# Rate_assignments.doc: Extensions and homework assignments designed to accompany this lesson.

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1 Rate of Change of Population Lesson Materials This file contains the Rate of Change of Population lesson as reported on in the lesson study. It incorporates the most successful aspects of the lesson as it was taught by members of this lesson study group. This file includes: This introductory page Student pages (3 pgs) Teacher s guide/notes (5 pgs) Rate of Change Graphs (1 pg) Sample Student Page (with completed graph) (1 pg) Other files associated with this lesson are: Rate_assignments.doc: Extensions and homework assignments designed to accompany this lesson. Rate_revised.doc: This lesson with additional student problems suggested by our experience during the last time we taught this lesson. Rate _samples.doc: Sample student work showing typical student work and errors.

2 Rate of Change of Population Worksheet page1 US Population Year Population (in millions) Change in population Change in time Rate of change in population Your first task is to figure out the change in population between each pair of successive years given in the table. The change between two years is associated with both the year before and the year after. Discuss where you should write the change of population in your table, and be prepared to explain your choice. Next, record the length of time between each pair of successive years given in the table. 3. a. Compute the rate of change of population for the same intervals, and record it in your table. b. Show your work for computing the rate of change of population between 1840 and c. Did you use the numbers you found in 1 and? Why or why not?

3 Rate of Change of Population Worksheet page Population Rate of Change of Population Decide with your group which year you will line up your rate of change numbers when completing the second graph, and then complete the graph. Be prepared to explain your choice. 5. Write in the appropriate units of measurement on the horizontal and vertical axis of both graphs.

4 Rate of Change of Population Worksheet page3 Use this close-up picture of the population graph between 1840 and 1860, with some line segments labeled. You may refer to this in answering questions 7-9. The x and y coordinates of the endpoints show the year and the population in that year. 30 c d b 0 a Instructions for 6-8: discuss and be prepared to explain why your answer makes sense. 6. What on the graph shows the change in population between the two years? 7. What on the graph shows the change in time between the two years? 8. What on the graph shows the rate of change between the two years? 9. The change in population between 1880 and 1900 is close to the change in population between 1960 and 1970, but the rate of change is much less. Why? 10. Choose an interesting rate of change (one that is smaller than its neighbors, or larger than its neighbors), and try to explain historically why the rate of change might have been particularly high or particularly low (e.g , , , ) 11. Could the rate of change of population over time ever be negative? What would it mean about the population if the change was negative? 1. What would it mean about the population if the rate of change was 0?

9 Teacher s Guide for Rate of Change of Population Lesson approximation works pretty well here). Actual census data is: 5.3 million in 1800, 6.5 million in 1980, 48.7 million in 1990 and 81.4 million in 000. Leave enough time to summarize the important concepts of the lesson, and to introduce whichever homework assignments you choose to use. Some topics to include in the summary are: Connections between the algebraic and numeric calculations of rate of change: p t p p 1 and the geometric rate of change: slope of the line. t t 1 Connections between the algebraic and numeric calculations of rate of change: p t p p 1 millions of people and the units of measurement of rate of change:. t t 1 year Connections between the graph of the population and the graph of the rate of change of population: that differences between rates of change of population correspond to steepness of the population graph. Connections between the units on the graph of the population and the graph of the rate of change of population. The rate of change of population between two years is our best approximation for the rate of change at any year between those two years (which corresponds to the slope of the line between those points). If we had a function for the population instead of a table, we could use calculus to find the rate of change at a specific time (which would correspond to the slope of a tangent line). Teacher page 5

10 Rate of Change Graphs

11 Sample Student Page Population Rate of Change of Population Decide with your group which year you will line up your rate of change numbers when completing the second graph, and then complete the graph. Be prepared to explain your choice. 5. Write in the appropriate units of measurement on the horizontal and vertical axis of both graphs.

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