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3-Act Math

# The Round Table Problem

## Introduction

Students can watch this video as many times as they would like.

## Part I: Entry

You can optionally ask students what they want, and come up with a shared question. Alternatively, you can use the question I already asked in the exit ticket. To modify the question, download the Word file. I am actually going to modify the question based on the responses I get from students. I feel like a better question would be, "What is the area of the table (in cm^2)?" or "If I take out one section of this table, how much surface area will be left (in cm^2)?"

## Part II: Attack

Ask students, "What resources do you need?". Click the first three links to the resources I have, or download the Word or PDF files to print them out in advance for students.

Resource 1

Resource 2

Resource 3

Resource 4

## Part III: Check, Reflect, Review, Extend

Students can check, reflect, and review in different ways. Ultimately, they need to conclude with something along the lines of, "My hypothesis was reasonably accurate, because ... However, I learned that ..." or "My hypothesis was inaccurate, because ... And I learned that ...". I don't usually deduct marks for an inaccurate hypothesis, so long as the student corrected their mistake along the way.

There are a couple possible extensions to this problem. At this point in the year, I expect students to be able to start generating these extensions themselves. However, in this case, I am going to get students to solve the following extension.

• Can I make a square table, with length and width 196 in. and with a square hole in the middle, that has the same surface area?

# The Pizza Problem

## Part I: Entry

For this problem, I brought in a Costco pizza and put it on the table at front. We started class with a number talk, and once students started to get curious about the pizza, I asked them what they wanted to know. The dimensions of the pizza and the pizza box are available in the captions of the images below, but if you do this problem in class, I would strongly recommend just bringing in a pizza.

We did this problem in the context of our "From Patterning to Algebra" unit. The grade 8 Curriculum and Connections expectation is to develop formulas for the circumference of a circle, area of a circle, volume for right prisms, and surface area of a cylinder.

## Part II: Attack

Students worked together in groups of 3 to solve this problem. They had a total of maybe three hours to work on this problem in class over the course of a week. The problem the students decided to work on was, "What are the area and circumference of the pizza?", and the extension was, "If we had 27 mini-pizzas (one for every student in the class), with the same total volume as this regular-sized pizza, what would the dimensions of each mini-pizza be?"

## Part III: Check, Reflect, Review

On Friday, we looked through student solutions together, and we found a rough estimate for the circumference, surface area and volume. The purpose of this activity was not to necessary find exact solutions for those dimensions, but rather to develop general formulas.

# The Watermelon Problem

## Prologue

Curriculum and Connections Expectations: Fractions

• use fraction skills in solving problems involving measurement

## Part 1: Entry

To introduce the problem, I just brought in a watermelon and sat it on the table until students had enough questions. Once they were ready, I put "What do you want to know?" on the board, and as they asked questions, I gave them markers so they could put their questions on the board.

Journal entry #1: The Watermelon Problem.

Based on these questions, students decided to study the problem, "How can you cut up the watermelon so that each of the 27 students can have an equal slice?"

What's The Watermelon Problem? Here are the questions the students asked.

## Part 2: Attack

Here are some resources for the watermelon we chose. It's honestly better to get your own watermelon though.

Here are some examples of student work and ideas. Many of them came up with some form of idea to do a third of a third of a third.

## Part 3(a,b): Review (Check, Reflect)

After finishing this problem, it's time for students to check if the solution worked the way they expected, and to reflect on what went well and what can be improved on. You can see in the video that cutting the watermelon into thirds would require a very high level of precision to do well. There are a whole bunch of hindrances. For the sequel, encourage students to find a better way to divide the watermelon. Can they add the precision necessary to cut the slices precisely equally? Can they find a better way to distribute the watermelon?

## Part 3(c): Review (Extend)

Once students are ready, they can try to find a strategy that can work easily for any number of students.

## Elements of Solutions

Here is one possible such strategy. The result is pretty easy to generalize to any number of students. However, there are still a couple of precision-related problems with this strategy.

Here is a different strategy. Still pretty easy to generalize to any number of students, but with different pros and cons.

# Fold and Cut

## Prologue

• Curriculum and Connections: Patterning and Algebra
• "Sort and classify quadrilaterals by geometric properties, including those based on diagonals, through investigation and using a variety of tools"
• Ontario Curriculum
• 8m40 (Overall Expectations) demonstrate an understanding of the geometric properties of quadrilaterals and circles and the applications of geometric properties in the real world
• 8m41 (Overall Expectations) develop geometric relationships involving lines, triangles, and polyhedra, and solve problems involving lines and triangles
• 8m43 (Geometric Properties) sort and classify quadrilaterals by geometric properties, including those based on diagonals
• 8m45 (Geometric Properties) investigate and describe applications of geometric properties in the real world

## Act I

• Is there a way to fold a triangle so that it can be cut out by using only one cut?

## Act II

• Suggest attempting a simpler problem. Ironically, both of the following are easier.
• Or maybe watch one failed attempt at cutting a triangle out in one cut

## Act III

• One possible solution for cutting out the triangle in one cut (view as image)

## Sequel

• Try cutting out a parallelogram in one cut. Try other polygons.
• Can you cut out the letters of the alphabet using this theorem?
• Check out Erik Demaine's awesome site
• Numberphile explains the "Fold and Cut Theorem"

# Cinnamon Toast Crunch

## Prologue

Play this video for students. You may have to play it more than once before they are ready to ask questions.

[wpvideo EdgORK0d]

## Act I

• Write, "What do you want to know?" on the board
• Allow students time to ask questions. All questions are valid. Don't answer any of them just yet.
• Optional: After each question ask, "Who wants to know the answer to this question?". Write that number beside the question.
• Answer all questions on the board.
• If a student asks for information about the price or the number of grams, give them access to the photos in Act II.
• If a student asks a good, math-related question, circle it and tell students you would like to know the answer to that. A good starter question is, "Which box of cereal is the best deal?", but there are many other problems that students can explore based on this video.
• Hopefully someone asked something like, "What does this have to do with math?" Give moving speech about how math is life. You get bonus points for moving kids to tears during this speech.

## Act II

Cinnamon Toast Crunch 1

Cinnamon Toast Crunch 2

Cinnamon Toast Crunch 3

## Act III and Sequel

[Placeholder text. Will be updated once students are finished the problem.]

# Stick or Switch? The Missing Ace Problem

## Prologue

[wpvideo Ytdf27NB]

## Act I

Is it better to stick or switch?

## Act II

1. What's your hypothesis: stick or switch?
2. What's the theoretical probability of your hypothesis?
3. How would you set up an experiment to test the experimental probability?
4. Will the experimental probability exactly match the theoretical probability?

## Act III and Sequel

Here is the problem in its original context. The history of this problem is fascinating. The Time Everyone "Corrected" the World's Smartest Woman

# Square Mural Problem

Credit Where Credit is Due:

Thanks, Mr. Brian Errey, for all your help trouble-shooting and putting this problem together.

## Prologue

Number Sense 8m15 Determine common factors and common multiples using the prime factorization of numbers

Measurement 7m1 Report on research into real-life applications of area measurements (This is a grade 7 overall expectation)

## Act I

[wpvideo SXgYClk5]

1. What are the dimensions of the smallest square mural you can create using copies of the original rectangle?

2. Will the mural fit in the wall space provided?

## Act II

3. What information do you need to get an answer?

Only provide the following information to a student if s/he asks for it.

Information 1

Information 2

Information 3

Information 4

## Act III

4. Here's one possible solution.

## Sequel

5. If you answered yes: How many sheets (not squares, but the big red, green, beige, pink and yellow sheets that you cut into the smaller squares) will you need, in order to make the entire mural?

6. If you answered no: Think of the dimensions of the individual squares used to create this entire mural. What size of little squares gives you the biggest square mural you can fit?

It isn't enough just to say yes or no. Students must be able to convince others of how they know.

# Factor Blocks

Credit for this problem goes to Brian Errey.

## Prologue

8m15 Determine common factors and common multiples using the prime factorization of numbers

## Act I

[wpvideo KOXr8PeV]

1. Do any of these models represent prime numbers or composite numbers?

2. Which of these best represents a prime factorization?

3. Do these numbers have a common multiple? How can you tell?

## Act II

4. What mathematical tools can we use to help us find the prime factors?

Students might suggest Factor Trees or Venn Diagrams. It can be helpful to draw these out on paper. Have linking cubes handy, in case students want to put together their own models and play around with them to figure out how they work.

## Act III

5. We'll discuss the solution together in class.

## Sequel

6. Is there an easy way to figure out the Greatest Common Factor of any two or three numbers? How about the Least Common Multiple?

7. Can you find the Greatest Common Factor or Least Common Multiple of any two 4-digit numbers? How about 5-digit numbers?

# Using Excel to Optimize the Surface Area of Cylinder

In the Chapter 11 assignment on surface area and volume of cylinders, the last question asked students to calculate the optimal dimensions of a can of soda in order to minimize the amount of aluminum used. Here is the original question:

You are in charge of creating a new soft drink can. This can needs to hold 200 cm3 of liquid.

(a) The radius of one cylindrical can is 2 cm. The radius of the other is 5 cm. What would the height and surface area of the can be in each case? Are the dimensions of the cans suitable? Explain your answer.

(b) In order to save money, the can needs to be designed so that it requires the smallest quantity of aluminum. Find the radius and height of the can that does this, to two decimal places.

This question can be done algebraically. However, when we did the question, students found it difficult to check their work. This is where Excel comes in. While there are other math programs out there that are specifically designed to do symbolic math, the advantage of Excel is its ubiquity. And if you can't find Excel where you are, there are a number of online spreadsheets that do everything we will need to do in this question.

The basic idea is this:

1. Put four columns into Excel: Radius, Volume, Height and Surface Area
2. Make the radius vary from 2 to 5, using whatever step you want (I used "0.1" on the left, then "0.01" on the right)
3. Calculate the height, based on the radius: $latex h = \frac{V}{\pi r^2}$
• For this, you can use the number 200, since you know the volume, or you can insert the cell value that contains the number 200
• My Excel formula for height looked like this: =B2/(PI()*A2^2)
• In that formula, "PI()" means $latex \pi$, "B2" and "A2" refer to their respective cells, and the "=" sign means to calculate the value
4. Calculate the surface area based on that.
• The surface area of a cylinder is $latex SA_{cylinder} = 2 \pi r^2 + 2 \pi r h$.
• In Excel, it looks like this: =2*PI()*A2*C2+2*PI()*A2^2
5. Once that's done, highlight the cells you want to continue, put your cursor in the bottom-left corner of the box, and drag it down. If done correctly, Excel should auto-fill all the values below.
6. Finally, if you want to make a graph out of it, highlight the "Radius" and "Surface Area" columns (use the Ctrl button to select the separate data sets), and click on "Insert", "Recommended Charts".

Once that's done, all that's left is to look through your data to see where the Surface Area hits its minimum.