My Other Math Sites
Lessons From the Classroom
Hotel Snap
[NCTM Illuminations has my blessings and thanks to re-write and feature this lesson on their site.]
Andrew Stadel and I recently presented this task at the 2013 CMC's North and South Conferences.
The Challenge
As a team, build a hotel that yields the highest profit [score].
Rules and guidelines for building the hotel
Each cube represents a hotel room. All 50 cubes must be used.
Hotel must stand freely on at least one side of cube. Here’s a non-example because it’s tilted.
Entire hotel is one piece. A non-example due to yellow cube not attached:
All rooms must have at least one window, a window is any exposed vertical side of cube. The white cube below has no window.
Building costs and tax (daily rate)
Land costs $400 per square unit.
Land refers to outline of top view of building.
All enclosed land is charged, for example, squares marked 8, 9, 10, and 11 in the left outline are open space, but because there's no access to the outside, you are charged for these 4 square units.
A roof costs $10 each, roof is any exposed top side of cube.
A window costs $5 each.
Tax on height of building is calculated by multiplying the tax rate for the highest floor by the total land cost.
Floors 1-10 —> 50%
Floors 11-20 —> 1000%
Floors 21-30 —> 2000%
Floors 31-40 —> 3000%
Floors 41-50 —> 5000%
Income from each type of room (daily rate)
The more windows, the more income.
4 windows, 1 roof = $600
4 windows, 0 roof = $500
3 windows, 1 roof = $300
3 windows, 0 roof = $250
2 windows, 1 roof = $200
2 windows, 0 roof = $175
1 window, 1 roof = $150
1 window, 0 roof = $125
Scoring
Your net profit/loss income will be checked for accuracy. A deduction of 50% of your error will be applied to the actual number. For example, your building nets a profit of $13,500, but your group submits a profit of $15,000, therefore you're off by $1,500. Then 50% of this error ($750) will be deducted from the $13,500 to give your team a score of $12,750.
If your calculations are right on, then your team's score will be awarded an extra $1,000.
Adapting this lesson
Change the number of cubes, as few as 10-15 cubes for younger kids, and maybe up to 100 cubes for high school students.
Also for younger kids, have the Excel file (more on this later) readily available on computers so kids can go back and forth between checking their profit margin and tweaking their hotel rooms — so no calculations needed on their part, they just need to be able to know how to count the different types of rooms.
Older students can create the spreadsheet, it's great practice for understanding how cells work and formulating equations.
Adjust the time for individual and group work based on your expectations.
Modify, take away, or add to the rules and guidelines.
Change any of the costs/income/tax numbers.
Change how you reward accuracy or penalize mistakes.
Ask each group to estimate and rank the profit margins of other teams' hotels just by looking at them (like on a -5 to +5 scale, -5 for biggest loss and +5 for biggest profit).
Ask, "What if all costs and tax stay the same, but now the incomes for the rooms are all reversed so that 4-window-1-roof earns only $125 while 1-window-0-roof earns $600? How would you build your hotel using the same rules?"
If I were to do this with my 6th graders, I'd first have everyone build the same 10-cube hotel with me, then we'd use this hotel to familiarize ourselves with the different types of rooms and tally them up. We could calculate the costs and income together for practice.
How I ran the lesson with 8th graders
(I did this lesson with two classes of 8th graders, one geometry and one algebra. We have 57-minute periods. This lesson took 2.5 periods. I'm so scripted here because the one thing that kids wished they had more of was time. Your teacher instruction needs to be tight and supplies distribution needs to be efficient to allow for all the student work time needed.)
State the challenge of task.
State the rules/guidelines.
Give each student a zip bag with 50 cubes — but I first took time to show them that how they see the cubes now is how they need to be put away when we clean up.
Give each student this cost_income_sheet:
Set timer for 15 minutes for individual work, reminding students that later they will be randomly assigned into groups of 3 to work on one hotel.
While students are working, I use Instant Classroom to put kids into groups of 3.
When the timer goes off, I give the following instructions:
I now need you to listen to directions for working with your teammates, and when I'm done giving these instructions, I will set the timer for 25 minutes for you to work. During this time, you'll need to do the following:
Share and discuss the best model to represent your team's hotel.
You may modify this chosen hotel, or you can even start from scratch, but watch your time.
Use this tally_sheet to record your room counts, land, roofs, etc. — remember this is just a tally sheet, so you want to do all your calculations on whiteboard.
On the large whiteboard, divide it into 4 quadrants, and your group will need to fill in 3 of the 4 quadrants answering these questions. Watch your time carefully because when the timer goes off, the hotel you have in front of you is the one you must keep as is.
Allow a couple of minutes for groups to get supplies and settle down together with their individual hotel models. One tally sheet needs to be passed out to each group.
Timer is set for 25 minutes. Monitor the groups, check for understanding, and listen in for building strategies. Keep counting down the time every 5 minutes.
When timer goes off, ask groups to break down the hotels that were not selected and put the cubes neatly back into the zip bags and return to the front.
Next day...
Kids get back into their groups with their whiteboard and hotel. Some groups may still need to finish filling in their tally sheets and whiteboards.
The question for the 4th quadrant is now asked: "If you could relocate just 5 cubes on your hotel, where would you place them?" Remember that you are no longer allowed to change your hotel, this question is just a what-if scenario.
Allow 10 minutes for groups to finish filling in their tally sheets and answer all 4 questions on whiteboard.
As indicated on bottom of tally sheet, groups are asked to bring up the tally sheets to teacher to check their calculations and arrive at the final score.
The Excel sheet
I have two files: one locked and one unlocked. The "locked" one so of course no one can inadvertently change the cells, especially wise if you give kids access to the file to use on their own - only the blue cells to enter data are open. The "unlocked" file is so you can edit as you please.
Notice that all the cells appear in the same order as the tally sheet, makes for quick entries with instant results. My husband gets most of the credit for this, I still claim credit for making it simpler.
Math Practices
I found this form 8_MP_questions_to_ask online and can't find it again to cite the source.
I'm making good use of this form after almost every task now. Students do this for homework. I ask them to highlight the questions that came up during the activity - from teacher, peers, and self.
We discuss this as a whole class the next day and generally come to a consensus of which math practices our task had fostered.
Reflections
We presented this lesson because it went really well in the classroom. The kids were completely engaged. They collaborated, talked a lot among themselves about what they were building and thinking, but they were pretty much dead silent during the initial individual work time. I love seeing the structural varieties — this was true too among the hotels built by math teachers.
To not spoil the fun, I won't post pictures of the high-profit hotels here, but I'd love to learn what some of your profit numbers are from your class.
Stack 'em High
My algebra kiddos are doing Dan Meyer's Stacking Cups because Andrew Stadel did the lesson and wrote glowingly about it. But Andrew used only one-size cups, I used three different sizes.
Just eyeballing
I ask students to look at me, look at this 12-oz Styrofoam cup that I'm holding, and estimate how many cups they would need to stack to reach my height. I tell them I will not answer any clarifying questions regarding this, "just make your estimate in whatever way you think I mean by this." (Their hands shoot up anyway, but I remind them I won't answer any questions right now. It's clear that they want me to define "stack.")
They write their answers on a quarter sheet of paper. Here are their 29 estimates, the median at 24 cups.
Given the heights
Then I tell them my height is 163 cm with the flat shoes that I have on. I carefully measure the cup's height in front of them, we get 11.25 cm.
Equipped with this knowledge, I ask the same question as above. Their 28 answers yield a median of 14 cups. Of course Dan had already anticipated this — most students just divided my height by the height of the cup.
Stack 'em like this
I now ask them what it was that they'd wanted to ask me earlier. Sure enough Eddie says, "By stacking do you mean bottom-to-bottom, then top-to-top, or... one inside another?" I give him some cups to show me. At this time at least 2/3 of the kids admit that they answered the previous questions believing that I meant to stack the cups the way Eddie just described formerly. (This is consistent with the low guesses we see in the top image.)
I ask the next question, "Okay, I need you to answer the same question again of how many cups it'll take to reach the top of my head, but you now know exactly what I mean by stack, and you also know my height and the cup's height. Here, I'll even stand on this table with 6 cups stacked at my feet so you can see. Go, give me a number.
Here are their 27 estimates, and 113 is the median.
We're just getting started
I randomly pair kids up. (Normally they are in groups of 3, but I think it's better to be in pairs for this activity.) I give each pair 6 cups. They have the last 25 minutes of class to figure out:
The equation for this problem
The number of stacked cups for my height
Some groups need help with finding the y-intercept. A few groups don't know where to begin. I ask them some questions and walk away. They plan to nominate me in June as their most non-helpful teacher. Whatever.
I like this group's drawing, even though the lip and body of cup do not add up correctly.
Using their equations
The majority of the groups figure that the lip of the cup is the slope, but many groups also think that the full height of the cup is the y-intercept. Using their equations to figure out the number of cups, they give me these numbers.
Not too shabby. My equation yields 102 stacked cups to reach top of my head, and that happens to be the mode and median.
And the actual number of cups is...
This is the moment they've been waiting for! I have 81 cups stacked already when the kids come in the next day, then they count out loud as each additional cup is added to reach my height. Our principal is paying us a visit today, and he keeps telling me how impressed he is with how engaged the kids are and how hands-on the lesson is. He helps with the countdown too and officially announces that it takes 100 cups to stack up to my height! (Three groups whose estimates of 99, 101, and 102 are having pizza with me next week!)
Getting our principal in on the fun
Working backwards. I tell the kids that it takes 116 cups to reach the top of our principal's head. How tall is he? Their answers give a median and a strong mode of 184 cm. His actual height is 183 cm!
Twice the volume and half the volume
Just when they think they're done, I pull out the 24-oz and the 6-oz cups. It's not often that I hear them shriek in delight to do more math! The pair of students now gets only 3 cups of each new size. They go to work.
My height is 40 24-oz cups and 126 6-oz cups. Their calculations are great for the big cups, not so much for the little ones.
This ranks up there as one of my favorite lessons. Thanks much to Dan for another fab activity. But due to us having already passed linear equations, I honestly would not have done this lesson now without Andrew's push.
Updated 02/06/13
Eight more kids will join me for a pizza lunch for getting the equations to the 24-oz cups correctly. And here's a pic of me and the 3 stacks. The boys are 6th graders helping me hold up the stacks.
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.