My Other Math Sites
Lessons From the Classroom
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:
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.
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?
It's good to work with solid objects instead of just flat polygons when learning similar shapes.
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.
How much does a 36-oz package cost?
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?
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.