© 2005 National Council of Teachers of Mathematics http://illuminations.nctm.org

Carousel Game The deck of cards used in the Carousel Game consists of 24 cards, twelve each of two types:

• Graphs of Functions: As the name implies, these cards have a graph of a rational, linear, quadratic, cubic, exponential, square root, or absolute value function.

• Descriptions of Problem Situations: The text on these cards describes a situation that can be represented by a rational, quadratic, linear, greatest integer or exponential.

If you only want to assess student understanding of rational functions, use only cards 1, 2, 5, 6, 7, 8, 14, 15, 18, and 23. The other cards can be removed from the deck. Materials:

• Deck of Carousel Cards • Transparencies with Carousel Cards • Answer Sheets (12 sheets of blank paper per team) • Markers • Timer • Overhead Projector

Object:

• To correctly determine the equation that corresponds to the problem situation or graph on the carousel card.

Set-Up:

• Prior to class, print the Carousel cards onto heavy cardstock, and cut the sheets as indicated. To preserve the cards for future use, have them laminated. Also, prepare a set of Carousel cards on overhead transparency sheets, for use at the end of the game.

Playing the Game:

• Divide the class into teams of four students each. • Give each team a marker, a pair of scissors, and twelve sheets of blank paper. • Each team should cut each blank sheet in half to form two 5½” × 8” sheets. The half-sheets

should then be numbered 1-24, placing a numeral in the upper left corner of each sheet. • Shuffle the Carousel cards, and deal two to four cards to each team. (Give each team the

same number of cards.) • If two cards were given to each team, allow teams two minutes to come up with equations

corresponding to the graph or problem description on each card. (If four cards were given to each team, allow four minutes.)

• Each Carousel card has a number written at the top. On the half-sheet answer paper with the proper number, students should write an equation for the graph or problem description. Students should write these equations very large and legibly, as they will hold these sheets up later so that you can assess their work.

• At the appropriate moment, call, “Time!” When time is called, students must stop working, collect the cards, and pass them to the team to their right.

© 2005 National Council of Teachers of Mathematics http://illuminations.nctm.org

• Continue this process until each team has seen all twenty-four cards. (This part of the game will take about 25-30 minutes to complete.)

• After all teams have seen all of the cards, place the Carousel cards on the overhead projector, one at a time. As each card is placed on the overhead, one member of each team should hold up their team’s answer sheet for that card number. Keep a record of which teams had the correct equation for each card. Teams earn two points for each correct answer. To keep interest for the entire game, do not reveal which teams are in the lead until the very end.

• As necessary, discuss the answers for each card, especially those cards for which many teams had difficulty.

• The winner is the team with the most correct answers. Answers to the Carousel Cards

1. 1 1

2y

x= −

+ 9 21500 20= −y x x 17.

1 ( 8)( 1)( 3)25

y x x x= + + −

2. 12

yx−

=+

10. ( )100 0.82= xy 18.4yx

=

3. 2( 2) 4y x= − − 11. 236= −y x x 19.4 4 53

y x= − −

4. 2( 1) 4y x= − + + 12. 765 17= −y x 20.3

8xy =

5. 3200 45

100+

=+

xyx

13. 22 4−= −xy 21.51 6

2

+⎛ ⎞= −⎜ ⎟⎝ ⎠

x

y

6. 1550

+=

+xyx

14.2 2yx

= + 22. 6 3y x= + −

7. 2400 20

50+

=−

xyx

15.3 32

yx−

= +−

23.1,000,000

=yx

8. 1000 20

10+

=−

xyx

16.4 15

y x= − − 24. 5= + ⎡ ⎤⎢ ⎥y x

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A beaker contains 50 ounces of a 30% acid solution, and x ounces

of pure acid are added.

Write an equation that expressesthe concentration of acid in the

combined solution (y) as afunction of the amount of pure

acid that was added (x).

Sometimes, I lie too long in my bath, and the water coolsdown. When this happens, I add more hot water.

Suppose my bath contained 100 quarts of water at 32ºC, and I added x quarts of water at 45ºC.

Write an equation that expresses the temperature (y)as a function of the amount of hot water added (x).

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The fixed costs of a summer camp — which includes rent,staff salaries, staff meals, and all equipment — are $2400. There is an additional cost of $20 per camper for food. Because some children might not be able to afford thecamp, 50 kids will be allowed to attend for free.

Write an equation that expresses the cost per payingcamper (y) in terms of the total number of campers (x).

The senior class is planning the Prom. The band costs $600,the rental of a hotel ballroom is $300, and the cost ofbeverages is $100. The hotel will charge an additional $20per person for food. Based on a lottery, ten couples will beallowed to attend the Prom at no charge.

Write an equation thatexpresses the cost perpaying student (y) in termsof the total number ofstudents (x).

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The class officers at Springfield Highfound that 500 students would buythe yearbook if it were sold for $50. They also learned that 100 morestudents would buy the yearbookfor each $5 reduction in price.

Write an equation that expressesthe total sales revenue (y), which isthe cost per book times the numberof books sold, as a function of thecost of each book (x).

SPRINGFIELDSPRINGFIELD

HIGHHIGH

A radioactive isotope decays 18% every hour. Currently, there are 100grams of the substance.

Write an equation thatexpresses the numberof grams (y) that remainafter x hours.

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Bart’s dog has been digging up the neighbor’s flower bed,and now he must keep her restrained. Bart has 36 metersof fencing with which to build her a rectangular pen.

Write an equation thatexpresses the area ofthe pen (y) as afunction of thewidth (x).

Malcolm borrowed some money from a friend to buya used motorcycle. He agreed to pay it back at $17 aweek. After 13 weeks, he still owes $544.

Write an equation that expresses the amount Malcolmowes (y) after x weeks.

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A group of friends plan to enter a competition witha $1,000,000 prize. They are thinking of inviting someother people to join their group.

Write an equation that expresses the amount of moneyeach person will receive (y) as a function of the numberof people in the group (x), assuming they win.

To deliver a package within the city, Springfield Courierscharge $5 plus $1 for each pound (or fraction of a pound)that the package weighs.

Write an equation that represents the amount charged (y)for a package that weighs x pounds.