4 rectangular surface: (6)(4) = 24 in2

Total surface area: 171 + 24 + 15 + 12 + 30 + 24 =

276 in2

3 in

4 in4 in

5 in

5 in