• The Write-Pair-Share activities may be used to pre-assess students’ understanding of ratios, rates, and unit rates.
• The Constant of Proportionality worksheet (M-7-3-1_Constant of Proportionality Practice and KEY.docx) may be used to assess students’ ability to identify the constant, k, in different proportional representations.
• The Proportion Practice activity sheet (M-7-3-1_Proportion Practice and KEY.docx) and Guided Practice activity may be used to assess students’ ability to apply proportional reasoning to problem-solving situations.
• Use the Lesson 1 Exit Ticket (M-7-3-1_Lesson 1 Exit Ticket and KEY.docx) to quickly evaluate student mastery.

Write-Pair-Share Activity 1

As an introduction to thinking about proportional relationships, ask students to define and provide examples of ratios, rates, and unit rates. Students should also discuss how the three terms are similar and different. Give students 2–3 minutes to record their ideas. Then, have partners share their ideas. After about 5 minutes, have one member from each pair share definitions and examples of the terms. Encourage discussion and debate.  

“Rates and unit rates are types of ratios. A ratio is simply a comparison of one value to another value. A rate is a comparison of two values, measured in different units. A unit rate is a type of rate in which the denominator is 1; in other words, a unit rate compares the value of one measurement to 1 of another type of measurement. Examples of ratios, rates, and unit rates are represented in the Venn diagram:”

Write-Pair-Share Activity 2

Ask students to think about customary and metric unit conversions they know, such as the fact that there are 12 inches in 1 foot. Give students 2–3 minutes to make a list of some conversions. Pair each student with a partner and give students time to share their lists. After about 5 minutes, one member from each group should come to the front of the room and write the conversions under appropriate categories, such as length/distance, weight, capacity, and time. When students have listed all they can think of, suggest any others they missed. Ask students to record the list of conversions in their notes or provide them with a prepared copy (M-7-3-1_Conversion Chart.docx).

Encourage a discussion on conversions and rates. Students should understand that the list of conversions represents rates, where a measurement in one unit is compared to a measurement in another unit. It is important that students understand that these rates may be used in proportional thinking.

Converting Values within the Customary System

After students have discussed ratios, rates, and unit rates, they may convert some values using proportional reasoning.

“We are now going to apply proportional reasoning to convert measurements within the Customary System.”

Present the following problems:

• 5 yd = ____ft              (15 ft)
• 48 oz. = ___lb             (3 lbs)
• 30 fl oz. = ____c         ( c)
• 4.5 lbs = ___oz.           (72 oz)

For each example, the following ideas should be presented:

1. Setting up the proportion.
2. Solving the proportion using one of the following methods:
   1. Fractional reasoning
   2. Inverse operations
   3. Cross-products
3. Identifying the constant of proportionality.
4. Writing the proportional relationship as an equation in the form y = kx.
5. Using the equation to identify more converted measurements.

“Let’s look at the first problem:

5 yd = ____ ft

“In order to find the number of feet in 5 yards, we will first set up a proportion. A proportion is an equation that sets two ratios (or fractions) equal to one another. For example,  is a proportion because the ratio on the left of the equal sign has the same value as the ratio on the right of the equal sign. The best way to set up a proportion is to think about the ratios you have been given.”

1. Setting up the proportion

“Our example, 5 yd = ____ ft, can also be written as the ratio . So we have one ratio, but we need two to complete the proportion:

“Can anyone think of another ratio that might deal with yards and feet?”
(There are 3 feet in 1 yard, or .)

“Many of you know that in every 1 yard, there are 3 feet. As a ratio this can be written as  or . Which version do you think we should use as the second ratio in our example?” (Use  so the units match up with the first ratio.)

“When writing proportions, we must remember that the two ratios are EQUAL, so it is very important that the numerator and denominator units of our two ratios match. For this reason, we must use  to complete the proportion.”

2a. Solving the proportion using fractional reasoning

“Notice, of course, that we still need to find x. In this case, finding x is easy as long as we remember that the two ratios in our proportion must have the same value. So if 1 yard is the same as 3 feet, then 5 yards must be the same as how many feet?” (15) “In other words, , which is an accurate proportion because  does indeed reduce to , meaning that the two fractions are equal.”

2b. Solving the proportion using inverse operations

“Sometimes a proportion may be too complex to get the right answer simply by reasoning. Fortunately, there are other ways to determine the unknown value in a proportion. A proportion is really just an algebraic equation, and you know by now that algebraic equations can be solved for a variable by using inverse operations. We can apply this method here. Follow along as I show the solution steps.

Explanation	Work
“First I will write the proportion without the units so we can focus purely on the numbers, operations, and inverse operations.”
“Next I will multiply both sides of the equation by x to remove the variable from the denominator.”
“Here the x is being multiplied by , so to isolate the x, we should divide both sides of the equation by .”
“Not surprisingly, we ended with the same value for x that we got using our first strategy of reasoning about the fractions.”

2c. Solving the proportion using cross-products

“There is yet another way to determine the unknown value in a proportion. This is by using something called cross-products. For any proportion, if  , then ad = bc, (where ad and bc are referred to as cross-products because we are literally multiplying on the diagonal across the equal sign). With this knowledge, we can easily rewrite any proportion so it appears as a statement stating that its cross-products are equal. Let’s use the cross-products to once again solve our example proportion.”

3. Identifying the constant of proportionality

“At this point, we have discussed how to construct a proportion, and then we looked at three different strategies we can use to solve the proportion. There is one other important idea about proportions we must discuss: the constant of proportionality. In a proportion, the constant of proportionality is essentially the value of the two ratios, given that one of the ratios is written as a unit rate. First of all, let’s review that a unit rate is a rate that has been simplified so the denominator is 1. Currently, the proportion we have been using is:

“In this proportion, can we say that either side of the equation represents a unit rate?” (No, because none of the denominators is 1.) “But there is an easy fix.  Watch what happens if we flip both ratios in the proportion upside down:

“Notice that the proportion is still true, as  Now, however, the second ratio can be called a unit rate because the denominator is 1. Because my proportion is now situated such that one of the two ratios has a denominator of 1, I am ready to determine the constant of proportionality. What is the value of each ratio in the proportion? (3) Thus, 3 is the constant of proportionality.”

4. Writing the proportional relationship as an equation in the form of y = kx

“Once we know the constant of proportionality, we can write our proportion in the form    y = kx, where k represents the constant of proportionality, and x and y represent our independent and dependent variables, as usual. In our example, the equation would be:

y = 3x

“So how does knowledge of the equation y = 3x help us? What can we do with this equation?” Provide time for discussion and debate.

“We can use the equation to find other converted measurements. We can state in words: y feet equals 3 times x yards. Alternatively, if we wish to find the number of yards in a given number of feet, we will substitute the number of feet for y, and solve for x.”

“For example, consider the problem:

___ yd = 30 ft

“Substituting 30 for y into the equation y = 3x, gives 30 = 3x. Solving for x gives x = 10. Thus:

10 yd = 30 ft

“Now let’s look at the next problem:

48 oz = ____ lbs

1. Setting up the proportion

“First, we will construct a proportion using the ratio we were given:

“Now, we must think of another ratio that compares ounces and pounds. There are 16 ounces in every 1 pound, so we can use this as the second ratio:

2. Solving the proportion

“Now that we have our proportion, we can solve it for x. If we use fractional reasoning, we may realize that 48 is 3 times more than 16. This implies that x must be 3 times more than 1 (to preserve the equality of the ratios). Therefore, x = 3.”

“We might also choose to use cross-products to solve the proportion. To do so, we would rewrite the proportion as a statement showing that the cross-products are equal:

“Now we solve the new equation for x:

“No matter the method we use to solve the proportion, it is clear that x = 3.

Thus:

48 oz = 3 lbs

and

3. Identifying the constant of proportionality

“With our proportion completed, we can now find the constant of proportionality, k, and write an equation in the form y = kx. Remember, to find the constant of proportionality, determine the value of each ratio in the proportion, given that there is a denominator of 1. Here, the second ratio is already written as a unit rate, and the value of each ratio is 16.”

4. Writing the proportional relationship as an equation in the form of y = kx

“Thus:

y = 16x

“We can state in words:

y ounces equals 16 times x pounds

“Now, you will try some examples with a partner.” Instruct pairs of students to follow the same steps as listed above (#1–4) with each of the next examples. The answers are provided in the table below. As students work, circulate the classroom to assess understanding and answer any questions.

Note: In each case, point out that the unit rate is the constant of proportionality. This idea will be revisited in Lesson 3.

Provide 5–10 additional examples for practice. Monitor students as they work to check for understanding.

Identifying the Constant of Proportionality in Tables, Graphs, Equations, and Verbal Descriptions

Now that students have had an opportunity to see the applicability of the constant of proportionality and develop a conceptual understanding of the term, provide various representations of proportional relationships and ask students to identify the constant.

“Consider the following proportional relationship: A driver drives at a speed of 65 miles per hour. The constant of proportionality is the described rate, which is 65. Let’s look at this relationship in a table:

“The constant of proportionality is represented by the ratio of the change in y-values per change in corresponding x-values. Since this table shows x-values that increase by 1, the change in consecutive y-values represents the rate of change, k. The constant of proportionality, or unit rate, is also represented by the y-value given for the x-value of 1.”

“Because we have identified that our constant of proportionality is 65, this proportional relationship can be represented by the equation y = 65x.”

“Now let’s look at a graph of this relationship:

“Notice that between any two consecutive points on the graph, there is a vertical difference of 65 and a horizontal difference of 1. Therefore, the ratio in the change of the y-values to the x-values is , or simply, 65. Again, since the point, (1, 65) is given, we can also discern from this one point that the constant of proportionality is 65.”

Provide students with a few more examples of tables and graphs, and ask them to identify the constant of proportionality from each representation. All representations do not need to be related. In other words, show a table, representing a particular proportion, a graph representing another proportion, an equation representing yet another proportion, and so on. The first example included representations of the same proportion, so students could easily make comparisons, regarding the presence of the constant in a description, table, equation, and graph. Include tables that do not show consecutive x-values. Provide students with the Constant of Proportionality worksheet (M-7-3-1_Constant of Proportionality Practice and KEY.docx).

Solving with Proportions

“We’ve used proportional reasoning to find converted measurements. Now, let’s look at some other applications of proportional reasoning.”

Go through examples similar to the following. Ask students to write a proportion for each and solve.

• If John ordered 4 pizzas for $28, how many could he get for $70? (; 10 pizzas)
• The cafeteria served 580 lunches. Forty-five of these were sandwiches. If 696 lunches will be served tomorrow, how many would be expected to be sandwiches?
  (; 54 sandwiches)
• Lisa ran 2.2 miles in 14 minutes. At this rate, how long would it take her to run 3.75 miles? (; about 23.9 minutes, rounded to the nearest tenth)

Go through additional examples if more instruction and practice is needed. When you are satisfied that students understand how to accurately set up and solve proportions of this type, introduce similar figures.

Similar Figures and Proportional Reasoning

“Another important use of proportional reasoning is to solve for missing values in mathematically similar figures. Similar figures have corresponding congruent angles and proportional corresponding sides. Thus, all pairs of corresponding sides have the same ratio. You can use the ratios to form a proportion to solve for missing side lengths.”

“These figures are similar. Find the missing side length using the ratio of sides to form a proportion.”

Give students time to write the proportion and find the solution.

“The following proportion may be used to solve for the missing height:

“Using cross-products, this proportion simplifies to 24x = 1008, where x = 42. Thus, the height of the larger triangle is 42 inches.”

“The proportional relationship may be represented by the equation, y = 1.5x, where 1.5 represents the constant of proportionality and x represents the dimension of the smaller triangle. Thus, the base of the smaller triangle, multiplied by 1.5, gives the base of the larger triangle. The height of the smaller triangle multiplied by 1.5 gives the height of the larger triangle.”

Guided Practice

Write 2–3 additional examples of similar figures on the board. Ask students to find the missing side length(s) in each example using proportions formed with the side length ratios. Monitor students as they work. Ask each student or small group questions about the process being used in order to gauge the level of understanding. Assist as needed.

Have students work with a partner to complete the Proportion Practice activity sheet (M-7-3-1_Proportion Practice and KEY.docx).

Proportional Reasoning and Scales

“Proportional relationships are also represented in scale drawings and maps. Suppose a map has a key where 1 inch = 60 miles.”

Have students answer the following questions.

• What do you think a 2-inch segment on the map would represent in real life?
  (120 miles)

• … a segment of  inches? (210 miles)
• …a segment of 7.5 inches? (450 miles)
• If two cities are 90 miles apart in real life, how far apart are they on the map?
  ( inches)
• How long would the segment need to be to represent the width of a state that is 300 miles across? (5 inches)
• If a mountain region is 45 miles long in real life, how long would it be on a map with this scale? ( inch)

Have students complete the Lesson 1 Exit Ticket (M-7-3-1_Lesson 1 Exit Ticket and KEY.docx) at the close of the lesson to evaluate their level of understanding.

Extension:

Use the Routine section for suggestions on ways to review lesson concepts throughout the school year. The Small Group section provides ideas for giving additional learning opportunities to students who may benefit from them. The Expansion section includes a challenge for students who are prepared to move beyond the requirements of the standard.

• Routine: During the school year, have students identify proportional relationships in the real world. For example, cumulative savings that increase by a constant rate each month represent a proportional relationship. In addition, when working with patterns, students should recognize sequences that are proportional. The connection of patterns to proportionality is very important. Such a discussion may be substituted for any part of this lesson or added as an extension. Students will get additional practice with examining proportional relationships when they determine proportionality in lesson 2, and interpret the meaning of points on a graph of a proportional relationship in lesson 3.
• Small Groups: Students who need additional practice may by pulled into small groups to work on the following activity: Small Group Practice (M-7-3-1_Small Group Practice and KEY.docx). Students can work on the problems together or work individually and compare answers when done.
• Expansion: Students who are prepared for a greater challenge may be given the Expansion Work worksheet (M-7-3-1_Expansion Work and KEY.docx). The worksheet includes more problems related to proportionality and the constant of proportionality, while also asking students to create their own representations of proportional relationships.