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Lesson 44

© 2012 MARS University of Nottingham

Mathematics Assessment Project

Formative Assessment Lesson Materials

Creating and Solving Equations

MARS Shell Center University of Nottingham & UC Berkeley

Alpha Version

Please Note: These materials are still at the “alpha” stage and are not expected to be perfect. The revision process concentrated on addressing any major issues that came to light during the first round of school trials of these early attempts to introduce this style of lesson to US classrooms. In many cases, there have been very substantial changes from the first drafts and new, untried, material has been added. 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].

Creating and Solving Equations Teacher Guide Alpha Version January 2012

© 2012 MARS University of Nottingham 1

Creating and Solving Equations 1

Mathematical goals 2

This lesson unit is intended to help you assess how well students are able to create and solve equations. In 3 particular, the lesson will help you identify and help students who have the following difficulties: 4

• Solving equations where the unknown appears once or more than once. 5

• Solving equations in more than one way. 6

Common Core State Standards 7

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

A-REI: Understand solving equations as a process of reasoning and explain the reasoning. 10

This lesson also relates to the following Standards for Mathematical Practice in the CCSS: 11

7. Look for and make use of structure. 12

Introduction 13

The unit is structured in the following way: 14

• Before the lesson, students work individually on an assessment task that is designed to reveal their 15 current levels of understanding and difficulties. You then review their work and create questions for 16 students to answer in order to improve their solutions. 17

• During the lesson, students work in pairs on two collaborative activities. They create equations for 18 each other to solve. In the first activity, students work with equations in which the unknown appears 19 once in the equation; in the second activity the unknown appears more than once. 20

• After a plenary discussion, students return to their original task, consider their own responses, then use 21 what they have learned to complete a similar task. 22

Materials required 23

• Each student will need a copy of the assessment task, Equations to Solve, and More Equations to Solve, 24 the cut-up sheet Creating Equations, the cut-up sheet Solving Equations, a sheet of paper, a mini-25 whiteboard, a pen, and an eraser. Some students may need extra copies of the sheets Creating 26 Equations and Solving Equations. 27

• There are some projector resources to support whole class discussions. 28

Time needed 29

Approximately 15 minutes before the lesson, a seventy-minute lesson, and 15 minutes in a follow-up lesson or as 30 homework. Exact timings will depend on the needs of your class. 31

32



Before the lesson 33

Assessment task: Equations to Solve (15 minutes) 34

Set this task, in class or for homework, a few days before 35 the formative assessment lesson. This will give you an 36 opportunity to assess the work, and to find out the kinds of 37 difficulties students have with it. You will then be able to 38 target your help more effectively in the follow-up lesson. 39

Give each student a copy of the assessment task Equations 40 to Solve. 41

Read through the questions and try to answer 42 them as carefully as you can. 43

It is important that students are allowed to answer the 44 questions without your assistance, as far as possible. 45

Students should not worry too much if they cannot 46 understand or do everything, as in the next lesson, they 47 will engage in a similar task that should help them to 48 progress. Explain to students that by the end of the next 49 lesson, they should expect to answer questions such as 50 these confidently. This is their goal. 51

Assessing students’ responses 52

Collect students’ responses to the task, and note what their 53 work reveals about their current levels of understanding 54 and their individual difficulties. 55

We suggest that you do not score students’ work. The research shows that this will be counterproductive, as it 56 will encourage students to compare their scores and distract their attention from what they can do to improve 57 their mathematics. Instead, help students to make further progress by summarizing their difficulties as a series of 58 questions. Some suggestions for these are given on the next page. 59

We recommend that you write a selection of questions on each piece of student work. If you do not have time, 60 select a few questions that will be of help to the majority of students. These can be written on the board at the 61 end of the lesson. 62

If your students are producing correct solutions for questions 1 and 2, you may decide miss out the first 63 collaborative activity and start the lesson with a whole-class discussion on equations in which the unknown 64 appears more than once. 65

66

Creating and Solving Equations Student Materials Alpha Version January 2012

© 2012 MARS University of Nottingham S-1

Equations to Solve Solve the following equations.

Show and explain all your steps.

1.

!

y + 1 3

= 2

2.

!

2 x " 4( ) + 2

3 = 10

3.

!

2w " 1 5

+ 5 = 2w



Common issues: Suggested questions and prompts:

The student applies an operation to only one side of the equation

For example:

2 x ! 4( ) + 2

3 = 10

Multiply by 3: 2 x ! 4( ) + 2 = 10

• Check your work. Are both sides of the equation always equal? How do you know?

The student incorrectly adds or subtracts a value from an expression

For example: In Q1, attempting to subtract 1 from both sides of the equation

!

y + 1 3

= 2 results in

!

y3

= 1.

Or: In Q2, attempting to subtract 2 from both sides of the equation

!

2 x " 4( )

3 + 2 = 10 results in

!

2 x " 4( )

3 = 10 + 2.

• Write the expression with the fraction bar as two fractions. Does it now make sense to subtract 1?

!

y 3

+ 13

"

# $

%

& ' = 2

• Are the two sides of the equation still equal? How

do you know?

The student incorrectly applies the distribution law

For example: In Q2, 2(x − 4) becomes 2x − 4.

• How would you say the expression in parentheses in words? Do these words match your expression on the other side of the equation?

The student incorrectly multiplies or divides part of the expression

For example: In Q3, when multiplying the expression

!

2w " 15

+ 5

by 5, the student writes 2w − 1 + 5 instead of 2w − 1 + 25.

• Have you multiplied everything in this equation by 5?

Student has solved all equations correctly • Now use a different method to solve equation 3.

67



Suggested lesson outline 68

Whole-class introduction: Equations in which the unknown appears once (20 minutes) 69

Give each student a mini-whiteboard, a pen, and an eraser. 70

During this introduction, ask students questions that provoke thoughtful answers. Treat these answers as building 71 blocks for further dialogue rather than end points in the discussion. This process should provide students with a 72 model for how they should work together on the collaborative activities. 73

We suggest you begin with a simple example showing the format for building and solving an equation, before 74 trying a more complex equation. 75

Building a simple equation 76

Choose a whole number between 1 and 10. This is your chosen value of x. 77

Write on the board or an overhead transparency: 78

!

x = 6

Ask students to choose an operation (+, −, ×, ÷) and an integer between 1 and 10. You will use these to build a 79 new equation. For example: 80

!

x = 6

Subtract 4

!

x" 4 = 2

Supply the notation, and explain it clearly. Ask students to explain how the new equation follows from the 81 previous equation. Check that students understand that the operation must be applied to both sides of the 82 equation: 83

Why is the equation not

!

x" 4 = 6? [The operation is used on both sides of the equation to keep the left 84 hand and right hand expressions equal.] 85

Ask students to extend the equation by choosing another operation and another integer between 1 and 10. 86 For example: 87

!

x" 4 = 2

Divide by 5

!

x" 45

=25

Again, supply the notation, and explain it clearly. In particular, encourage a student to suggest a division 88

operation and discuss how (for example) (x ! 4)÷ 5 is written as x ! 45

89

Checking the equation 90

Ask students to use their mini-whiteboards to check by substitution that the original value of x still satisfies the 91 equation. 92

We started by choosing x = 6. On your whiteboard, check whether the equation is true for x = 6. 93

After a minute or so, ask one or two students to explain why the equation is true for this value of x. 94

95



Solving the equation 96

Now erase all the steps written on the board except the final equation. 97

!

x" 45

=25

Ask students to recall each operation used to build the equation in order. Write these on the board. 98

This equation tells the story of ‘a day in the life of x.’ 99

What happened to x first? How can you tell by just looking at the equation? 100

What was the last thing that happened? 101

Then ask students to solve the equation by undoing what they did when creating it, working in reverse order. 102

What number will you end up with if you solve this equation? [x =6.] 103

How do you undo dividing by 5? [Multiplying by 5.] 104

What is the inverse of subtracting 4? [Adding 4.] 105

Why do you undo multiplying by 5 first? 106

As they do this, uncover the preceding equations one by one, and write the corresponding operation to the right 107 of each equation. For example: 108

!

x = 6

Subtract 4.

!

x" 4 = 2 Add 4.

Divide by 5.

!

x" 45

=25

Multiply by 5.

109 Summarize the activities students have worked through: 110

You chose a value for x, built an equation, checked the equation was true for your value of x, then 111 solved the equation. 112

Explain that in this lesson, students will be building equations for each other, but they will be more complicated 113

equations using each of the four operations, +, −, ×, and ÷ and integers between 1 and 10. 114

Show students a more complicated example using all four operations. For example, take x = 5: 115



Add 3

Divide by 2

Subtract 1

Multiply by 4

x = 5

x + 3 = 8

!

x + 32

= 4

!

x + 32

" 1 = 3

!

4 x + 3

2 " 1

#

$ %

&

' ( = 12

As students suggest each operation, again, supply the notation and make sure it is explained carefully. For 116 example, ask students to explain how to use parentheses to show a whole expression is being multiplied. 117

Research has shown that it is best not to simplify the left side of the equation at any stage. For example, if 118

students suggest +5, −2, ÷4, ×3, then write

!

3 x + 5( ) " 2

4

#

$

% %

&

'

( ( not 3

x + 34

#

$ %

&

' ( 119

Checking the Equation 120

As before, ask students to use their mini-whiteboards to check by substitution that the original value of x still 121 satisfies the equation. 122

4 5 + 3

2 ! 1

"

#$

%

&' = 4

82

! 1"

#$

%

&' = 4 4 ! 1( ) = 4(3 = 12 123

After a few minutes ask a couple of students with different answers, to justify them. 124

Solving the Equation 125

Erase all the steps except the final equation and ask students to recall each operation in sequence. 126

Here is a story of ‘a day in the life of x.’ 127

What happened to x first? How can you tell by just looking at the equation? 128

What then? What then? What was the last thing that happened? 129

Then ask students to solve the equation by undoing what they did when creating it. As they do this, uncover the 130 preceding equations one by one and write the corresponding operation to the right of each equation. 131



Add 3

Divide by 2

Subtract 1

Multiply by 4

x = 5

x + 3 = 8

!

x + 32

= 4

!

x + 32

" 1 = 3

!

4 x + 32

" 1#

$ %

&

' ( = 12

Subtract 3

Multiply by 2

Add 1

Divide by 4

Collaborative activity 1 (15 minutes) 132

Organize the class into groups of two students. Give each student the sheets Creating Equations and Solving 133 Equations. The slide, Working Together, in the projector resource summarizes how students are to collaborate. 134

Your first job is to build and check two equations. Use your sheet, Creating Equations. Each equation 135

should use all four operations +, −, ×, and ÷ and four different integers. Make sure the order of the 136 operations is different for each equation. 137

Check that each equation works by substituting the original value for x into it. 138 Then write your equations at the top of the sheet Solving Equations. 139

Your second job is to solve each other’s equations. 140

Give your sheet to your partner and ask them to solve the equations. 141 Help your partner if they become stuck. 142

If your partner’s answers are different from yours, ask for an explanation. If you still don't agree, 143 explain your own thinking. 144

It is important that you both agree on the answers. 145

The purpose of this structured group work is to make students engage with each other’s explanations, and take 146 responsibility for each other’s understanding. 147

While students are working in small groups you have two tasks: to note how students approach the task, and to 148 support student reasoning. 149

Note different student approaches to the task 150

Notice how students make a start on the task, where they get stuck, and how they respond if they do come to a 151 halt. Notice any errors. Students may make calculation errors when substituting into equations. They may use 152 non-standard notation. Students may forget to perform operations on both sides of the equation, or not use the 153 distribution property accurately when solving equations. Students may multiply or divide just one term in the 154 expression, instead of the whole expression on both sides of the equation. 155

You can use this information to focus the whole-class discussion towards the end of the lesson. 156

157

158



159

Support student reasoning 160

Try not to make suggestions that resolve errors and difficulties for students. Instead, ask questions to help 161 students to reason together to identify and resolve issues. 162

Did your partner find the value of x you started with? Does that matter? 163

Can you explain what you wrote here? 164

How could you write ‘divide the whole of the left-hand side expression by 3’ in algebra? 165

Explain how you know which operation to undo first. 166

Can you find a different way of writing this expression? 167

How do you know these two expressions are equal? 168

The questions in the Common Issues table may also be helpful. 169

Encourage students who quickly complete the two sheets to create more challenging equations without using the 170 structured sheets. 171

If students struggle with this activity you may want to spend the rest of the lesson on it and leave the second 172 activity for another time. 173

Whole-class discussion 1: Equations in which the unknown appears more than once 174 (10 minutes) 175

Creating and Checking an Equation 176

When students have had time to work with at least two equations, explain to your class that they will now build 177 equations that include steps that use letters as well as numbers. 178

For example, you might add 2x to both sides of the equation: 179

Add 2x

Multiply by 3

Subtract 1

x = 4

3x = 4 + 2x

9x = 3(4 + 2x)

9x – 1 = 3(4 + 2x) − 1

Again, ask students to check that the value for x makes the original equation true. 180

181



Solving the Equation 182

Write the equation on the board. Ask students to solve the equation using two methods. 183

9x – 1 = 3(4 + 2x) − 1 184

For example: 185

Add 1

Divide by 3

Subtract 2x

9x − 1 = 3(4 + 2x) − 1

9x = 3(4 + 2x)

3x = 4 + 2x

x = 4

OR

Distribute the 3

Simplify

Add 1

Subtract 6x

Divide by 3

9x − 1= 3(4 + 2x) − 1

9x − 1 = 12 + 6x − 1

9x − 1 = 11 + 6x

9x = 12 + 6x

3x = 12

x = 4

Collaborative activity 2 (15 minutes) 186

Again ask students to create two equations, making sure at some stage the unknown is on both sides of each 187 equation. Once they have checked their equations by substitution, they are to ask their partner to solve the 188 equation in two different ways. Do not give students the structured sheets, but ask students to work on blank 189 sheets of paper. 190

Support the students as in the first collaborative activity. 191

Whole-class discussion (10 minutes) 192

Organize a discussion about what has been learned. Depending on how the lesson went, you may want to focus 193 on the common mistakes students made, review what has been learnt, or you may want to extend and generalize 194 the math. 195

Throughout this plenary encourage students to justify their answers. Try not to correct answers, but encourage 196 students to challenge each other's explanations. 197

Write this equation on the board: 198

!

5

b " 12

" 1#

$ %

&

' ( + 2b

3= 5 + b

199

Show me a method for solving this equation. 200

After a few minutes ask students to show you their whiteboards. Ask two or three students with different 201 answers to justify them to the rest of the class. 202

203

204



For example, this is one possible method: 205

5 b ! 1

2 ! 1

"

#$

%

&' + 2b

3= 5 + b

Multiply by 3 5 b ! 1

2 ! 1

"

#$

%

&' + 2b = 15 + 3b

Distribute the 5 5b ! 52

! 5 + 2b = 15 + 3b

Add 5 5b ! 52

+ 2b = 20 + 3b

Subtract 3b 5b ! 52

! b = 20

Multiply by 2 5b ! 5 ! 2b = 40

Simplify 3b ! 5 = 40

Add 5 3b = 45

Divide by 3 b = 15

206

207 Ask students to critique each other’s solution methods. 208

Does anyone disagree with this method? 209

Does anyone have a different method? 210

Does anyone have a more efficient method? 211

Follow-up lesson: More Equations to Solve (15 minutes) 212

Return to the students their original assessment task: Equations to Solves as well as copy of the task More 213 Equations to Solve. 214

If you have not added questions to individual pieces of work, then write your list of questions on the board. 215 Students should select the questions they think are appropriate to their own work. 216

Look at your original responses and think about what you have learned this lesson. 217

Carefully read through the questions I have written. 218

Spend a few minutes thinking about how you could improve your work. 219

You may want to make notes on your mini-whiteboard. 220

Using what you have learned, try to answer the questions on the new task Solving Equations (again). 221

222



Solutions 223

Assessment Task: Equations to Solve 224

There are several ways to solve each equation. Below are some examples. 225

1. 226

y + 13

= 2

Multiply by 3 y + 1 = 6Subtract 1 y = 5

227

228

2. 229

2 x ! 4( )3

+ 2= 10

Subtract 22 x ! 4( )

3 = 8

Multiply by 3 2 x ! 4( ) = 24

Divide by 2 x ! 4 = 12

Add 4 x = 16

230

3. 231

2w ! 15

+ 5 = 2w

Subtract 5 2w ! 15

= 2w ! 5

Multiply by 5 2w ! 1 = 10w ! 25

Add 25 2w + 24 = 10w

Subtract 2w 24 = 8w

Divide by 8 w= 3

OR

2w!15

+ 5 = 2w

Multiply by 5 2w!1+ 25 = 10w

Simplify 2w+ 24 = 10w

Subtract 2w 24 = 8w

Divide by 8 w = 3

232

233



Assessment Task: More Equations to Solve 234

There are several ways to solve each equation. Below are some examples. 235

1. 236

y + 32

= 5

Multiply by 2 y + 3 = 10

Subtract 3 y = 7

237

238

2. 239

3 x ! 2( )

4+ 5 = 8

Subtract 53 x ! 2( )

4 = 3

Multiply by 4 3 x ! 2( ) = 12

Divide by 3 x ! 2 = 4

Add 2 x = 6

240

3. 241

3w + 52

! 4 = 3 + w

Add 4 3w + 52

= 7 + w

Multiply by 2 3w + 5 = 14 + 2w

Subtract 5 3w = 9 + 2w

Subtract 2w w = 9

242



Equations to Solve Solve the following equations.


1.

!

y + 1 3

= 2

2.

!

2 x " 4( ) + 2

3 = 10

3.

!

2w " 1 5

+ 5 = 2w



Creating Equations Operations Operations

x = y =

This is Equation 1 This is Equation 2 Check Check

Solving Equations Operations Operations

Equation 1 Equation 2



More Equations to Solve Solve the following equations.


1.

!

y + 3 2

= 5

2.

!

3 x " 2( )

4 + 5 = 8

3.

!

3w + 5 2

- 4 = 3 + w

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

Working Together

1

1.  Build and check two equations. Use your sheet Creating Equations. Each equation should use each of the four operations +, −, ×, and ÷ and four different integers. Make sure the order of the operations is different for each equation.

2.  Check that each equation works by substituting the original value into it.

3.  Write your equations at the top of the sheet Solving Equations.

4.  Give your sheet to your partner and ask them to solve the two equations.

5.  Help your partner if they become stuck.

6.  If your partner’s answers are different from yours, ask for an explanation. If you still don't agree, explain your own thinking.

It is important that you both agree on the answers.

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