Checkpoint - Course 1, Unit 6
Exponential Models

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.

Possible Responses to Unit Summary Checkpoint
The words in bold represent vocabulary and concepts that students should have mastered in this unit.
a. In deciding whether an exponential model will be useful, what hints do you get from
  • The patterns in data tables?
    The patterns in the data tables show that the differences in the values of the dependent variable are not increasing or decreasing at a constant rate when differences in the x values are constant. The y values should increase by larger and larger amounts (or decrease by smaller and smaller amounts). More specifically, if the differences in the x values are constant, then the ratio of any two successive y values should be constant. In a data table for a linear model, the y values increase (or decrease) by adding a constant increment, whereas for an exponential model the y values increase (or decrease) by multiplying by a constant factor.
  • The patterns in graphs and scatterplots?
    For exponential increase, the patterns in the graphs or scatterplots show curves approaching the x-axis as the x values decrease, and rising more and more rapidly as the x values increase. For exponential decrease, the curve approaches the x-axis as x increases. (In Course 3, students apply their knowledge of translations to examine graphs which look exponential but are asymptotic to horizontal lines other than the x-axis.)
  • The nature of the situation and the variables involved?
    In a setting where the value of one variable depends on a previous value of that same variable, the situation is likely to involve exponential growth or decay. For example, the growth of a population in one year depends on the size of the population the previous year. If this increase involves a constant factor, then an exponential model would be appropriate.
b. Exponential models, like linear models, can be expressed by an equation relating x and y values and by an equation relating NOW and NEXT y values.
  • Write a general rule for an exponential model, y =.
    y = a(bx).
  • Write a general equation relating NOW and NEXT for an exponential model.
    NEXT = NOW x b, starting at a. This format of the equation shows clearly that each y value is produced by multiplying the last y value by the factor b.
  • What do the parts of the equation tell you about the situation being modeled?
    The a is the starting value, that is, the value of y when x is zero. The b is the factor of increase or decrease in the situation. If 0 < b < 1, then the situation is an exponential decay situation. If b > 1, then the situation is an exponential growth situation.
c. How can the rule for an exponential model be used to predict the pattern in a table or graph of that model?
By looking at the rule, students can predict the start value, the value of y when x is zero. This will also be the y-intercept of the graph. The value of b tells whether the y values will be increasing (b > 1) or decreasing (0 < b < 1) in the table, and whether the curve will be rising (b > 1) or falling (0 < b < 1) to the right. The b value tells the factor of increase or decrease.
d. How are exponential models different from linear models?
The graph of a linear model is a line, while that of an exponential model is a curve. For constant changes in x values, the table of a linear model shows the y values increasing by constant additive amounts, while the table of an exponential model shows the y values increasing by constant multiplicative amounts. The rule for a linear model shows x being multiplied by the rate of increase, indicating how many times this constant increase should be added (y = a + bx). The rule for an exponential model shows x as the exponent of the factor of increase, indicating how many times this constant factor should be multiplied (y = a(bx)).
e. What real situations would you use to illustrate exponential change for someone who did not know what those patterns are like and used for?
Some investments increase by a factor that reflects a constant interest rate, compounded over some period of time. You start with a certain amount of money, and then multiply this by the interest rate. The next time you figure the interest, you multiply the new, larger amount by the same interest rate, and because the base amount has increased, the additional interest is now larger than at the first calculation. The interest amount increases. The amount increases at an increasing rate. Populations often increase (or decrease) by a factor that reflects birth and death rates. Radioactivity decreases by a constant factor, the rate of decay varying from one situation to another.

If you would like to see specific problems from Course 1, Unit 6, a link is provided to Examples of Tasks from Course 1, Unit 6. 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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