Modeling: Rolling Cups - kentoncountymdc - home · Modeling: Rolling Cups ... Model with mathematics. 14 This lesson gives students the opportunity to apply their knowledge of the

Lesson 017

© 2011 MARS University of Nottingham

Mathematics Assessment Project

Formative Assessment Lesson Materials

Modeling: Rolling Cups

MARS Shell Center University of Nottingham & UC Berkeley

Alpha Version

Please Note: These materials are still at the first draft stage. They are intended for initial trials by our team of robust teachers and will go through several cycles of revisions and corrections before being released for general use. We suggest that you check with the Nottingham team before releasing any of this material outside of the core project team.

If you encounter errors or other issues in this version, please send details to the MAP team c/o [email protected].

Rolling Cups Teacher Guide Alpha Version January 2012

© 2012 MARS University of Nottingham 1

Modeling: Rolling Cups 1

Mathematical goals 2

This lesson unit is intended to help you assess how well students are able to: 3

• Choose appropriate mathematics to solve a non-routine problem. 4

• Generate useful data by systematically controlling variables. 5

• Develop experimental and analytical models of a physical situation. 6

Common Core State Standards 7

This lesson relates to the following Mathematical Practices in the Common Core State Standards for 8 Mathematics: 9

1. Make sense of problems and persevere in solving them. 10 2. Reason abstractly and quantitatively. 11 3. Construct viable arguments and critique the reasoning of others. 12 4. Model with mathematics. 13

This lesson gives students the opportunity to apply their knowledge of the following Standards for Mathematical 14 Content in the CCSS: 15

F-BF: Build a function that models a relationship between two quantities. 16 G-MG: Apply geometric concepts in modeling situations. 17 G-SRT: Prove theorems involving similarity. 18

Introduction 19

• Before the lesson, students watch a video of Rolling Cups. Students then work on the Rolling Cups 20 scenario in a task designed to assess their current approaches to modeling. You collect and review their 21 work, and write questions to help students improve their models. 22

• During the lesson, students first work individually, reviewing their initial work. They then work in 23 pairs to compare their approaches, and choose a strategy to produce an improved model together. They 24 work to implement that strategy. Students next review some sample solutions to the task, and use what 25 they have learned to produce a poster showing their own improved models. After reading each other’s 26 posters, there is a whole class discussion in which students identify the strengths and weaknesses of 27 different approaches to the modeling task. 28

• After the lesson, students review and improve their work on the assessment task. 29

Materials required 30

• Each individual student will need a copy of the task sheet, Modeling Rolling Cups. 31

• It is desirable that each pair of students has access to a computer with the applet Modeling with 32 Geometry: Rolling Cups. If this is not possible, then the lesson may still be conducted with one class 33 computer and a data projector. 34

• Each pair of students will need a fresh copy of the task sheet, Modeling Rolling Cups, copies of each of 35 the Sample Solutions, a large sheet of paper, and felt-tipped pen. 36

• A backup supply of paper and plastic cups, rulers, rough paper, graph paper, protractors, string, chalk, 37 glue, scissors, and calculators for students who choose to use them. 38

• For the whole class discussions, you will need a computer with data projector, and two plastic/paper 39 cups that have different base diameters and heights. 40

Time needed 41

Thirty minutes before the lesson, an 80-minute lesson (or two shorter lessons), and ten minutes after the lesson. 42 All timings are approximate, and will depend on the needs of your students. 43

44



Before the lesson 45

Introduction to Modeling: Rolling Cups (20 minutes) 46

(i) Practical demonstration: 47 Show students a cup and make sure students understand the terms, slant height, perpendicular height, narrow 48 diameter, wide diameter. 49

Show the students a paper cup, and indicate the slant side of the cup. 50

I’m going to lay this cup on its slant side and push it. 51 What do you think will happen? 52

Roll the cup. 53

Tell me what you see. 54 Was this what you predicted? 55

Take a second cup. Ask students to make predictions about the roll radius, the radius of the circle the outside of 56 the cup rolls. 57

Do you think the circle this cup rolls will have a bigger or smaller radius? Why? 58

Check students’ predictions by rolling the cup. 59

(ii) Using the computer data projector Introduce the task using the video extracts on the MAP website: http://map.mathshell.org.uk/pilot/lesson_support/cups_task.htm

Display the introductory slide, The Task. Ask students to make some predictions:

Which of these objects do you think will roll the largest circle? Why? What about the second largest circle? What will happen when the soup can is rolled?

Accept all student answers without comment.

Show the four videos: Short glass, Plastic cup, Tall glass, Soup can. 60

As I play each video, try to guess what will happen! 61 Were you surprised by any of the results? 62

Note: You do not need to show the Cup rolling Calculator – 63 this is for the main lesson. 64

Individual work on the assessment task: Modeling 65 Rolling Cups (15 minutes) 66

Explain the task. Give each student a copy of the task, 67 Modeling Rolling Cups. 68

I would like you to think about how to solve this 69 problem. Think about what you’ve seen so far. 70

Explain what you want students to do. 71

I would like you to spend fifteen minutes working on 72 this task. Make sure you explain your work clearly. 73

At the top of the sheet is the data you saw in the four 74 videos together with some more data. 75

Rolling Cups Student Materials Draft 2 version December 2011

© 2010 Shell Center/MARS University of Nottingham UK S-1

Modeling Rolling Cups

Here is a reminder of the data you saw in the video with a few extra cups added.

1. Describe how each of the three lengths on the picture affect the roll radius. Show how you used the data to explain your ideas.

2. Show how you can use math to predict the radius of the circle rolled by any size of cup. Show all your reasoning, including any diagrams and calculations.

Cup Dimensions in inches

Wide diameter

Narrow diameter

Slant Length

Roll radius

A 3½ 3 3¾ 26¼

B 3 2 3½ 10½

C 2½ 2 5¾ 28¾

D 3 3 4¼ Infinite!

E 3 2 6 18

F 3½ 2 3¾ 11¼

G 3¾ 3 3¾ 18¾

!"##$%&'(")*+*#

,(-*&'(")*+*#

./"0+&1*(23+&



The slides Modeling Rolling Cups and the data from the videos may be projected as you make these points. 76

It is important that students are allowed to answer the question without assistance, as far as possible. If students 77 are struggling to get started then help them understand what is required, but make sure you do not do the task for 78 them. The first few questions on the Common issues table, on the next page, may be useful. 79

Students who sit together often produce similar answers. When they come to compare their work, they often 80 have little to discuss. For this reason we suggest that, when students do the task individually, you ask them to 81 move to different seats. Then at the beginning of the formative assessment lesson, allow them to return to their 82 usual seats. Experience has shown that this produces more profitable discussions because students have varied 83 approaches to discuss. 84

When all students have made a reasonable attempt at the task, tell them that they will have time to revisit and 85 revise their solutions later. 86

Do not be too concerned if you found this work difficult, as we will have a lesson tomorrow that should 87 help you. You will have a chance to improve your solutions after that lesson. 88

Assessing students’ responses 89

Collect students’ responses to the task. Make some notes on what their work reveals about their current modeling 90 strategies. 91

We suggest that you do not score students’ work. The research shows that this will be counterproductive, as it 92 will encourage students to compare scores, and distract their attention from what they can do to improve their 93 mathematics. 94

Instead, help students to make further progress by summarizing how to develop their modeling strategies as a list 95 of questions and prompts. Some suggestions for these are given the Common issues table. We suggest that you 96 make a list of your own questions, based on your students’ work, using the ideas below. You may choose to 97 write questions on each student’s work. If you do not have time to do this, select a few questions that will be of 98 help to the majority of students. These can be written on the board at the beginning of the lesson. 99



Common issues: Suggested questions and prompts:

Student makes no use of data (Q1&2)

The student makes no reference to the data in his discussion of the problem.

• You have data. How might that be helpful? • Could you reorganize the data from the video to

show any patterns? How might that help you come to a decision about this problem?

Student does not consider all the variables (Q1&2)

For example: The student has only considered the Slant Height in her hypothesis about which cup rolls the larger circle. Or: The student omits reference to one variable.

• Which measures occur in the data from the video?

• Are there any other measures that might be useful?

Student has some idea of the effect of the variables but has not considered how they interact. (Q1) For example: The student says that “if the wide diameter is made bigger, then the cup rolls in a smaller circle.”

• Is this true whatever the value of the narrow diameter?

• What if the narrow diameter is equal to the wide diameter?

Student does not control the variables. (Q1)

For example: The student draws a conclusion by comparing two cups with no dimensions in common.

• What do you think would happen if you fixed the narrow diameter and slant height and varied the wide diameter?

Student has no modeling strategy (Q2) For example: The student has written/drawn little.

• What do you already know? • What are you trying to find? • Can you draw the cup as it is rolling? • Where is the roll radius on your drawing? • How is it related to the measures in the

problem?

Student requests further information For example: The student says that she cannot reach a decision because there is insufficient information to solve the problem.

• What data do you want in order to solve this problem? How will that information help you solve the problem?

• Describe the experiments you want to do to find more data.

Student draws a diagram For example: The student makes a scale drawing. Or: The student sketches the cup, identifying key variables.

• How will you use your diagram/scale drawing to solve the problem?

Student provides a model For example: The student has used scale drawings to predict which cup will roll the larger circle. Or: The student has written a formula.

• You predict that … How could you check this? • How do you know your model is correct? • Does your model make a correct prediction for

each of the cups in the table? • Would your model work for any cup? How do

you know?

100

101



Suggested lesson outline 102

Individual work (10 minutes) 103

Remind students of their work on Modeling Rolling Cups. 104

Your work today will be to produce a mathematical solution to Modeling Rolling Cups. 105 There are lots of different ways of doing this! 106

Give the students their scripts from the assessment task. If you have not written your questions on individual 107 scripts, display them now on the board for all the class to read. 108

I read your solutions from [last lesson], and wrote some questions that might help you to improve that 109 work. Spend a few minutes answering my questions. Then decide what will you do next to solve this 110 problem. 111

Ask students to read through your questions carefully, spend ten minutes answering them, and make a decision 112 about a plan of action. 113

Introduce the Cup Rolling Calculator (5 minutes) 114

Show students the Cup Rolling Calculator using a data 115 projector. 116

To improve your models, you could try rolling 117 some cups. We also have a calculator that will 118 help you produce more data if you need it. 119

Demonstrate how to use this calculator. 120

What values could I choose for the narrow 121 diameter? Where do I put that? 122

Which number or numbers do you want me to 123 change? 124

What is the dependent variable? 125

Small group work: Collaborative modeling (30 minutes) 126

There are two possible ways to work on the task: using a practical method, and/or using the computer calculator. 127 The practical method has the advantage that students can more readily visualize the 3-dimensional aspects of the 128 situation and this will help them move towards an analytical method. The computer calculator generates data 129 rapidly and helps students to find patterns and make generalizations. Both approaches have their difficulties, 130 however. Students may find it difficult to record on paper how the cup rolls and to make accurate measurements. 131 The computer approach may lead to students getting buried in data and will not help them develop explanations. 132

It is worthwhile for students to try both approaches where possible. Everyone should have the experience of 133 rolling one cup and gathering data from it, if possible. The computer can then be used to gather additional data. 134

Ask students to form pairs. 135

Give each pair a fresh copy of the lesson task, Modeling Rolling Cups, a paper cup, a ruler, a protractor, a pencil, 136 and a large sheet of paper and a felt-tipped pen with which to write their ideas. 137

You’re going to work in pairs today. To begin, take turns to explain your work to the other student in 138 the group. When listening to your partner’s solution, ask questions so you understand. 139

When your pair has shared solutions, choose a strategy to work on together. 140

Work together to produce a joint solution to the problem that is better than either of your individual 141 solutions. 142



I have given you a paper cup and some equipment that you can use to help you visualize the problem 143 and collect some data on one cup. 144

There is also the Cup Rolling Calculator that you can use to collect additional data. Plan your data 145 collection before you start, so that you don’t collect too much! 146

If there is only one computer and data projector, explain that when students have discussed their ideas in their 147 pairs for a while, they should write down any additional data they want collected. Then you will collect this data 148 for them using the computer. 149

During small group work, you have two tasks: to notice students’ approaches to the modeling task, and to 150 support their modeling activity. 151

Notice students’ approaches to the task 152

What kind of models do the students attempt to make? Do they produce an empirical model based on data or an 153 analytical model by analyzing the practical situation? 154 155

• When using the computer calculator, do students control the variables, keeping some fixed and varying 156 others systematically? Do they consider extreme cases, (such as wide diameter = narrow diameter; or 157 narrow diameter = 0?) Do they choose useful representations, such as tables or graphs? Do they try to 158 identify a general rule or formula from their data? 159 160

• When using a practical approach by rolling a cup, do students attempt to draw scale diagrams? Do they 161 produce an analytical geometrical model using similar triangles? Do they distinguish slant height and 162 perpendicular height? Do they recognize which variables height plays which role in their geometrical 163 models? 164

Support students’ modeling activities 165

There are many ways to tackle this task. Try not to prompt students towards using a particular approach. Instead, 166 ask questions that help students develop and clarify their thinking whilst following their chosen modeling 167 strategy. 168

Why did you choose these values for the narrow/wide diameter? What other data might be useful? 169

What happens if… you make the narrow diameter much bigger than/close in size to/smaller than the 170 wide diameter? …you increase the wide diameter but keep the narrow diameter fixed? 171

Can you see any patterns in the data? 172

Use your patterns to predict new results. Check to see if you are correct. 173

How else could you represent your data to help understand the relationships? (Graphs, tables, 174 diagrams, scale drawings.) 175

How do think the roll radius depends on the narrow diameter? The wide diameter? The slant length? 176

Try to combine the patterns you have learned about into a formula to help you predict the size of any 177 cup roll radius. 178

How can you check that your answer is correct? 179

Think about what you know about real cups rolling. Does your formula make sense? 180

If your lesson is of short duration, we suggest you break at this point. You can resume with the next activity in 181 the following lesson. 182

183



Small group work: Analyzing Sample Solutions (20 minutes) 184

The sample solutions will help students to think further about their own approaches and confront them with 185 approaches they may not have yet considered. 186

Give each small group of students a copy of all three Sample Solutions. 187

Choose one of these Sample Solutions. Read it together. Work together to answer the questions. 188

When you have finished your work on one Sample Solution, choose another. 189

It is more important for students to work through a couple of the Sample Solutions in depth, to see contrasting 190 approaches, than to cover all of the material. 191

Whilst students are analyzing the Sample Solutions, continue to support their thinking as before. 192

What approach is Gerry taking? 193

How might Heather use that data to complete a formula? 194

How does Judi’s diagram connect with Gerry’s? 195

196

Whole-class discussion: Comparing different approaches (15 minutes) 197

Ask students to compare the work they’ve seen on this problem, and to identify the different strengths of the 198 various methods. There are some slides of the Sample Solutions to support this discussion. 199

Describe one strategy for solving the problem. 200

Did any of you solve the problem in that way? 201

What are the strengths of that method? Were there any difficulties with that method? 202

Describe another strategy for solving the problem. 203

Which method did you prefer? Why? 204

Ask students to review their work on the modeling problem across the lesson. 205

What was your initial strategy for solving this problem? 206

Has your strategy changed during the lesson? How? Why? 207

What advice would you give to someone just starting on this problem? 208

209

Follow-up lesson: Review individual solutions (20 minutes) 210

Ask students to sit in the pairs/small groups in which they worked the previous lesson. Give students their initial 211 individual responses to the assessment tasks, their small group work, and their posters. Ask students to read 212 through all their work, beginning with their individual solutions. 213

Think about the story of your own progress on this task. 214

Write some advice to someone just starting on this problem. 215

216



Solutions 217

The Rolling Cups problem can be modeled using a range of strategies. Three possibilities are shown in the 218 Sample Solutions. Your students may devise other, equally valuable approaches. 219

Assessment task. 220

1. This question is designed to see whether students can 221 make deductions from the data by controlling variables. 222

They can deduce that: 223

If the wide diameter is increased (and other variables held 224 constant), then the roll radius will decrease. 225 This may be deduced from the data for cups A and G. 226

227

If the narrow diameter is increased (and other variables 228 held constant), then the roll radius will increase. 229 This may be deduced from the data for cups A and F. 230

231

If the slant height is increased (and other variables held 232 constant), then the roll radius will increase. 233 This may be deduced from the data for cups B and E. 234

235 2. This question is best answered using a geometrical construction. 236 The diagram shows a cup ABCD on its side. It rolls in a circle radius r about a point O The wide diameter = w The narrow diameter = n The slant height = s The roll radius = r Triangles AOD and BOC are similar, since <DAO = <CBO (AD is parallel to BC) and <AOD is common. Therefore: rw=r ! sn

" rn = rw! sw" sw = r(w! n)

" r = sww! n

This result shows that that the roll radius is proportional to the slant height. It also shows that when n and w are 237 nearly equal then the roll radius becomes very large (as is shown by the tin can!). 238

239

w

n

sr

A

B

CD O


Wide diameter

Narrow diameter

Slant Length

Roll radius

A 3½ 3 3¾ 26¼

B 3 2 3½ 10½

C 2½ 2 5¾ 28¾

D 3 3 4¼ Infinite!

E 3 2 6 18

F 3½ 2 3¾ 11¼

G 3¾ 3 3¾ 18¾



Judi’s Solution 240

Judi draws a diagram showing the cup in two dimensions as a pair of similar triangles. She says that the cup 241 scenario can be modeled this way, but gives no explanation of how to move from a cup to the diagram. 242

Throughout, Judi uses capital letters to represent variables. While not incorrect, this may be confusing as the 243 labels for vertices are also capitals. It is more conventional to use lower case for variables. 244

Judi claims that Triangle ACE is similar to Triangle BCD but does not explain how she knows this. Overall, 245 although she has chosen quite sophisticated mathematics to use, she provides almost no explanation of her work. 246

The argument for similar triangles could be made as follows: 247

Angle BCD is common to both triangles. 248

Slicing the cone perpendicular to the ground, the face is a triangle. The wide diameter and narrow 249 diameter included in that triangle are parallel because they are in the same plane (slice) and because the 250 top of the cup is (assumed) to be parallel to the base. So W is parallel to N. 251

Then Angle EAB = Angle DBC. 252

Since the sum of the angles in a triangle is always 180º, the third angles must also be congruent. 253

So as the three angles are equal, the triangles must be similar. 254

Since the triangles are similar, corresponding side lengths are in the same ratio. 255

Hence

!

RW

=R " S

N 256

Judi makes a standard error in the manipulation of this formula: 257

!

RN =WR " S 258

Judi’s working can be corrected, and her method used to find a formula for R. 259

!

RN =W (R " S ) and

!

RN =WR "WS . 260

Thus

!

R(W " N ) = SW and

!

R =SW

(W " N ). 261

A full solution would include making sense of the formula in the context, by, for example, showing what 262 happens for values of W and N that produce the extreme cases of the cone and cylinder. 263

Gerry’s Solution 264

Gerry uses a simple, accessible strategy that gives a partial solution to the problem. Given any cup, Gerry could 265 draw a scale diagram that would allow him to make an approximate measure of the roll radius, depending on the 266 accuracy of the scale drawing. However, Gerry’s approach must be used each time; it is not a general solution. 267

Gerry provides a little explanation of how to move from a cup to a cone, and from a cone to a triangle. He does 268 not explain how he knows that his triangle is isosceles. 269

Asking students to use Gerry’s method enables them to engage with his construction problem: how, given these 270 measures, you construct an isosceles triangle with an embedded isosceles trapezium. The value of R found given 271 the values of N, W, and S should be about 4½". 272

The advantage of Gerry’s method is that it is easy to understand. Some difficulties with Gerry’s method are that 273 the construction method needs to be used accurately to produce a reliable measure for the roll radius, and that the 274 method does not produce a general solution: a scale drawing must be produced for each cup. As a result, his 275 solution is not easily reversible, e.g. it would be difficult constructing the triangle knowing the roll radius to find 276 specific values of N, W and S. 277



Heather’s Solution 278

Heather uses the computer calculator to generate data. She generates the data by fixing two variables, W and N, 279 and varying S to see the impact on R. She analyses some of her data to find patterns, and uses the patterns to 280 define aspects of the formula for R. She interprets some of the formula and data in the context of the problem. 281 The way Heather sets out her data on the page is important. She aligns values so that it is possible to see how 282 systematic changes in N and W result in patterned change in R. These are strengths in her solution. 283

Heather might usefully continue with her systematic exploration of values of N and W by looking at N=2, W= 3, 284 N=3, W=4, and so on. She might then be able to see patterns in the resulting values of R. Graphing values of S, 285 R for sets of values of N with fixed W, or W with fixed N might be useful 286

However, Heather could complete the formula using her data on N=0. 287

She claims that

!

R =S " ?

W # N. 288

She knows that when N=0, the cup is a cone and it doesn’t matter what W is. In that case, the formula must 289

make

!

R =S " ?

W # 0= S gives

!

R =SW

W " 0. 290

The formula is thus

!

R =SW

W " N. 291

Some of Heather’s argumentation is strong: she links the real cups she’s seen with empirical data, patterns in the 292 data, and the formula. However, her presentation is a bit disorganized and she does not bring all the pieces of her 293 argument together. 294

Modeling with Geometry: Rolling Cups Student Materials Alpha version January 2012

© 2012 MARS University of Nottingham UK S-1






Wide diameter

Narrow diameter

Slant Length

Roll radius

A 3½ 3 3¾ 26¼

B 3 2 3½ 10½

C 2½ 2 5¾ 28¾

D 3 3 4¼ Infinite!

E 3 2 6 18

F 3½ 2 3¾ 11¼

G 3¾ 3 3¾ 18¾

!"##$%&'(")*+*#

,(-*&'(")*+*#

./"0+&1*(23+&



Sample Solutions: Heather



What values for the narrow diameter and wide diameter do you think Heather should choose next?

Justify your answer.

Heather has described some patterns in the data.

Find and describe a new pattern in the data.

What are the strengths of Heather’s solution?

What would you do to improve Heather’s solution?

Explain your answer.



Sample Solutions: Gerry



Use Gerry’s method to find the roll radius of this cup:

Wide diameter = 3" Narrow diameter = 1" Slant length = 3"

Gerry says he can use his method to find the roll radius of any cup. Is this correct? Explain your answer.

What are the strengths of Gerry’s approach?

What could Gerry do to improve his solution?



Sample Solutions: Judi



Judi says Triangle ACE and Triangle BCD are similar.

Is Judi correct?

Explain your answer.

Judi draws a triangle diagram.

Explain how this represents a cup.

Judi has made a mistake in her formula.

Find and correct the mistake.

Now find a formula for R, the roll radius.

© 2012 MARS University of Nottingham Alpha January 2012 Projector Resources:

Modeling Rolling Cups Rolling Cups Student Materials Draft 2 version December 2011

© 2010 Shell Center/MARS University of Nottingham UK S-1






Wide diameter

Narrow diameter

Slant Length

Roll radius

A 3½ 3 3¾ 26¼

B 3 2 3½ 10½

C 2½ 2 5¾ 28¾

D 3 3 4¼ Infinite!

E 3 2 6 18

F 3½ 2 3¾ 11¼

G 3¾ 3 3¾ 18¾

!"##$%&'(")*+*#

,(-*&'(")*+*#

./"0+&1*(23+&

1


Sample Work: Heather

2


Sample Work: Gerry

3


Sample Work: Judi

4

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Modeling: Rolling Cups - kentoncountymdc - home · Modeling: Rolling Cups ... Model with mathematics. 14 This lesson gives students the opportunity to apply their knowledge of the