Content Area: Math

 

Index: 4.3B Grade 8 CPI 2

 

Standard: 4.3 - Patterns and Algebra

 

Strand: B - Functions and Relationships

 

Cumulative Progress Indicator: 2 -   The student will recognize and describe the difference between linear and exponential growth, using tables, graphs, and equations.

 

Grade: 8

 

Sample Activities:

 

·        Students develop a table showing the sales tax paid on different amounts of purchases, graph their results, note that the graph is a straight line, and recognize that this situation represents a constant rate of change, or linear growth.

 

·        Students make a table showing how much money they would have at the end of each of eight years if $100 was invested at the beginning and the investment grew by 10% each year. They note that the graph of their data is not a straight line; this graph represents exponential growth.

 

·        Students make a table showing the value of a car as it depreciates over time. They note that the graph of their data is not a straight line; this graph represents "exponential decay."

 

·        Students are presented stories which represent real life occurrences of linear and exponential growth and decay over time, and are asked to construct graphs which represent the situation and indicate whether the change is linear, exponential, or neither.

 

·        Students measure the height of water in a beaker at five second intervals as it is being filled, being careful to leave the faucet on so that the water runs at a constant rate. They make a table of their results and generate a graph. They note that this is a linear function.

 

·        Students investigate patterns of exponential growth with the calculator, such as compound interest or bacterial growth. They make a table showing how much money is in a savings account after one quarter, two quarters and so on for ten years, if $1000 is deposited at 5% interest and there are no further deposits or withdrawals. They represent their findings graphically, noting that this is not a linear relationship, although in the case of simple interest, where the interest does not earn interest, the graph is linear.

 

·        Students obtain a table showing the depreciated value of a car over time. They graph the data in the table and observe that it is not a straight line. The value of the car exhibits "exponential decay."

 

·        Students compare different pay scales, deciding which is a better deal. For example, is it better to be paid a salary of $250 per week or to be paid $6 per hour? They realize that the answer to this question depends on the number of hours worked, so they create a table comparing the pay for different numbers of hours worked. They make a graph and decide at what point the hourly rate becomes a better deal.

 

·        Students predict how many times they will be able to fold a piece of paper in half. Then they fold a paper in half repeatedly, recording the number of sections formed each time in a table. Students find that the number of folds physically possible is surprisingly small (about 7). The students try different kinds of paper: tissue paper, foil, etc. They describe in writing any patterns they discover and try to find a rule for the number of sections after 10, 20, or n folds. They also graph the data on a rectangular coordinate plane using integral values. They extend this problem to a new situation by finding the number of ancestors each person had perhaps ten generations ago and also to the situation of telling a secret to 2 people who each tell two people, etc.