Writing Mathematical Expressions and Equations
Writing Mathematical Expressions and Equations
Objectives
Students work with equivalent expressions and recognize how to apply properties (commutative, associative, and distributive) to show that expressions are equivalent. Students will:
- write and evaluate mathematical expressions and equations that correspond to given situations.
- use expressions and formulas to solve problems.
- understand and use variables that represent numbers for which exact values are unknown.
- understand that expressions written in different ways can be equivalent.
Essential Questions
How are relationships represented mathematically?
How can data be organized and represented to provide insight into the relationship between quantities?
How can expressions, equations, and inequalities be used to quantify, solve, model, and/or analyze mathematical situations?
How can mathematics support effective communication?
How can recognizing repetition or regularity assist in solving problems more efficiently?
How is mathematics used to quantify, compare, represent, and model numbers?
What does it mean to estimate or analyze numerical quantities?
What makes a tool and/or strategy appropriate for a given task?
- How is mathematics used to quantify, compare, represent, and model numbers?
- How are relationships represented mathematically?
- How can expressions, equations and inequalities be used to quantify, solve, model and/or analyze mathematical situations?
- How can recognizing repetition or regularity assist in solving problems more efficiently?
Related Unit and Lesson Plans
Related Materials & Resources
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Materials haven't been entered into the unit plan.Formative Assessment
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Multiple-Choice Items:
- Which of the following equations represents the statement “twice a number is thirty-six”?
A
2 + n = 36
B
2 • n = 36
C
n ÷ 2 = 36
D
n − 2 = 36
- Which symbolic equation could be used to represent the following problem?
Lani is x years old. In 12 years, she will be 36.
A
12 + 36 = x
B
12 x = 36
C
12 + x = 36
D
36 ÷ x = 12
- Use the information below. Express the cost of a sundae with an unknown number of toppings.
A plain ice cream sundae costs $3.50. Each topping costs 25¢.
A
0.25t = 3.50
B
3.50 − t = 0.25
C
3.50 ÷ 0.25t =
D
3.50 + 0.25t =
- Which of the following expressions is equivalent to n + n + 2b?
A
n + 2bn
B
2b + (n + 2)
C
b + b + 2 + n + n
D
2n + 2b
Use the figure below to answer question 5.
- Which pair of equivalent expressions shows the perimeter of the figure?
A
a • 2c • b + b = a • 2c • 2b
B
2b + ac + c = 2b + a + 2c
C
a + 2cb = a + 2c + 2b
D
c + c + 2b + a = 2c + 2b +a
- Which of the following equations is equivalent to 3l + w = 87?
A
w + (3 + l) = 87
B
w + (2 • l) + l = 87
C
2l + ( w • l ) = 87
D
3lw = 87
- Which of the following equations is true?
A
(16 − 2) × 2 = 32 − 16 ÷ 4
B
8 × 2 − 4 = 4 + 2 × 2
C
36 ÷ (9 × 4) = 5 × 2 + 6
D
(10 + 5) + (2 • 3) = 10 • 4 + 11
- Which of the following equations is true?
A
5 • □ = □ • 5
B
5 − □ = □ − 5
C
5 + 5 = □ + □
D
5 ÷ □ = □ ÷ 5
- Which of the following equations is true?
A
3 • 5 • (4 − 1) = 2 • 5 + 5 • 4 – 1
B
3 • 5 • (4 − 1) = 3 • 5 • 4 – 1
C
3 • 5 • (4 − 1) = (2 • 5 + 5) • (4 − 1)
D
3 • 5 • (4 − 1) = (3 • 5 • 4) – 1
Multiple-Choice Answer Key:
1. B
2. C
3. D
4. D
5. D
6. B
7. A
8. A
9. C
Short-Answer Items:
- Rewrite the following equation using words: 2x − 10 = 4.
- Write three different expressions that are all equivalent to the expression below.
4 + m + m + n + n
- Explain the order of operations in words. Be sure to include the correct sequence.
Short-Answer Key and Scoring Rubrics:
- Rewrite the following equation using words: 2x − 10 = 4.
Answers may vary.
Twice a number decreased by ten is four.
Two times a number minus ten is four.
Ten less than a number multiplied by two is four.
Points
Description
2
- Written equivalent is complete, accurate, and detailed.
- Student demonstrates thorough understanding of translating equations into words.
1
- Written equivalent is correct but brief or simplistic.
- Student demonstrates partial understanding of translating equations into words.
0
- Written equivalent is incorrect or missing.
- Student demonstrates little or no understanding of translating equations into words.
- Write five different expressions that are all equivalent to the expression below.
4 + m + m + n + n
Possible answers:
m + m + 2n + 4 2m + n + 4 + n 4 + (2 × n) + (2 × m)
4 + 2n + 2m 2(n + m) + 4 2m + n + n + 4
Points
Description
2
- Work is complete and detailed.
- Student demonstrates thorough understanding of creating equivalent expressions.
- Answers are all mathematically correct.
1
- Written explanation or work is correct but brief or simplistic.
- Student demonstrates partial understanding of creating equivalent expressions.
- Most answers are mathematically correct.
0
- Written explanation or work is incorrect or missing.
- Student demonstrates no understanding of creating equivalent expressions.
- Most answers are incorrect or missing.
- Explain the order of operations in words. Be sure to include the correct sequence.
The order of operations is a set of rules used to simplify expressions. The sequence used in the order of operations is parentheses first, then exponents, then multiplication/division left to right, then addition/subtraction left to right.
Points
Description
2
- Written explanation is complete, accurate, and detailed.
- Demonstrates thorough understanding of order of operations.
1
- Written explanation is correct but brief or simplistic.
- Demonstrates partial understanding of order of operations.
0
- Written explanation is incorrect or missing.
- Demonstrates little or no understanding of order of operations.
Performance Assessment:
Modeling and Solving Problems Involving Perimeter
Part 1
Part A: Write an expression for the perimeter of the figure.
Part B: If a = 1 foot and b = 2 feet, find the perimeter of the figure in feet.
Part 2
The chart below is part of a 200s multiplication chart.
154
155
156
157
158
164
165
166
167
168
174
175
176
177
178
184
185
186
187
188
194
195
196
197
198
Look at the squares that have been highlighted. Write an equation using variables that represents the sum of the three numbers when x = 187.
Write an equivalent equation:_______________________________________________
Shade in any four adjoining squares on the chart below.
154
155
156
157
158
164
165
166
167
168
174
175
176
177
178
184
185
186
187
188
194
195
196
197
198
Pick one of the numbers you shaded in and rename it s. That number is _______. Write an equation using variables that represent the sum of the four numbers.
Write an equivalent equation: ______________________________________________
Part 3
Evaluate the following equations and determine if they are true or false. Give evidence to support your answer.
16 ×
+ 5 = 16 × ( + 5) 3s + s + 2w + 6 = 4s + w + (2 + 6)
9 × (100 ÷ 25) + 4 = 4 + 6 • 6
In words, describe how you can evaluate any equation like the ones above to determine if it is true or false.
______________________________________
______________________________________
______________________________________
Performance Assessment Key:
Modeling and Solving Problems Involving Perimeter
Part 1
Part A: Write an expression for the perimeter of the figure.
Answers will vary.
a + a + a + a + b + b 4a + 2b
2a + 2a + 2b 4a + b + b
b+ a + b + 3a 3a + 2b + a
b + a + b + a + 2a 2a + 2b + a + a
2(2a + b) a + a + 2a + b + b
Part B: If a = 1 foot and b = 2 feet, find the perimeter of the figure in feet.
The perimeter is 8 feet.
Part 2The chart below is part of a 200s chart.
154
155
156
157
158
164
165
166
167
168
174
175
176
177
178
184
185
186
187
188
194
195
196
197
198
Look at the squares that have been highlighted. Write an equation using variables to represent the sum of the three numbers when x = 187.
187 = x x + x − 11+ x + 11 = 561
176 = x − 11
198 = x + 11
Write an equivalent equation: Answers will vary: 3x = 561; 2x + x = 561
Shade in any four adjoining squares on the chart below.
154
155
156
157
158
164
165
166
167
168
174
175
176
177
178
184
185
186
187
188
194
195
196
197
198
Pick one of the numbers you shaded in and rename it s. That number is ______. Write an equation using variables to represent the sum of the four numbers.
Answers will vary. Be sure students accurately represent the numbers as variable expressions in relation to the number represented by s.
Write an equivalent equation: Answers will vary.
Part 3
Evaluate the following equations and determine if they are true or false. Give evidence to support your answer.
16 ×
+ 5 = 16 × ( + 5)False, because the grouping changes. In the equation on the left, you would multiply 16 ×
first; in the equation on the right, you would add + 5 first.3s + s + 2w + 6 = 4s + w + (2 + 6)
False, because in the equation on the left, you have 2w and 6 wholes; in the equation on the right, you only have 1w and 8 wholes.
9 × (100 ÷ 25) + 4 = 4 + 6 • 6
9 × 4 + 4 = 4 + 36
36 + 4 = 4 + 36
True, because both sides of the equation equal 40.
In words, describe how you can evaluate any equation like the ones above to determine if they are true or false.
Answers will vary. You need to remember the order of operations when solving a numerical expression. That means you do parentheses first, then exponents, then multiplication/division left to right, then addition/subtraction left to right. Also remembering number properties like the commutative and associative property can be helpful.
Performance Assessment Scoring Rubric:
Points
Description
4
- Five or more correct expressions for perimeter are stated in Part 1, with no incorrect expressions for perimeter stated.
- All equations given for Part 2 are correct.
- Four squares are shaded and the number selected for s is stated in Part 2.
- Student demonstrates advanced understanding of the mathematical ideas and processes related to finding perimeter and writing expressions with variables to express relationships in Parts 1 and 2.
- Student correctly determines if equations in Part 3 are true or false.
- Student adequately supports conclusions about true/false equations in
Part 3. - Student thoroughly explains the importance of using the order of operations and that failing to do so accurately will result in incorrect conclusions in Part 3.
3
- Four or more correct expressions for perimeter are stated in Part 1, with one or no incorrect expressions for perimeter stated.
- One equation for Part 2 is correct or both expressions (without the total) are correct (e.g., 4x + 22).
- Four squares are shaded but the number selected for s is missing in Part 2.
- Student demonstrates solid understanding of the mathematical ideas and processes related to finding perimeter and writing expressions with variables to express relationships in Parts 1 and 2.
- Student correctly determines if at least two of the three equations in Part 3 are true or false.
- Student adequately supports conclusions about two true/false equations in Part 3.
- Student partially explains the importance of using the order of operations and that failing to do so accurately will result in incorrect conclusions in Part 3.
2
- Three or more correct expressions for perimeter are stated in Part 1, with one or two incorrect expressions for perimeter stated.
- At least 1 expression or equation for Part 2 is correct.
- Some squares are shaded in Part 2.
- Student demonstrates some understanding of the mathematical ideas and processes related to finding perimeter and writing expressions with variables to express relationships in Parts 1 and 2.
- Student correctly determines if at least one of the equations in Part 3 is true or false.
- Student adequately supports conclusion about one true/false equation in Part 3.
- Student explains the importance of using the order of operations and that failing to do so accurately will result in incorrect conclusions in Part 3. Explanation mostly correct although may be lacking in thoroughness or clarity.
1
- At least one correct expression for perimeter is stated in Part 1.
- At least one part of one equation or expression for Part 2 is correct.
- Some squares are shaded in Part 2.
- Student demonstrates very little understanding of the mathematical ideas and processes related to finding perimeter and writing expressions with variables to express relationships in Parts 1 and 2.
- Student incorrectly determines whether equations in Part 3 are true or false.
- Support for true/false equations in Part 3 is inadequate but not completely incorrect.
- Student correctly explains something about the importance of using the order of operations in Part 3.
0
- No correct expressions for perimeter are stated in Part 1.
- All equations and equivalent equations for Part 2 are incorrect or missing.
- No squares are shaded in Part 2.
- Student demonstrates no understanding of the mathematical ideas and processes related to finding perimeter and writing expressions with variables to express relationships in Parts 1 and 2.
- Student fails to make any determination about whether equations in Part 3 are true or false.
- Support for true/false equations in Part 3 is missing or incorrect.
- Student’s explanation about the importance of using the order of operations is missing or incorrect in Part 3.
Final 05/03/2013
