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The activity was written in response to a teacher-centered lesson I saw presented in our fifth grade classroom. It is similar to Show Me 48, but is considerably more in-depth. I would anticipate that it could take 2-4 class sessions to complete.
Needed Materials: Grid paper, scissors, tape or glue stick, and a writing utensil for each student; calculators for Day 3. Newsprint or butcher paper and wall space large enough to display all the results is important for comparison and contrast. You may want to have the butcher paper prepared ahead of time with the numbers 1-100 equally spaced along its length.
Directions: Day 1
1. Begin the activity by placing these words on the board or overhead: factors, multiples, prime, composite, area, perimeter, dimensions, rectangle, square. Brainstorm with the students to determine what ideas they already have concerning these words. Tell them the activity they’re going to complete is going to involve all these terms and that the class will brainstorm again when the activity is finished.
2. Tell the students you are contemplating building a new dog pen for your beloved pet and would like lots of options to choose from. For instance, if you want the pen to contain 24 square feet, what are all the possible rectangular pens you could build? Guide the students to think of all the rectangles they could create that contain 24 square units. (If you know the students did “Show Me 48” in third or fourth grade, referencing that activity may bring the idea back to them and will help them see that math is connected between grades.) Make a systematic list on the board; then have students create the rectangles on grid paper and cut them out. (With a large class, you might want to discuss several numbers, with lists on the board, so that each student has a unique rectangle to make.) As the students make their rectangles, circulate to ascertain the methods they are using to verify for themselves the number square units contained. (Counting one by one, skip counting, counting rows and multiplying by the number is each row, etc.) This is an appropriate place to discuss “scale models” since the grid paper is obviously not made out of square foot grids.
3. Have the students attach the rectangles under the appropriate number on the butcher paper in a systematic way (perhaps like the systematic list on the board).
4. Working in pairs, have the students create the rectangle sets for all integer areas, 1 to 100. You will probably want to let the student teams choose a range, completing the task more quickly by utilizing “divide and conquer”. For many numbers, they will need to splice the grid paper in order to get pieces big enough to complete all the rectangles. This will probably take the remainder of the time for this first day spent on the activity.
5. Possible homework: Let students pick a number (or numbers) from 101-500 to analyze at home. They need not cut the rectangles out, but simply draw them on grid paper. This could be a great problem solving opportunity, as students will either need to splice paper or do the drawing using a scale other than 1:1. You could hint at this as a possibility without actually telling them how to do it.
Day
2
6. Still working in pairs, have the students analyze the class results posted on the butcher paper. Tell them that as a mathematician, sometimes you notice interesting patterns that distract you momentarily from the task at hand—in fact, sometimes those “distractions” lead to important discoveries. You’ll get back to building the dog pen, but right now you’re intrigued by some rather interesting patterns in the pens they’ve done that you’d like them to see as well. Have them look for patterns and write down any observations they have. Allow an appropriate amount of time, then have the teams “think, pair, share” with another team. Finally, conduct an all-class discussion regarding their findings.
7. Questions that might help them get started:
a) What numbers have only 1 rectangle? (This will be what you use to introduce prime. You’ll need to discuss what “2 unique sides” means in order to prevent the students from including 1 in the list of primes.)
b) Are the 1 rectangle numbers always odd? (All except 2—why is that?)
c) What numbers have an even number of rectangles? (“All evens are composite but are all composites even?”)
d) What numbers have an odd number of rectangles? (Introduction to perfect squares)
e) What numbers contain a square among their rectangles? (Will tie in with their odd number rectangles observation and help explain why we call the numbers “perfect squares”.)
f) Is there a particular dimension that ALL the pens share? (All numbers share 1 as a factor)
g) Do any of the pens have duplicate rectangles with another pen? (12 has a 2 by 6 pen—do any of the other numbers? The factorization of a number is unique.)
h) What numbers have rectangles that all share a common dimension? (E.g., what numbers all have a rectangle with one side 4 units long, or 5 units long, etc.?)
i) …what other questions have you found helpful?…email me and let me know!
8. Back to the dog pens…you’re thinking about making the pen out of chain link fencing. Discuss which pens the class thinks would contain a reasonable area in a reasonable shape for a dog pen (e.g., 72 square feet is ample space for a kennel, but it isn’t very practical in a 1’x72’ configuration). This should include discussing the breed of the dog to determine size and exercise needs. Another consideration is whether the doghouse is contained within the pen or situated adjacent to one side. (Determining an appropriate size is a critical part of number sense. Since the paper models are not square foot grids, the students may have difficulty visualizing what a 4’x12’ pen—or another size—really looks like**. If the floor of your school is tiled in square foot tiles, you’ve got an excellent tool already at your “toe tips”. Otherwise, have the students use a tape measure and actually mark off different sized pens to give them experience with the actual dimensions.) For the smaller pens obviously not big enough for a dog, you might discuss what kind a pets for which they could be used. For example, a pet tarantula would be fine in a 1’x1’ pen—but you might want to consider something other than chain link fence! This discussion can be really fun and tie into cross-curricular ideas in science—this is the stuff “real life math” is made of!
**By the way, don’t miss the opportunity to discuss the traditional dimensional notation, “4x6” or “2x10”, etc. We see a times sign, but we say “by”. Ask the students if they see a relationship between the “lumberyard” notation and the “mathematics” notation. This is an excellent opportunity to discuss the differences and similarities between “dimensions” and “area”. I have found that my junior high and high school students have difficulty when problems ask for the dimensions of an object—they don’t know what to give or what kind of labels to use. Experiences like this at the middle school level should help alleviate that problem.
9. Chain link fence costs $112.78 for 50 feet (5’ tall) or $135 for 50 feet (6’ tall) at a local hardware store. (Check with a local hardware or farm supply for reasonable costs in your area). Working in pairs, have the students find the one pen that would be most economical to fence for each of the reasonable-area pens they chose. Remind them they must buy the fencing in 50’ lengths. Be sure to work the term perimeter into your discussion.
10. If you build the pen on dirt, it will soon be a muddy mess as you clean it and when it rains. One option if the pen is in a permanent location is to pour a concrete floor. Another option, if the pen is temporary, is to simply lay outdoor carpet so that it goes under the fence and beyond about a foot on each side. (Honestly—I used this temporary “floor” for 5 years under my boxer, moved it, and used it another 3 years! It’s easy to clean with a water hose and an occasional sprinkling of Pinesol, and it’s not rough on dog knees and elbows like concrete.) Obtain the price for carpet like this and have the students calculate the cost for covering the pens they are considering. (Pay close attention to how the carpet is sold—width, priced per foot or yard, will they sell a specific size chunk or do you have to buy the width of the roll whether you need it or not, etc. Remember, it needs to stick out past the fence so the dog doesn’t chew on the edges.) Challenge Extension involving volume: Concrete costs about $70 a (cubic) yard when ordered from a concrete company. Since you don’t have a concrete mixer, you would need to hire them to mix and deliver it. You can order fractions of a cubic yard. Typical concrete depth is 4 inches. You would need to build a frame to pour the concrete in, but you think you can find enough scrap lumber to do that. What will it cost to pour the floor for each of the pens under consideration?
11. You may want to split the class so that half work on fencing and half work on flooring. As homework, you could ask the members of each group to verify the findings of the other group.
The big question: What size dog pen should I build? How much money do I need to budget for this dog pen? (Budgeting itself should include allowing for a little extra—a good place to talk about estimates that round up. Included in their decision might be a discussion on waste of materials—what am I going to do with leftover materials if I have any?)
12. A few more math “distractions”—what common characteristics do the pens that cost the least to fence share? (Rectangular shapes closest to a square contain the most area for the least perimeter. After students discover this fact, you might want to tell them the apocryphal story of the founding of ancient Carthage.) When looking at overall cost, is it possible for a larger pen to cost less?
13. Finally, brainstorm the same vocabulary list you used on the first day of the activity. For assessment, have each student write about the terms, explaining what they understand about each and expressing any further questions or curiosities they might still have.
The mathematical goals are numerous in this performance task. Students will think about numbers and how they are composed (multiplicatively). The will use the terms factor, multiple, prime, composite, area, perimeter, dimensions, rectangle, and square in context of a problem situation. They will see an example of real world notation and how it relates to mathematics notation. They will learn that systematically approaching a problem is a valid problem-solving skill. They will have practice minimizing and maximizing while making decisions regarding money.
This activity has not yet been classroom tested. Like the “How Many Laps” activity, it is a performance task—and performance tasks usually teach the teacher as much as the students the first time it’s used! Please feel free to tell me how it worked in your classroom by contacting me at rokershner@alumni.okbu.edu