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.
1. For each part below, explain how the measure of the unknown angle can be found without using a protractor.
a. Find the measure of ∠D.
b. In this figure, Q, R, and S lie on a line. Find the measure of ∠QRT.
c. Q, R, and S lie on a line, as do P, R, and T. Find the measure of ∠PRS.
2. Mike drew some two-dimensional figures.
Sketch the figures and answer each part about the figures that Mike drew.
- He drew a four-sided figure with four right angles. It is 4 cm long and 3 cm wide.
What type of quadrilateral did Mike draw?
How many lines of symmetry does it have?
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.
1. Model the number 8.88 on the place value chart.
a. Use words, numbers, and your model to explain why each of the digits has a different value. Be sure to use “ten times as much” and “one tenth of” in your explanation.
b. Multiply 8.88 x 104. Explain the shift of the digits, the change in the value of each digit, and the number of zeroes in the product.
c. Divide the product from (b) by 102. Explain the shift of the digits and how the value of each digit changed.
2. Annual rainfall total for cities in New York are listed below.
Rochester 0.97 meters
Ithaca 0.947 meters
Saratoga Springs 1.5 meters
New York City 1.268 meters
a. Round each of the rainfall totals to the nearest tenth.
3. The following equations involve different quantities and use different operations, yet produce the same result. Use a place value mat and words to explain why this is true.
4.13 x 103 = 4130 413,000 ÷ 102 = 4130
4. Dr. Mann mixed 10.357 g of chemical A, 12.062 g of chemical B, and 7.506 g of chemical C to make 5 doses of medicine.
- About how much medicine did he make in grams? Estimate the amount of each chemical by rounding to the nearest tenth of a gram before finding the sum. Show all your thinking.
- Find the actual amount of medicine mixed by Dr. Mann. What is the difference in your estimate and the actual amount?
- Round the weight of one dose to the nearest gram. Write an equation that shows how to convert the rounded weight to kilograms and solve. Explain your thinking in words.
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.
1. Tell the volume of each solid figure made of 1-inch cubes. Specify the correct unit of measure.
a.
b.
2. Juliet wants to know if the chicken broth in this beaker will fit into this rectangular food storage container. Explain how you would figure it out without pouring the contents in. If it will fit, how much more broth could the storage container hold? If it will not fit, how much broth would be left over? (Remember 1 cm3 = 1 mL.)
3. A rectangular container that has a length of 30 cm, a width of 20 cm, and a height of 24 cm is filled with water to a depth of 15 cm. When an additional 6.5 liters of water is poured into the container, some water overflows. How many liters of water overflow the container? Use words, pictures, and numbers to explain your answer. (Remember 1 cm3 = 1 mL.)
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.
1. Jack found the volume of the prism pictured to the right by multiplying 5 × 8 and then adding: 40 + 40 + 40 = 120. He says the volume is 120 cubic inches.
a. Jill says he did it wrong. He should have multiplied the bottom first (3 × 5) and then multiplied by the height. Explain to Jill why Jack’s method works and is equivalent to her method.
b. Use Jack’s method to find the volume of this right rectangular prism.
2. If the figure below is made of cubes with 2-cm side lengths, what is its volume? Explain your thinking.
3. The volume of a rectangular prism is 840 in3. If the area of the base is 60 in2, find its height. Draw and label a model to show your thinking.
4. The following structure is composed of two right rectangular prisms that each measure 12 inches by 10 inches by 5 inches, and one right rectangular prism that measures 10 inches by 8 inches by 36 inches. What is the total volume of the structure? Explain your thinking.
5. a. Find the volume of the rectangular fish tank. Explain your thinking.
b. If the fish tank is completely filled with water, and then 900 cubic centimeters are poured out, how high will the water be? Give your answer in centimeters, and show your work.
6. Miguel and Jacqui built towers out of craft sticks. Miguel’s tower had a 4-inch square base. Jacqui’s tower had a 6-inch square base. If Miguel’s tower had a volume of 128 cubic inches, and Jacqui’s had a volume of 288 cubic inches, whose tower was taller? Explain your reasoning.
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.
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?
1. The most common women’s shoe size in the U.S. is reported to be an 8 ½. A shoe store uses a table like the one below to decide how many pairs of size 8 ½ shoes to buy when they place a shoe order from the shoe makers.
Total number of pairs of shoes being ordered / Number of pairs of size 8 ½ to order50 / 8
100 / 16
150 / 24
200 / 32
a. What is the ratio of the number of pairs of size 8 ½ shoes they order to the total number of pairs of shoes being ordered?
b. Plot the values from the table on a coordinate plane, and draw a straight line through the points. Label the axes. Then use the graph to find the number of pairs of size 8 ½ shoes they order for a total order of 125 pairs of shoes.
2. Wells College in Aurora, New York was previously an all-girls college. In 2005, the college began to allow boys to enroll. By 2012, the ratio of boys to girls was 3 to 7. If there were 200 more girls than boys in 2012, how many boys were enrolled that year? Use a table, graph, or tape diagram to justify your answer.
3. Most television shows use 13 minutes of every hour for commercials, leaving the remaining 47 minutes for the actual show. One popular television show wants to change the ratio of commercial time to show time to be 3:7. Create two ratio tables, one for the normal ratio of commercials to programming and another for the proposed ratio of commercials to programming. Use the ratio tables to make a statement about which ratio would mean fewer commercials for viewers watching 2 hours of television.
4. Loren and Julie have different part time jobs after school. They are both paid at a constant rate of dollars per hour. The tables below show Loren and Julie’s total income (amount earned) for working a given amount of time.
Loren
Hours / 2 / 4 / 6 / 8 / 10 / 12 / 14 / 16 / 18Dollars / 18 / 36 / 54 / 72 / 90 / 108 / 162
Julie
Dollars / 36 / 108 / 144 / 180 / 216 / 288 / 324
a. Find the missing values in the two tables above.
b. Who makes more per hour? Justify your answer.
c. Write how much Julie makes as a rate. What is the unit rate?
d. How much money would Julie earn for working 16 hours?
e. What is the ratio between how much Loren makes per hour and how much Julie makes per hour?
f. Julie works 112 hours/dollar. Write a one or two-sentence explanation of what this rate means. Use this rate to find how long it takes for Julie to earn $228.
5. Your mother takes you to your grandparents’ house for dinner. She drives 60 minutes at a constant speed of 40 miles per hour. She reaches the highway and quickly speeds up and drives for another 30 minutes at constant speed of 70 miles per hour.
a. How far did you and your mother travel altogether?
b. How long did the trip take?
c. Your older brother drove to your grandparents’ house in a different car, but left from the same location at the same time. If he traveled at a constant speed of 60 miles per hour, explain why he would reach your grandparents house first. Use words, diagrams, or numbers to explain your reasoning.
6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.
1. L.B. Johnson Middle School held a track and field event during the school year. The chess club sold various drink and snack items for the participants and the audience. All together they sold 486 items that totaled $2,673.
a. If the chess club sold each item for the same price, calculate the price of each item.
b. Explain the value of each digit in your answer to 1(a) using place value terms.
2. The PTA created a cross-country trail for the meet.
a. The PTA placed a trail marker in the ground every four hundred yards. Every nine hundred yards the PTA set up a water station. What is the shortest distance a runner will have to run to see both a water station and trail marker at the same location?
Answer: hundred yards
b. There are 1,760 yards in one mile. About how many miles will a runner have to run before seeing both a water station and trail marker at the same location? Calculate the answer to the nearest hundredth of a mile.
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.
c. 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.