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The activity was developed as a review for grade 4 and as a lead-in to fraction area models. I believe this kind of activity (with variations) can and should be used in grade 3 to visually develop the concept of multiplication as groups of like quantities (repeated addition). The area model idea is not original by any means, but it should never be assumed that students automatically understand it.
Needed Materials: Grid paper and a writing utensil for each student, overhead grid and projector or large easel grid (big enough for all to see).
Directions:
1. Initially, the teacher should pick the numbers so that those chosen have several different representations. In Show Me “48”, 48 is simply one example of a number to use. Younger students should begin with smaller numbers such as 12.
2. Direct the students to draw a rectangle that they know contains 48 square units. Circulate to ascertain the methods they are using to verify for themselves how many square units are contained. (Counting one by one, skip counting, counting rows and multiplying by the number is each row, etc.)
3. When ready, have the students compare their pictures with a classmate. The students should discuss whether their rectangles are the same or different. Be prepared to discuss orientation of the rectangles if needed (the commutative property works for multiplication). If different, have each student verify that their partner’s rectangle does indeed contain 48 square units.
4. Call the class together and have volunteers record their answers on the overhead or the large grid. Question the class as to whether they think they have discovered EVERY possible rectangle. Help them develop systematic ways to determine if all possibilities are discovered—let them try for themselves if they are uncertain about whether a particular number is a factor or not.
5. Do other numbers in class as needed for conceptual understanding. For homework, have students do all the rectangles for a series of numbers, e.g., all the factors of the numbers 1-20 or 30-40, etc. Possible variations would be to challenge them to find the number with the most rectangles within a certain range or to challenge a student to find all the rectangles within a given area range with one side a given length (e.g., all the rectangles with one side 6 units long and an area less than 50). For each assignment, the students should write a paragraph telling what patterns they see in the rectangles. Follow-up class discussions could use questions like “What numbers had only 1 possible rectangle?” or “What numbers had an odd number of rectangles?”, etc. to develop the concept of prime, composite, and perfect squares.
6. I believe the project could be expanded to 3-dimensions by having student build prism models of numbers using blocks or any of the several kinds of connecting-cube manipulatives available.
The mathematical goals are for the student to think about numbers and how they are composed (multiplicatively). This activity could be used to introduce the term “factor” and “multiple” or as a review of the terms. The activity also provides an opportunity to develop the skill of approaching a problem systematically.
Comment on the overall effectiveness of this activity. Then use the form below to reflect on the activity from the perspective of both the teacher and the student. Give the lesson points based on how well it went--each question is worth up to 4 points.
When I introduced the activity, I may not have said “draw a rectangle”—or maybe I did, and Mitchell wasn’t listening—whatever the reason, his first drawing lead to an interesting discussion, even though it wasn’t a part of the plan. He drew a plus sign with a 4x4 center and 4-2x4 “arms”. Although he wasn’t giving factors of 48, he did a beautiful job of breaking down the 5 sections by factors, then combining them to make 48. If I’d been thinking fast enough, I would have made a special assignment for him to find what areas are possible from different sizes of plus signs. (I didn’t though, and he reluctantly joined the others in drawing ordinary rectangles—oh, the teaching opportunities we miss!) His plus sign would have added depth and challenge to later discussions about shading fractions of the areas drawn.
The students seemed to enjoy this activity, especially because they got to use grid paper. (I did suggest they start drawing in the top left corner so they’d be able to use more of the paper. I was suspicious that without that instruction, the first rectangle would have been smack in the middle of the paper!) Early comparisons of shapes led to discussions of whether a 2x24 rectangle was different from a 24x2 rectangle. I had the students hold their papers up to the light, turn the papers, and line up the rectangles to see if they were the same. I introduced the term “congruent” but without a great deal of discussion. We could have cut the rectangles out, but that would have taken longer than I had to spend with the students that day. We discussed the “systematic list” idea by deciding to start by always listing the “skinniest” rectangle possible. They were quick to conclude there would always be a 1x__ rectangle. I wondered if anyone would suggest cutting the square units in half, but no one did.
Unfortunately, I did not get to follow up with the writing assignment. Because I only met with the students 1 time a week, the regular teacher collected any assignment I gave and used it for assessment as she saw fit. We discussed the results, but too much time lapsed between my meeting with the students for a “What patterns did you see?” discussion. The lack of continuity was frustrating for me. However, I know that in a regular class setting this activity has good potential for developing number sense regarding multiplication and problem solving skills.
_4_ Was the purpose of the activity clear to me?
_4_ Was the purpose of the activity clear to my students?
_3_ Did the activity support a range of learning styles?
_4_ Were the students engaged during this activity?
_15_ Total points (out of 16)
The purpose (objective) of the activity was…
I communicated to the students why we were doing the activity by…
…telling them we wanted to look for some interesting number patterns while we practiced multiplication. We talked about how much easier it is to do something if you’re organized instead of using a “hit and miss” method.
The part of the lesson that was student-centered was…
…the drawing of the rectangles and comparing them in small groups and whole class discussions.