Carpet Caper

Task

How many rolls of carpeting do I need to buy to cover my floor? Remember carpeting is very expensive (100 square feet including the pad is on sale at a local carpet company for $200), so I want to buy the least number of rolls needed to do the job.

How should I cut the 100 square foot rolls of carpet so that I get the least number of seams? (A seam is where 2 pieces of carpet come together.)

This task was presented to students following a unit on multi-digit multiplication. Students were taught several different models of multiplication, resulting in a deeper understanding of the process. I began by having students use Base-10 pieces to create rectangular arrays of equations. Students were then introduced to the concept more abstractly using Base-10 paper. Next, students were shown an algebraic approach (A + B)(C + D)= Area. Finally students were shown the more traditional algorithm. This approach better allowed students to understand how the algorithm is derived.

What This Task Accomplishes

Not only does this task accomplish an assessment of a student's ability to perform multi-digit multiplication in context, but it also requires them to go beyond computation and evaluate their solutions, often by rounding to the nearest whole number of carpets. The task also assesses spatial sense. Students need to cut the carpet resulting in the least number of seams. This problem can be solved in many different ways, using traditional calculation, but depends on the creation of a model and/or diagram to obtain a complete solution. The task lends itself well to mathematical observations and the use of a wide variety of mathematical terms, symbols and notations.

What the Student Will Do

Students will realize that three rolls of carpet are needed. The tricky part comes when figuring how to cut the carpet. Many students are unable to realize that the 9 x 1 foot strip needs to be cut into three 3 x 1 foot strips in order to fit in the space. Some students in my class applied Pick's Theorem, a theorem we discovered weeks earlier when studying geoboards. Students who applied this prior knowledge to a practical situation provided more sophisticated, verified solutions.

Time Required for Task

1-2 hours

Interdisciplinary Links

This task could be given for a unit on careers (carpet layers). In fact one of my student's parents was a carpet layer and was fascinated with the task. There are many children's books with carpet as a theme and students could even write fictional stories about "magic carpets". In art, students can weave carpets and students could study different types of carpets such as oriental carpets or Japanese mats.

Teaching Tips

After my students had worked on the problem for a while, we brainstormed together a list of mathematical terms, symbols and notations that they might consider using when communicating their solutions. This was posted on the wall for reference. We also brainstormed a list of possible extensions, observations and connections students could make about their solution. For instance, students thought of finding the total cost, total area and perimeter, as well as noticing patterns and relationships. I required each student to address at least five of the extensions so they could practice this skill of looking above and beyond the task for extensions and observations.

I used the following assessment for grading purposes:

Possible Points	Criteria
	Understood the Problem
5	Chose an appropriate strategy
5	Understood both parts (# rolls and # seams)
	Decision Making
5	Correct
10	Verifies solution by solving two ways
5	Gives reasons
	Math Representation
5	Appropriate
5	Accurate and descriptive title
5	Communicates clearly
5	Labels and keys where necessary
	Math Language
7	Variety of terms, symbols and notation
5	Relied on to communicate
8	Accurate
	Extensions, Observations, Generalizations, "I noticed..."
20	Must have five. (Four points each) You have #___
	Overall Presentation
3	Detailed
3	Organized
2	Neat
2	All parts connected
100	Total: Your Grade

Suggested Materials

• Base-10 pieces


• Base-10 paper


• Calculators


• Yard sticks


• Markers


• Stencils


• Sticky notes

Possible Solutions

There would be three rolls of carpet with one square foot of carpeting left over.

Benchmark Descriptors

Novice
A solution that shows a limited understanding of the problem and/or shows no attempt of how to begin. Work shows little evidence of leading to an appropriate solution.

Apprentice
A solution that shows partial understanding of the problem (either understanding that the number or rolls or the number of seams is needed). A solution that shows appropriate use of numbers, but has a weak or random explanation of the strategy used to achieve the solution.

Practitioner
A solution that demonstrates understanding of both aspects of the problem (number of rolls and number of seams). A solution that demonstrates appropriate equations (29 x 13), has a correct solution and has a described strategy.

Expert
A solution that shows a sophisticated interpretation of the problem and/or experiments successfully to create multiple solutions. For instance, the student who can interpret 100 square feet in more than one way (10 x 10 and 50 x 2). A solution that generalizes from a previous mathematical experience (For instance, the student who multiplies 29 x 13 = 299, then divides 299 by 100 to get 2.99 rolls of carpet needed, then interprets the remainder).