Checkpoint - Course 1, Unit 2
Patterns of Change

In each unit, there is a final lesson and Checkpoint that helps students summarize the key ideas in the unit. The final Checkpoint will generally be discussed in class, with the teacher facilitating the summarizing, and students making notes in their Math Toolkits (teachers may just refer to this as "notes") of any points they need to remember, adding illustrative examples as needed. If your student is having difficulty with any investigation in this unit, this Checkpoint and the accompanying answers may help you recall the concepts involved, and give you the big picture of what the entire unit is about. If your student has completed the unit, then a version of this should be in his or her notes or toolkit. Students should also have Technology Tips in their toolkits, which are useful for this unit.

Possible Responses to Unit Summary Checkpoint
a. Make sketches of at least five different graphs showing different patterns relating change in two variables.
In this unit, students investigated linear, periodic, exponential, and quadratic patterns. These are not intended to be complete investigations at this stage, only to get some general graphical patterns and their accompanying symbolic forms available to students. The focus is on how the y-value changes in the table, at a constant rate or an increasing or decreasing rate, and how that appears in the graph. Students use NOW-NEXT equations which will probably be unfamiliar to you. This language is the precursor to work with sequences and series in Course 3, Unit 7, and focuses on how one y-value is related to the next, when x is an integer. Students also use "y = ..." equations, which you will probably find familiar. The situations investigated in the unit were:
  • Price charged and predicted number of customers - a linear relationship
  • Weight and stretch of rubber band - a linear relationship
  • Time into ride on a Ferris wheel and height of rider - a periodic relationship
  • Time and population - an exponential relationship
  • Time worked and money earned - a linear relationship
  • Number of payments and unpaid balance - a linear relationship
  • Number of tickets sold and dollars of income - a linear relationship
  • Tickets sold and profit in dollars - a linear relationship
  • Time and distance traveled - a linear relationship if speed is constant
  • Time and height of a ball - a quadratic relationship
  • Price and profit - a quadratic relationship
  • Time and fuel remaining - a linear relationship
See sample graphs, possible equations, and descriptions below
b. For each graph write a brief explanation of the pattern of change shown in the graph and describe a real life situation that fits the pattern.
See sample graphs, possible equations, and descriptions below.
1. The graph is likely to look somewhat linear, with price on the horizontal axis and number of customers on the vertical axis. As the price rises, the number of customers falls.
2. The graph is likely to look somewhat linear, with stretches reported for each weight. The table will show approximately a constant rate of increase. The axes of the graph should be labeled with units of weight on the horizontal axis and units of length on the vertical axis.
3. This graph shows how the rider rises and falls as the Ferris wheel turns. The rider's height cycles through the same up and down pattern as the wheel continues to turn.
4. Because populations tend to grow at a percentage rate, meaning that the actual increase depends also on the number of people present at any time, the growth is likely to be exponential growth, slow to begin with then faster and faster as time passes.
5. This graph shows a constant rate of change. For each additional hour worked, the money earned increases by the same amount.
6. Assuming the payments are all equal, the balance will fall at a constant rate.
7. Since each ticket sells for the same price, the income will rise at a constant rate.
8. The profit may start as a negative until enough tickets have been sold. The profit increases at a constant rate.
9. Since speed is constant, the distance traveled increases at a constant rate.
10. The height of the ball increases and then decreases. It starts out moving fast, slows down as it reaches its maximum height, and then speeds up again as it falls back down.
11. As the ticket price goes up the profit goes up, but when the ticket price becomes expensive the number of tickets falls so the rate of increase of the profit becomes less. Eventually the ticket price becomes so high that the number of tickets sold drops off enough to make the profit fall. Profit depends on ticket price, but also on number of tickets sold.
12. Assuming the speed stays constant, the gasoline consumption will remain constant and the graph will decrease linearly.

If you would like to see specific problems from Course 1, Unit 2, a link is provided to Examples of Tasks from Course 1, Unit 2. If you are interested in following up on the Algebra strand in general, then the Scope and Sequence will help you see where different concepts are introduced. On the Algebra page, you can read an explanation of the main algebra concepts as they are developed in all four courses.

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