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.
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
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.
Rate of Change Activity Sheet
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
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
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
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.
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.]
- 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?