Problem: Your employer manufactures circuit boards in a
variety of convex shapes. The board must be convex so that any two
points on it can be joined by a straight segment of wire lying entirely
on the surface of the board. What circuit board shapes can be designed
using all 7 pieces of the tangram set?
Work with a partner using all 7 pieces of a tangram set to design
as many convex polygon shapes as you can for circuit boards.
Use uncooked spaghetti to check that your shapes are convex:
Connect any two points on the polygon with a piece of spaghetti.
If the spaghetti that lies between the two points lies entirely within
the polygon, it is convex. If part of the segment of spaghetti between
the two points falls outside the shape, it is concave.
Record your convex shapes by tracing around their borders.
Cut out your convex shapes and sort them by the number of sides they have.
Make sure all of your shapes are different. Be ready to
discuss the shapes you have found.
Math Topic/Concept: Comparing polygons: concave, or convex
Materials: tangrams, 1 set per student, uncooked spaghetti, scissors
Classroom Use: (Introductory/Developmental)
Classroom use comments*: Use this activity to discuss strategies to design convex shapes, different ways to check that all were different, properties of polygons, etc.
Grade: 7-8
Grade Cluster: (MS-Jr.High)
Illinois Goal: 9: Geometry
Standard: 9A3c, 9B3
Applied? (1-4): 2
Source: Super Source: 7-8, Geometry , Cuisenaire, 1998 ISBN: 1-57452-011-3
Answer: There are 13 unique convex polygons, using all 7 pieces of a tangram set.
Strategies Listed: Use a model, use manipulatives, Guess and Check.
Solution: various
Extensions or related problems*: Students may explore whether there are more or fewer concave polygons that can be made with a 7-piece tangram set. They may also extend to using two sets of tangrams.
Intended rubric or assessment method: Assess: were all polygons convex? Did students work together to find as many polygons as possible? Did students sort polygons by number of sides? Could students describe properties, and any patterns or relationships noted?
Notes*: Journal writing entries might respond to: “Describe the difference between a convex and concave polygon. State two methods to determine whether a polygon is concave or convex.
Write-up submitted by: M.K. Robbins
Title: Reflections
Problem: The point (4, 3) is reflected about the x-axis to a point P. Then P is reflected about the y-axis to point Q. What is the sum of the coordinates of Q?
Math Topic/Concept: Graphing/Symmetry
Materials: Graph Paper
Classroom Use: Developmental
Classroom use comments: This problem helps the student visualize what happens when a point is reflected across an axis.
Grade: 7th
Grade Cluster: (EarlyElem/LateElem/MS-Jr.High/EarlyHS/LateHS)
Illinois Goal: 9
Standard: 9.A.3c
Applied? (1-4): 1
Source: Mathematics Teacher
Answer: -7
Strategies Listed:
Solution: Graph the point (4, 3). Reflect across x-axis to get (4, -3). Reflect that point across the y-axis to get (-4, -3). Add these two coordinates to get –7.
Other solution methods (if any):
Extensions or related problems: Construct a rectangle, then a solid is constructed when you rotate around an axis.
Intended rubric or assessment method: This is a developmental problem that can serve as a sponge activity. Therefore, I don’t believe in placing a big emphasis on the amount of points. I would grade this problem mainly on effort. So you could grade this problem using a general rubric which can be based on a point system with an 0 given for no effort, 1 given for effort and partial understanding, and 2 given for effort and complete understanding.
Notes (if any):
Write-up submitted by: Dana Kimberley
James R. Olsen, Western Illinois University
E-mail: jr-olsen@wiu.edu
updated:
December 16, 2001