MATH 613—M Explorations Math Tasks
Geometry and Measurement: K - 8 Learning and Teacher Practices
M Math Task 1: Perimeters
Use square tiles to explore these ideas. Use actual tiles (photograph your work) or use virtual color tiles (or squares from pattern blocks). Other displays that make sense are also OK.
Task
Exploring perimeter and area
Questions for you (please answer)
1.
What is the perimeter of Polygon A? Dashed lines are guides, not part of the polygon.
1 inch
2.
Using the rule that at least one side must touch,
Polygon A
OK NOT OK
Give three (if possible) examples each for parts a) – c) answering the following question.
What new perimeters can you make by adding: a.
One square tile to Polygon A? b.
Two square tiles to Polygon A? c.
Three square tiles to Polygon A?
For parts d) – g), if the answer is yes, give a single illustration of this, if the answer is no, briefly explain why. d.
Can you reduce the perimeter of Polygon A by adding square tiles according to the rule? e.
Can you keep the same perimeter of Polygon A by adding square tiles according to the rule? f.
Can you increase the perimeter of Polygon A by odd amounts by adding square tiles according to the rule? g.
Can you increase the perimeter of Polygon A by even amounts by adding square tiles according to the rule?
3.
a.
Add tiles to Polygon A until you reach a perimeter of 16 inches. What is the area of your new polygon? b.
Repeat, finding all extensions of Polygon A with a perimeter of 16 inches. c.
Which configuration with perimeter 16 inches has the maximum area?
M Math Task 2: Cylinders and Prisms

MATH 613—M Explorations Math Tasks
Geometry and Measurement: K - 8 Learning and Teacher Practices
Explore how using the same material for the faces of a prism or cylinder, but changing the shape of the base changes the surface are and volume of the resulting solid.
Tasks
1.
Make prisms and cylinders Use scratch paper, scissors and tape to make six cylinders and prisms. For each, make top and bottom “lids”. You can use a wrapping method (like plastic wrap on a bowl), or precisely measure and cut your bases. You choose. i.
Tall Cylinder Curl the 11" edge of a sheet of 8.5” × 11” paper and tape it into the face of a cylinder (don't overlap the edges). ii.
Short Cylinder Fold and then cut a sheet of 8.5” × 11”paper in half (cut perpendicular to 11” side). Tape the two pieces together into a 5.5”  17” rectangle. Curl the 17" edge and tape it into the face of a cylinder (don't overlap the edges). iii.
Square Prism Fold a sheet of 8.5” × 11” paper into four identical rectangles; the fold lines should be perpendicular to the 11” side.
Tape the paper into the shape of a square prism.
11" edge iv.
Equilateral Triangular Prism Fold a sheet of 8.5” × 11” paper into three identical rectangles; the fold lines should be perpendicular to the 11” side. Tape the paper into the shape of an equilateral triangular prism. v.
Pentagonal Prism Fold a sheet of 8.5” × 11” paper into five identical rectangles; the fold lines should be perpendicular to the
11” side. Tape the paper into the shape of a pentagonal prism.
11" edge vi.
Hexagonal Prism Fold a sheet of 8.5” × 11” paper into six identical rectangles; the fold lines should be perpendicular to the
11” side. Tape the paper into the shape of a hexagonal prism.
11" edge
2.
Visually order your prisms and cylinders by volume
Using visual inspection only, no calculations or measurements, put your prisms and cylinders in order by volume. Write this down for your Exploration forum post.
3.
Use filler to physically order your prisms and cylinders by volume i) Use a volume fill material like rice, beans, popped popcorn (no butter), puff balls, packing peanuts, or a material of your choice to fill each of your prisms and cylinders.
Put your prisms and cylinders in order again by comparing the amount of fill used (still no calculations). Write this down for your Exploration forum post. ii) TAKE A PHOTO of your measurement system to post.
This task is not meant to be complicated, keep it simple and fun.

MATH 613—M Explorations Math Tasks
Geometry and Measurement: K - 8 Learning and Teacher Practices
4.
Use math to measure the volumes and surface areas of each of the prisms and cylinders
 Show you work for all calculations, including sketches of bases with measurements labeled on the sketch.

For your bases, sketch (fairly) precise polygons and circles using a compass and protractor or technology. See the “how to draw a regular polygon” link on the M
Resource page for protractor directions.

If you use technology (GSP, GeoGebra, or? See VG Explorations) to sketch the polygons, feel free to use technology area tools to check your work, but also show your personal calculations in your write up.

Hint for iv) and vi): a 2 + b 2 = c 2 .

Hint for v): On your lid (or sketch of your lid), measure to estimate the height of the triangles that make up the base (r on the image). You need trigonometry to precisely determine this height, and that is not required for this course.
5.
Compare your methods and post your results
Using the M Explorations forum, post:
1) Your initial visual ordering list (e.g. tall cylinder, equilateral triangular prism, …)
2) Your “using fill” ordering list and the photo of your measuring system. Ideally use the
Moodle insert/edit image button to insert the photo into your post.
Make sure the image is a good size (drag image corners to resize).
If this doesn’t work, use the attach feature to attach the image.
3) Your calculation ordering list
Questions for you (please answer)
Included in the above math task components