8.2.3 Algebraic Expressions
Evaluate algebraic expressions, including expressions containing radicals and absolute values, at specified values of their variables.
For example: Evaluate πr2h when r = 3 and h = 0.5, and then use an approximation of π to obtain an approximate answer.
Justify steps in generating equivalent expressions by identifying the properties used, including the properties of algebra. Properties include the associative, commutative and distributive laws, and the order of operations, including grouping symbols.
Overview
Students need to move from relational thinking about equivalence to using algebraic properties to generate equivalent expressions. It is essential for students to be able to use the properties of algebra to manipulate numerical and mathematical expressions. It is often difficult for students to make meaning of these properties, so it is essential that they experience a variety of situations in which the property is necessary and the mathematics behind the property makes sense. Students need real situations and context that allows them to express the same idea in multiple ways. This allows students to discuss their point of view and compare it to others. This comparison helps students to see the algebraic properties "in action" and examine this idea of equivalence. Students need to make connections between these expressions and other representations in context (verbal, graphical, tabular) in order to make sense of the variable.
All Standard Benchmarks - with codes:
8.2.3.1 Evaluate algebraic expressions, including expressions containing radicals and absolute values, at specified values of their variables.
For example: Evaluate $\pi{r}^{2}h$ when r = 3 and h = 0.5, and then use the approximation of $\pi$ to obtain an approximate answer.
8.2.3.2 Justify steps in generating equivalent expressions by identifying the properties used, including the properties of algebra. Properties include the associative, commutative and distributive laws, and the order of operations, including grouping symbols.
Benchmark Group A
8.2.3.1 Evaluate algebraic expressions, including expressions containing radicals and absolute values, at specified values of their variables.
For example: Evaluate $\pi{r}^{2}h$ when r = 3 and h = 0.5, and then use the approximation of $\pi$ to obtain an approximate answer.
8.2.3.2 Justify steps in generating equivalent expressions by identifying the properties used, including the properties of algebra. Properties include the associative, commutative and distributive laws, and the order of operations, including grouping symbols.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Evaluate an algebraic expression by replacing the variable with the defined values and using order of operations to find the value of the expression.
- Understand the properties of algebra and be able to recognize when they have been used.
- Evaluate expressions involving radicals and absolute value.
- Generate equivalent expressions using algebraic properties such as the commutative, associative, and distributive properties.
Work from previous grades that supports this new learning includes:
- Perform operations on integers
- Simplify integer/numeric expressions
- Use and understand absolute value
- Explore order of operations
NCTM Standards:
Represent and analyze:
In grades 6-8 all students should-
- develop an initial conceptual understanding of different uses of variables;
- use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships;
- recognize and generate equivalent forms for simple algebraic expressions and solve linear equations
Understand meanings of operations and how they relate to one another.
- use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals;
CCSS:
7. EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients
Misconceptions
Student Misconceptions and Common Errors:
- Students will sometimes combine unlike terms such as 2x + 5 combines to 7x or 2x + 3x2 combines to 5x2 and in these examples the terms are not alike and should NOT be combined;
- PEMDAS (Order of Operations) is often used improperly and students believe multiplication should occur before division and addition should occur before subtraction when in reality multiplication and division are one step (from left to right) along with addition and subtraction;
- Not all calculators follow correct order of operations. Non-scientific calculators perform operations in the order entered;
- Most students remember the process/rules of the properties but sometimes either get the names confused or forget the names altogether;
- Students recognize the distributive property when the multiplier is written first as 6(2x + 3), but sometimes overlook the distributive property when the multiplier is written after the parenthesis (2x + 3)6 even though it means the same thing;
- When students learn order of operations they learn PEMDAS and often don't realize the first step of P includes ALL grouping symbols, not just the parenthesis, such as bracket, brace, and absolute value;
- Students sometimes mistakenly interchange the mathematical words "expression" and "equation";
- When evaluating x2 for a negative value, students will evaluate using their calculator without using parentheses and therefore will not have the correct sign. Ex: Evaluating x2 for x = -3 should result in 9, but entering -32 in a calculator will result in -9.
Vignette
In the Classroom
Teacher: I want you to clear everything off of your desk, including your pencil.
Take a look at the square on the board. I want you to notice that the whole square is a 10 by 10 tiled grid. I also want you to notice that the tiles on the border of the square are shaded. Without counting; without writing; and without talking, how many of the tiles are shaded.
Austin: I got 40 tiles because since it's a 10 by 10 there are 10 tiles on each side of the square.
Becca: I got 36 tiles because the corners get counted twice so I subtracted the 4 corners from 40.
Austin: Oh yeah. I guess those do get counted twice. So I agree that there are 36.
Cort: I got 36 too, but I just counted 9 on each side. So then you don't have any tiles that are counted twice. Can I show you on the board?
Teacher: Did anyone see it in a different way?
David: It's the same answer. I saw it like 10 across the top, 10 across the bottom, and 8 leftover across each of the sides. So that's 10 +10 + 8 + 8 which is 36.
Eric: I looked at the areas. The whole area is 100 because 10x10 is 100. Then I subtracted the inside square that isn't shaded. The square is an 8x8 so the area is 64. So if I take out the middle square I am left with 36 tiles, which is the outside. This was kind of harder than the other examples. David's idea was much easier. I would do it like that the next time.
Teacher: Any other ideas? So let's get this thinking up on the board.
After recording the original thinking, make a second column and ask "Is there another way to record this thinking?"
Becca:
10 + 10 + 10 + 10 - 4
4 (10) - 4
Cort:
9 + 9 + 9 + 9
4(9)
David:
10 + 10 + 8 + 8
(10) + 2(8)
Eric:
10x10 - 8x8
102 - 82
Teacher: Now I need you shrink this to a 6 tile by 6 tile square. Again the border is shaded. How many tiles are shaded in the 6 by 6 grid?
Student: Well I thought about it like Cort. I knew there would be 4 sides with 5 tiles so I got 20 tiles. (Teacher records Cort's thinking in a third column.)
Student: I got 20 tiles too, but I knew there were 6 on the top and bottom then 4 left on each side. So 6 + 6 + 4 + 4 is 20. It's like David's. (Teacher records David's thinking in a third column.)
Teacher: How would Becca think about this 6 by 6 grid?
Student: Well, she would say that there are 6 on a side but we have to subtract 4 because the corners are counted twice. (Again teacher records in third column next to Becca's work)
Teacher: How would Eric think about this 6 by 6 grid?
Student: He would find the area of the 6 by 6 and then subtract the 4 by 4. So 36 - 16 is 20 tiles. (Again teacher records with others.)
Teacher: In groups, you will be finding the number of tiles bordering two more different size squares. I want you to show how Becca, Cort, Eric and David would think about each of your squares.
(Students work in groups to work through 2 more squares. Groups record and post how each of the students would think about the new squares. Post the work from all the groups.)
Teacher: OK... so let's look at all these different size squares. What do you notice?
Student: I notice that for all the squares, Becca's thinking they take the side length of the square and multiply it by 4, then subtract 4 for the double counted corners.
Student: I notice that if we think like Eric, every time we have a new square we just find the area of the whole square and then subtract the inside square. The side length of the inside square is always 2 tiles less than the whole square.
Abby: Cort's thinking always cuts one tile off the side length and then multiplies that by the 4 sides.
Frank: I thought Cort cut one of the tiles off and then added all those lengths 4 times.
Teacher: Are those two statements different? Let's check it out. How could we write a rule for any size square based on the two statements for Cort's thinking?
Student: The way Abby described Cort's thinking I would write: 4(n - 1) . n would be the side length of the size of the square.
Student: Frank would write n - 1 + n - 1 + n - 1 + n - 1 to show Cort's thinking.
Teacher: Are these expressions the same or different?
Student: Well they look different but they would be the same answer. Frank added n - 1 four times and Abby basically did the same thing. When you add something four times it's just easier to multiply it by four. So I think they are the same.
Teacher: How could we write an expression to represent Becca's thinking for any size square?
Student: Well, I would write 4n - 4 because 4n would be 4 times the side length and then minus four is for the corners.
Teacher : Are these three expressions equivalent?
4(n - 1)
n - 1 + n - 1 + n - 1 + n - 1
4n - 4
Student: Isn't that what we just said?
Teacher: So how could you prove it? How could you convince someone else that these three ways of thinking about the square are equivalent?
Student: We could test it out for other side lengths and plug in the side length into each expression.
Student: I would simplify each expression by combining like terms and the distributive property. Then they would all be 4n - 4 so they are all equivalent.
Teacher: What if I write 2n + 2(n - 2) ? Who's thinking could be modeled by this expression?
Student: Well David wrote 10 + 10 + 8 + 8.
Which is kind of like 2(10) + 2(10 - 2) so that's like 2n + 2(n - 2) If you look at the rest of the squares and David's thinking for those the same thing happens.
Student: He always adds the total side lengths twice for the top and bottom. Then the two left over sides were two tiles less. So I could write the expression n + n + (n - 2) + (n - 2).
Teacher: Is this equivalent to the rest? How do you know?
Student: Yes! If I combine like terms again, it simplifies to be 4n - 4.
Student: But how does 2n + 2(n - 2) simplify to be 4n - 4 ?
Student: You just have to keep simplifying. You can use the distributive property for
2(n - 2) to get 2n - 4 then you add the 2n and get 4n - 4.
Teacher: All of you just showed us four ways of thinking about the number of shaded tiles around different size squares. You all came up with the same number of tiles but you wrote different expressions.
*note: Students can usually also write an expression to represent Eric's thinking. They see the pattern of n2 - (n-2)2 However, this is very difficult for students to show symbolically it is equivalent to the others. Depending on their previous experience with quadratic functions and multiplying binomials, this could be an interesting challenge for some students.
Resources
Teacher Notes:
- Students must learn to evaluate expressions that contain multiple operations and variables. Therefore students must follow the correct order of operations when evaluating the expressions.
- Students have a tendency to want to not take the time to show all the steps when evaluating an expression. Teachers should help students see that not showing all steps might lead to incorrect calculations, and provide students with opportunities to find the mistake or see that different order produces different results
- Help students develop word associations to remember the correct names of the properties. Ex: 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 job. To commute might mean to go from Long Island to NYC and then later come back, so the position switches back and forth. (The order of the terms change in commutative property) To DISTRIBUTE means to pass something out. The distributive property passes whatever term is outside the parenthesis to each term within. Your IDENTITY is who you are. The multiplicative and additive properties perform the operation, but don't change the identity of the answer.
- It is helpful for students to understand why we use the commutative and associative properties. For the commutative property, think about 3 + 56. It is easier to start at 56 and add 3 than start at 3 and add 56. For the associative property, think about 5 x (2 x 53). It may be easier for students to multiply 5 x 2 as a group first to get 10 and then multiply by 53. Thinking about these properties as another way to chunk numbers to make them "work" for the student will give the properties more purpose for students.
- Students may have inadvertently learned order of operations and think that multiplication needs to be done before division and not realize that they should be done in order from left to right (the same with addition and multiplication).
A common technique for remembering the order of operations is the abbreviation "PEMDAS," which represents the phrase "Please Excuse My Dear Aunt Sally." It stands for Parentheses, Exponents, Multiplication, Division, Addition and Subtraction, which also identifies the rank of operations. Parentheses outrank exponents, which outrank multiplication and division (but multiplication and division are at the same rank), and these two outrank addition and subtraction (which are together on the bottom rank). When you have multiple operations of the same rank, you operate from left to right. For instance, the expression 15 ÷ 3 × 4 is not 15 ÷ 12 but is rather 5 × 4, because going from left to right, the division operation is first. You may test it by entering the expression into a graphing calculator which has been programmed with the Order of Operations hierarchy.
- Students should understand the difference between a mathematical expression and a mathematical equation. An equation has an equal sign while an expression does not.
- Scientific/Graphing Calculators follow a strict order of operations. For example, if evaluating x2 for x = -3, when entering -32 the calculator will first square the 3 and then make it negative, resulting in an incorrect answer. In order to get a correct answer of 9, parentheses are necessary. Without a calculator students will give a correct answer of 9, but they will inevitably believe the calculator.
- Fairly simple equivalences can be involved. For example, the cost (in dollars) of using Keep-in-Touch phone company can be expressed as y = 0.10x + 20, as y = 20 + 0.10x, as 20 + 0.10x = y, or as 0.10x + 20 = y. Complex symbolic expressions also can be examined, such as the equivalence of 4 + 2L + 2W and (L + 2)(W + 2) - LW when representing the number of unit tiles to be placed along the border of a rectangular pool with length L units and width W units.
Instead of calling numbers to play Bingo, you 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. Can also be updated to where the expressions to be evaluated involve variables to be substituted in first. An activity could also be that the students need to develop the expressions that need to be evaluated to get each of the bingo numbers when solved.
Additional Instructional Resources:
- Cool math
- Very important for teachers to use the area models to teach distributing and factoring.
- Distributing and Factoring Using Area NCTM Illuminations Lesson
- Illuminations Lesson Algebraic Transformations
- This lesson uses shapes to explore and understand Identity, Inverse, Commutative Property, and Associative Property
- A scope and sequence of teaching properties of real numbers.
Vocabulary:
simplify: To use the rules of arithmetic and algebra to rewrite an expression as simply as possible. In this standard the focus is on simplifying expressions with variables by combining like terms.
evaluate an algebraic expression: To find the value of an expression by replacing each variable in an expression with numbers.
associative: The way you group three or more numbers when adding or multiplying does not change their sum or product. For any numbers (a + b) + c = a + (b + c) and (ab)c = a(bc)
commutative: The order in which you add or multiply numbers does not change their sum or product. For any numbers a and b, a + b = b + a and 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. For example, a(b+c)= ab + ac or 7(3 + 5) = 7(3) + 7(5) or (b + c)a = ba + ca
identity: The identity property for addition tells us that zero added to any number is the number itself. Zero is called the "additive identity." The identity property for multiplication tells us that the number 1 multiplied times any number gives the number itself. The number 1 is called the "multiplicative identity."
Addition
5y + 0 = 5y
Multiplication
2c × 1 = 2c
inverse property: Inverse properties state that when a number is combined with its inverse, it is equal to its identity. There are two types of inverses of a number: Additive Inverse and Multiplicative Inverse. -a is said to be the additive inverse of a if a + (-a) = 0. $\frac{1}{a}$ is said to be the multiplicative inverse of $a$ if $a\times \frac{1}{a}=1$.
order of operations: The rules to be followed when simplifying expressions.
When you are simplifying expressions, operations should be performed in the following order to ensure accuracy:
1. Parentheses (and other grouping symbols)
2. Exponents
3. Multiplication and Division (in order from left to right)
4. Addition and Subtraction (in order from left to right)
grouping symbols: Grouping symbols include parentheses ( ), brackets [ ], braces { }, and fraction bars in $\frac{(5+7)}{2}$. Evaluate expressions inside grouping symbols first when using order of operations.
radical: The √ symbol that is used to denote square root or nth roots.
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
- As a result of today's lesson, what evidence do you have of students' learning?
- How do you know when students have reached understanding of this learning?
- What are next steps, including for students who do not understand?
- What are the most common misconceptions students have regarding the order of operations? What can be done to break those misconceptions?
- What examples were most helpful in getting students to understand the order of operations? What other examples would help students to better understand the order of operations?
Materials
Boaler, J.; Humphreys, C. Connecting mathematical ideas: Middle school video cases to support teaching and learning. This text comes with a DVD and is a great resource to watch and discuss as a Professional Learning Community (PLC).
- Vignette adapted from Connecting Mathematical Ideas: Middle School Video Cases to Support Teaching and Learning by Jo Boaler and Cathy Humphreys.
- AIMS Education Foundation. Triangles of Squares and Pythagorean Puzzle.
- Focal Points Grade 8, NCTM.
Good Questions: Great Ways to Differentiate Mathematics Instruction Marian Small (NCTM).
Boaler, J., & Humphreys, C. (2005). Connecting mathematical ideas: middle school video cases to support teaching and learning. Portsmouth, NH: Heinemann.
Focus in grade 8: teaching with curriculum focal points.. (2010). Reston, VA: National Council of Teachers of Mathematics.
Friel, S. N. (2008). Navigating through problem solving and reasoning in grades 6-8 . Reston, VA: National Council of Teachers of Mathematics.
Illuminations: Algebraic Transformations. (n.d.). Illuminations: Welcome to Illuminations. Retrieved June 13, 2011, from http://illuminations.nctm.org/LessonDetail.aspx?id=U157
Properties of Real Numbers. (n.d.). T/TAC Online. Retrieved June 13, 2011, from www.ttaconline.org/d/sol/Mathematics/Math7_OT02LN01.pdf
Silver, Z. (n.d.). Illuminations: Order of Operations Bingo. Illuminations: Welcome to Illuminations. Retrieved June 13, 2011, from http://illuminations.nctm.org/LessonDetail.aspx?id=L730
Solving Absolute Value Equations and Inequalities - Cool math Algebra Help Lessons - Solving Absolute Value Equations . (n.d.). Cool math .com - An amusement park of math and more! Math lessons, math games, math practice, math fun!. Retrieved June 13, 2011, from http://coolmath.com/algebra/18-absolute-value-equations-inequalities/02…
Tran, A. (n.d.). Illuminations: Distributing and Factoring Using Area. Illuminations: Welcome to Illuminations. Retrieved June 13, 2011, from http://illuminations.nctm.org/LessonDetail.aspx?id=L744
Assessment
1. Which property is used in the equation mg + mh = m(g + h)?
A. Associative
B. Commutative
C. Distributive
D. Identity
Taken from MN MCA III test specs
Correct answer: C
2. What is the value of -3│-2x - y│when x = -4 and y = 5?
A. -27
B. -9
C. 9
D. 27
Taken from MN MCA III test specs
Correct answer: B
3. Is the equation 3(2x -4) = -18 equivalent to 6x -12 = -18?
A. Yes, the equations are equivalent by the Associative Property of Multiplication.
B. Yes, the equations are equivalent by the Commutative Property of Multiplication.
C. Yes, the equations are equivalent by the Distributive Property of Multiplication over Addition.
D. No, the equations are not equivalent.
Correct answer: C
4. Which of the following equations illustrated the inverse property of multiplication?
A. 5 x $\frac{1}{5}$ = 1
B. 5 x 1 = 5
C. 5 x 0 = 0
D. 5 x 5 = 25
Taken from CA assessment test
Correct answer: A
5. Which expression is equivalent to 3x - 3y?
A. 3xy
B. 3(x - y)
C. 3x - y
D. x - 3y
Correct answer: B
6. Which equation is equivalent to 3[7x - 4(x - 3)] + 1 = 16?
A. 9x - 2 = 16
B. 9x + 37 = 16
C. 17x - 2 = 16
D. 17 +13 = 16
Correct Answer: B
7. Which equation is equivalent to 5x - 2(7x + 1) = 14x?
A. -9x - 2 = 14x
B. -9x + 1 = 14x
C. -9x + 2 = 14x
D. 12x - 1 = 14x
Correct Answer: A
8. What is the perimeter of the figure shown below, which is NOT drawn to scale?
A. 5x + 33
B. 5x3 + 33
C. 8x + 30
D. 8x4 + 30
Correct answer: C
9. If x = 4 and y = 3, then xy-2x =
A. 4
B. 6
C. 19
D. 40
Correct answer: A
10. The formula below can be used to find S, the sum of all integers from 1 to n, where n is any positive integer.
$S=\frac{n(n+1)}{2}$
What is the value of S when n = 50?
A. 1250
B. 1275
C. 2500
D. 2550
Taken from Massachusetts Comprehensive Assessment Released Exam
Correct answer: B
11. Evaluate the radical expression $\frac{8+2\sqrt{b}}{a}$ when a = 2 and b = 4.
Choices:
A. 9
B. 8
C. 7
D. 6
Correct Answer: D
12. Evaluate the radical expression $\sqrt{3x+5y}$ when x=5 and y=2
Correct answer: 5
Differentiation
Struggling Students:
- If students are struggling with the distributive property teachers could let them use algebra tiles to demonstrate 5(x +2), or use the distributive property with real world items. 5(2 apples + 6 oranges) means you have 5 groups of 2 apples and 6 oranges for a total of 10 apples and 30 oranges.
- For students struggling with evaluating expressions, sometimes giving them a sheet of paper with step by step instructions can help. For example:
Procedure
To evaluate an algebraic expression, replace the variables by the given values, and then follow the rules for the order of operations.
- Replace the variables by the given values.
- Evaluate the expression within the grouping symbols such as parentheses, always simplifying the expressions in the innermost groupings first.
- Simplify all powers and roots.
- Multiply and divide, from left to right.
- Add and subtract, from left to right.
- Review all the ways we can indicate to multiply: 3(x), 3x, 3 * x, 3(9), 3 * 9, 3 times sign 9. Describe the expressions as being equivalent and reinforce that math vocabulary (equivalent) from the word wall.
- The teacher can add the vocabulary words to a class word wall.
- Create a vocabulary graphic organizer to explain the relationships of the vocabulary and/or their meaning (e.g., Frayer model, concept web).
- Students can add these vocabulary words to a student dictionary that has the definitions and illustrations for each word.
- Small groups of students can create a matching activity to use with other students. Give each group a stack of index cards. List the names of the properties on index cards. On matching index cards, list the corresponding numerical examples. The cards are mixed, turned over, and a memory matching game is played (match the property to the numerical example).
- Provide a chart or bulletin board with pictorial, numerical, and written examples for each property of real numbers.
- Have students do a "number jumble". Put 4 random numbers on the board ranging from 1-9 and give them a random target number to try and reach by using each of the four original numbers exactly once. They may use any mathematical operation, but must use correct order of operation. Variations: use more or less numbers to start with. (This is also like the game 24).
- Super Chocolates are arranged in boxes so that a caramel is placed in the center of each array of four chocolates, as shown below. The dimensions of the box tell you how many columns and how many rows of chocolates come in the box. Develop a method to find the number of caramels in any box if you know its dimensions. Explain and justify your method using words, diagrams, or expressions.
Parents/Admin
Administrative/Peer Classroom Observation:
Students are: (descriptive list) | Teachers are: (descriptive list) |
evaluating a variety of algebraic expressions for given values of the variables. | teaching distributive property using an area model. |
recalling the different algebraic properties and recognizing where they have been used. | providing students with opportunities to find the mistake in order of operations or see that different order produces different results.
|
justifying their steps when they simplify algebraic expressions. | making meaningful connections to real life meanings of associate, commute, and distribute to help students understand what the associative, commutative, and distributive properties mean in mathematics. |
working with absolute value and radicals (square roots) when evaluating algebraic expressions. | expecting students to show their steps as they simplify expressions, writing equivalent expression along the way. |
using models to make sense of algebraic properties. | giving situations in which students must evaluate a negative value for a squared variable. |