Greenies

Our black lab Mandy is 3.5 years old, weighs a ton, and her breath used to smell like death. Until we started giving her one Greenies a day. My husband orders them from Amazon, he also gets them for our neighbor's small dog Bailey.

Although both boxes weigh the same 36 ounces, Bailey gets 130 treats in the Teenie size and Mandy gets only 24 in the Large size. This caught my attention which led to this task with my 8th graders who happen to be working with similar shapes. (Like I had planned this all along.)

Greenies are sold in various size packages. I'm interested in the 27-oz and 36-oz.

To launch the task, I hold up the 2 treats: 1 Large and 1 Teenie. I tell them that there are 24 Large ones in a 36-oz package, and I want them I guess how many Teenie ones are in a 36-oz package.

Then I give the students these:

• The photo of the 2 packages (so they know only the highlighted information in the table above).

• Each group of 3 students get two real treats: 1 Large and 1 Teenie.

In return, the students need to give me these:

• A 2-dimensional outline (with dimensions labeled) of what a Petite, a Regular, and a Jumbo may look like. For example, these are the actual outlines of the Large and Teenie. They may write down the thickness also.

• A completed table with the missing counts filled in.

Highlights of this task:

1. Kids use some known information to construct new information. They use modeling to figure out what the other sizes may look like and how many of them would fit in a 27-oz or 36-oz package.

2. It's kinda messy and weird. While the kids can measure whatever lengths of a treat, how do these numbers translate into the mass of each treat?

3. It's good to work with solid objects instead of just flat polygons when learning similar shapes.

4. The reveal (Act 3) of something like this is always a lot of fun. Not only the reveal in the count per package, but also how close their outline sketches are to the actual treats when I bring in the Petite, Regular, and Jumbo.

5. How much does a 36-oz package cost?

6. Is the Jumbo a shot in the dark? Would kids think to ask me for the size of the dogs? How does this help, if at all?

Prices, Proportions, Percents

I was in Garden Grove with my son on Sunday, and he insisted that I try this smoothie place called Tastea. With Jamba Juice and Blenders and all the other juice bars around town, I was skeptical that this joint's concoctions would be anything different. He ordered a taro milk tea and I got a Thai tea, both with boba. Just one sip and I said to him, "Let's order another round! It's a long drive home!" Soooo delicious.

While there I saw a math lesson brewing, so I picked up their menu with prices. This is the lesson with my two classes of 6th graders.

Me: (I tell them about Tastea and how I wish it were closer.) Okay, let's start with something you might be more familiar with, Starbucks. I love that now I live within walking distance from one! Do you know the different sizes that they have there?

Class: (When I refer to "class," I don't mean the whole class, of course, but somebody in the class joins in on the conversation.) Tall, grande, venti.

M: Do you know exactly how much liquid each size holds? (They make various guesses. I bring out the 3 sizes that I got from Starbucks so they have a visual.) I normally order a tall mocha frappuccino, let's say the price is $3.50. Do you think the venti, which is twice the volume of the tall, would cost twice as much, or $7.00?

C: No.

M: Why not?

C: You normally get a better deal with a bigger size.

M: What do you mean by a "better deal"?

C: (All their answers show me that they understand the idea of more bang for your buck. Then finally someone says...) Lowest unit price!

M: Right! That's why so many families go to Costco. Buying in bulk normally saves us money because the item has the best unit price. Well, we're talking about Starbucks now, so buying more is a better deal, but drinking more is not so good for our body. Let's fill in this sheet. (I pass out this handout. And here’s the key.) How do we calculate unit price? What place value should we round it to? How do you write thirty-one-cents-per-ounce?

M: Tastea has three sizes: mini, gigantic, and even more. Their teas can also be purchased by the "partea jug," which holds a gallon. I've given you the prices of the 10-ounce minis for the three different types of drinks, your job is to figure out the prices of the other sizes. You'll work in small groups to figure out these out. So, do you think the gigantic will cost twice as much as the mini because it holds twice as much?

C: No. It'll cost less.

M: How much less? Well, that's your group's job to come up with the best estimate. We have Starbucks' prices for their three sizes, you could look at how they price their drinks. But here's the sweet deal for you. You and your group mates do the math that you need to, then write down your first estimation right here in this column. Bring your paper up to me (only the "captain's" paper), give me a few seconds to figure out the percentage that your estimation is off by, and I'll write it in this column and give you back your paper. What percent do you want to see, large or small? What if your estimation were the actual price — what percent would I write there?

I tell them that they could figure out the actual price of the drink if they knew how I calculated the percentage of error. So, work work work. Think think think. What makes sense? Oh, I remind them that the percentage does not indicate if their estimation is too high or too low. So, again, what makes sense?

We also note that prices generally end in a 0, 5, or 9. So, even if the calculation tells them the price should be $4.23, they might want to change that to $4.25 or $4.20.

When they bring up their paper again with the second estimation, all I do is write their estimation again in pen and circle it — this is so they can't change their answer and I know that I've seen it. I do NOT fill in the "Actual Price" column at this time because the groups are working at different rates, and in a crowded room, it's easy for kids to see each other's papers, even inadvertently, and the game of estimation is over if they saw the actual price beforehand.

They simply move on to the next size to make a first estimation again. We repeat the process.

When all groups are done with estimations for the first type of juice — smoothies — I tell them what the actual prices are.

Now, it's their turn to figure out the percentage of error. I give the groups about 10 minutes to do so without help from me. At the end of the 10 minutes, either there's at least one group that knows how to do so and can show it to the class, or no group knows how, then I'll walk them through the calculation by asking them questions to figure this out.

They continue in the same manner for the Slushy Freeze and Specialteas on page 2. This time hopefully they'll be able to work backward from the error percentage that I give them after their first estimation.

Reasons I'm proud of this lesson:

1. It's about proportions, but many priced items in real life are not directly proportional. The kids knew this coming in because they've been consumers.

2. We get to talk about business strategies that entice people to buy the larger sizes while still make a profit. (Starbucks calls it "tall" because it rhymes with "small," but clearly the word tall naturally elongates the imagination.)

3. Students get to make estimations throughout, but they know these aren't "wild-ass guesses." They start with the calculation of proportions and adjust the prices accordingly. They get to critique and argue with their group mates to come up with the best estimations.

4. I get kids to think about percentage in a context that they can wrap their heads around. And they want to know how because their second estimation could be dead on if they knew.

5. It's fun that the error percentage does not indicate if their estimation is too high or too low. A few groups do go farther in the wrong direction. Oh, well — good to learn that now.

6. It'd be fun for me to get Starbucks or Jamba Juice for the group with the lowest total in percentage errors.

Updated 04/05/14

I got some thoughtful reflections on this lesson, I'll just share two:

I learned how to work backwards with percentages and try to get the number spot on. I also learned how business would price things by dropping the price by the perfect amount. My number sense got a lot better from all the multiplying, dividing, and reasoning. It was very difficult, which I'm very happy about. The teamwork was probably the hardest part of the project. M and I are very competitive, and we got different answers a lot. I learned how to work together with others a lot better, and it doesn't move your team along to place blame and argue. I'm really grateful we did this project because it was very hard and worthwhile. It was a great use of three days!

I learned how to use different data to get answers. Also, we have to see a pattern. This Tastea assignment was really fun. I enjoyed it and look forward to another. Teamwork is really important even though people can't agree, you got to support it. If your group gets it wrong, but your answer was right, you can't blame someone or put them down because probably they will get some right for you. So always stay positive to your teammates and encourage them.

Equilateral Triangles

Thank you to Dan Meyer for this great task idea on equilateral triangles.

Act One is this video which asks for a ranking of how well each teacher had drawn his equilateral triangle.

But as soon as I saw the video, two thoughts came up:

1. We're in a 0:1 classroom right now, and having the whole class come up to the big screen just isn't efficient.

2. I want to be in the competition! The kids will all want to do this.

So I knew this would have to be a pencil-to-paper activity in my class.

Eyeballing

My instructions to twenty 8th-grade geometry students:

Here's a blank sheet of paper for you to do this task. Be sure your pencil is sharpened. Put your name in upper-right corner.

Using only your eyeballing skill and your pencil, mark three dots in the shape of an equilateral triangle.

Gabe normally asks amazing questions that make my heart sing. Today he asked, "So you want us to draw the best equilateral triangle?" I replied, "No, Gabe, I want the crappiest one you can draw."

Now, connect the dots with a straightedge.

Pass your papers forward, I'm going to make a photocopy of your drawing so I can have a clean copy of it just in case.

While I'm making the copies, I want you to think about how you are going to decide which triangle is most equilateral.

I dashed quickly to the copy room a few doors down. I wish you wouldn't tell anyone that I'd left the children unattended for 3 minutes.

I'm now going to randomly pass the papers back, meaning you should have someone else's drawing to work with. Write your name on their paper as the "tester."

Okay, I want to know which one of you drew the bestest equilateral triangle. To do that, we need to come up with some kind of criteria, a way to test it, a way to score it. Talk to me.

They said that an equilateral triangle had to have three congruent sides or three 60-degree angles. So, I said:

I guess you'll be measuring the sides and angles. Then you get these six numbers. Do you need all six? What are you going to do with the numbers?

We know what perfection looks like: 3 congruent segments, 3 congruent angles. Let us safely assume that no one drew a perfect triangle. So how far from perfect is it? What score would it get? How fair is your test?

I need you to work quietly by yourself for now. Get your tools: ruler, protractor, compass, calculator, whatever. Then figure out a way to test for equilateralness.

Working alone

They worked diligently and carefully. I appreciated seeing this student use her ruler to extend the side length to help her spot the angle measure more accurately. Julia asked, "Measure in centimeters, right? To the tenth?"

Working in small groups

Now I'm going to randomly put you in groups of three. In your small group, share your scoring strategy. Fight about it. Defend your methods. Eventually I will ask you to choose the best method from your group to present to the class.

By the way, just because someone in your group is in possession of a drawing that you know is far from perfect doesn't mean that his/her method for testing it should be dismissed.

Oh, hey, should a larger triangle deserve more points in your scoring system? I mean, is it harder to throw down three dots that are spaced farther apart?

I moved from group to group, listening to their discussions and observing their calculations.

Watching them, listening to them, asking them questions — I didn't want to be anywhere else.

Presenting to the whole class

One by one, the groups were eager to share. They questioned each other: Why 100? Why 30? Why divide by 3

Summary of what they shared on the board. You can see that 4 of 7 groups used either side lengths or angle measures and not both.

Voting on the best one

The scoring method from Gianna's group got the most votes with 9. Gabe volunteered, "None of these is spot on. But I don't know what the best way is either." I said, "Thanks for saying that, Gabe. Me neither. But I love what you guys are all coming up with!"

Over two days, no one mentioned using perimeter or area. And I vowed not to say anything — I wanted the kids to drive this entire lesson to wherever it needed to go.

Okay, Gianna's famous now, we'll refer to her group's method as "Gianna's formula" from now on. I need everyone to go back to the triangle you have and use this formula to find a score for it.

Testing another triangle

I made another set of copies from the originals (during my prep this time) and randomly passed these out. This very diligent work was still human, and I just felt each triangle deserved another pair of fresh eyes on it.

Now, you're going to apply Gianna's formula to another triangle. This way each person's triangle gets tested by two different classmates. Record your numbers on the board.

About 7 of the 20 sets of numbers had enough variance that I had to ask both scorers of each set to re-do their calculations and/or measurements. I then took the average of the two scores.

The results

I went back to the kids' original drawings and measured all the sides (with a ruler, thanks), applied Dan's formula using this calculator and here are the results. The names highlighted in yellow share the same rankings via Dan's and Gianna's formulas. The greens are off by just one.

We thought this lesson was pretty great. Maia said, "Our way was not too bad at all.”

Last Math Lessons

On the first day of school, I promised all my students that I'd do my best to make math relevant and challenging. I also promised to never waste their time, therefore they could expect to do math every day in my class, including the last day of class.

I did not promise them that math would be fun because doing math is always fun to me. I gave out math puzzles at my kids' birthday parties when they were younger, also at a baby shower that I gave, and of course at my own wedding.

Math 6

I called this lesson How Many Regions? (Adapted from AIMS)

I gave each student a piece of 4-by-4-inch paper and a handful of 6-inch long thin strips of construction paper. Each strip placed on the square paper would represent a "cut."

Question: What is the maximum number of regions that you can divide the square into using n number of "cuts"?

We did the first two cuts together as a class, and each student kept track of the data in their journal.

Because the goal was to get the maximum number of regions from the cuts, the kids learned quickly that the cuts needed to intersect. For example, two non-intersecting cuts created only 3 regions. A kid yelled out, "Parallel cuts!" Gotta forgive kids who blurt out academic language!

This class has been working with a pattern worksheet almost every Monday in my class, so they know to look for a recursive rule and try to find the equation for the nth term. Recursively, they saw the pattern, so it didn't take long for them to figure out that for any n (cuts), the maximum number of regions is the sum of n and the number of regions from the previous n.

With 30 seconds left of class, I told them my gift to them was to try and figure out the equation. Matt asked, "Can I email you then over the summer?"

Algebra 1

Our 8th graders' last day of classes was yesterday (Wednesday). They'll show up this evening for their promotional ceremony, dressed to the hilt.

I needed a quick one-period lesson, so I had them make 2012 Clocks. This is a common assignment: using all 4 digits in "2012" and each digit only once, they had to create expressions that would equal to the hours of 1 through 12 on a clock.

Geometry

I've seen different versions of this problem; the first time I worked on it was when it involved 3,000 bananas and 1 camel traveling 1,000 miles. I know there's a perfectly good strategy called "solve a simpler problem," but we could also start with a simpler one!

I gave my kids Desert Crossing, also from AIMS:

You live in a desert oasis and grow miniature watermelons that are worth a great deal of money, if you can get them to the market 15 kilometers away across the desert. Your harvest this year is 45 melons, but you have no way to get them to the market, except to carry them across the desert. You have a backpack that holds up to 15 melons, the maximum number that you can carry at a time. To walk across the desert, you need a certain amount of fluid and nourishment that is supplied by the melons you carry. For each kilometer you walk (in either direction), one melon must be eaten.Your challenge is to find a way to get as many melons as possible to market.

As I type this, Slater does not know that I will be awarding him the Math Excellence Award at the Promotional Ceremony this evening.

Here is Slater's work that shows one way to get the correct answer of 8 watermelons.

I did this a few weeks ago and forgot all about it; so here's a little blurb on it.Each student quickly constructed these two prisms from two same size papers.

(Dan Meyer folded them into cylinders.)

Question: Pretend the two prisms have bottoms on them, which one holds more popcorn? Take a look... Okay, grab the one you think holds more, or grab both if you think they're both equal.

And this was how they grabbed:

Then they measured and calculated the volume.

My favorite student comment: The tallness didn't make up for the fatness.

Estimating volume is a funny thing.

Visualizing Volume is Tricky!

This was one of those lessons that I think I gained a lot more than my 6th graders did. It was meant as a one-period activity, but I kept going off on different tangents and brought the students along for the ride.

Over a month ago Andrew Stadel tweeted me a picture that he took of a William Sonoma display of their cupcake mixes. I was at our local mall last Sunday and saw a similar display. We both thought about buying the mixes to make cupcakes for our kids, but it was $15 a can, and we'd need three, so the poor teachers said no can do.

I projected the images above and asked the kids to give me a guess of how many dozens or how many individual cupcakes can the large container make when the small [real] container can make 1 dozen or 12 cupcakes.

Their guesses were all over the place, ranging from 47 to 994 cupcakes. (We were very careful whether the submitted guesses were in "dozens"or in "individual cupcakes.")

So I did the only thing I knew. I replicated the two cans so the kids could see them physically in the room instead of just on a still photo. Granted the large "can" made from butcher paper was pretty awful.

But, before I asked for another guess at the number of cupcakes, it occurred to me that I wanted to know if kids were better at guessing one and two-dimensional items.

Question 1: How many times taller is one segment than the other?

The two segments below are proportional to the heights of the two cans.

Their estimates:

Question 2: How many times longer is one circumference than the other?

The two circles are proportional to the cans' tops/bottoms.

Their estimates:

Question 3: How many times larger is the area of one circle than the other?

The two circles are proportional to the cans' tops/bottoms.

Their estimates:

Then, I let the kids — row by row — come up to get a visual check at the paper replicas of the cans. But they may not manipulate the models because I was more interested in seeing the difference between their guesses of the still photo and the physical models.

Their estimates:

If I wrote this post as Dan's 3-Act Lesson, then it was time for Act 2: figure out the volumes of the two cans using their measurements. (Yes, I carry a measuring tape wherever I go now.)

I tried to show the kids equivalent measures whenever possible. We worked a lot with centimeter cubes this year, so this was a rare time that we measured in cubic inches.

And here are some estimates from grown-ups who only saw the left image at the top of post:

(Got more guesses after I printed this: 1,000, 182, 600, 576.)

The straight lines [heights] seemed easiest to estimate. It got a little bit tougher when these lines bend into circles [circumferences] — and there was a large number of over-estimates here. I thought area estimates were pretty good, average of all 33 student guesses was 28.3 (calculated was 26).

The volume estimates, from kids and adults, remained well under the calculated numbers. I don't know what to make of all this. But I kept wondering: Are boys or girls better at making volume estimates? (From my small sample of 33 students, the girls were closer.) How about science teachers? The 3D models helped overall; and I bet if I let the kids do everything but measure the cans, their numbers would be closer. Interestingly, at one point I'd placed the smaller can inside the larger can, and kids who stood nearby kinda gasped. One said, "Oh, you can fit a lot inside."

Thank you, once again, to Andrew for sending me that tweet that started all this! He did a wonderful presentation of 3-Act Lessons to his staff.

Area of a Circle

After finding the formula for the circumference of a circle, my 6th graders were ready to work on finding the area of a circle.

I asked them to draw a circle on notebook paper, any size, but not too small. Then I gave each a centimeter cube to trace one face onto their paper to remind them how much the area of one square centimeter covered. (You can see it drawn on top right pic below.)

Question 1: Give me a guess — only by looking — what is the area of your circle, in square centimeters? Please write that number on your paper, label it "guess."

Question 2: Now use whatever tools you need, give me a better answer. I know some of you already know the formula for the area of a circle, but you may not use it unless you can tell me where it came from. (No one even tried.)

I can't tell you how happy I was to see all the different ways the the kids had tried to approximate area. Their perseverance humbled me. A few students drew triangles, knew to use "base times height divided by two," but erroneously used radius as height.

Last week I was reading Mimi's post about estimating area of circles, and Sue VanHattum reminded me of the rectangle model in her comment.

I had the kids fold their circles like this.

They cut out the pieces — turning every other piece 180 degrees — and glued them together. Cristian said, "First we had a circle, then triangles, then a rectangle. That's crazy!"

Never once did I answer any of the questions. I just asked them. I began with, "What is the area of this rectangle or parallelogram?" Step by step, as a class, the kids walked this equation all the way to Area = pi x radius^2.

This morning, two days after the activity, I did a My Favorite No to see if they remembered: 31 out of 33 students got the correct circumference formula; 24 of 33 got the correct area formula. Bonus if they showed me how the rectangle model helped explain the area of a circle.

Why Wait for Calculus?

(I just had my very best lesson yesterday, on a Friday, thank you. I feel almost brilliant right now. And I only feel like this once every 47 years, so please stay and read this post!)

My own kids tell me they will stock up their dorm rooms and apartments with junk food and soda when they move out to make up for these years of deprivation. (And this is supposed to make me feel bad.) So when I intentionally bring home a snack, like this bag of kettle corn, I usually find it empty within 24 hours. But seeing the empty bag made me think of a volume activity that I could do with my 6th graders with all these other bags of Orville Redenbacher's popcorn.

But the activity I had in mind — maximizing the volume of a box — is commonly done in a pre-calc or calculus class. These are my 6th grade babies. But didn't we do okay with approximating the volume of a torus via my doughnut lesson? So, why can't we do this too? I have to get rid of the popcorn.

I randomly assigned the kids in pairs, gave each pair two sheets of white copy paper. I told them to use one paper at a time to make a box — the goal is to make the box as big as possible so it'll hold the most popcorn. But the box must be made simply like this: cut off 4 corners from the the paper, then fold up the sides and tape them together. I used a half-sheet (so they couldn't duplicate mine) to demonstrate what I meant.

They quickly went to work. A few students were NOT cutting off square corners, so the top edges of the sides didn't line up. Two groups folded in their papers, in addition to cutting off corners, so they had to re-do.

Ryan and Annamaria wanted to make a shallow box. Ryan said, "... it doesn't matter how high it is."

Rapha and Cristian made the biggest corner cuts that I saw in the first round.

Mike's and Roy's first box was the shallowest in the class, but they changed their mind for their second box.

With 10 minutes left in our first hour together, I asked the kids to measure the box and find the volume. They had no trouble with this since we did the doughnut. They recorded the volume inside each box, and I tacked them on the board. (The butter seeped through in few of the boxes.)

Well, that was fun. I pointed out that two of the bigger boxes were over 1,000 cubic centimeters. The bell rang. I said, "We'll wrap up this afternoon."

I didn't know what I was going to do to "wrap up" the lesson. The microwave actually overheated — my room stank of greasy popcorn.

There was a confidence in me, however, that the kids would help me figure out how/where this lesson could go next. I began the afternoon hour by going over what they'd learned in the morning. They said:

The four corners must be of the same size. (I never told them this in my instruction.)

Each corner must be a square. (I didn't tell them this either. Not everyone was convinced of this, so I cut non-square corners to show them.)

There was a limit to how big the square could be. (I loved this! And this made me ask, "Is there a minimum to the size of the square?" Their eyes squinted, almost as if they were trying to "see" how small these corner squares could get. One kid said, "No. Technically, no.")

The volume numbers that people wrote down could be wrong.

By then they understood the different boxes and their volumes depended on the size of the corner squares that would get cut off. We focused on this. I asked them to draw a 10 x 12 rectangle in their math journal. We removed 1 x 1 corners from this rectangle and found the volume. I guided them through the next 2 x 2 corners. They continued on their own.

Then I gave each kid another white piece of copy paper. We measured the length and width of the paper and agreed that the paper was 28 cm x 21.5 cm. I asked them to build a systematic table like the one they just did in their journal. I said something like this, "Because you now know how to figure out the volume without actually cutting and making the boxes, see if you could figure out what size square the corners should be to maximize volume."

I saw kids high-fiving each other, "The corner has to be 4 by 4!" Rapha and Cristian beamed after congratulating each other, "That was one of the boxes we made!" We ended class with that. I swore I felt myself tearing up.

On Monday we'll play around with this applet.

And we'll ask Wolfram Alpha to take the first derivative of the volume for us. (I'm pretty sure the class could write this equation V = (28-2x)(21.5-2x)(x) for me to enter into WA.) Well, I actually just did it, and WA gives the side of our corner square as approximately 4.01965. My kids got 4 — pretty damn good for 6th grade brute-force math. Now that I'm writing this, however, I am really most proud of how well the kids had worked together. I randomly paired them up — a handful of the pairs were like the odd couples: high/low, shy/outgoing, squirrelly/quiet, jock/nerd, princess/cowboy. There was not a whisper of whine when their names were called to pair up. How did I get so lucky?

I am a doughnut.

I cannot eat a doughnut without thinking of Eddie Izzard. Language caution in this video clip.

Ich bin ein Berliner aside, I needed a calorie-laden lesson for my 6th graders to welcome them back into the classroom after two wonderful weeks of spring break.

(We are also two short weeks from state testing and should be reviewing for the test. But who wants to do that when one can eat a doughnut instead?!)

I got the idea for this lesson from NCTM.

Estimating Volume

I held up one doughnut and a centimeter cube and asked each student to write down an estimate for the doughnut's volume in cubic centimeters. But first, I needed to tap the kids' prior knowledge of volume.

Me: So, what is volume? What is the volume of this doughnut?

Julia: Length times width times height.

Me: Hmm... Where is the length of this doughnut? Or its width?

Julia: Oh...

Matt: It's the inside of something.

Sam: But that's area.

Rapha: It's the whole thing. The doughnut itself.

Matt: No, I mean the space inside.

Julia: Oh... it's how many of those cubes fit inside the doughnut!

I always dread hearing kids give me a formula for anything unless I specifically asked for a "formula," but this time I didn't break into hives.

I asked if it would be okay that I just counted the centimeter cubes in each of the solids below to find volume. The kids immediately pointed out the empty space inside and one said, You have to add like 5 to your answer to make up for the space. Another student said, More like add 10.

The class agreed that the cylinder held more cubes than the sphere. They also agreed that the doughnut would be bigger than the cylinder. I told them there were 69 green cubes in the cylinder and allowed them to change their doughnut volume estimate if they wanted to.

When a centimeter cube was placed on top of the doughnut box, almost everyone said, Wow! (This made me happy because it showed they were really thinking of volume as volume and not as length-times-width-times-height.) They recorded their estimates for the box's volume.

Estimating Surface Area

What the kids said when I asked them about SA: the outside, all around, total area, all the surfaces, doughnut skin. I added that the amount of glaze could also represent SA.

I told them there were 22 by 18 rows or 396 square centimeters on this grid paper and asked for their estimate of the doughnut's surface area.

They also recorded their estimate of the box's surface area.

Measuring the Doughnut

To manage this potentially messy lesson, the doughnut had to stay on construction paper and all measurements were done with a paper ruler.

Students took as many different measurements of the doughnut as they thought necessary to figure out volume and surface area.

They ate the doughnuts, and Cristian wanted me to take a picture of him.

We'll wrap up the lesson tomorrow and talk more about approximating for volume.

No, none of my 6th graders brought up any of these -us words: torus, annulus, calculus. But we did a lot of estimating and measuring with visual and concrete models. We each gained 200 calories from this. 

Updated 04/17/12

I was telling @Peter_Price in my comment about making a cylinder similar to the doughnut, then I found this roll of masking tape that was really close in size!

My 6th graders are familiar with area, so surface area was easy to grasp. (I felt smart when I could just peel off the tape to show the side SA.) I showed them my set of vinyl nets to the geo solids also.

I needed the students to figure out how to calculate volume on their own. I used these two models because of both the CDs and the patty paper sheets are very thin.

Student A: Find the area of one sheet and multiply by how many sheets there are.

Me: There are 1,000 sheets here, so area times 1,000 gives me volume?

Student B: No, that just gives you the area of 1,000 sheets, kinda like their surface area, but you're still doing area. Aren't you?

Me: It would seem so.

I told the class that finding the area of one sheet was correct. And they also knew that it was the same area as the base of the box. Then I poured out the stack of CDs and started filling its container back up one by one. (Easier than stuffing the patty sheets back in.) I said quietly, Volume is three-dimensional.

The class watched and soon the hands went up. Before I could call on anyone, a few students couldn't contain themselves and violated our classroom rule of waiting to be called on before talking, two kids yelled out, The height! Then more chimed in to say, Multiply by the height!