Cutting Cube Conundrum:
Goal: SWBAT describe different 2-dimensional cross sections that will result from the slicing of a cube.
Domain Standard: Mathematics 7.G.3: Describing 2-dimensional cross sections resulting from cubes and rectangular prisms.
Develop Understanding Task:
Cutting Cube Conundrum
Given your modeling clay, mold it into the shape of a cube. The dimensions need not be exact, just make it as similar to a cube as you can. Using the dental floss, cut a cross section out of the cube and examine the cut surface. Sketch and label the shape you see with identifying information. Find as many shapes as you can.
Launch:
· Prepare a cube out of modeling clay to show students.
· Place students in groups of no less than 3 and no more than 5 (4 is ideal).
· Hand out modeling clay for molding the cube and dental floss, fishing line, or plastic knives for cutting cross sections.
· Say: You are to make a cube out of your clay (show a model) this cube will help you to do your task. (hand out the Cutting Cube Conundrum task sheet)
· Note that colored modeling clay dyes may come off on students hands and having wet wipes or other means of cleaning off dyes may be necessary.
Possible Anticipated Responses: Cutting Cube Conundrum
1. Make the possible shapes:
a square
an equilateral triangle
a rectangle that is not a square
a triangle that is not equilateral
a pentagon
a hexagon
2. Limiting the number of shapes to triangles, or squares, etc.
3. Arcing your cut instead of a straight cut.
4. Naming shapes incorrectly.
5. Listing a shape that is not possible.
Possible issues in completing task:
1. Difficulty in one person cutting through the modeling clay.
2. Difficulty in shaping the cube.
3. Difficulty in visualizing the shape that is cut.
4. Cube should be at least 2” by 2” for students to easily visualize shapes.
5. Students may take too much time making “perfect” cubes.
6. Directing off-task behavior.
Solidifying Task: Students investigate the following extensions:
· Record a list of shapes you can not create from the cube and why you can not create them.
· Investigate the same task with a right rectangular prisms.
o Can you make all the same shapes as you made with the cube? Are there more shapes you can make? Describe how you know.
Possible anticipated responses:
1. Shapes with curves.
2. Shapes with more sides than 6.
3. Shapes that are concave/convex.
4. May or may not make a square.
Possible anticipated issues:
1. Creating a right rectangular prism.
2. Similar issues listed above in the Develop Task.
Practicing Task: Cut Cube Conundrum Practice
For problem 1, choose all of the possible shapes that can be made by intersecting a rectangular prism.
For problems 2-8, draw the cross section formed when the plane indicated intersect the shape.
a. a squareb. an equilateral triangle
c. a rectangle (not a square)
d. a triangle (not equilateral)
e. a pentagon
f. a hexagon
g. an octagon
h. a parallelogram (not a rectangle)
i. a circle / 2.
3. / 4.
5. / 6.
7. / 8.
Anticipated Responses for Practice Phase:
1. Correct is a, b, c, d, e, f, h,
Possible issues #1:
· A only works if there is a square face
· Choose only equilateral or non-equilateral triangle but not both.
2. Correct response is a drawn triangle.
3. Correct response is a drawn rectangle or parallelogram.
4. Correct response is a drawn triangle.
5. Correct response is a rectangle or parallelogram.
6. Correct response is a pentagon.
7. Correct response is a rectangle.
8. Correct response is a square or a rectangle.
Monitoring Chart:
Response / Student / Sequence1. All possible shapes: square, equilateral triangle, rectangle (not square), triangle (not equilateral), pentagon, hexagon.
2. Limiting shapes (just one triangle? Just one quadrilateral?)
3. Arcing the cut
4. Naming shapes incorrectly.
5. Listing an impossible shape.
6. Other
Questions to have in our pocket:
· Are there other triangles you can make?
· Is there a limit to the number of sides on the shape? How does that relate to the faces on the cube?
· Try to make a 3 or 4 or 5 … sided figure.
· Does it matter which way you cut it?