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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?

5 Teacher s Guide for Rate of Change of Population Lesson Outline: Students work in groups to discuss and complete the worksheets, and the whole class discusses answers to most of the worksheet questions. Preparation: Copy a set of worksheets for each group. Alternately, you may want to make a copy of the worksheets for each individual, so they will have it as notes to take home. Make an overhead projector copy of the worksheets to display during the class discussion. Also make an overhead projector copy of the page: Rate of Change Graphs (which follows the Teacher s Guide in this document) Decide how many students you will allow per group. (In some large groups, students are productive and communicate well, and in others a few people do all of the work and others are observers. You may want to do something to ensure that everyone participates such as having each student in the group initial next to the problems that they wrote the answers for.) Read the teachers guide, and decide at what points you plan to stop the class to discuss their work so far. This guide assumes that you will stop to discuss at the end of each page. Make copies of whatever additional problems you choose to assign as homework. Our experience is that we tend to think that we can do more in a 55-minute class period than we actually can. Most of the Rate of Change Assignments are things that one or another of us intended to do in class in addition to the population activity, but we did not have time. If you expect you will have extra time, please choose an activity that you can use flexibly, either in class if time permits, or as homework. Student sheet 1 contains US census data, along with prompts to compute the change in population, change in time and rate of change of population. You should prepare a data sheet for each group to complete, and overhead copies of the sheets to write on during discussions Direct the students attention to the table on sheet one, and discuss with the class how to complete the table for a pair of values somewhere in the middle before giving the groups the task of discussing and completing sheet 1 (1880 to 1900 is a choice that has friendly mentalarithmetic numbers). A typical exchange might be as follows: Teacher: Please look just at the years 1880 and Can anyone tell me the change in population between 1880 and 1900? Student 1: 6 Teacher: Good, how did you get that? Student 1: I subtracted Teacher records p p p 1 on the overhead copy of sheet 1, or on the board. Teacher: Good. Can anyone tell me what the change in time is between 1880 and 1900? Student : It s , which is 0 years Teacher records t t t 1 on the overhead copy of sheet 1, or on the board. Teacher: Good. Now, can anyone tell me what the rate of change of population would be? What is it; how is it different from the change in population? Student 3: It s change over time. It s change in population per year. Teacher: Good, and how would you compute that? Student 4: You divide the change in population by the time Teacher: [repeats or paraphrases] You divide the change in population by the change in time. Teacher page 1

6 Teacher s Guide for Rate of Change of Population Lesson Teacher records p/t on the overhead copy of sheet 1, or on the board Teacher: Does everyone think they know what they are doing?... OK, please work together on this sheet, and please read the instructions, because I have some thinking questions for you to discuss as you go along. We ll stop and discuss page 1 in about 10 minutes. While students are working on sheet 1, the teacher should circulate and address individual problems that come up. Expect it to take about 10 minutes for students to complete sheet 1, perhaps a little longer. Some errors to watch for are: Some groups may calculate the rate of change of population per year, but report the rate of change as a percent rather than as a number. Remind them that percent implies that it is a percent of something, like percent of the current population, and point out that their rate of change should say that the population increases by a number of millions of people per year, and not a percent of the population, because they divided number of people by number of years, not by the previous population. Encourage students to make the connection between number of people per year and the calculation of dividing a number of people by a number of years. Some groups may report a change and rate of change at 180 as 0. They should be reminded that they don t know the population at 1800, and so they can t figure a change between 1800 and 180. For example, you could prompt: This says 0 is a change in population. What two population numbers is it a change between? Some groups may compute the difference between the population at 180 and whatever year they are working with (and/or the time between 180 and whatever year they are working with. This happens less often when you choose a pair of years from the middle as your example, than when you choose the first pair of years as your example. This error comes from not understanding the goal, which is to know what is happening over the shortest time frame available. You might restate the goal as: what I really want to know here is our best calculation for how fast the population is growing in 1850, so I want to know what s happening over the short term and 1860 are the closest numbers I have to tell me about short term population growth, so I want you to use those two instead of 180 and Some student work to look for: What are different ways that groups are recording their calculations in the table? Which groups are writing the differences in line with the second data value (like a spreadsheet)? Which groups are writing the differences between the two values? Which groups used columns one and two to get column 3? Did any groups do the subtractions from scratch to figure out the rate of change? Discussion of page 1: When most groups are done with page 1, conduct a follow-up discussion: If some groups did their calculations by subtracting from scratch, invite someone from a group that used the change in population and change in time columns to get rate of change to show how they did it, and then show how that is the same as the longer calculation, but doing it the short way shows the connection between the values in the different columns. Show the different ways of recording the calculations used by people in the class, and ask two groups with different recording strategies why they chose the way of recording that Teacher page

7 Teacher s Guide for Rate of Change of Population Lesson they did. Tell the class that they will be graphing those rates of change on the next page, and they should think about those same issues as they graph. Do they line the rate of change up with the year previous? The year following? Halfway in between? Preview question 5 by pointing out that they will also be putting the units on the graphs. Remind students that units in a contextual problem are things like hours or miles or years. Student sheet involves graphing the rate of change of population data, and labeling the graphs with their correct units of measurement. Allow students time to complete page. You may want to let the first group that is done to plot their graph on your overhead of page to share with the class. Expect page to take about 5 minutes to complete Errors to look for: Adding an extra dot at the point that looks like the origin, or otherwise adding an extra dot to the beginning or end and connecting it to the rest of the graph. The graphs should either not include 180 or not include 1970 or both, depending on students choice of how to line up the graph. If students have an extra dot, you can help them track it down by asking what data in their table it corresponds to (and if they have an extra line of data in the table, ask what pair of years it is finding the rate of change between). Abbreviating the units so that they are less clear (for example, the population axis might be labeled millions or people rather than millions of people.) Prompt students to be more specific: Millions of what? So there were 9.6 people? Etc). Student work to look for: You may want to have the group that finishes first copy their graph on the overhead page for the class discussion. Discussion of page : Show a graph of Rate of Change of population. This could be either one drawn by students, or the Rate of Change Graphs page. What choice did this group make about what year to line up with? (First year/last year/middle year.) Is this a correct graph? (Do the points match up with the correct table of rate of change values?) Show the Rate of Change Graphs page and ask how they are different. Choose a sample decade such as and show where that rate of change shows up on each graph. On the first graph it looks like that rate of change was most typical of 1930, on the second graph it looks like that rate of change was most typical of 1940, and on the third graph it looks like that rate of change was most typical of Emphasize to students that they need to understand what the points on their graph correspond to in order to answer questions about those rates of change correctly. Show the overhead copy of page or the Sample Student Page (which follows the Rate of Change Graphs page) and ask a group what units of measurement to put on each axis. Ask the class: Do you agree with this group s units of measurement for the graph? Try to elicit specific units units of measurement: millions of people (not just people, and not just millions); years (or year) and millions of people per year. Discuss how the units: millions of people per year on the rate of change graph corresponds to the units millions of people and years on the population graph. Discuss how millions of people per year can be written as millions of people/year and that this corresponds to the formula p/t. Teacher page 3

8 Teacher s Guide for Rate of Change of Population Lesson Student page 3 asks students to make connections between the calculations and the graphs, and to interpret rates of change in terms of historical events. Have students discuss the questions on page 3 in their groups. Allow at least 10 minutes for whole class discussion, and try to give all groups time to answer at least problems 6-9. Errors to watch for: The most common errors on this page are ones of omission (you may even want to discuss your expectations with the class as a whole), so on problems 6-8, when you see the answer a, that group should be challenged to say what it is about a that makes it the answer (is it the length? The direction? How do you know a is the answer?) Student work to watch for: Watch for which group is discussing which decade so you can call on appropriate groups to give their explanations. Discussion of page 3: Ask a group for their answer to #6. Students may answer just b and c. If so, follow up with a question of why those are the right answers, which is likely to get an intelligent response about the vertical axis measuring population. If students do not specify, explain that it is the length of lines b and c, which represents the change in population. [Note: the length of segment b and the length of segment c both show the change in population.] Ask a group for their answer to #7. Prompt students to add to the initial response with questions like what is about line a that shows the answer? and how do you know that line a shows a change in time? to get a complete response such as: the length of line a shows change in time because it is measuring horizontally along the time axis. Ask a group for their answer to #8. Prompt students to add to the initial response with questions like what is about line d that shows the answer? and how do you know that line d shows a rate of change? to get a complete response such as: the slope of line d shows rate of change over time because rise/run is the same as population/time. Ask a group to explain their answer to #9. Students generally have the right idea (that the rate of change is different because of the difference in change in time), but they need to be prompted to make connections: how is the rate of change different if the time interval is smaller? Encourage students to share what they know about what was going on during various decades, and how that affected the rate of change. Point out that during those time periods, the rate of change was always positive, so the population always increased, and indicate how that behavior is reflected on the graphs on page. Ask students to explain their answer to # 11 and 1. Point out that the population would still be positive even though the rate of change was negative. The lesson to this point will take most of a 55 minute class period. If there is sufficient time, you may want to have students discuss some extra questions such as: Predict, using the graphs, what the population might have been in 1800 (a straight line approximation here would yield a population of just above 0 most students will recognize this as being too small), and what the population might have been in 1980 (a straight line Teacher page 4

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

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