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Our
principal and elementary secretary prompted this activity. The elementary building has an interior
corridor that goes completely around the building. The adults often try to walk laps during free time as a means of
getting some exercise. They wondered
how many times they needed to walk in order to get in a mile and asked me if I
thought the fourth graders could figure it out. It sounded like a great idea to me. This was one of my early projects with the fourth grade, so I
learned a whole bunch more than the students—and not particularly about
math! We tackled the project during my
scheduled Wednesday time with the students.
Description
Needed Materials: A large area around which to walk, yardsticks, scale drawing of the area (optional).
Directions:
1. Define the task, estimate an answer.
2. Collect data.
3. Collaborate data, discuss differences, and use estimation to reach a consensus.
4. Calculate the number of laps in one mile.
The main math goals are measurement and relating multiplication to repeated addition. However, as with most performance tasks, there were several math ideas used throughout.
This lesson ended up taking 3 weeks (working 1 day each week). I introduced the lesson by telling the students which adults needed us to help them. We talked about the benefits of walking for exercise especially as people get older and can’t do more strenuous activities. I told them we know that it takes 22 laps around the gym at the high school to make a mile. But how many laps would it take if we walked around the inside of our elementary building? I asked them to estimate--would it take more or less laps? The students agreed that the elementary school was bigger than the high school gym, so it would take fewer laps. But how many? Just how far is a mile?
[Day 1]
[Day 2]
[Day 3]
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.
This performance task was rich
with math ideas. Not knowing what the
students could and couldn’t do made planning and facilitating their tasks more
difficult. However, I believe the
project is worthwhile and worthy of doing again with next year’s class. A regular fourth grade teacher would not have
experienced many of the pitfalls I experienced or would have at least known to
expect them! The important principle is
that the students were DOING tasks that involved math without the math being
pigeon-holed in a worksheet or textbook.
It is critically important for elementary students to grow accustomed to
using what they have learned and learning from what they use. I believe mathematics will become far more
meaningful for students to whom this style of learning is familiar.
These fourth graders had never
attempted a “large” multiplication problem by trial and error with repeated
addition. I believe this approach to
multiplication builds number sense far better than the traditional rote method
of multiplying from right to left.
Computation methods based on number sense need to be introduced before
traditional paper and pencil algorithms in order to enhance their understanding
of the paper and pencil method.
Otherwise it seems the students think you are simply tacking on another
method they have to learn.
I made several mistakes in the
multiplication discussion. I had
specific number sense skills in mind that didn’t match the developmental level
of the students. When they wanted to
add repeatedly, perhaps I should have let them solve the problem that way. I know that in the least I should have let
them add on from the multiple of 10 instead of forcing the multiple-of-5
strategy. “Forcing” number sense is not
possible; rather it must be developed individually.
_4_ Was the purpose of the activity clear to me?
_3_ Was the purpose of the activity clear to my students?
_4_ Did the activity support a range of learning styles?
_3_ Were the students engaged during this activity?
_14_ Total points (out of 16)
The purpose (objective) of the activity was…
I communicated to the students why we were doing the activity by…
…giving them a purpose (helping the principal and secretary) and then reminding them as the steps unfolded of what our final goal was (How many laps should they walk to walk 1 mile?) I did not focus on telling them the “math” objectives, but rather looked at the purpose as solving an actual problem.
The part of the lesson that was student-centered was…
…the whole thing except the introduction and the instruction on using repeated addition in conjunction with benchmark multiplying. If the students had already developed the sense of multiplication, the teacher-centered discussion would not have been necessary.
The elementary floor is tiled with 12” square tiles. The students knew that 12” = 1 foot. I told them that 1 mile = 5280 feet. I asked them if we lined up 5280 of these tiles out on the road, would it reach to Adrian (the nearest town, about 13 miles away)? At first two or three students thought it would, then they said, “No, because Adrian is more than 1 mile away.” How far would 5280 tiles reach? No one had any idea, so I named a landmark about a mile down the road that the students would recognize. But back to the original question--how many times would we have to walk around the elementary school to have walked a mile? How could we figure it out?
One of the students was quick to suggest that we count the number of tiles around the corridor. After congratulating him on a great idea, we talked about whether it would matter where a person walked. The students agreed that if you walk closer to the classroom walls, you walk further. We decided to always stay 2 blocks away from the walls. I asked them if they thought we had a workable plan. One of the students mentioned some permanent obstacles that would keep them from staying 2 blocks from the wall. We decided how to deal with that. After some more reflection, I pointed out that there was a short section where the walkers have to “cut across” and not move in a line with the blocks. The students were not aware that counting those blocks like the others would not work. I told them we’d measure that length with yardsticks and combine it with our block count. Off we went to count blocks. This is where the adventure began!
Who would think that simply counting blocks would be a difficult and time-consuming task? I had suggested that the students take a piece of paper to record different sections so they wouldn’t have to count so high and risk getting lost, but even with that it was mass chaos! I started the students at different corners so they wouldn’t all be bunched up together and throw each other off counting, but that didn’t help much either! First problem--do you count corner blocks once or twice? Second problem--one of the classrooms had temporarily moved a table out in the hall in the way of the normal 2-block-from-the-wall path. Some students came to me and said they couldn’t count that side of the path. I watched another student crawl under the table to count the blocks! I asked two other students how they dealt with the surprise obstacle; they told me they just skipped that whole wall because of the difficulty! Oh my--fourth graders really ARE different from high school students! What a wild variety of data we collected. Time was up for the day, so I had a week to contemplate before we would meet again.
[Day 2]
The next week I presented them with a simplified scale drawing of the elementary. Each block represented 2 tiles on the real floor (and thus gave me an opportunity to witness their ability to skip count). I told them we’d compare this “map result” with their results from the week before. Again the problem of counting corner blocks surfaced. To my surprise, rather than counting a side by twos, posting the answer, and combining the results, the students wrote 2, 4, 6, 8, … beside each square all the way around the drawing. It took much longer than I expected. We divided up the sides of the school to double-check each other’s counting, added all the results, and finally agreed upon 252 feet. The only part that remained was to measure the angled section of the path. That we would do in our final project session the next week.
[Day 3]
Prior to our session the following week, I put masking tape down on the floor to help the students measure a “straight-line” distance. Equipped with a yardstick for every 2 students, I reviewed with the students what we had done so far. Today was the final leg--we would soon have our answer to how many laps Mr. Hargrave had to walk to travel 1 mile. I asked them if they had ever used a yardstick before and they answered in the affirmative. “Go measure--take turns--don’t get in each other’s way” were my directions. (Note: When a fourth grader tells you he can use a yardstick, make him prove it!) None of the students thought to simply count how many times the yard stick could be placed on the length of masking tape, and then multiply by 3. Only one of the pairs could successfully skip count by 3s to get the correct answer. One pair was trying to add 36 and 36 and 36 and…. The other two pairs had trouble accurately laying the yardstick down, then moving it to measure again. However, we muddled through and came to an agreement of about 11 yardsticks long--33 feet--and added that to our total of 252 feet.
The path around the elementary school was 285 feet. How many 285s make 5280? (The fourth graders were just beginning multiplication by 2 digits and had not yet done any long hand division.) One student suggested we keep adding 285 until we got to 5280. I told her that was an excellent idea. I wondered out loud “How long is that going to take?” and another student said “Too long”! I asked if there was another way to do addition over and over again, and they replied, “Multiplication”. “But we don’t know what to multiply by!” “Guess”, I suggested. One student instantly responded “35”. I wrote 5280 X 35 (in vertical format) on the board. “We don’t know how to do that yet.” several students remarked. I told them this would be a nice time for a calculator, but dog gone it, I didn’t have one with me. Not to worry--we could still do this! I asked them if they could multiply by 10. About half the class remembered how to do that. I suggested we mix the ideas of multiplying and adding and see what we could find. 285 X 10 = 2850. Would ten trips around be enough? “No.” How about 20 trips? “Twenty trips would be too much because that would be 5700.” (They didn’t do that mentally, but rather used paper to add.)
Now we know he should walk more than 10 times around, but less than 20. What now? Several students had begun repeatedly adding 285 to 2850. One student suggested multiplying by 5 and adding that to 2850. I did that on the board with their help. 285 X 5 = 1425 and 2850 + 1425 = 4275. How many trips is that? “15.” Is it enough? “No.” So we know it’s more than 15, but less than 20. Time was almost out. We started adding 285 (frantically) to 4275…1 time, another time, another--not quite enough--another. Aha! Too much! So, how many trips? 15, 16, 17, 18…more than 18, but not quite 19. One student had already figured that if Mr. Hargrave walked 19 times around he would have walked 135 feet too far. As the students moved on to their next activity, I congratulated them on finding an answer for Mr. Hargrave’s problem. I returned to my Geometry classroom at the high school, exhausted!
