Nonlinear Models in a Data Context

Examples, solutions, and videos to help Grade 8 students learn how to give verbal descriptions of how y changes as x changes given the graph of a nonlinear function.
Students draw nonlinear functions that are consistent with a verbal description of a nonlinear relationship.

New York State Common Core Math Grade 8, Module 6, Lesson 12

Worksheets for Grade 8

Lesson 12 Summary

When data follow a linear pattern, the rate of change is a constant. When data follow a non-linear pattern, the rate of change is not constant.

Lesson 12 Classwork
Example 1: Growing Dahlias
A group of students wanted to determine whether or not compost is beneficial in plant growth. The students used the dahlia flower to study the effect of composting. They planted eight dahlias in a bed with no compost and another eight plants in a bed with compost. They measured the height of each plant over a 9-week period. They found the median growth height for each group of eight plants. The table below shows the results of the experiment for the dahlias grown in non-compost beds.

Exercises 1–7
1. On the grid below, construct a scatter plot of non-compost height versus week.
2. Draw a line that you think fits the data reasonably well.
3. Find the rate of change of your line. Interpret the rate of change in terms of growth (in height) over time.
4. Describe the growth (change in height) from week to week by subtracting the previous week’s height from the
current height. Record the growth in the third column in the table below. The median growth for the dahlias from Week 1 to Week 2 was 3.75 inches (i.e., 12.75 - 9 = 3.75).
5. As the number of weeks increases, describe how the weekly growth is changing.
6. How does the growth each week compare to the slope of the line that you drew?
7. Estimate the median height of the dahlias at 8 1/2 weeks. Explain how you made your estimate.

Exercises 8–14
The table below shows the results of the experiment for the dahlias grown in compost beds.
8. Construct a scatter plot of height versus week on the grid below.
9. Do the data appear to form a linear pattern?
10. Describe the growth from week to week by subtracting the height from the previous week from the current height.
Record the growth in the third column in the table below. The median growth for the dahlias from Week 1 to Week 2 is 3.5 in. (i.e.,13.5 - 10 = 3.5).
11. As the number of weeks increases, describe how the growth is changing.
12. Sketch a curve through the data. When sketching a curve do not connect the ordered pairs, but draw a smooth curve that you think reasonably describes the data.
13. Use the curve to estimate the median height of the dahlias at 8 1/2 weeks. Explain how you made your estimate.
14. How does the growth of the dahlias in the compost beds compare to the growth of the dahlias in the non-compost beds?

Exercise 15
15. When there is a car accident how do the investigators determine the speed of the cars involved? One way is to measure the skid marks left by the car and use this length to estimate the speed.
The table below shows data collected from an experiment with a test car. The first column is the length of the skid mark (in feet) and the second column is the speed of the car (in miles per hour).
a. Construct a scatter plot of speed versus skid-mark length on the grid below.
b. The relationship between speed and skid-mark length can be described by a curve. Sketch a curve through the data that best represents the relationship between skid-mark length and speed of the car. Remember to draw a smooth curve that does not just connect the ordered pairs.
c. If the car left a skid mark of ft., what is an estimate for the speed of the car? Explain how you determined the estimate.
d. A car left a skid mark of ft. Use the curve you sketched to estimate the speed at which the car was traveling.
e. If a car leaves a skid mark that is twice as long as another skid mark, was the car going twice as fast? Explain.

Lesson 12 Exit Ticket

The table shows the population of New York City from 1850–2000 for every 50 years.

1. Find the growth of the population from 1850–1900. Write your answer in the table in the row for the year 1900.

2. Find the growth of the population from 1900–1950. Write your answer in the table in the row for the year 1950.

3. Find the growth of the population from 1950–2000. Write your answer in the table in the row for the year 2000.

4. Does it appear that a linear model is a good fit for this data? Why or why not?

5. Describe how the population changes as the number of years increases.

6. Construct a scatter plot of time versus population on the grid below. Draw a line or curve that you feel reasonably describes the data.

7. Estimate the population of New York City in 1975. Explain how you found your estimate.

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