Title:  Penguin Fun

Problem:  Pose this problem: Three of [Mrs. Waters’] penguins escaped from the classroom and hopped the Amtrak to Marriott’s Great America.  As they traveled to Chicago, the three stuffed birds were so happy to be free, they decided to give themselves new names: Perky, Waddles, and Tux.  When they got to the amusement park, they paddled around until they found the best ride: the Antarctic Ice Float.  Unfortunately, there was a weight limit of 110-pounds posted at the entrance. The three penguins wanted to ride all together but they did not know how much they weighed.  Waddles said he thought he might weigh about 40 lbs., but he wasn’t sure.  The scale could not hold all three of them at once, but they could get on the scale 3 times.  However, penguins are very, very shy, so they were afraid to weigh one at a time.  After they talked about it, they agreed to weigh like this:

First Perky and Waddles got on the scale.
       Together, they weighed 90 pounds.
Second, Waddles got off and Tux got on with Perky.
       Together, they weighed 68 pounds.
Third, Perky got off and Waddles got on with Tux.
       Together, they weighed 52 pounds.

Although Perky, Waddles, and Tux have been sitting on the shelf in Mrs. Waters’s room for a long, long time, they have not learned very well how to figure out problems.  They need your help to show them how to figure out how much they all weigh together so they know whether or not they ride the Antarctic Ice Float.  Can you help?

Math Topic/Concept: algebraic and analytical reasoning and strategies

Materials:  paper & pencil  (optional: manipulatives or pictures to represent 3 penguins)

Classroom Use: (Introductory)

Classroom use comments*: Prior to this lesson, review strategies for developing equations and number sentences with unknowns and review T-charts (and other models) for managing patterns in analysis.  Also remind students to look for what is KNOWN, what is NOT KNOWN and what are the RESTRICTIONS.

Grade:  4

Grade Cluster: (LateElem)

Illinois Goal: 8

Standard:  8.A.2b and 8.D.2

Applied? (1-4):  2

Source:  http://mathforum.com/pow/   (adapted)

Answer:  P = 53, W = 37, T = 15, so altogether they weigh 105.  Yes, they can ride.

Strategies Listed:  guess & check, logical thinking, computation, making charts or tables (calculators are optional and provided)

Solution:  Students may work with partner or group (or independently) to try strategies for solving.  They should show their thinking processes on paper and identify the final solutions with appropriate labels.  They will write a response (in math journals or on problem paper) to describe their thinking processes and how they finally solved the problem.

Other solution methods (if any)*:  Students might need to use visual aids or manipulatives.  They might use a balance scale and a variety of weights.

Extensions or related problems*:  Role play the weighing with three students, one from a kindergarten class, one from 2nd grade and one in 5th.  Without knowing any individual weights, students should be able to apply the same processes to the real-life example.  Prove the results with final weighing of each individual.

Intended rubric or assessment method:  Analytical Scoring Scale (Jim Olsen, WIU)

Write-up submitted by:  Rebecca Cummins (Westmer CUSD 203)

Problem: Luke is very excited. His parents have decided that he should have his very own bedroom. They have also said that Luke can decorate it any way he wants to so long as he does not go over his budget of $350.00.

Luke really likes the way the room looks already except for the floor. He decides that he would like to tile the floor in his two favorite colors: alien green and midnight black. In a catalog, he has found a set of 9-inch-square floor tiles in his two favorite colors.  Luke needs to figure out how many tiles he will need for his 12 ft. by 15ft. room and whether he can afford to buy them. Question: How many tiles will Luke need?

Math Topic/Concept: Algebraic Reasoning, fractions, units, measurement.

Materials: Calculators Optional

Classroom Use: (Developmental)

Classroom use comments*: NA

Grade: 4/5

Grade Cluster: (LateElem)

Illinois Goal: 8

Standard: 8A2a, 8B2, 8C2, 8D2

Applied? (1-4): 3

Source: http://mathforum.com/elempow/solutions/solution.ehtml?puzzle

Answer: 320 tiles

Strategies Listed: Multiplication, Division, Units of Measurement, Monetary Units

Solution: I decided to convert 12 feet and 15 feet into inches because 9 does not go into either one evenly. Since there are 12 inches in a foot and there are 12 feet, I took 12 x 12 = 144 inches. I then divided 9 into 144 to see how many tiles it would take and 144 ÷ 9 = 16 tiles.  I repeated the same process for the side with 15 feet.  I took 15 x 12 = 180 inches. Then, to find the amount of nine-inch tiles, I took 180 ÷ 9 = 20 tiles.  The final step is to take the 20 tiles x 16 tiles = 320 total tiles.

Extensions or related problems*: If the tiles are sold in packages of 12 for $12.95 each (including tax), can Luke afford the tiles?

To figure out the cost, I took the answer, 320, and divided it by 12, for the package of 12 tiles. We got 26 remainder 8. So, you must add another pack to take care of the remainder, even if it is only eight. The answer is 27 packages. I then multiplied 27 by the cost of each pack, $12.95, and got  $349.65.  Luke could afford it.

Intended rubric or assessment method:
Assessment ISAT Mathematics Grade 5 Student-Friendly Rubric
www.isbe.state.il.us/isat/rubric5.htm

Write-up submitted by: Carl Carlson – Westmer School

Problem:  There are two taxi cab companies in Small Town and they have different rates for determining how much to charge for a ride.
  Speedy Cab Service      $3.00 for the first mile
       $0.50 for each additional mile

Whirlwind Cab Company  $1.00 per mile
With which cab company would you pay less for a 3-mile ride?
Which cab company would be less expensive for a 10-mile ride?
For what distance ride will the two companies charge exactly the same amount?
Explain how you find your answers.
 

Math Topic/Concept:  Number sense, addition, currency

Materials: paper, pencil

Classroom Use: (Developmentalaluation)

Grade:  5

Grade Cluster: (LateElem)

Illinois Goal: 6.B.2 , 7.A. 2b  and 8.D.2

Standard:  6.B.2 ,7.A.2b and 8.D.2

Applied? (1-4):  3

Source:  Explain It!  Grades 5-6 from Creative Publications  ISBN 0-7622-1598-4

Answer: Whirlwind cab would be less expensive for a 3-mile ride.  They would charge $3.00, while the same distance would cost $4.00 with Speedy Cab.
For a 10-mile ride, it would be less expensive to use Speedy Cab.  They would charge $7.50 rather than the $10.00 you would be charged by Whirlwind.

Strategies Listed: While it is a more tedious way to find the comparative rates for the first 10 miles, making a chart as done in Solution 1 makes it easy to compare the rates for any distance quickly.  All students should understand this approach
 
 

Solutions:
Solution 1: You can make a chart showing what the two cab companies would charge
for each distance up to ten miles.  The chart shows that Whirlwind Cab Company is less expensive for traveling distances less than 5 miles, and Speedy Cab is less expensive for distances of more than 5 miles.  They are the same price for exactly 5 miles.

Other solution methods (if any)*:  Solution 2:  Since Speedy Cab Company charges $0.50 for each mile, except for the first mile, which is 3 dollars, their charge is $3.00 + $0.50 x the total miles minus 1.  For Whirlwind Cab, the charge is $1 x the number of miles.  Speedy Cab costs less for a 3-mile ride, but they charge more for a 10-mile ride.
For a 3-mile ride:
Speedy Cab = $3+ ($0.50x{3-1}) = $3+$1=$4
Whirlwind Cab = $3.00
For a 10-mile ride:
Speedy Cab = $3.00 + ($0.50 x{10-1})= $3+$4.50 =$9.00
Whirlwind Cab = $10.00
If there is a place where the charge is the same for both companies, it should be between 3 miles and 10 miles.  By checking different distances, you find that at 5 miles they charge the same rate- $5.00.

Intended rubric or assessment method:  Grade 5 "Student-Friendly" Mathematics Scoring Rubric found at http://www.isbe.state.il.us/isat/rubric5.htm

Write-up submitted by:  Ann Hulsizer , 5th Grade, Monmouth

James R. Olsen, Western Illinois University
E-mail: jr-olsen@wiu.edu
updated June 27, 2001