CC Course 1 — Beta Version
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· Introduction
· Chapter 1
· Chapter 2
· Chapter 3
o 3 Opening
o 3.1.1
o 3.1.2
o 3.1.3
o 3.1.4
o 3.1.5
o 3.1.6
o 3.2.1
o 3.2.2
o 3.2.3
o 3.2.4
o 3 Closure
· Chapter 4
· Chapter 5
· Chapter 6
· Chapter 7
· Chapter 8
· Chapter 9
· Reference
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· Lesson
· Answers
· Teacher Notes
· Sharing
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· In mathematics, a ratio is used to express certain relationships between two or more quantities. You were working with ratios when you expressed portions as percents and fractions. You also used ratios when you compared the portion of raisins or peanuts to the whole mix in the trail-mix problem.Today you will extend what you know about fractions to investigate more general ratios. You will learn how they can be used to compare the trail-mix ingredients and much more. As you work with your team, keep the following questions in mind.
· How do the quantities compare?
· What quantities am I comparing?
· How can I represent the relationship?
· Can I represent it in another way?
· 3-78.Quinn just had a large pond dug on his farm and wants to stock the pond with fish. He was researching the best way to go about this when he saw the following posting on the PondPerson blog:
· “The biggest factor to keep in mind is the predator-prey ratio: Stock the pond with 1 predator fish for every 3 prey fish. Predator fish are large-mouth bass or walleye, while prey fish are perch, bluegill, or sunfish.”
· Quinn chose walleye for the predator species and bluegills for the prey fish. He knows he will need a little over 1000 fish total to fill a pond the size of his. With your team, discuss what you think the predator-to-prey ratio in this situation means. Then estimate how many of each fish (walleyes and bluegills) Quinn should buy to stock the pond while keeping the ratio of predators to prey correct. Be prepared to share your ideas with the class.
· 3-79. ON THE TRAIL AGAIN
· Rowena and Polly are investigating their trail-mix problem again. Rowena took a handful of her mixture and counted her raisins and peanuts. She found she had 8peanuts and 32raisins in her sample. Rowena drew the following diagram to represent her sample.
·
· You can use a ratio to compare the number of peanuts or raisins in this sample to the total. If you find what percent of the sample is made up of raisins (or peanuts), you are writing a special kind of ratio. It describes how many raisins or peanuts would be present if the whole sample contained 100 raisins and peanuts combined. In fact, you can use aratio to express the relationship between any two quantities in the mix.
1. For the sample shown above, identify what each of the following ratios are comparing. For example, for “A ratio of 40 to 32,” you would write, “total to raisins,” because the ratio is comparing the total number, 40, to the number of raisins, 32.
1. A ratio of 8 to 40
2. A ratio of 8 to 32
3. A ratio of 32 to 8
4. A ratio of 32 to 40
2. Use what you have learned about portions to describe what portion of Rowena’s sample is peanuts and what portion is raisins. Express each answer as a fraction and as a percent.
· 3-80.Rowena and Polly remembered the diagrams they used in the “Handful of Pennies” Lesson from Chapter1. They recalled how the diagrams helped them quickly see the number of pennies in a group. They drew the diagrams below to represent their sample.
· P P P P P P P P
R R R R R R R R R R R R R R R R
R R R R R R R R R R R R R R R R
· P P P P P P P P
R R R R R R R R R R R R R R R R
R R R R R R R R R R R R R R R R
o Then they wrote the following ratios:
o 4 to 16,
o 2 to 8,
o 8 to 2,
o 16 to 4, and
o 1 to 4.
7. What do you think each of these ratios represents? How do these ratios compare to the ratios in problem 3-79?
8. Rowena thinks that saying there are 32 raisins for every 8 peanuts is the same as saying there is 1 peanut for every 4 raisins. She claims these ratios are the same. Is Rowena correct? How are these ratios the same? How are they different?
· 3-81. WAYS TO WRITE A RATIO
· Just as you can express portions in multiple ways, you can write a ratio in any of three forms.
o With the word “to,” such as: The ratio of raisins to peanuts is 4 to 1.
o In fractionform, such as: The peanuts and raisins have a ratio of
o With a colon (:), such as: The ratio of peanuts to raisins is 1: 4.
4. Sidra has a sample of trail mix containing 22 raisins and 28 peanuts. Use the bulleted list above to write the ratio of peanuts to raisins in her sample using three different methods. What would you have to change to write the ratio of raisins to peanuts?
5. Ratios, like fractions, can be written in simplified form. The ratio of 32 to 8 can be written equivalently as 4 to 1. Simplify your answer to part (a) in each of the three ratio forms.
6. Find the percent of Sidra’s trail mix that is peanuts. Can you use the ratios you found in parts (a) and (b)? Explain.
7. Find the percent of Sidra’s trail mix that is raisins.
· 3-82. Ratios can be particularly useful when you want to keep the percent of an ingredient or the ratio of ingredients the same, but you want to change the total amount.
· For example: Rowena is not very fond of peanuts. So she is pleased that the number of peanuts is quite small compared to the number of raisins in her sample from problem 3-79. She would like to keep the same ratio of peanuts to raisins when she mixes up a large batch of trail mix. Rowena and Polly decided to use ratio tables to describe all the relationships in the trail mix. The table will help them make sense of the ratios so they know how much of each ingredient to purchase.
1. Analyze the tables below. Why did Rowena and Polly record different numbers? Did one of them make a mistake? Why or why not?
Rowena’s Table
1 / 2 / 3 / 4 / 5 / 6 / 7 / 84 / 8 / 12 / 16 / 20 / 24 / 28 / 32
Polly’s Table
1 / 2 / 3 / 4 / 5 / 6 / 7 / 85 / 10 / 15 / 20 / 25 / 30 / 35 / 40
2. With your team, recall the definition of “percent.” Whose table would help you find most easily the percent of the trail mix that is peanuts?Why?
· 3-83.Marina found a quilt pattern that she wants to use to make a quilt. Her pattern is shown at right. The large square is made of 9small squares. Some of the small squares contain a design pattern (she calls them “pattern squares”), and some small squares do not (she calls them “plain squares”). Marina is trying to determine how much material she will need to make a quilt.
1. Create a ratio table comparing the total number of pattern squares to the number of plain squares.
2. Marina measured her material and found that she had enough material to cut out 100 plain squares. How many pattern squares will she need to create? How many large squares will she be able to make? Explain your reasoning.
3. Use three different methods to express the ratio of pattern squares to the number of plain squares.
·
· Graphing Points on an xy-Coordinate Grid
· Numerical data that you want toput on a two-dimensional graph is entered on the graph as points.
· The graph has a horizontal number line, called the x-axis, and a vertical number line, called the y-axis. The two axes cross at the origin (0, 0) which is the 0 point on each axis.
· Points on the graph are identified by two numbers in an ordered pair. An ordered pair is written as(x, y).The first number is the xcoordinate of the point, and the second number is the ycoordinate.
· To locate the point (3, 2) on an xy-graph,first start at the origin. Go 3units to the right (to the mark 3 on the horizontal axis). Then, from that point, go 2units up (to the mark across from 2 on the vertical axis).
· The example graphatright shows one of the four regions of the
xy-coordinate graph.
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· 3-84. Walter is mixing cement for his new patio. He knows he needs to use a water-to-cement ratio of 20 to 30. What percent of his total mixture is water? Homework Help ✎
· 3-85. David wants to find and is wondering if using decimals can help him make sense of adding fractions. Homework Help ✎
1. How could be written using decimals? What is the sum as a decimal?
2. How could your answer from part (a) be written as a fraction?
3. Rewrite as a fraction that could be added easily to.
· 3-86. Use the data and axes below to create a histogram for Mr. Nguyen’s class grades. Homework Help ✎
· 50, 55, 57, 60, 62, 65, 78, 80, 82, 85, 88, 89, 90, 91, 93, 95, 96, 98, 99
·
· 3-87. If you walk forward 5 feet and then walk backward 5 feet, you will end up exactly where you started. For each of the actions below, describe an action that will get you back where you started. Homework Help ✎
1. Walk up 10 steps.
2. Earn 8 dollars.
3. It gets 5 degrees warmer.
4. Lose 6 dollars.
5. Travel south 3 kilometers.
6. Run backward 9 steps.
· 3-88. The diagram at right is made up of Base Ten Blocks. Use the diagram to answer the following questions. 3-88 HW eTool (CPM).Homework Help ✎
1. Find the area and perimeter of the shape.
2. Draw a Base Ten Block shape with a value of 126 using the fewest number of blocks possible. Find the perimeter of the shape that you drew.
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