Personal Reflection:
Finding photos like this on popular "student" sites like 9gag.com always makes me happy! I know, right away, that this will get them to "wonder mathematically" and dive into exploring skills and concepts that they might not have ever wanted to consider in the past. My "wondering" and wandering led me to questions like: How much would it cost to tile that area? Is using pennies the cheapest? How does that compare to normal tile? I hope that your students will wonder mathematically in their own ways!
Grade Level: 4-9
Course: Math, Pre-Algebra
Standards: 5.NBT.5, 5.NBT.7, 5.MD.1, 6.PR.3d, 6.NS.1, 6.NS.3, 6.EE.1, 6.EE.2, 6.EE.7, 6.G.1
SMP: MP1, MP2, MP3, MP4, MP5, MP6
Skills: Area, Unit conversions, exponential notation, estimation
How to use this as a mad minute:
You have 60 seconds. How much do you think it would cost to cover the floor of our classroom in pennies? Estimate accurately and be prepared to justify your estimate.
How to use this as a warm up:
You could ask the students to consider one of the following:
1. How much do you think it would cost to tile our floor in pennies? Estimate accurately, show your work and steps and be prepared to explain.
2. Do you think covering the floor with pennies would cost MORE or less than covering it with nickles? dimes? quarters? dollar bills? Why?
3. Do you think covering the floor with pennies would cost more or less than using a traditional floor covering such as tile or carpet? Why?
4. Is there a currency in the world that would be better to use than pennies? If you can't think of one, can you think of what properties it would need to have to be a better choice?
How to use this as a mini-lesson:
If I had 20 minutes to spend on this lesson, and the goal was for kids to estimate, use critical thinking skills, use their knowledge of area, and compute, this is what I'd do!
Show them the photos! Let them talk with their friends about them for less than 60 seconds.
Give them a challenge. Tell them that you will offer (____reward_____) for the student who is able to BEST estimate the cost of covering ONE DESK (or table) in your room in pennies.
"Best" estimate, as determined by me, would not only be accurate, but would be clearly justified and explained. It would specifically address concepts such as measurement, how to compute area, etc. I would NOT equally reward a group that "counted" how many pennies fit along each side and multiplied and a group that used measurements, staggering (to fit more in), repeated trials and samples, etc. Although both may be right, one shows much more depth of analysis and thinking. The first would be perfect for an answer in 5 minutes or less, and the second would be appropriate for a 20 minute exploration.
I'd offer them a selection of supplies:
- Baggies of 10 or fewer pennies
- Rulers or meter sticks
- Tape
- Paper
It would be up to the students what supplies they might use.
How to use this as a full lesson?
Use the
EQUATE thinking routine. Give them the photos with NO guidance. Let them Explore, Wonder, and Question.
Record
their questions and clarify! Develop depth to the questions by
extending them, drawing out detail and asking how they might go further.
For
example, a student might ask, "How much would that cost?" This is a great
"wonder" but not enough to explore mathematically. What are they going
to investigate? How can they extend this question? As you draw out
these ideas, make sure to ask them to list what information they need to
answer their questions. Feel free to share answers or to encourage
them to FIND THEIR OWN answers. This is a great time to use technology
to your advantage. Students can research on phones and iPods, or you
can nominate a class researcher who will research the answers to these
questions while you continue to work with the rest of the class. Narrow
down your questions to your top two or three questions. Tell them they
can choose from "These" deep understanding questions to answer in their next steps. It could be something from the suggested warm ups with more depth, it could be calculating the cost of covering the floor of your room, their bedrooms, the hallway, etc. For students who are advanced, ask them to compare the prices of using pennies to nickels and dimes with only a ruler and one of each coin.
Establish expectations for products, time frames, behaviors, and jobs.
Set
them free! Help them question and explore and apply their mathematical
knowledge to solve their problems and answer their questions.
How to use this as an assessment?
You know your
students best, and you know if you will have prepared them for this. I know many teachers ask challenge questions such as, "How many boxes of Kleenex will it take to fill this classroom?" as assessment questions. However, many other background experiences are necessary before students can attack these kinds of problems independently on an assessment.
If you feel you've provided similar learning experiences that would help a student successfully approach such a problem, go for it!
- Ask a specific and clear question.
- Be sure you know what kinds of answers you will accept.
- Determine how much written and verbal guidance you will provide during the assessment.
- Make sure you provide clear rubrics or standards for achievement. Students need to know what will earn a passing, or excelling, grade!
- Make notes of what works and what doesn't so you can improve it for next time.
Every school grades differently, sometimes based on standards, some using IB or pre-IB rubrics, some on critical thinking and creativity, some on a strictly points-based system. While I cannot help every one of you, in the future I will update this with my own question, instructions, and rubric!
Please feel free to use any of these ideas and
modify them to meet your needs. However, please acknowledge the
original source of the items and my own lesson outlines.
©NatalieRSprigg 2013
Here's the thing: I think that in order to create TRUE mathematicians, kids need to "wonder mathematically" much more often. Sure, they might laugh at the joke, but do they ever "wonder" what those numbers actually mean? Do they have any number intuition? Can we GET kids to wonder mathematically? My EQUATE thinking model asks kids to do just that. Unfortunately, I'm not sure that this is cut out to be an EQUATE type of problem.
Standards: 4.NBT.1, 4.NBT.5, 4.NBT.6, 5.NBT.6
