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Section 2.5 Proportional Reasoning Every Day

Subsection 2.5.1 Overview

We have seen that algebra arises in making introductions, the 100s chart, number and magic number puzzles, children's literature, and Aesop's fable. Now we examine contexts that arise in our daily experiences.

Student Page 2.5.2 Recipes and Proportional Reasoning

A proportional relationship that frequently arises in daily life is scaling recipes. As you work on Recipes and Proportional Reasoning, think about these questions: What does slope mean in this context? How can we find the slope in each representation, context, table, graph, and equation? How are the slopes for each of the ingredients in the guacamole recipe related?

1.

(a)

Have you had to make more or less of a recipe? Why did you alter it?

(b)

What did you do to rescale the recipe?

Guacamole.

  • 2 medium avocados

  • 1 teaspoon sea salt

  • 2 tablespoons lemon juice (~ half a lemon)

  • \(1/4\) cup onion (~\(1/8\) a large onion)

  • 1 medium tomato

  • \(1/2\) cup fresh cilantro leaves

Blend on low for 15 to 20 seconds. Do not over mix. Leave chunky. Garnish with diced tomato and parsley.

Yields 1.5 cups.

2.

Use the Guacamole recipe. Determine the amount of each ingredient you need for the number of recipes of guacamole shown in the first column. Complete Table 2.5.2.1.

Table 2.5.2.1. Guacamole Ingredients
  Ingredients
Amount needed for
\(n\)recipes with \(n =\)
Avocados Tomatoes Lemons Onions
\(\frac{1}{2}\)
1 2 1 \(\frac{1}{2}\) \(\frac{1}{8}\)
2        
3        
4        
\(n\)

3.

What patterns do you notice in the table?

4.

Graph the data by plotting the points, (Number of recipes, amount of ingredient needed). Use Figure 2.5.2.2

Figure 2.5.2.2. Plot for Guacamole Ingredients
(a)

What scale should you use for each axis? How did you decide what the scale should be?

(b)

Label the scale on each axis. Write a title on each axis showing what it represents.

(c)

Graph each ingredient using a different color. Label each graph with the ingredient it represents.

5.

Compare the graphs. What do you notice? List as many relationships as you can.

6.

For each ingredient, find an equation that will predict the amount you will need for an unknown number of recipes. Write the equations in the last row of Table 2.5.2.1.

7.

You have 5 avocados.

(a)

How many recipes of guacamole can you make and use all of them?

(b)

How can you tell from the graph?

(c)

How can you tell from the table?

(d)

How can you tell from the equation?

8.

You have 1 onion but want to save half of the onion for salsa.

(a)

How many recipes of guacamole can you make?

(b)

How can you tell from the graph?

(c)

How can you tell from the table?

(d)

How can you tell from the equation?

9.

(a)

What does slope mean in the context of number of recipes versus amount of ingredients?

(b)

How can you find the slope from a table? From a graph? From the equation? From the recipe?

(c)

How are the slopes related for the ingredients in the guacamole recipe?

Student Page 2.5.3 Where Else Do You Use Proportional Reasoning? Proportional Linear Function Example

We have seen that the relationship between distance and time is proportional when speed is constant. We have also seen that proportional reasoning arises in rescaling recipes. What other relationships are proportional? Work with your group to identify at least 3 other proportional relationships. Complete Where Else Do You Use Proportional Reasoning? Proportional Linear Function Example for each relationship your group identifies.

Once you have completed the above, look for common properties among the examples you found. With your group, answer these questions:

  • What is a proportional relationship?

  • What properties do all proportional relationships have in common?

  • For a proportional relationship, how can you find the slope from:

    • A table?

    • A graph?

    • An equation?

    • A context?

Choose one or two of your group's proportional relationships to share with the class. Post them as directed by your teacher. Groups will cycle through them and indicate with a checkmark if they agree with your work. They will draw a circle if they have a question about your work.

Title:

Description

Independent variable (description and variable):

Dependent variable (description and variable):

1.

Create a table as demonstrated in Table 2.5.3.1. Briefly describe the independent variable in the cell above \(x\text{.}\) Briefly describe the dependent variable in the cell above \(y\text{.}\) Choose eight reasonable \(x\)-values (not all consecutive); write them in the first column of the table. Find the corresponding \(y\)-values; write them in the second column of the table.

Table 2.5.3.1. Table for a Proportional Linear Function
  
  
  
  
  
  
\(x\) \(y\)
             
             
             
             
             
             
             
             
Figure 2.5.3.2. Blank Graph for a Proportional Linear Function

2.

Plot the data on Figure 2.5.3.2. Label accurate scales for both \(x\) and \(y\) so that you use most of the graph. Write titles on each axis to show what quantity each axis represents.

3.

Find an equation.

(a)

How can you get the value of the dependent variable from the value of the independent variable?

4.

Answer these questions on the back of this page.

(a)

What is the slope of the graph?

(b)

How is the slope of the graph related to the equation?

(c)

How is the slope of the graph related to the table?

(d)

How is the slope of the graph related to the context?

Homework 2.5.4 Homework

1.

A step conversion chart indicates that if you walk 4 miles per hour, you would take approximately 140 steps per minute. The same chart shows that if you cycle at 15 miles per hour, you would take the equivalent of 160 steps per minute.

Table 2.5.4.1. Step Conversion Chart
  Number of steps taken
Number of Minutes Walking at 4 mph Cycling at 15 mph
1 140 160
2    
3    
4    
5    
10    
20    
60    
\(m\)    
(a)

Complete the table for one of the sports.

(b)

Write an equation that fits your data in the last row of the table. Tell how you know your equation is correct.

(c)

Plot the data. Label scales and titles for each axis.

(d)

For your choice of exercise, how many minutes would you have to exercise to take the equivalent of 10,000 steps? Show your work

2.

The step conversion chart in Table 2.5.4.2 showed the following information:

Table 2.5.4.2. Step Conversion Chart: Cyclying
Cycling at \(C\) miles an hour Equivalent number, \(N\text{,}\)
of steps per minute of exercise
5 55
10 93
15 160
20 200
(a)

Comparing \(C\) to \(N\text{,}\) is the relationship linear or non-linear?

(b)

How can you tell from the table?

(c)

How can you tell from a graph?

3.

When you eat at a sit-down restaurant, it is customary to leave a tip to reward the server for good service. Suppose you always leave a 15% tip.

(a)

Write in words how you would determine how much to tip a server if the amount of your check is $30.

(b)

What if your check is $40?

(c)

What if your check is \(D\) dollars?

(d)

Write an algebraic expression to compute the amount of the tip to leave the server if the tip is 15% of the check and your check is \(D\) dollars.

(e)

How can you quickly estimate a 15% tip?

(f)

What changes in the expression you wrote in Task 2.5.4.3.d if you leave a 20% tip?

(g)

How can you quickly estimate a 20% tip?

4.

A dog's life span is a fraction of time compared to that of humans. Table 2.5.4.3 shows a how a dog's age might be adjusted to compare to a human's age.

Table 2.5.4.3. Dog Age vs. Human Age
Number of actual
years of life, \(y\)
1 2 3 4 5 6 7 8 \(y\)
Comparative human age
(dog years), \(d\)
6.5 13 19.5 26          
(a)

Find the pattern and fill in the missing values in the table.

(b)

What expression can you use to find a comparative human age for a dog that is \(y\) actual years old?

(c)

Would you call a dog middle-aged if she is 7 actual years old? Why or why not?

5.

Most of the rest of the world uses kilograms rather than pounds. Note: 1 kg = 2.2 pounds = 2.2 lbs.

(a)

When traveling in the United Kingdom, I bought 2 kg of bananas. How many pounds of bananas is this?

(b)

Fill in Table 2.5.4.4.

Table 2.5.4.4. Kilograms vs. Pounds
Kilograms 0.5 1 1.5 2 2.5 3 3.5 4 10 \(k\)
                    \(p\)
(c)

Write an equation you can use to convert kilograms to pounds exactly.

(d)

When you buy bananas, how many pounds do you usually buy? (Three medium bananas weigh approximately a pound.) How many kg of bananas is this number of pounds?

6.

JoAnn Fabrics regularly runs sales. A recent flyer included a coupon for 60% + 15% off custom framing. How much would you pay for a framing bill of \(x\) dollars? Explain.

7.

Ashley loves Salted Caramel Mocha coffee. She orders her favorite with either 2% or coconut milk and occasionally also orders whipped cream. The number of calories for each type of drink depends on the number of ounces of coffee in the size of cup she buys.

Table 2.5.4.5.
Number of ounces
of coffee
Number of calories, \(C\)
With 2% milk With 2% milk
and whipped cream
With coconut milk
8 180 280 160
12 270 370 240
16 360 460 320
20 450 550 400
30      
\(x\)      
(a)

Plot all three sets of data on the same coordinate plane. Connect the points for each data set. Label each graph to match the data set: 2% milk, 2% milk with whipped cream, or coconut milk.

(b)

For each proportional relationship, in the last row of the table, write equations that fit the data. Tell how you know your equations are correct based on the context.

(c)

Suppose the coffee shop offered a 30-ounce cup of coffee. If Ashley ordered this size, what would be the number of calories? Include this information in the table.

(d)

Is there a data set that is not proportional? If so, which one? How do you know?

8.

Each of the relationships can be modeled by a linear equation. For each set of data in Exercise¬†2.5.4.1‚Äď2.5.4.7:

(a)

Graph the data on a separate coordinate plane. Find the slope of the graph. Write the equation under the graph.

(b)

How does the slope of the graph relate to the equation?

(c)

How can you find the slope of the graph using the table?

9.

(b)

Think about the examples above, scaling recipes, and relationships discussed in class. What are some general properties of linear relationships?

(c)

Think of another linear relationship that arises in your life. Describe the relationship. Why do you think it is linear?

10.

Practice using Desmos to graph the examples in Exercise¬†2.5.4.1‚Äď2.5.4.7. If you need more help, click on the ‚Äú?‚ÄĚ icon (Help) in the upper right corner of the Desmos graphing screen. Choose Tables for a tutorial on entering and graphing data. Choose Sliders to see what these do. Use a slider with the equation, \(y = mx\) to fit your data. Explain why the equation you find makes sense for your data.