lesson 2: equivalent ratios - flagstaff unified school ... 6-7... · shanni and mel are using...

Lesson 2 Equivalent_Ratios_Math_67_WP_Summary.notebook August 24, 2014

Jan 88:18 AM

Lesson 2: Equivalent RatiosIn Lesson 2, we learned how to use ratio tables and tape diagrams to find equivalents ratios. Equivalent ratios are ratios that have the same value. First, we built a tape diagram from the ratio relationship:

Shanni and Mel are using ribbon to decorate a project in their art class. The ratio of the length of Shanni’s ribbon to the length of Mel’s ribbon is 7:3.

Then we created equivalent ratios using tape diagrams.

Next, we used tape diagrams to solve equivalent ratio problems.Mason and Laney ran laps to train for the long‐distance running team. The ratio of the number of laps Mason ran to the number of laps Laney ran was 2 to 3. a. If Mason ran 4 miles, how far did Laney run? Draw a tape diagram to demonstrate how you found the answer.

b. If Laney ran 930 meters, how far did Mason run? Draw a tape diagram to determine how you found the answer.

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Jan 88:18 AM

Lesson 2: Equivalent RatiosThen we made the connection between tables and tape diagrams for making equivalent ratios.

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Jan 88:18 AM

Lesson 2: Equivalent Ratios Next, we learned that you can determine if two ratios are equivalent by identifying whether there is a constant, c.

In the example above, the ratios are not equivalent because the quantity in the first ratio is not multiplied by the same number in the second quantity. This can be fixed by changing the second ratio to 42:77 so that the constant is 7 (c = 7).

It can also be fixed by changing the second ratio to 48:88 so that the constant is 8 (c = 8).

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Jan 88:18 AM

Lesson 2: Equivalent Ratios We also learned how to justify whether ratios in a tape diagram were equivalent using this method.In a bag of mixed walnuts and cashews, the ratio of number of walnuts to number of cashews is 5:6. Determine the amount of walnuts that are in the bag if there are 54 cashews. Use a tape diagram to support your work. Justify your answer by showing that the new ratio you created of number of walnuts to number of cashews is equivalent to 5:6.


Jan 88:18 AM

Learning Targets By the end of this lesson, you will be able to answer the following questions:(1) How can tables be used to find equivalent ratios?(2) How can tape diagrams be used to find equivalent ratios?(3) How can tape diagrams be used to determine if two ratios are equivalent?(4) How can you determine if two ratios are equivalent using a constant, c?


Jan 88:18 AM

Learning Targets Why do you need to know this?Ratios can be used to solve all types of real world problems. We use ratios to decide what items have the best price when we shop, which cars get the best gas mileage, and many other real world problems.


Aug 115:57 PM

ClassworkExercise 1Write a one‐sentence story problem about a ratio.

Write the ratio in two different forms.


Aug 115:57 PM

Exercise 2Shanni and Mel are using ribbon to decorate a project in their art class. The ratio of the length of Shanni’s ribbon to the length of Mel’s ribbon is 7:3.

Read this description. In a few seconds, I am going to take it away. I want you to describe in as much detail what the problem is about without looking at the problem.


Aug 116:00 PM

Describe in as much detail what the problem is about without looking at the problem.


Aug 115:57 PM


Let's represent this ratio in a table.


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Draw a tape diagram to represent this ratio:


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What does each unit on the tape diagram represent?Each unit represents one length of ribbon.

What if each unit on the tape diagram represents 1 inch? What are the lengths of the ribbons?Each unit represents one inch of ribbon.

What is the ratio of the lengths of the ribbons?


Aug 115:57 PM

Exercise 2Shanni and Mel are using ribbon to decorate a project in their art class. The ratio of the length of Shanni’s ribbon to the length of Mel’s ribbon is 7:3.What if each unit on the tape diagrams represents 2 meters?

What are the lengths of each of the ribbons?

How did you find that?If each unit is 2 meters, you put 2 meters in each unit and add or multiply to find the total for each person's length of ribbon.What is the ratio of the length of Shanni's ribbon to the length of Mel's ribbon now?


Aug 115:57 PM


What if each unit represents 3 inches? What are the lengths of the ribbons?

If each of the units represents 3 inches, what is the ratio of the length of Shanni's ribbon to the length of Mel's ribbon?


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We just explored three different possibilities for the length of the ribbon. Did the number of units in our tape diagram ever change?No, the number of units in our tape diagram always stayed 7: 3. All we did was change the value of each of the units.


Aug 115:57 PM


What did these 3 ratios, 7:3, 14:6, 21:9, all have in common?


Aug 115:57 PM


7:3, 14:6, 21:9

Mathematicians call these ratios equivalent.


Aug 116:09 PM

Exercise 3Mason and Laney ran laps to train for the long‐distance running team. The ratio of the number of laps Mason ran to the number of laps Laney ran was 2 to 3. a. If Mason ran 4 miles, how far did Laney run? Draw a tape diagram to demonstrate how you found the answer.

Mason ran 4 miles. That means his total is four. Since he has 2 units representing his part of the ratio, we divide 4 by 2 to find the value of each unit. Each unit it 2 miles. This means that each unit in Laney's part of the ratio is equal to 2 miles. We write 2 in each of the units of Laney's part of the ratio and use that to find her total. Since 2 + 2 + 2 = 2 x 3 = 6, Laney ran 6 miles.


Aug 116:09 PM

Exercise 3Mason and Laney ran laps to train for the long‐distance running team. The ratio of the number of laps Mason ran to the number of laps Laney ran was 2 to 3.

b. If Laney ran 930 meters, how far did Mason run? Draw a tape diagram to determine how you found the answer.

Laney ran 930 meters, so we label her part of the tape diagram with 930 meters. Then we divide 930 by 3 since there are three units in Laney's part of the tape diagram. The quotient is 310, which means each unit of the tape diagram is 310 meters. Since all units are the same size, we write 310 in each of Mason's units, and add or multiply to find the total meters Mason runs.


Aug 116:09 PM

Exercise 3Mason and Laney ran laps to train for the long‐distance running team. The ratio of the number of laps Mason ran to the number of laps Laney ran was 2 to 3.

c. What ratios can we say are equivalent to 2:3? The two ratios we found to be equivalent are 4:6 and 620:930.


Aug 116:09 PM

Exercise 4Josie took a long multiple‐choice, end‐of‐year vocabulary test. The ratio of the number of problems Josie got incorrect to the number of problems she got correct is 2:9.

a. If Josie missed 8 questions, how many did she get right? Draw a tape diagram to demonstrate how you found the answer.


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b. If Josie missed 20 questions, how many did she get right? Draw a tape diagram to demonstrate how you found the answer.


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c. What ratios can we say are equivalent to 2:9?

d. Come up with another possible ratio of the number Josie got wrong to the number she got right.


Aug 116:09 PM

Exercise 4Josie took a long multiple‐choice, end‐of‐year vocabulary test. The ratio of the number of problems Josie got incorrect to the number of problems she got correct is 2:9. e. How did you find the numbers?


Aug 116:09 PM


f. Describe how to create equivalent ratios. Equivalent ratios are created by taking the original ratio and multiplying both quantities by a constant (the same) number. You can do this in a tape diagram by putting the same number in all of the unit boxes, or you can do it by creating a table of equivalent ratios.


Aug 116:09 PM

Example 1

The morning announcements said that two out of every seven

6th graders in the school have an overdue library book. Jasmine said, “That would mean 24 of us have overdue books!” Grace argued, “No way. That is way too high.” How can you determine who is right? You can determine who is right if you know the total number of students in sixth grade. Then you can make a tape diagram to figure out if the ratios are equivalent.

If there are 84 total sixth graders, then Jasmine if correct. If there aren't 84 total sixth graders, then she is not correct. Grace is correct if there are fewer than 84 sixth graders.


Aug 116:09 PM

Exercise 5Decide whether or not each of the following pairs of ratios is equivalent.§ If the ratios are not equivalent, find a ratio that is equivalent to the first ratio.§ If the ratios are equivalent, identify the positive number, , that could be used to multiply each number of the first ratio by in order to get the numbers for the second ratio.


Aug 116:20 PM

Exercise 6In a bag of mixed walnuts and cashews, the ratio of number of walnuts to number of cashews is 5:6. Determine the amount of walnuts that are in the bag if there are 54 cashews. Use a tape diagram to support your work. Justify your answer by showing that the new ratio you created of number of walnuts to number of cashews is equivalent to 5:6.


Aug 106:38 PM

CLOSINGAnswer the Essential Questions in your math journal.Answer the following questions:(1) How can tables be used to find equivalent ratios?Equivalent ratios are created by taking the original ratio and multiplying both quantities by a constant (the same) number. In a table, you can repeat the pattern of the original ratio by adding the original ratio each time to create new (equivalent) ratios in the table.(2) How can tape diagrams be used to find equivalent ratios?Draw the tape diagram with the original ratio. You can create equivalent ratios by putting the same number in each of the unit boxes and then adding or multiplying to find the value of the units in each line of the tape diagram. (3) How can tape diagrams be used to determine if two ratios are equivalent?Create a tape diagram and figure out what number to put in each of the unit boxes after you are given one of the quantities in the equivalent ratio. If the totals pass the "justification test" where the number in each box is the same, then the two ratios are equivalent.

(4) How can you determine if two ratios are equivalent using a constant, c?If you can multiply each number in the first ratio by a constant, c, to get the second ratio, then the two ratios are equivalent. If the number multiplied is not the same, then there is no constant, c, and the ratios are not equivalent.


Aug 106:42 PM

HomeworkProblem Set Lesson 2

lesson 2: equivalent ratios - flagstaff unified school ... 6-7... · shanni and mel are using...

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