Some Math on the Farm By: Kimberly Mullan

Some Math on the Farm By: Kimberly Mullan

“Hi. My name is Ethan, and this is my friend Paul.” Ethan Paul “He always tells me that I should pay attention to my math teacher in school. Especially if I want to take care of a farm like he does someday.” “Once I asked him “What would you ever use math for on your farm?” “Paul looked at me and said…

What do you think?”

“Well, just to take care of cows we came up with this list of things we needed to understand.” Multiplication Area Perimeter

“It’s spring time on the farm, and the cows are having calves. Each cow has one calf.”

“Paul showed me his 12 cows.” “He always has to keep track of how many cows he has. So, he asked me to help him figure out how many cows he will have after all the calves are born.” 2 1 3 4 5 6 11 12 7 8 9 10

24! “I counted one by one.” 2 1 3 4 5 6 14 15 16 17 13 18 11 12 7 8 9 10 23 24 19 20 21 22

3 7 9 11 5 1 4 6 8 10 12 4 cows 6 cows 8 cows 10 cows 12 cows 2 cows 2 13 15 17 19 21 23 14 16 18 20 22 24 14 cows 16 cows 18 cows 20 cows 22 cows 24 cows “Paul told me that there was a quicker way to count the cows. Instead of counting one by one, I could count them in groups. So, I tried putting them in groups of two like this.”

1 2 “Paul showed me how to write my counting down like this.” This is the number of cows in each group. This is the number of groups. 2 X 12 = 24 This is how many cows I counted!

6 cows 1 2 3 4 5 6 6 cows 7 8 9 10 11 12 6 cows 13 14 15 16 17 18 6 cows 23 24 19 20 21 22 “I can also group the cows this way.”

“So Ethan, how would you write down your counting this time?” “Well, I put 6 cows in each group. So, the first number is 6.” “I multiply that by the number of groups. I made four groups.” And guess what? I still get 24! 6 X 4 = 24

So…. How many multiplication ways can you think of to count Paul’s 24 cows? 12 X 2 = 24 2 X 12 = 24 6 X 4 = 24 4 X 6 = 24 3 X 8 = 24 8 X 3 = 24 24 X 1 = 24 1 X 24 = 24

“Ok Paul, you’re right. You do use math to count your cows. We put the cows into groups and multiplied to see how many there were. But you can count just about anything that way.” “What other math do you do?” “Well, I have to figure out how many bales of hay to feed them, how many fence posts I’ll need to build a fence around the field, how much space the whole group needs to get enough grass, the area of the field…”

“Um… what’s area again?” “Area is a number that tells how many times a certain kind of unit can fit inside of a shape.” “Let me show you what I mean.”

“My favorite field is in the shape of a rectangle. I need to figure out how many cows can live in this size field at one time. So, let’s say that I want to give each cow a 1 acre sized square of grass. This unit is called a square acre.” 1 acre “Just by looking at this picture, how many 1 acre squares do you think will fit?” “To find out how many cows can live in the field let’s count how many square acres will fit in the rectangle.” 24 1 acre “This means that the area of my favorite rectangle shaped field is 24 square acres. So, could all 24 cows live in this field?” 4 7 10 13 16 19 22 1 5 8 11 2 14 17 20 23 3 6 9 12 15 18 21 24

“You figured out that there was enough room for 24 cows to live there because the area of the rectangle field is 24 square acres. But there is another way you could’ve found that out without counting the square acres one by one.” “Look at the picture again.” 3 acres 8 acres 1 acre “Remember that one square acre is the unit we are using. It’s how much space each cow needs. I labeled the picture counting the acres instead of cows because I am measuring my field in acres, not cows.” 1 acre

“Area is a measurement of how many times a certain kind of unit can fit inside of a shape.” “We can still write our counting like this.” This is the number of square acres in each column. This is the number of square acres in each row. (A column looks like this.) (A row looks like this.) 3 X 8 = 24 This is how many square acres we counted! “We counted the square acres in the rectangle by multiplying the number of squares in each column and row of the rectangle.” “So, the field has an area of 24 square acres!”

“You’re right Ethan. What if I wanted to give each cow a bigger square? That way they would be able to eat more grass and grow much bigger.” “Let’s give each cow 4 square acres. That should be plenty of space.” 2 acres 2 acres 2 X 2 = 4 Each cow gets 4 square acres.

2 acres “Let’s look at the same rectangle field again.” “How many units can the rectangle field hold? Remember our unit this time is four square acres, not just one.” 2 acres “Look at the picture first and estimate how many cows can live here now.” “Well, let’s find out.”

“Uh oh, Paul. We have room for 4 cows, but what do we do with the room left over?” “Well, let’s look at the 1 acre squares again.” “We’ve said that we now need four of these 1 square acre units to have enough room for 1 cow.” 1 acre X 4 = “But these 4 square units don’t have to be in this big square shape.” 1 acre “So, that means that the squares could look like this?” “Yes Ethan. That works because you still have 4 square acres. You haven’t changed how many you have for a group, just rearranged the shape of those 4 square acres.”

“Try the new arrangement in the rectangle field.” “Ok. I don’t have tons of room left, but I’ll bet one or two more cows can fit.” = 4 square acres = 1 cow space

“So, six cows can fit in the rectangle field now.” 1 2 3 4 5 6

“But that means that all the cows can’t stay in the same field! What did I do?” “You didn’t do anything wrong Ethan. You’re exactly right that all the cows won’t fit in the rectangle field now. Each cow takes up more space, but did that change the area of the field?” “No, because there’s still 24 squares. I just put them into groups of 4 instead of counting one by one. Right?” “Go ahead and check, just to make sure.”

“Ok, I’m counting again to make sure there are still 24 square acres.” “Well, there’s still 24 square acres. I just counted the long way again.” 1 5 7 9 11 13 15 2 6 8 10 12 16 3 4 14 17 18 19 20 21 22 23 24

“How else could you have answered your question Ethan?”

“I could’ve multiplied!” 4 square acres per cow 6 cows X = 24 square acres = 24 4 X 6

“Now you’re thinking like a farming mathematician.” “But there’s still not enough room for all of the cows in this field.” “You’re right again. So, what could we do about that?” “Can you build a whole new field? Or just make this one bigger?”

“I can do either of those things Ethan. But in order for anything like that to happen, I have to build a fence first.” “Otherwise the cows would run away?” “Yes, that’s why I need the fence. Before I can begin building it though, I have to figure out how long of a fence I need.” “So, you need to know how far it is all the way around the field.” “Exactly. The distance around a shape is called the perimeter. You can find it by adding up the lengths of all the sides.”

“Let’s look at my favorite rectangle shaped field again.” “To find out how long of a fence I have to build I need to measure the distance around the outside of the field. In other words, I need to know the perimeter.” “Let’s divide the rectangle into square acres again. Now we can count each acre along the outside edge to find the perimeter.” “The perimeter of the field is 22 acres.” 1 acre 2 3 4 5 6 7 8 22 9 10 21 20 11 19 18 17 16 15 14 13 12

“But let me guess, there’s another way to count the perimeter?” “Yes there is Ethan. Let me show you.” “Ok. I know that since it’s a rectangle the opposite sides have the same length.” “ Let’s add up the total acres on each side first.” “That is true. Now you can add the numbers together.” 8 + 3 + 8 + 3 = 22 or The perimeter is 22 acres. 11 + 11 = 22 8 3 3 8

“Now, to build this 22 acre long fence I need fence posts.” “We’re in luck because Tractor Supply Company is having a sale right now. We can get all the fence posts we need for a one acre long fence at a very good price. We’ll talk money later.” “How many fence posts do you need for each acre?” “Well, for now, I’ll use 38 fence posts for one acre.” “How could you find out how many posts I should buy?” “Multiply!” “Yes!”

“Paul, can I use a calculator for this?” “Well, for these big numbers I guess that’s ok. Just make a guess about what you think the answer might be close to first.” “Ok. I’m multiplying the number of acre lengths I have by the number of fence posts you need for each acre. That should tell me how many fence posts I need for the whole fence.” X 38 = 836 fence posts 22 “Wow!”

“You’ve done a lot of math today Ethan.” “But there’s plenty of math for another day too.”

Bye!

Special Thanks To… • Ethan Cokerl • Paul McPherson • The McPherson cows • Clip art - cow and fence (slide 4, 31,32)

Some Math on the Farm By: Kimberly Mullan