# 7.2.3 Numerical and Algebraic Expressions

7
Subject:
Math
Strand:
Algebra
Standard 7.2.3

Apply understanding of order of operations and algebraic properties to generate equivalent numerical and algebraic expressions containing positive and negative rational numbers and grouping symbols; evaluate such expressions.

Benchmark: 7.2.3.1 Properties & Equivalent Expressions

Use properties of algebra to generate equivalent numerical and algebraic expressions containing rational numbers, grouping symbols and whole number exponents. Properties of algebra include associative, commutative and distributive laws.

For example: Combine like terms (use the distributive law) to write 3x - 7x + 1 = (3 - 7)x + 1 = -4x + 1.

Benchmark: 7.2.3.2 Evaluate Algebraic Expressions

Evaluate algebraic expressions containing rational numbers and whole number exponents at specified values of their variables.

For example: Evaluate the expression $\frac{1}{3}\left ( 2x-5 \right )^{2}$ at $x=5$.

Benchmark: 7.2.3.3 Order of Operations & Calculators

Apply understanding of order of operations and grouping symbols when using calculators and other technologies.

For example: Recognize the conventions of using a caret (^ raise to a power) and asterisk (* multiply); pay careful attention to the use of nested parentheses.

## Overview

Big Ideas and Essential Understandings
##### Standard 7.2.3 Essential Understandings

Students at this level are continuing to develop the ability to generalize numerical relationships and express mathematical ideas concisely using expressions and equations (e.g., three more than a number as x + 3, doubling as 2n, commutativity as a + b = b + a). Concrete models and pictorial representations of algebraic expressions are used to develop understanding that the commutative, associative, and distributive properties and order of operations apply in the same way that they did for numeric expressions. Students use these properties and the order of operations to generate equivalent expressions and evaluate expressions that involve all rational numbers and positive exponents.

##### All Standard Benchmarks - with codes

7.2.3.1
Use properties of algebra to generate equivalent numerical and algebraic expressions containing rational numbers, grouping symbols and whole number exponents. Properties of algebra include associative, commutative and distributive laws. For example: Combine like terms (use the distributive law) to write
3x - 7x + 1 = (3-7)x + 1 = -4x + 1.
7.2.3.2
Evaluate algebraic expressions containing rational numbers and whole number exponents at specified values of their variables. For example: Evaluate the expression 2x-7, at x = 5.
7.2.3.3
Apply understanding of order of operations and grouping symbols when using calculators and other technologies. For example: Recognize the conventions of using a caret (^ raise to a power) and asterisk (* multiply); pay careful attention to the use of nested parentheses.

Benchmark Cluster
##### 7.2.3 Numerical and Algebraic Expressions

7.2.3.1
Use properties of algebra to generate equivalent numerical and algebraic expressions containing rational numbers, grouping symbols and whole number exponents. Properties of algebra include associative, commutative and distributive laws. For example: Combine like terms (use the distributive law) to write 3x - 7x + 1 = (3-7)x + 1 = -4x + 1.
7.2.3.2
Evaluate algebraic expressions containing rational numbers and whole number exponents at specified values of their variables. For example: Evaluate the expression 2x-7, at x = 5.
7.2.3.3
Apply understanding of order of operations and grouping symbols when using calculators and other technologies. For example: Recognize the conventions of using a caret (^ raise to a power) and asterisk (* multiply); pay careful attention to the use of nested parentheses.

##### What should students know and be able to do [at a mastery level] related to these benchmarks?
• Apply correct order of operations without use of a calculator;
• Master the order of operations which now includes parentheses and positive exponents;
• Substitute values into an expression;
• Identity properties given numerical expressions and also verbal descriptions;
• Apply the order of operations to generate equivalent numeric expressions involving rational numbers;
• Identify commutative, associative and distributive properties used to generate equivalent numeric and algebraic expressions;
• Recognize order of operations, grouping symbols and exponents when using calculators and other technologies;
• Use appropriate symbolism, including exponents and variables;
• Evaluate expressions through numerical substitution;
• Generate equivalent expressions;
• Master adding and subtracting expressions.
##### Work from previous grades that supports this new learning includes:
• Know basic exponent rules (using only positive exponents);
• Know how to combine like terms;
• Know basic functions and buttons of a scientific calculator and how to use them;
• Understand the concept of variables;
• Manipulate variables and constants separately;
• Add, subtract, multiply and divide;
• Know order of operations, including multiplication and division before addition and subtraction.
Correlations
##### NCTM Standards

Understand meanings of operations and how they relate to one another:

• Understand the meaning and effects of arithmetic operations with fractions, decimals, and percents;
• Use the associate and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals.
##### Common Core State Standards (CCSS)

6.EE (Expressions and Equations) Apply and extend previous understandings of arithmetic to algebraic expressions.

• 6.EE.3. Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

7.EE (Expressions and Equations) Use properties of operations to generate equivalent expressions.

• 7.EE.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

## Misconceptions

Student Misconceptions
##### Student Misconceptions and Common Errors
• Students may have difficulties with order of operations problems; they often want to do multiplication before division, but they only can if multiplication appears first going from left to right; the same is true with addition and subtraction. For example, in the problem 8-32 / 4 x 5, students want to multiply 4 x 5, then divide 32 by the product.
• The distributive property may present challenges for some students. For example, a student may rewrite the problem 4(3 - 2x) as 4 (3) -4(2x), multiplying 4(3) to be 12, but then say he can't multiply the 4 and 2 to make 8 because the 4 doesn't have an x with it, while the 2 does. As a result, the student gets 12 - 2x for his answer, forgetting to distribute the 4 with all the terms in parentheses.
• Students may not completely distribute a number through a whole problem. For example, in the problem 4(3 + x), students simplify it to 4(3) + x, forgetting to multiply the 4 by the x.
• Students may not recognize brackets as a form of parentheses.
• Up to this point, many problems students have encountered have had integers for answers, so students will think they have the incorrect answer if they produce a non-integer answer.
• Students sometimes do not understand how to input exponents into a calculator.
• Students sometimes fail to realize that the distributive property needs to be applied in situations where the multiplying number is written after the parenthesis, such as

(2x - 3)5.

## Vignette

##### In the Classroom

This vignette shows one way that students can practice using the order of operations.

Teacher: Your task is to take these four cards and arrange them in a way, connected by any mathematical symbols you want, to equal the largest quantity. The cards you will use are: 4, -6, -1, 5. Make sure you use the correct order of operations.

As the teacher walks around the room, he sees students trying many different approaches. Michael writes 4x(-1) x 5x(-6). He comes up with 120.

Teacher: While that is a large number, Michael, I think there is a larger number. Think about what else you could do to make the number even larger.

Going over to Jordin's desk, the teacher sees that Jordin multiplies the first two numbers to get -24 and the last two numbers to get -5, and then adds the numbers to yield -29.

Teacher: Ok, Jordin, I see you have used the correct order of operations. Good job. Remember our task here was to get the largest number possible though. Do you think -29 is the largest number that you could have gotten? Think about the value of negative numbers. Are they greater than or less than positive numbers?

Jordin: Smaller.

Teacher: Yes. So is there something you could do to your expression to at least make it a positive number?

Jordin: Can I change the order of the numbers?

Teacher: The only directions I gave you were to use any mathematical symbols to make the largest value. Did I say you couldn't change the order?

Jordin: I guess not.

Teacher: Good. Don't limit yourself. Don't put any restrictions on yourself that I haven't. Try changing the order and see what you get.

Jordin sits down and changes the order of the numbers, putting the two positive numbers together, multiplying them, and then putting the two negative numbers together, multiplying them as well.

Jordin: OK, look what I got now. I multiplied -6 x -1 and got 6, and multiplied 4 and 5 and got 20. I'm going to multiply those two, instead of just adding them like I did last time. And I get 120. I think that's the biggest number.

Teacher: Why did you multiply the two values instead of adding them?

Jordin: Because I know when you multiply it increases faster because you are basically adding lots of values. What I mean is by multiplying 20 and 6, I really have 20 6's, or 6 20's. That's going to be a lot bigger than just adding 20 and 6, because that would only be 26.

Teacher: True. While I agree that your answer of 120 is indeed bigger than your previous number, I think you should try other strategies as well. Think about all the orders of operations. What are they again?

Jerry: Parentheses, exponents, multiplication, division, addition and subtraction.

Teacher: Right. Has anyone used parentheses or exponents yet?

Since no one had thought of that, the teacher gave the students more time to manipulate the numbers.

Tom: OK. I threw in parentheses around the 5 and -6, like this: (5x - 6)4+(-1). I got -30 to the 4th power, which is -30 x -30 x - 30 x -30. That is 810,000. Then add a -1, and you get 809,999. That is a pretty big number!

Teacher: Wow, Tom, that is! Nice job. Good use of exponents. Does everyone see how by adding that exponent in there, the number got bigger so much faster?

Barry: So I wrote down 4 -(-1) x 5 + (-6) and got 19. Jerry says he got 3. Who is right? I am confused.

Teacher: "OK, can someone tell me what Barry did?

Jordin: I think he did the problem in order from left to right. 4 - (-1) is 5, and 5 x 5 is 25. Now you have 25 plus -6, and you end with 19, so that's what Barry did.

Teacher: So, Barry just went left to right?

Jordin: Yes. That's not the correct order of operations. We have to multiplication and division first!

Barry: That's right. I guess I just forgot.

Teacher: So how can we ALL remember the order of operations so we don't make the same mistake that Barry did?

Tom: PEMDAS: Please Excuse My Dear Aunt Sally. I write it on my paper as soon as I get it so I don't forget!

Teacher: Good. That's a great suggestion. Another thing to do would be to write down ALL your steps and double check your work. Also, by just slowing down a bit, you may catch those errors. Nice job today!

##### Vignette.2

This vignette walks a student through the definition of the order of operations, how to use it, and how to remember the order.

Student: If I have the problem 2 + 5 * 2, will the answer be the same if I start with 2 + 5 and if I start with 5 * 2?

Teacher: No, your answer will not be the same. By adding 2 + 5 you get 7 and then you are multiplying the number 7 by 2 to get 14. However, if you start by multiplying 5 * 2 then you get 10 and adding 2 would give you 12.

Student: I see how it can make a difference! Well, how do I know which way is the right way to solve the problem?

Student: The order of operations sets rules so that you know which operations you should do first.

Student: What operations are included in the order of operations?

Teacher: The order of operations tells you the order to do problems with exponents, problems with parentheses around specific operations, and problems that involve addition, subtraction, multiplication and division.

Student: OK, how do I learn which to do first?

Teacher: Here is the order that you must work through a problem:

1. First, you must do any operations that are in parentheses.
2. Second, you must take care of the exponents.
3. Third, you must do all the multiplication and division operations. (Start from the beginning of the math problem and work your way towards the end doing anything that is multiplication or division as you go. In other words, work from left to right.)
4. Fourth, you must do all addition and subtraction operations. (Start from the beginning of the math problem and work your way towards the end doing anything that is addition or subtraction as you go. In other words, work from left to right.)

Student: That is a lot to remember! So if I were to represent the order using math symbols it would be:

1. ( )
2.  ^
3. x and ÷
4. + and -

Teacher: That is right! There are also some fun ways to remember the order. One is to remember the rule as an acronym: P for parenthesis, E for exponents, M for multiplication, D for division, A for addition and S for subtraction. The acronym is PEMDAS.

Student: PEMDAS, I don't know if I can remember all of those letters.

Teacher: Another way to remember the order of operations is to memorize this saying: Please Excuse My Dear Aunt Sally. Each of those words begins with the letter of the acronym in the correct order.

Student: So, I guess now I can go back and figure out which method for solving

2 + 5 * 2 was correct.

Teacher: That is right. Using the order of operations, you can solve for the correct answer.

Student: OK, well, the problem 2 + 5 * 2 includes multiplication and addition. According to the order of operations, multiplication comes before addition. Therefore, I should first multiply 5 by 2 to get 10, and then after doing that I should add 2 to the 10, which would equal 12.

Teacher: Great work! Let's do a more complicated one to make sure you understand all of the operations and their order. Try solving this problem: 19 - (3 + 1) ÷ 4 + 2^3 * 2.

Student: Wow, that problem looks hard.

Teacher: Just remember to go step by step and you'll do fine.

Student: OK, the first thing I should do is solve any problems that are in parentheses. There is a problem in parentheses: 3 + 1. I know that 3 + 1 = 4. Now the problem is:

19 - 4 ÷ 4 + 2^3 * 2

Next, I should solve any problems with exponents. There is a number with an exponent: 2^3. I know that 2^3 is 8. Now the problem is:

19 - 4 ÷ 4 + 8 * 2

Now I need to solve any problems that are multiplication or division. There are two problems that are multiplication and division: 4 ÷ 4 and 8 * 2.

Teacher: Remember, you should start with the operation at the beginning of the problem and then work your way towards the end so that you are working from left to right.

Student: Right, so first I would do 4 ÷ 4 since it is closer to the front. I know 4 ÷ 4 = 1. Now the problem is:

19 - 1 + 8 * 2

Next I will do the multiplication problem 8 * 2. I know 8 * 2=16. Now the problem is:

19 - 1 + 16

Teacher: It looks a lot better now!

Student: Lastly, I look for addition and subtraction problems. I see two. Since I am supposed to start at the beginning of the problem with these, I will start with the subtraction problem 19 - 1. I know 19-1 = 18. Now the problem is:

18 + 16.

This leaves only one operation left! I need to add 18 and 16. 18 + 16 = 34. Now there is nothing left of the problem. The final answer is 34!

Teacher: Great job! Just remember to always go step by step through the order of operations when you have a problem with multiple operations!

Source: adapted from http://www.shodor.org/interactivate/discussions/OrderOfOperations/

## Resources

Instructional Notes
• The order of operations was established in order to prevent miscommunication, but PEMDAS can generate its own confusion; some students tend to apply the hierarchy as though all the operations in a problem are on the same level, but often those operations are not equal. Many times it helps to work problems from the inside out, rather than left-to-right, because often some parts of the problem are "deeper down" than other parts. For example, to simplify 4 + [-1(-2 - 1)]2, the nested parentheses shouldn't be worked on from left to right; rather, they should be dealt with from the inside out. The subtraction inside the curved parentheses should be simplified first, then inside the square brackets, and only then take care of the squaring. After that is done, the 4 can be added:

4 + [-1(-2 - 1)]2

= 4 + [-1(-3)]2

= 4 + [3]2

= 4 + 9

= 13

Source: Adapted from http://www.purplemath.com/modules/orderops.htm

• Students may need support in further development of previously studied concepts and skills.
• Be sure students understand that each time a variable occurs in an equation or expression, it must be replaced by the same number.
• Be sure students understand that multiplication and division need to be calculated inorder from left to right as well as addition and subtraction.
• For example, in the problem 6 - 4 + 2, the correct answer is 4, but if students add first, they would compute the answer to be 0.
• Be deliberate in showing students how to input expressions into a calculator, including using the caret key (^) for exponents. But also point out that not all calculators have ^ and they might need to use the yx key to evaluate exponents. Also work backwards and show students calculator notation and see if they can evaluate what is typed, including recognizing exponents, nested parenthesis and different notation for multiplication.
• When introducing the distributive property, use areas of rectangles to help explain the property. The area of a rectangle is length x width (l x w). When simplifying the expression 2(3 + 5), one length of the rectangle is 2, and the other is (3 + 5). To find the area of the whole rectangle, one could multiply 2 x (3 + 5), or one could separate the rectangle into two parts: one part with dimensions 2 x 3, and the other with dimensions
2 x 5. Have the students figure out the areas of each, and compare them. They will see that they are the same value.

• Help students develop word associations to remember the correct names of the properties. For example, Students associate in groups in the hallway (Associative property is when the grouping of the terms (generally with parentheses) changes). Sometime parents commute to their jobs. To commute might mean to go from Long Island to New York City and then return later, so the position switches back and forth, just as the order of the terms changes in the commutative property. To distribute means to pass something out. The distributive property passes whatever term is outside the parenthesis to each term within.
• Students get confused on substituting values for variables and forget their computation rules, have them write out all steps so they can see what they are doing and be less likely to make errors.
Instructional Resources
• Order of operations Bingo
Instead of calling numbers to play Bingo, the teacher can call (and write) expressions to be evaluated for the numbers on the Bingo cards. The operations in this lesson are addition, subtraction, multiplication and division. None of the expressions contain exponents. As an extension to this lesson, have students write their own expressions that include exponents for the answers on the bingo board.
• Distributing and factoring using area
In this lesson, expressions representing area of a rectangle are used to enhance understanding of the distributive property. The concept of area of a rectangle can provide a visual tool for students to factor monomials from expressions.

#### Additional Instructional Resources

• Substituting values
This website shows a way to use manipulatives to demonstrate substituting (plugging in) values into an expression and evaluating.
• Frank, M. (2009). Warm Up With Garfield: Math Conundrums. Nashville, TN: Incentive Publications.
This book offers warm- ups with cartoons as short, challenging learning experiences requiring the use of higher level thinking skills.
• Brainpop video on order of operations (subscription)

This website provides examples that explain how to simplify expressions with parentheses.

New Vocabulary
• evaluate: to perform operations to obtain a single number or value.
• substitute: to replace one element of a mathematical equation or expression with another.
• simplify: to make less complex or complicated; to make plainer or easier
• equivalent: equal in value.
• exponent: the number of times a number or expression (called base) is used as a factor of repeated multiplication; also called the power.
• associative property: a property that denotes that an operation is independent of grouping.

Examples:

(a + b) + c = a + (b + c)
(ab)c = a(bc)

• commutative property: a property that denotes that an operation is independent of the order of combination.
Examples:
a + b = b + a
a * b = b * a
• distributive property: the sum of two addends multiplied by a number is the sum of the product of each addend and the number.

Examples:
a(b + c)= ab + ac
7(3 + 5) = 7(3) + 7(5)
(b + c)a = ba + ca

• like terms: terms in an algebraic expression that have the same variable raised to the same power. Only the coefficients of like terms are different.
• expression: a collection of numbers, variables and symbols such as +, -, x and /. An expression can contain grouping symbols such as parentheses.
• order of operations: a procedure for evaluating an expression that has more than one operation, in the following order: Parenthesis, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right;
often referenced by the mnemonic: Please Excuse My Dear Aunt Sally.
Professional Learning Communities
##### Reflection - Critical Questions regarding the teaching and learning of these benchmarks
• How can the properties of positive exponents be used to simplify expressions?
• How can it be determined if two expressions are equivalent?
• How are expressions simplified and evaluated?
• What steps are involved in generating equivalent expressions?
• What properties are used in generating equivalent expressions?
References

## Assessment

1. Which is a verbal description of the associative property of addition?

a. The sum of zero and any number is equal to that number.
b. The sum of two numbers added to a third number is the same as adding the first number to the sum of the second and third number.
c. The order of two numbers being added can be switched without affecting the sum.
d. The product of a number and the sum of the two other numbers is equal to the sum of the products of the first number with each of the other two numbers.

2. Which of the following is the best way to simplify this expression? 47 x 512

a. (40 + 512) x (7 + 512)
b. (40 x 500) + (7 x 500)
c. (47 x 500) + (47 x 10) + (47 x 2)
d. (47 + 500) x (47 x 10) x (47 +2)

3. Which property does this equation demonstrate? 8 x 5 x 6 = 5 x 6 x 8

a. associative property
b. distributive property
c. identity property
d. commutative property

4. Evaluate the expression for the given values of the variables: -3x + 2y

A) What is the value when x = -3 and y = 3?

B) What is the value when x is 4 and y is -1/2?

Answers: A) 15; B) -13

5. Simplify the following expression: 3(2.25)2
Source: MN MCA Grade 7 Item sampler

6.  What is the value of 4t2 + 6r - tr when t =-3 and r = 5?
Source: MN MCA Grade 7 Item sampler

7.  Simplify 8 - 2(n + 4)(-3)2.

A.  -2n - 9
B.  -18n
C.  -18n - 64
D.  36n - 216

Source: MN MCA Grade 7 Item sampler

## Differentiation

Struggling Learners
• Make sure the classroom is equipped with ample sets of manipulative materials and supplies.
• Teachers and students should have access to appropriate resource materials from which to develop problems and ideas for explorations.
• All students should have a calculator to check their work when finished.
• Every classroom should have at least one computer available at all times for demonstrations and student use. Additional computers should be available for individual, small group and whole class use.
• Model alternative ways to solve the problem, and model more than once.
English Language Learners
• Encourage the use of manipulatives and model how to use them.
• Model alternative ways to solve the problem, and model more than once.
• Homework assignments must reinforce ELLs understanding of the instructional objectives. Appropriate homework can enhance communication with parents, but such homework should not be dependent upon the parents' skills in mathematics.
Extending the Learning
• Extend evaluating mathematical expressions at a given value to using function notation. For example, evaluate the expression 2x + 5 when x = -5 could be written as if f(x) = 2x + 5 then find f(-5).
• Use a wide variety of assessment measures beyond standardized achievement tests which limit mathematics to low level computation.
• Give students a wide variety of rich, inviting tasks that require spatial as well as analytic abilities.
• Encourage students to persist in solving mathematical problems.
• Expect students to not only solve problems posed by others but to pose and solve new problems of their own.
• Try this exercise about doubling numbers (exponential growth).
Some growth patterns follow a pattern that can be described using an exponent. A doubling of population every few years is an example of exponential growth.
• You are studying population growth patterns. You find that the population of a town is doubling every 5 years. The population is 15,000 now. In how many years will the population reach 500,000?

Solution: Make a list or table

 Number of years from now Population 0 15,000 5 15,000 · 2 = 30,000 10 15,000 · 22 = 60,000 15 15,000 · 23 = 120,000 20 15,000 · 24 = 240,000 25 15,000 · 25 = 480,000

Suppose the population of a town of 15,000 tripled every 10 years. When would the population be greater than 1,000,000? (Answer: in a little less than 40 years)

Ask students to solve this problem:
Aki multiplied this expression. Did she get the right answer? Explain your thinking.
(w - 5x - 4y)(z + 7) = wz - 4yz + 7w - 2815,000 · 2

Classroom Observation
##### Administrative/Peer Classroom Observation
 Students are: (descriptive list) Teachers are: (descriptive list) correctly using the properties of algebra. making sure students are evaluating a variety of expressions using correct order of operations. using correct order of operations. allowing students to verbally explain the mathematical properties being used when simplifying expressions. using strategies to remember order of operations, such as Please Excuse My Dear Aunt Sally. including expressions that involve whole number exponents for students to evaluate. writing all steps down to see work being done. reminding students to show their steps when simplifying a mathematical expression. using calculators and understanding when a number has been rounded or when it has been truncated.