When you take time for yourself, good things follow. In this case, it was some REALLY AWESOME MATH.
Friday morning, I missed the bus (oops) and was able to drive to work at a legal speed.
I’ve been thinking a lot about Dan Meyer commenting on how we spoon feed each step to a problem solving situation, and so today, I went out on a skinny limb and used this graphic to help us work on our measurement skills. I wasn’t sure where our work would take us, but we’re early on in the unit, so many of my students are still working to measure accurately using a ruler.
I showed them the graphic, and Samuel helped me pronounce all the players’ names. He was our resident expert. Then I opened the floor to mathematical questions.
Here’s what we brainstormed as our big questions.
Then, people started asking more “nitty-gritty” questions, which we identified as being the “questions along the way” you had to answer to get to your big ideas. We kept this poster up as we worked. I stayed near my computer so I could capture students’ comments.
“You need to know how big the field is,” Savanah spoke up. I handed her my iPad so she could find the field size. She paused. “Do I need to know like, how BIG it is or how long the sides are?” “I think you’re asking me whether you need the area or the perimeter?” “Yeah… ohhhh, I need the length of the sides.” Here’s the information she found.
After checking another site to verify the accuracy of her information, we added the dimensions of the field to the poster. (Yes, I know I could have taken a screen shot of the iPad, and I did, but I couldn’t get the image sent to my computer. Hrmph.)
“But what’s a yard?” “Who can answer that?” “It’s three feet,” Ivy answered. “How can you check to see if you agree?” “Well, I could look in my math book, but I remember what yard sticks last year look like, and I know there are three rulers.” (I knew we’d need to convert from yards to feet to inches so they’d be able to convert the lengths they measured on their papers into the actual lengths)
“Well, then you need to multiply by three to get the length – 120 times three.” “Woah. How’re we going to do that?” “Use a known fact, 12 x 3.” “36?” “Yeah, 36.” “So it’s 360 feet.”
They did the same for the other side. Then a group of students wanted to determine the linear distance the ball traveled for each player. I asked how many inches long their picture was, and Marcos stopped us all.
Marcos: WAIT. You blew up the picture from your newspaper article. So our picture isn’t the same size as yours and the distances will be all different. (I photocopied the graphic at 121% so it’d be easier to read than my original copy of the newspaper.)
Me: Nice. That would be a problem if the image were STRETCHED like a rubber band and warped, but since it was enlarged to scale, we’ll be okay AS LONG AS you don’t let me use my original copy, okay?”
Marcos: Okay. So the field is 11 inches long.
“You know, if they would have just included a map scale on this picture, we wouldn’t have to do ANY of this measurement.” “I guess that’s why Miz Houghton wants us to be able to use map scales in social studies.”
Then a few of us worked to create this poster.
We knew the field was 11 inches in our image, but we wanted to know how far just ONE inch would be because then we could find out how far Jone Samuelson’s 6-inch kick actually went. We also knew how long an actual field was, so we tried to find the relationship between the two.
Using a fact family (the triangle drawn above) helped us figure out the ratio. Or. What I initially THOUGHT was the ratio. DO YOU SEE MY GLARING ERROR??? I didn’t notice until lunch. I neglected to convert the 240 feet into inches so the units matched. Drat. I frantically called AP Calculus teacher James Brown to make sure I didn’t make any further errors.
So after lunch we converted 240 feet into inches, THEN used the ratio and found out that one inch in our picture equalled approximately 33 feet.
Some students switched to using calculators for these larger computations, which gave us a chance to talk about how calculators represent 1/2, equivalent fractions (5/10), etc. Above, Alejandra calculated how many feet David Villa kicked the ball (5 inches, according to her measurements, making the kick 165 feet). I asked her about the “33 in. in a inch” she wrote, and she said, “Oh no no no, it’s not 33 INCHES or that would be like a mini soccer field.” So she was also looking at reasonableness of answers.
Another group wanted to know how far the balls would have gone if they were kicked on the moon. Again, I told them to ignore the parabolic motion and just look at linear distance. I know the physics of this aren’t entirely correct, but I didn’t think it hurt the integrity of the original problem situation.
Oh, actually! Selam originally asked how far the ball would go in SPACE, but Maya pointed out that if the they were in space, the player and ball would both push off each other and the ball would never land (AMAZING INSIGHT, RIGHT???). So we clarified that the ball would be kicked on the moon, where there was still a force acting on the ball, but a lesser force than what we’d find on Earth.
Adam went to the classroom library to find out what the gravity was on the moon. Here’s the passage he found, from the DK Eyewitness Book UNIVERSE.
Eayn: It says the gravity is one-sixths of Earth!
Me: So the gravity is 1/6 of the gravity on the Earth. So if we are converting from the moon, what would we have to do to the distance we calculated for the ball kicked on Earth?
Adam: Multiply it by three?
Me: Where did you get three from?
Adam: I dunno.
Milena: Multiply it times five.
Me: Five? Where did you get that from?
Milena: If the moon’s gravity is 1/6, then the rest of the fraction that’s left is 5/6.
Me: Ohhh, I think I see what you’re picturing in your head. But think of the gravity on the Earth as being one whole, and the gravity on the moon being 1/6 of that whole. You’re not looking at the other 5/6ths.
Vy: You’d multiply it times six.
Me: Where did you get six from?
Vy: If it’s dividing by six to get the pull on the moon, then you’d multiply by six to show how much further the ball would go when it has a sixth of the gravity slowing it dowwn.
Me: So you’re saying that fractions can be a way of dividing.
Vy: Yep. And then the opposite, er, inverse, is multiplying, so you times by 6.
(It is perhaps worth noting that Vy has not voluntarily spoken in front of the class in the past year and two months)
Wow. So now that we knew how to find distances on Earth and on the moon, we plugged away, with at least three people needing to agree on their measurements to the nearest half-inch before we would post the results. (reviewing our estimation and rounding unit from earlier in the year)
As we approached second recess, we posted what we’d come up with so far.
We also reflected on what we’d learned over the course of the day, and on the math we used.
As you can see, we didn’t finish everything, so some students asked if they could finish the calculations during Math Daily Five. UM, YES OF COURSE.
What suggestions or modifications do you have to offer me and my students? Where can we take things from here? Other thoughts?