## Growth Rate

- Lesson

Given growth charts for the heights of girls and boys, students will use slope to approximate rates of change in the height of boys and girls at different ages. Students will use these approximations to plot graphs of the rate of change of height vs. age for boys and girls.

Students learn the formula for the slope of a line when they study algebra. However, most students have only used this formula with linear graphs. During this lesson, students use the formula for slope of a line to approximate the rate of change at a single point on a non-linear graph. To accomplish this, students will use the slope formula to calculate the rate of change between two points very near to the desired point.

The graphs that students will be using are the Height vs. Age charts that were developed by the National Center for Health Statistics (NCHS), part of the Centers for Disease Control and Prevention (CDC). The graph for girls age 3 to 20 is shown below.

There are two things to note about this chart. First, the chart uses the term *stature*
to refer to height. You may want to point this out to students as they
begin the lesson. Second, the chart also includes graphs for Weight vs.
Age. This portion of the chart will not be used for this lesson. Only
the curves in the top portion will be used.

Here is an example of how students will use these charts to approximate the rate of change in height (i.e., growth rate). To approximate the growth rate for a 3‑year‑old girl at the 50th percentile, use the height vs. age graph to determine the height for a girl at the 50th percentile at 2.5 and 3.5 years of age. The heights for these ages are approximately 35.5 and 38.5 inches, respectively. The slope of the line that passes through these points is: .

Keep in mind that this is not the only approximation that can be used for the growth rate of a 3‑year‑old girl at the 50th percentile. A student could instead use the data for 2 and 4 years of age, which leads to: .

Notice that the results are slightly different. It is important to note that there can be small differences in student’s answers depending on the approximations that they use as well as how they read the graph. To get the best approximations, students should use values that are close to the desired age. To assure reasonable results, you may want to encourage students to choose ages that are within ½ year of the desired age and to estimate heights to the nearest tenth of an inch.

To introduce the lesson, have students read the introductory information about Height vs. Age charts included on the Rate of Change Activity Sheet. Provide students time to examine the growth charts, which appear on the last two pages of the activity sheet, and ask them why such charts are important.

Students will notice many graphs for both girls and boys. Reinforce that each graph represents a different percentile. Explain to students that for the purposes of this lesson, they will use only the 50th percentile, which has a distinctive bold line.

To ensure that students understand how the charts work, conduct a brief discussion about them. Some questions you could ask are:

- Does a boy grow as fast at age 11 as they do at age 2? [No, he will grow faster at age 2.]
- How can you approximate how much a girl grows from age 6 to age 7? [Determine height each year, and subtract.]

Before students begin calculations, explain that they will be using the formula for slope in a different way than they have before. They may be uncertain as to how to use the slope formula with a non‑linear graph whose points are not clearly marked. Explain that they will be using slope to approximate rates of change at particular ages on the graphs. Discuss that they should find the rate of change between two ages that are close to the desired age.

You may want to show students an example of calculating the growth rate for one point on the chart, so that students understand the process. Inform students that they can use the desired age in the slope formula. Reinforce to students the importance of using two ages that are no more than one year apart. Encourage students to use half-years in their calculations, and to estimate heights to the nearest tenth of an inch.

During the activity, some students may attempt to find the growth rate by simply dividing the height by the age. For instance, a girl at the 50th percentile is 37 inches tall at age 3, so a student might argue that her growth rate is 37 ÷ 3 ≈ 12.33 inches per year. (Students may be convinced that this method is correct because the units are correct.) If students use this method, discuss how it might find an approximation, but probably not a good one. In particular, use this as an example of an incorrect method that coincidentally gives correct results, but getting a correct answer does not validate the method.

Begin the activity using the girls chart (Questions 1‑4). Allow students to work individually to complete the tables in Question 1, then have them compare their calculations with a partner. To review, have each pair record their calculations on the board. This allows the entire class to examine the similarities and differences in calculations.

Continue by allowing students to work on Question 5, which requires them to create a similar table using the boys chart. Again, allow students to work individually to complete this table, then compare their results with a partner. Working together, students can complete Questions 2, 3, 6, and 7. After partners have compared their responses, discuss the results as a class.

Students should recognize that for girls, the time of fastest growth occurs around age 3. For boys, there seem to be two times of fastest growth, around age 3 and again around age 13. For both girls and boys, as they approach age 20, their growth rates approach zero, because they stop growing.

After students discuss the written responses, have them plot the results of their tables (Questions 4 and 8). Students can plot these results on the graph provided, or they can use technology to create the graphs. Students can also enter the data into lists on a graphing calculator and create a scatter plot of the data. Have students write about what is happening to growth rates as girls and boys move from age 3 to age 17. For both girls and boys, they begin growing quickly, and then their growth rates decrease until puberty, when their growth rates spike again. After puberty, the growth rates drop off quickly until they reach zero.

The results included in the lesson represent sample student results for both girls and boys.

- Rate of Change Activity Sheet
- Scientific or Graphing Calculator
- Sample Solutions

**Assessment Options**

- Have students complete the activity for girls or boys in a different percentile.
- Have students complete the activity for girls in the 10th, 25th, 75th, and 95th percentiles. Instruct students to compare and contrast the growth rates for children in the different percentiles. For example, does a girl in the 95th percentile grow in a different way than a girl in the 10th percentile, or does a girl in the 95th percentile just grow at a faster rate at each age?

**Extensions**

- When students plot approximate growth rates for girls and boys, they are sketching approximate graphs of the derivates of the height graphs for girls and boys. Emphasize the connections between the slopes of the height graphs and the values of the growth rate graphs.
- Using a spreadsheet program or a graphing calculator,
have students place the graphs for both boys and girls on the same set
of axes. Have students compare the graphs and explain what the
differences mean in terms of growth patterns.

**Questions for Students**

1. Why is it important to use a small interval when using the slope formula to approximate girls’ and boys’ growth rates at different ages?

[A large interval may distort the slope.]

2. How does a girl’s growth rate change as she moves from ages 3 to 17?

[The growth rate decreases from age 3 until about age 9, when she hits a growth spurt. The growth rate then increases until about age 11. The growth rate decreases through the teen‑age years, until it eventually slows to zero around age 20.]

3. How does a boy’s growth rate change as he moves from ages 3 to 17?

[The growth rate decreases from age 3 until about age 9, when he hits a growth spurt. The growth rate then increases until about age 13, and the increase during this period is much greater than the growth spike for girls around the same time. The growth rate decreases through the teen‑age years, until it eventually slows to zero around age 20.]

4. What is similar about the growth rates of boys and girls? What is different?

[Both grow quickly when they are very young, slow down until puberty, hit a growth spurt during puberty, then slow down steadily until they stop growing around age 20.]

5. Why do the growth rates of boys and girls approach zero as their ages get closer to 20?

[Because they stop growing.]

6. How could you use the growth rate graphs to predict the growth rate of a girl or boy at ages 4, 6, 8, 12, 14, or 16?

[Find the age along the horizontal axis, and find the corresponding point on the graph. The y‑coordinate of that point is the growth rate at the given age.]

**Teacher Reflection**

- How did the lesson extend students’ understandings of rate of change?
- Did you set clear expectations so that students knew what was expected of them? If not, how can you make them clearer?
- What were some of the ways that the students illustrated that they were actively engaged in the learning process?
- How did you integrate and use technology effectively in instruction and assessment?

### Learning Objectives

Students will:

- Use slope to approximate the rate of change in height for boys and girls at different ages.
- Use approximations to plot graphs of the growth rate vs. age for boys and girls.

### NCTM Standards and Expectations

- Judge the reasonableness of numerical computations and their results.

- Approximate and interpret rates of change from graphical and numerical data.

- Analyze precision, accuracy, and approximate error in measurement situations.

- Apply informal concepts of successive approximation, upper and lower bounds, and limit in measurement situations.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP5

Use appropriate tools strategically.

- CCSS.Math.Practice.MP7

Look for and make use of structure.