How Big Is A Mile?

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SWBAT use place value strategies to figure out how many times they would have to walk around the gym to equal one mile. They are introduced to the term perimeter.

Big Idea

How far do I have to walk in order to make 1 mile? What if I want to walk 2 miles? We can measure, add and figure it out.

Doing the Measuring

20 minutes

I tell students that we will be walking a mile or more over the course of the next few weeks! They will be measuring a mile. Can they estimate how many times they might have to walk around the gym to equal a mile in distance? We record our list of estimates on the board.

I tell students that we will go into the gym and measure the distance around it in yards. I ask first for estimates of how many yardsticks they think it will take to equal one side of the gym. I record their answers so we can check later.  I assign 4 groups to measure using yardsticks. Each group will measure and keep track of the distance along 1 side of the gym. They have paper, pencil, clipboards and 4 yardsticks/group. I remind students that it is important to attend carefully to their measurements. They should try to make sure that they measure carefully (attend to precision MP6) and use the yard stick correctly (MP7 - use appropriate tools correctly)

I show students what line they will be measuring on and then let each group complete their measurements. 

When students have finished with their measurements, we head back to the room to figure out the size of the gym.

Introducing Perimeter

15 minutes

While using perimeter is a third grade Common Core standard, it is being introduced today as we talk about the size of the gym. It is not essential to know what perimeter is here, but it is a perfect opportunity to introduce the term to students. I don't shy away from a concept with students just because it is not in the second grade Common Core standards, but I do not worry about mastery of the term at this point. 

I draw a large rectangle on the board. I explain that this is like a map of the gym. We now need to figure out how big the gym is around the outside so we can figure out how many times we need to walk around it to equal a mile.

I ask each group to tell us which side they measured and how many yards long it was. I write the amounts around the outside. The students came within a half yard of each other for parallel sides of the gym, so we were able to go to the nearest whole yard and come up with matching measurements for parallel sides. I praised students for how carefully they attended to their measuring task (MP6) when they measured to come out with the same measurements for parallel/opposite sides. (The gym is a 15 yard by 17 yard rectangle).

Now I ask students what we might do to figure out the distance all the way around the gym. I tell them there is a word for the distance around the outside of something and it is called the perimeter. 

I take suggestions and encourage those that suggest adding up all the amounts. If no one suggests this, I try to ask questions to lead their thinking in that direction. Such as, would we get a bigger or smaller number than these for the perimeter if each of these is only one side? 

Once we have decided to add up the numbers, I ask students to try this on their own. I give them a chance to work and then we compare strategies and answers. I ask for several students to share how they solved the problem. 

We decide on one answer of how many yards there are going around the gym.

Yards to Miles

20 minutes

We have found the number of yards around the gym. Now we need to see how many times around the gym we would need to go to get a mile. 

There are 1760 yards in a mile. I ask student how we can figure out how many times around the gym we need to walk, when we know the perimeter of the gym. I tell them that because 1760 is such a big number, we might want to figure out 1/2 a mile. That would be 880 yards. 

I am hoping to get them to think of using repeated addition or subtraction to solve the problem. If they suggest adding or subtracting, I will ask if we can do that just once to get our answer. 

Once we have decided on several alternatives (repeated addition or subtraction) I show them how they could use the  tens frames and hundred's frames as a way of keeping track of what they are doing. I show them that if the gym were 65 yards around (I helped children realize that while we had 64 yards, several groups had half yards in their measurements, and counting by 5s is easier for us than 4s so we are going to use the 65 instead of 64 today) and I need to get to 880 yards, I might take 6 tens frames and 5 ones, and put them in 2 piles. I ask what I could do next? (repeat the process adding 6 more tens and 5 ones and add that up to see that I have 12 tens or 120, and 10 more, or 130 with 2 sets). Is that enough? (no). So what might I do next?  I ask for volunteers to come up and help me with the next step such as repeat the first or  take a one hundred frame and 3 tens and tally 2 more sets.) Now I count and am at 260 with 4 sets (4 times around). Are we there yet? (no) What can I do now? Again I ask for suggestions hoping students will begin to see the pattern by looking for and making use of the structure we have started here (MP7). Now I check for understanding of the structure we are using and ask for a thumbs up from students who believe they can continue on with what we have started?  I pass out plenty of hundreds, tens and ones frames for students to use to complete this work.

(Resource for tens frames and hundreds grids that can be hundred's frames.)

I tell them we will work in partners to solve the problem and then we can use a calculator to check our work. We won't use the calculator first. 

I partner students up so they can support each other's thinking. I want to make sure that students who may struggle with the application of place value strategies to do repeated addition with larger numbers will have a support person to help them throughout the process. I remind students that they should not do the work for their partner, but that they could help them fill a tens frame, or add the frames together. I remind them to talk to their partner about what they are doing so their partner understands the process. 

I provide help as needed,especially by helping students work together as partners. 

When groups finish, I tell them to get the calculator and check to see how they did.