Number of Cups 1 2 3 4 5 6 7 8
Height of Stack (cm)
1. Graph the data with the inputs (number of cups) on the horizontal axis and the outputs (height of stack) on the vertical axis.
2. Use the information you have gathered to estimate the height (to the nearest tenth of a centimeter) of 10 stacked cups.
3. Now construct a stack of 10 cups. Measure the stack and compare it with your estimate.
Was your estimate reasonable? If not, why not?
4. Estimate the height of 16 cups. Is a stack of 16 cups twice as tall as a stack of 8 cups?
Justify your answer.
5. Estimate the height of 20 cups. Is a stack of 20 cups twice as tall as a stack of 10 cups?
Justify your answer.
6. Describe the formula or method you used to estimate the height of a stack of cups.
7. If your shelf has clearance of 40 centimeters, how many cups can you stack to fit on the shelf? Explain how you obtained your answer.
1. Describe several ways that relationships between variables may be represented. What are some advantages and disadvantages of each?
2. Obtain a graph from a local newspaper, magazine, or textbook in your major field. Identify the input and output variables. The input variable is referenced on which axis? Describe any trends in the graph.
3. You will be graphing some data from a table in which the input values range from 0 to 150 and the output values range from 0 to 2000. Assume that your grid is a square with 16 tick marks across and up.
a. Will you use all four quadrants? Explain.
b. How many units does the distance between tick marks on the horizontal axis represent?
c. How many units does the distance between tick marks on the vertical axis represent?
Cluster 1 What Have I Learned?
Exercise numbers appearing in color are answered in the Selected Answers appendix.
1. You are a scuba diver and plan a dive in the St. Lawrence River. The water depth in the diving area does not exceed 150 feet. Let x represent your depth in feet below the surface of the water.
a. What are the possible replacement values to represent your depth from the surface?
b. Would a replacement value of zero feet be reasonable? Explain.
c. Would a replacement value of be reasonable? Explain.
d. Would a replacement value of 12 feet be reasonable? Explain.
2. You bought stock in a company in 2007 and have tracked the company’s profits and losses from its beginning in 2001 to the present. You decide to graph the information where the number of years since 2007 is the input variable and profit or loss for the year is the output variable. Note that the year 2007 corresponds to zero on the horizontal axis. Determine the quadrant in which, or axis on which, you would plot the points that correspond to the following data. If your answer is on an axis, indicate between which quadrants the point is located.
a. The loss in 2004 was $1500.
b. The profit in 2008 was $6000.
c. The loss in 2010 was $1000.
d. In 2003, there was no profit or loss.
e. The profit in 2001 was $500.
f. The loss in 2007 was $800.
3. Fish need oxygen to live, just as you do. The amount of dissolved oxygen (D.O.) in water is measured in parts per million (ppm). Trout need a minimum of 6 ppm to live.
The data in the table shows the relationship between the temperature of the water and the amount of dissolved oxygen present.
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Cluster 1 How Can I Practice?
Temp (°C) 11 16 21 26 31
D.O. (in ppm) 10.2 8.6 7.7 7.0 6.4
Exercise numbers appearing in color are answered in the Selected Answers appendix.
0 0 2 4 6 8 7 9 10
5
3
5 10 15 20 25 30 35 40 Time (years)
Amount (thousands of dollars)
45 50 1
b. What general trend do you notice in the data?
c. In which of the 5° temperature intervals given in the table does the dissolved oxygen content change the most?
d. Which representation (table or graph) presents the information and trends more clearly?
4. When you were born, your uncle invested $1000 for you in a local bank. The following graph shows how your investment grows.
a. Represent the data in the table graphically. Place temperature (input) along the horizontal axis and dissolved oxygen (output) along the vertical axis.
a. Which variable is the input variable?
b. How much money did you have when you were 10 years old?
c. Estimate in what year your original investment will have doubled.
d. If your college bill is estimated to be $3000 in the first year of college, will you have enough to pay the bill with these funds? (Assume that you attend when you are 18.) Explain.
e. Assume that you expect to be married when you are 30 years old. You figure that you will need about $5000 for your share of the wedding and honeymoon expenses.
Assume also that you left the money in the bank and did not use it for your education. Will you have enough money to pay your share of the wedding and honeymoon expenses? Explain.
5. A square has perimeter 80 inches.
a. What is the length of each side?
b. If the length of each side is increased by 5 inches, what will the new perimeter equal?
c. If each side of the original square is increased by x inches, represent the new perimeter in terms of x.
6. Let x represent the input variable. Translate each of the following phrases into an algebraic expression.
a. 20 less than twice the input b. the sum of half the input and 6
c. 2k + 3h, for k = - 2.5, h = 9
7. Evaluate each algebraic expression for the given value(s).
a. 2x + 7, for x = - 3.5 b. 8x - 3, for x = 1.5
Cluster 2 Solving Equations
The sales tax collected on taxable items in Allegany County in western New York is 8.5%.
The tax you must pay depends on the price of the item you are purchasing.
1. What is the sales tax on a dress shirt that costs $20?
2. a. Because you are interested in determining the sales tax given the price of an item, which variable is the input?
b. What are its units of measurement?
3. a. Which variable is the output?
b. What are its units of measurement?
4. Determine the sales tax you must pay on the following items.
The rule you followed to calculate sales tax (the output) for a given price (the input) can be described either verbally (in words) or symbolically (in an algebraic equation).