Problem:  You work for a company that manufactures jigsaw puzzles for the visually impaired.  Your job is to write step-by-step instructions for assembling the puzzles.  The instructions are then translated into Braille for the visually impaired.
· Sit back-to-back with your partner so that you cannot see each other’s work.  Make a puzzle design using 5 to 7 pieces of a tangram set.
· Record your design by tracing the pieces on paper.  This will be your answer key.
· On a second sheet of paper, write step-by-step instructions for placing your tangram pieces into the puzzle you designed.  Be concise, using mathematical terms without drawings.
· Exchange instructions with your partner.  Follow your partner’s instructions to make his design.
· When finished, compare the finished puzzle to the answer key.  Discuss how the designs compare and how the written instructions might be improved.

Extension problem:
 You are promoted at the puzzle factory.  Your new job is to write instructions for the jigsaw machine to cut the frame of the finished puzzle design.  The jigsaw will follow your instructions, moving around the border of your puzzle to make appropriate cuts.
· Sit back-to-back with your partner so that you cannot see each other’s work.  Make a puzzle design using 5 to 7 pieces of a tangram set.
· Make an answer key by tracing the frame or outline of your puzzle on a piece of paper.
· On a second sheet of paper, write the commands you would use to program the jigsaw to cut the frame of your puzzle.  Select one vertex as your starting point.  Use a ruler and protractor as measuring tools to write a series of steps that will generate the path around your design, ending back at the starting point.
· Exchange instructions with your partner.  Follow your partner’s instructions for making his or her puzzle frame.
· When finished, compare to the answer keys, and discuss how the written instructions might be improved.

Math Topic/Concept:   properties of polygons,  Measurement of angles and distances, Writing/communicating mathematical descriptions and instructions

Materials:  1 set of tangrams per student.    Protractor, ruler, pencil and paper

Classroom Use: (Introductory/Developmental/Evaluation)

Grade:  7 - 8

Grade Cluster: (EarlyElem/LateElem/MS-Jr.High/EarlyHS/LateHS)

Illinois Goal:  9 & 7  Geometry and Measurement

Standard:  9B3, 9C3a, 7A3a, 7B3

Applied? (1-4):  3

Source:  Super Source for 7-8 Geometry, Cuisenaire: 1998, ISBN: 1-57452-011-3

Answer:  varies, depending on student design

Strategies Listed:  Make a diagram

Solution:  varies

Extensions or related problems*:  Students might use 3-D solids to build towers and describe them to a partner who then must build a copy of the tower.

Intended rubric or assessment method:  Were the instructions complete, concise, and accurate?  Did instructions include appropriate mathematical language?  Were measurements accurate?

Notes*:    Although the Braille jigsaw puzzle premise may be a bit contrived, technical writing and following technical instructions is a highly applicable skill in daily life.

Write-up submitted by:  M. K. Robbins

Problem:  Zac is going to his friend’s house on the way to the basketball game. To save time and gasoline, how should Zac plan his route so that he stays on the streets and does not backtrack? How many different routes are there from his house to his friend’s?
 
 
 
 
 

Math Topic/Concept:  Spatial Reasoning

Materials:  Map

Classroom Use: (Introductory)

Grade: 6-8

Grade Cluster: (MS-Jr.High)

Illinois Goal:  9

Standard:  9.C.3

Applied? (1-4):  3

Source:  Mathematics Teaching in the Middle School (Jan. 1999)

Answer:  16 ways

Strategies Listed: Guess and Check

Intended rubric or assessment method:  Does their solution meet the criteria?  Use class discussion to talk about other solutions.

Write-up submitted by:  Denise Mann and Jenni Robinson

Problem: When the snow stopped falling, Kelly, Miguel, and Neha rushed outside to build a snow statue. Kelly made one large snowball for the head. Miguel made a larger snowball for the middle section. Neha made a giant snowball for the bottom section.
   As they were getting ready to put the snow statue together, the children noticed that the circumference of the smallest section was about 2/3 the circumference of the middle section. The middle section was about 3/4 the circumference of the largest section. When they measured the smallest snowball, they found that it had a circumference of about 80 centimeters.
     How tall will the snow statue be when the sections are piled one on top of the other? Assume that the  snow does not compact.

       Bonus: The group would like to make another snow statue with a total height of at least two meters. If all other conditions are kept the same, what circumference will they need for the smallest section?

       Note: Please be sure to use the appropriate units of measurement, and state your final answer in a complete sentence.

Math Topic/Concept:  measurement: circle circumference/ diameter

Materials:  calculator, pencil, paper

Classroom Use: (Developmental)

Classroom use comments*:  Students may need help with appropriate rounding, and making sense of answers rather than depending on calculator output of long decimal numbers.  Using 3.14 or 22/7 for pi work equally well.  The pi key on a calculator may give less user-friendly results.

Grade:  7 - 8

Grade Cluster: (MS-Jr.High)

Illinois Goal:  7: Measurement & 9: Geometry

Standard:  7C3b & 9C3b

Applied? (1-4):  3

Source:  www.forum.swarthmore/libr
  Swarthmore University’s past problems of the week

Answer:  The snow statue will be about 115 cm tall.  Bonus: the smallest snowball will need a circumference that is about 140 cm.

Strategies Listed:  drawing, algebraic formulas

Solution:  Small snowball has C = 80 cm.  Middle snowball C = 120 cm, since the small is 2/3 of the middle.  The middle is 3/4 the size of the large, so the large snowball's C = 160 cm.  The diameter of each = C / pi. Using pi = 3.14, you get diameters of about 25.478, 38.217, and 50.956 cm.  Stacked as a snowman, the height is about 115 cm.

Extensions or related problems*:  How many rotations will your 26-inch bicycle tire make in a fifteen mile bicycle trip?

Intended rubric or assessment method:  ISAT “student friendly” rubric

Write-up submitted by:  M.K. Robbins

Problem:  When Pat came home from the carnival, she couldn’t wait to tell her mother about the Ferris wheel.  Her mother asked how many cars the Ferris wheel had.  Although Pat didn’t know, she thought she could figure that out because when she and her friend Judy had gotten into their car, she noticed it was car number 5.  As they were going around, the car directly across from them was number 17.  How many cars did the Ferris wheel have?  Explain how you find your answer.

Math Topic/Concept:  Spatial Reasoning

Classroom Use: (Introductory)

Classroom use comments*:  Use to introduce circle properties and diameter

Grade:  7-8

Grade Cluster: (MS-Jr.High)

Illinois Goal:  9

Standard:  9C3

Applied? (1-4): 2

Source:  “Explain It”  (Creative Publications)  Grade 7-8  ISBN  0-7622-1599-2

Answer:  24 cars

Strategies Listed:  Make a table or Draw a picture

Solution:
   1     2    3    4     5    6     7     8     9   10   11   12
 13   14   15   16   17   18   19   20   21   22   23   24
 

Other solution methods (if any)*:  Draw a circle for the Ferris wheel and draw a diameter across the circle, with 5 at one end and 17 at the other.  There are 11 numbers between 5 and 17, so there would be 11 cars between them on one half of the wheel.  There would be the same number of cars on the other half of the wheel.  That would make 22 cars between cars 5 and 17, and then add on cars 5 and 17, there are 24.
 
 

Extensions or related problems*:  Lower level:  make a smaller Ferris wheel.
Higher level: Cars are only numbered with even numbers.

Related problem:  James is sitting on the ski lift, which is a continuous loop with chairs that are numbered consecutively.  Exactly halfway up the ski slope, he notices that chair number 9 passes him on the way down.  He is sitting on chair 81.  How many chairs are on the ski lift?

Intended rubric or assessment method:  Solve a similar problem on their own.

Write-up submitted by:  Jenni Robinson and Denise Mann

Problem:  Augie, Oscar, and Rupert are married to Lucy, Suzie, and Mary, not necessarily in that order.  Four of the six people are playing mixed doubles tennis.
· Oscar never plays tennis.
· Suzie’s husband and Augie’s wife are partners.
· Mary’s husband and Lucy are partners.
· No married couples are partners.
Who is married to whom?  Who are partners?

Math Topic/Concept:  Logical Reasoning—Deductive Reasoning

Classroom Use: (Introductory)

Grade:  6-8

Grade Cluster: (MS-Jr.High)

Illinois Goal:  9

Standard:  9.C.3

Applied? (1-4): 2

Source:  Mathematics Teaching in the Middle School  May 1999

Answer:  Married couples are Suzie and Rupert, Mary and Augie, and Lucy and Oscar.        Tennis partners are Lucy and Augie and Mary and Rupert.

Strategies Listed:  Make a Table

Solution:  Could make two logic tables.  One to figure the married couples and one to figure the tennis partners.

Intended rubric or assessment method:  Solve similar logic puzzles

Write-up submitted by:  Jenni Robinson and Denise Mann
 
 

James R. Olsen, Western Illinois University
E-mail: jr-olsen@wiu.edu
updated Aug. 20, 2001