## Increasing Portions, Expanding Waistlines: Exploring the Relationship Between Calories Consumed and Weight

• Lesson
9-12
2

This activity uses a 2009 study based on the book The Joy of Cooking that found portion sizes in the iconic cookbook had expanded 60% since 1936 and 33% since 1996. Students use linear functions to create models of weight as a function of calories and time, then use the models to make projections about what impact increased portion sizes may have on weight.

Discuss with students the problems with rising obesity rates in America. Draw from current events as appropriate to help illustrate the point. Some suggested resources are listed under Related Resources, although other, timelier, resources might be available for your classroom.

Discuss with students the causes for obesity. Keep in mind (and point out to them) that there are many causes for obesity, some of which may or may not be within a person's control, such as genetics and medical conditions. Keep an eye out for comments from students that might imply obesity is entirely an obese person's fault and address them if they come up. Have them list causes that they can think of. Steer them towards considering rising portion sizes as one cause. Explain that while there are many components of a healthy diet, just like there are many causes for rising obesity levels, today in class you will examine one: portion size. Remind students that this is an oversimplification of a complex issue (refer to the list to imply just how complicated it is) but it is one important element.

Have students read the Increasing Portions, Expanding Waist Lines Article independently. Then, hold a brief class discussion on what they read and learned.

Distribute the Increasing Portions, Expanding Waist Lines Activity Sheet. Have students work through it, perhaps in pairs or groups of three. If you don't want to stop and discuss the activity periodically with the class, you should circulate and discuss different elements with groups, or combine the two. You want to help push and refine student thinking, so even students that have correct answers can be pushed to deepen their thinking.

In Question 4, there is intentionally a mismatch between units. Students wrote their function in terms of weeks in Question 3 and they have to answer Question 4 in terms of months and years. Help them work flexibly through the many ways they can convert units to make it work.

In Question 5, caution students to use an appropriate viewing window when graphing the function. Help them figure out what a reasonable viewing window would be for each function.

The calorie information in this activity was obtained from the calorie calculator at MayoClinic.com. Adjust the information so that it is slightly different for each group, or have students make their own hypothetical person, if you prefer.

After groups have had a significant amount of time to work (let them be at least halfway done) distribute overheads or poster paper for them to record their work and share with the class, either via presentations or a gallery walk. (During a gallery walk, groups walk around and observe each other's work. This may be a good opportunity for students to assess each other’s work.)

As a closing activity, have students write or journal about their own eating habits. Remind students that unhealthy weights can be too low as well as too high. Give students a safe place to reflect on their relationship with food. Be aware that you might have some students for whom this can be a delicate issue. Keep an eye out for anyone who might be bothered by this activity. If you are uncomfortable or feel you don't know your students well enough to support them through a potentially sensitive journal exercise, consider something more objective, like having students analyze the calories in different school lunch choices or a restaurant menu and write about what healthier choices are (or are not) available. Another alternative is to have students create a one day healthy menu for breakfast, lunch, and dinner that provides 2160 calories.

Assessment Options

1. Have groups create posters with illustrations of different portion sizes (small fast food cup sizes vs. large fast food cup sizes, small fry vs. supersize fry, etc) and comparisons of calories. Other images could be pictures of meals simply scaled up or down using a basic image editing software. Posters could also include recipes like the ones mentioned in the Joy of Cooking study. Graphs and equations similar to the ones used in the activity can be used to illustrate the impact of larger portion sizes.
2. Have students make a brochure or video explaining to their peers how portion size can impact weight, using appropriate mathematics to make their point.

Extensions

1. The Increasing Portions, Expanding Wait Lines Activity Sheet states that a 19 year old women who is 5'7'' tall and weighs 165 pounds needs 2160 calories a day to maintain her current weight. Have students do research to find if this is a reasonable statement. Reading about the Harris Benedict Equation is a good place for students to start.
2. Have students keep a food journal and record their own or a friend or family member's eating over the course of a week. Use a calorie tracker program to calculate average calories per day and have students create models for their own weight over time. Have them reflect on their eating habits and exercise routines and think about changes they could make to lead a healthier lifestyle. You could also provide an eating/activity journal for a fictional person and have students analyze their calorie consumption and make recommendations.
3. Bring activity into the discussion and have students adjust their models to account for different activity levels or period of aerobic activity throughout the day.

Questions for Students

Opening Activity

1. If obesity rates rise from 34% of the population to 42% of the population (as implied in the LA Times article linked above), what percent increase is that?

2. The LA Times article also claims a non-obese person has a 2% chance of becoming obese each year, but that probability increases by .4% for each obese person you know. If you have 5, 10, and 15 obese friends or family members, what's your probability of becoming obese each year?

[4%, 6%, and 8%]

3. How many calories do you think a person eats in a year?

[Obviously answers may vary, someone on a 1500-to-2500 calorie/day diet would consume between 550,000 and 900,000 calories/year.]

4. Assume that the average person eats 2,000 calories per day for a year. Organic apples costs $2.47/lb and have 52 calories per 100 grams (there are about 450 grams in a pound). A McDouble costs$.99 and has 390 calories. If the average person were to consume all of your calories a year in apples vs. McDoubles, what would be the difference in cost?

[365×2000cal×(100gm/52cal)×(1lb/450gm)=3120 pounds of apples. 3120lbs×($2.49/1lb)=$7769.
365×2000cal×(1McD/390cal)=1872 McDoubles. 1872McD×($0.99/1McD)=$1854. The difference is  \$5915 more for apples.]

Main Activity

1. What should be the relationship between weight and calories consumed? Should weight increase as you consume more calories or decrease? Does your model reflect that?

[The more calories you eat the more you should weigh, the relationship should be a direct variation.]

2. Does your model seem reasonable, or does it suggest you would be gaining or losing way too much weight or far too quickly?

[If students wrote the correct model it should seem reasonable (within limits). One thing to point out to students is that this model projects linear growth forever. In reality, weight gain might level off after awhile or even increase more and more quickly.]

3. How much weight is she gaining each week, according to the two models?

[1.43 pounds and 2.59 pounds (the slopes).]

4. What are reasonable boundaries for this graph? How can you tell?

[Encourage students to think about how far into the future their model would work. This is a judgment call. Have them look for obviously unreasonable predictions about weight and then "back up" to a number they think is reasonable. This will differ some from group to group.]

5. Do you think the weight gain predicted in Question 4 is reasonable?

[It's probably high. At some point she would probably go on a diet before she hit 900 pounds. As she gains weight she will need more calories to maintain her weight, too, so it's probably predicting too much weight gain.]

Closing Activity

1. Why might people choose to eat food like McDoubles over food like apples?

[It's cheaper and it still fills you up (see opening activities).]

2. If you consumed your regular number of calories (whatever you needed to maintain your current weight) but then started stopping after school to get a McDouble for a snack every day, how much weight would you gain in one 180 day school year?

[Assuming "one school year" is 180 days, 390 calories per McDouble, that's about 70,200 extra calories, so around 20 pounds.]

3. You decide to start eating healthier, so you get the 6 piece chicken nuggets now instead of the McDouble, which have about 250 calories. How much weight will you gain if you eat those everyday after school?

[250 × 180 is about 45,000 extra calories, so around 13 pounds.]

4. The following year you quit eating fast food altogether and start taking your dog for a walk for 30 minutes a day every school day, which burns about 100 extra calories. How long will it take you to lose the weight you gained?

[About two years to burn off the 20 pounds from the McDoubles (you need to burn off 70,000 extra calories at 100 calories/day), and about 1 year and 3 months (1.25 years) to burn off the 13 pounds from the chicken nuggets.]

Teacher Reflection

• What strategies did you observe students using to come up with the various models? What were the common errors you observed? How did you help your students without "giving them the answer"?
• Were students able to adjust their models or did it seem like they were starting from scratch with each one? If they were not able to adjust their models, how would you change your instruction to help students see the similarities of the situations?
• How well were students assessing the reasonableness of their answers?
• Describe the quality of the student's graphs? Did they label/scale axes? Was an appropriate scale chosen?
• Student weight is a touchy subject for students. Describe how you met the needs of those students who may have been uncomfortable with this lesson due to their being overweight. Were you able to identify the parts of discussion where students were uncomfortable? What would you do differently in the future to make this lesson objective and factual to help students analyze the information?

### Learning Objectives

Students will:

• Write a linear model describing weight as a function of calories consumed and elapsed time.
• Use a linear model to make predictions about what will happen over time.
• Compare and contrast linear models.

### NCTM Standards and Expectations

• Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships.
• Develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases.
• Judge the reasonableness of numerical computations and their results.
• Understand relations and functions and select, convert flexibly among, and use various representations for them.
• Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships.
• Use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts.
• Draw reasonable conclusions about a situation being modeled.
• Approximate and interpret rates of change from graphical and numerical data.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.