Problem:  On the map 3 centimeters stand for 20 meters.  Use the clues.  Write the name of each ride next to each vertex on the map.
Clues:
a. The distance from the ferris wheel to the loop-the-loop is 40 meters.
b. The merry-go-round is 30 meters from the entrance.
c. A round trip from the ferris wheel to the roller coaster is 60 meters.
d. The water slide is 50 meters from the roller coaster.

 
 

Math Topic/Concept:  scale drawings, logic, proportions

Materials:  problem hand-out, pencil, paper

Classroom Use: (Introductory/Developmental/Evaluation)

Grade:  7-8

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

Illinois Goal:  6, Number and 7, Measurement

Standard:  6C3a, 7C3a

Applied? (1-4):  3

Source:  Greenes, Findell.   Groundworks: Algebra Puzzles and Problems, Grade 7. 1998 Creative Publications. ISBN: 0-7622-0560-1

Answer:

Strategies Listed:  Use proportions, use logic to follow the clues.

Solution: Use proportions to convert each map distance in centimeters to actual distance in meters.  Using clues:
· the merry-go-round is one of the 2 points that are 4.5 cm from the entrance.
· Only the ferris wheel and roller coaster are 4.5 cm apart.
· The ferris wheel and loop-the-loop must be 6 cm apart. This locates the ferris wheel and the loop-the-loop. With that information, locate the roller coaster and merry-go-round.
· The water slide is 7.5 cm from the roller coaster.

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

Write-up submitted by:  M.K. Robbins

Problem:  You have studied cubism in art appreciation class.  Your assignment now is to create sculptures made with 16 cubes.  The visible faces of each cube will be painted a different color.  How many different colors of paint will you need?
· Work with a partner.  Use linker cubes to design models of several different sculptures, each containing 16 cubes.
· Determine the number of colors needed to paint each sculpture: remember, each face must be a different color.
· Draw your sculptures on isometric dot paper.  Record the volume and surface area of each sculpture, using the edge of one block as one unit.
· Try to make one model for at least one sculpture for every possible number of visible faces.  Be ready to discuss your findings.

Math Topic/Concept:  surface area, and volume

Materials:   linker cubes, about 70 per pair of students
  Isometric dot paper

Classroom Use: (Introductory)

Classroom use comments*:  This might be used to introduce concepts of surface area and volume.  Assessment in that case would be informal, with a journal writing entry for individualized assessment.

Grade:  7 - 8

Grade Cluster: (MS-Jr.High)

Illinois Goal:  7: measurement

Standard:  7C3b

Applied? (1-4):  2

Source:  Super Source: 7-8, Measurement; Cuisenaire, 1999. ISBN: 1-57452-172-1

Answer:  There are 14 possible surface areas using 16 cubes ranging from 40 to 66 square units.

Strategies Listed:  building models, drawing models

Intended rubric or assessment method:  Journal write: Suppose you were building sculptures made from 20 cubes.  Describe how you would build the models with the greatest and the least surface areas.  Explain how you would know that these were the greatest and the least.

Write-up submitted by:  M.K. Robbins

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:  You work for Gourmet Chocolate Company.  Each piece of candy is packaged in a single cube shaped box.  Two dozen boxes are packaged in each rectangular shaped carton.  How many different carton designs are possible?
· Work with a partner. Each linker cube represents a box with one piece of candy.  Build three different cartons, each containing two dozen candy boxes.  The cartons must be rectangular.
· Find the volume and surface area of each carton.  The length of the edge of one cube is 1 unit.
· Draw each of your cartons and record its volume and surface area on isometric dot paper.
· Use snap-cube grid paper to draw a one-piece pattern that can be folded to wrap each carton.  Use dotted lines for fold lines, and do not allow for any overlaps.
· Cut out your patterns and fold them to check that they wrap around your cartons properly.  Make needed changes to your patterns so that they work.
· Record the total number of square units of wrapping paper needed to make each of your patterns.
· Note any patterns in your data and be ready to discuss your findings.

Math Topic/Concept:  surface area, volume, nets

Materials:  linker cubes, isometric dot paper, snap cube grid paper, scissors

Classroom Use: (Developmental)

Grade:  7-8

Grade Cluster: (MS-Jr.High)

Illinois Goal:  7: measurement

Standard:  7C3b

Applied? (1-4):  3

Source:  The Super Source: 7-8, Measurement; Cuisenaire, 1999. ISBN: 1-57452-172-1

Answer:  There are 6 different rectangular cartons with a  volume of 24 cubic units.  Surface areas range from 52 to 98 square units.

Strategies Listed:  physical models, drawings

Solution:  Record data from all groups in the class, 1 design per group until all are listed.

Extensions or related problems*:  Do the same activity, but do not require cartons to be rectangular prisms.  Include nets to wrap odd-shaped solid stacks of cubes.

Intended rubric or assessment method:  Informal class discussion:
· How did you determine surface area and volume of cartons?
· How do we know whether we have found all possible carton shapes?
· How did you go about designing the wraps for your cartons?
· What patterns did you notice in our class data?
· How are the measurements you found related to the dimensions of each box?

Notes*:  Journal entry: Suppose you know the length, width, and height of a rectangular solid, but do not have a model.  Explain how you would design a net for the solid, and how you would determine how much paper would be needed to make the net.

Write-up submitted by:  M.K. Robbins
 
 

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