TEACHER GUIDE TO CLARIFICATION
Properties of Operations
This standard is building conceptual and application knowledge regarding equivalent expressions in order to apply algebraic reasoning in later grades.
For this standard, we expect students to play (and we mean really play) with the properties of mathematical operations.
In order to apply the properties of operations, students need to be familiar with them. Combining like terms is also. Only after they can recognize and identify these properties will they be able to play around with them freely.
As students use these properties to create equivalent expressions, stress that these expressions are equivalent because no matter how we evaluate them, they'll always end up equaling each other. This is true of numerical expressions and variable expressions; that the two expressions should evaluate to give the same result every single time.
The beauty of generating equivalent expressions is that there is literally an infinite number of ways to write the same expression. There might be two or three obvious choices, like rewriting y + y + y as 3y, but we can also write the same expression as y(1 + 1 + 1) or 3y + 0 or even ½y + ½y + ½y + ½y +½y + ½y.
Take the time to allow students to explore equivalent expressions. Time spent developing this skill will provide deeper algebraic understanding.
My friend is building a stair case. Pat wants 26 steps in the staircase.
The pattern looks like these blocks. The picture below represents 5 steps and Pat used 15 blocks. How many blocks will Pat use to make 26 steps? 351
Extension: How many blocks will Pat use to make n steps?
Extension Sample Answer
Students can see an expression many different ways, allow them to explore and share why they are equivalent when representing different expressions
6.EE.3 Students use the distributive property to write equivalent expressions. For example, area models from elementary can be used to illustrate the distributive property with variables. Given that the width is 4.5 units and the length can be represented by x + 3, the area of the flowers below can be expressed as 4.5(x + 3) or 4.5x + 13.5.
When given an expression representing area, students need to find the factors. For example, the expression
10x + 15 can represent the area of the figure below. Students find the greatest common factor (5) to represent the width and then use the distributive property to find the length (2x + 3). The factors (dimensions) of this figure would be 5(2x + 3).
Students use their understanding of multiplication to interpret 3 (2 + x). For example, 3 groups of (2 + x). They use a model to represent x, and make an array to show the meaning of 3(2 + x). They can explain why it makes sense that 3(2 + x) is equal to 6 + 3x.
An array with 3 columns and x + 2 in each column:
Modeling with variables in an array will show a visual representation and allow students to interpret the expressions
Students interpret y as referring to one y. Thus, they can reason that one y plus one y plus one y must be 3y. They also use the distributive property, the multiplicative identity property of 1, and the commutative property for multiplication to prove that y + y + y = 3y:
y + y + y = y x 1 + y x 1 + y x 1 = y x (1 + 1 + 1) = y x 3 = 3y
Provide opportunities for students to write equivalent expressions, both numerically and with variables. For example, given the expression x + x + x + x + 4•2, students could write 2x + 2x + 8 or some other equivalent expression. Make the connection to the simplest form of this expression as 4x + 8. Because this is a foundational year for building the bridge between the concrete concepts of arithmetic and the abstract thinking of algebra, using hands-on materials (such as algebra tiles, counters, unifix cubes, "Hands on Algebra") to help students translate between concrete numerical representations and abstract symbolic representations is critical
Provide expressions and formulas to students, along with values for the variables so students can evaluate the expression. Evaluate expressions using the order of operations with and without parentheses. Include whole-number exponents, fractions, decimals, etc. Provide a model that shows step-by-step thinking when simplifying an expression. This demonstrates how two lines of work maintain equivalent algebraic expressions and establishes the need to have a way to review and justify thinking.
Provide a variety of expressions and problem situations for students to practice and deepen their skills. Start with simple expressions to evaluate and move to more complex expressions. Likewise start with simple whole numbers and move to fractions and decimal numbers. The use of negatives and positives should mirror the level of introduction in Grade 6 The Number System; students are developing the concept and not generalizing operation rules.
Great examples of how and why to take this standard deeper!
Many of the misconceptions when dealing with expressions stem from the misunderstanding/reading of the expression. For example, knowing the operations that are being referenced with notation like, x3, 4x, 3(x + 2y) is critical. The fact that x³ means x·x·x, means x times x times x, not 3x or 3 times x; 4x means 4 times x or x+x+x+x, not forty-something.
When evaluating 4x when x = 7, substitution does not result in the expression meaning 47. Use of the “x” notation as both the variable and the operation of multiplication can complicate this understanding.
Kansas Association of Teachers of Mathematics (KATM) Flipbooks. Questions or to send feedback: firstname.lastname@example.org.
Retrieved from Math Flipbooks.
Mathematical Practices # 2 Reason abstractly and quantitatively
This standard correlates well with Math Practice 2. Look at the characteristics and student behaviors below.
Make sense of quantities and their relationships.
Able to decontextualize (represent a situation symbolically and manipulate the symbols) and contextualize (make meaning of the symbols in a problem) quantitative relationships.
Understand the meaning of quantities and are flexible in the use of operations and their properties.
Create a logical representation of the problem.
Attend to the meaning of quantities, not just how to compute them
For example, students can move from 3a + 3b to (a + b) + (a + b) + (a + b) to 3(a + b)
Let’s think of equal expressions in terms of perimeter.
A fourth grader might see
3 + 3 + 5 + 5 or 3 + 5 + 3 + 5
A fifth grader might see
2 x 3 + 2 x 5
A sixth grader may see
2(3 x 5)
At this grade level students can be using whole numbers, decimals and fractions when working with equivalent expressions.
Write and equivalent expression for
¾ (12k + 16) = 9k + 12
(0.25 x 20) + (0.25 x 20) + (0.25 x 20) + 4 = 3(0.25 x 20) + 4
We know that the area of a triangle is ½ bh. Write 2 different expressions to find the area of a triangle.
½ AB x BC
½ BC x BA
Students should be able to know that
x + x = 2x and not x2. Have a discussion.
As students move from numerical to algebraic work the multiplication and division symbols × and ÷ are replaced by the conventions of algebraic notation. Students learn to use either a dot for multiplication, e.g., 1•2•3 instead of 1×2×3, or simple juxtaposition, e.g., 3x instead of 3 × x (during the transition, students may indicate all multiplications with a dot, writing 3 • x for 3x). A firm grasp on variables as numbers helps students extend their work with the properties of operations from arithmetic to algebra. MP2 For example, students who are accustomed to mentally calculating 5 × 37 as 5 ×(30 + 7) = 150 + 35 can now see that 5(3a + 7) = 15a + 35 for all numbers a. They apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x and to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y). 6.EE.3
Common Core Standards Writing Team. (2011, April 22).
Progressions for the Common Core State Standards in Mathematics(draft). 6-8, Expressions and Equations. Tucson, AZ: Institute for Mathematics and Educations, University of Arizona.
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 +2x; 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.
Coherence and Connections: Need to Know
PARCC Model Content Framework:
Example of Key Advances from Grade 5 to Grade 6
Students begin using properties of operations systematically to work with variables, variable expressions, and equations (6.EE).
Examples for Opportunities for Connections between Standards, Clusters, or Domains
Students use their skill in recognizing common factors (6.NS.4) to rewrite expressions (6.EE.3).
Examples for Opportunities for In-Depth Focus
By applying properties of operations to generate equivalent expressions, students use properties of operations that they are familiar with from previous grades’ work with numbers — generalizing arithmetic in the process.
PARCC Model Content Frameworks: Mathematics Grades 3-11 (version 5). (2017, November).
Retrieved from https://files.eric.ed.gov/fulltext/ED582070.pdf
Illinois Assessment of Readiness Mathematics Evidence Tables https://www.isbe.net/Documents/IAR-Grade-6-Math-Evidence-State.pdf
Evidence Statement Text
Base explanations/reasoning on the properties of operations.
Content Scope: Knowledge and skills articulated in 6.EE.3, 6.EE.4
i) Tasks should not require students to identify or name properties
Also check out Student Achievement Partners Coherence Map.
1a. Write more than one expression that is equivalent to 4x – 24.
4(x – 6) this answer is likely to be common. Ask students to continue to find different equivalent expressions.
2(2x – 12) Distributive Property
½ (8x – 48) Distributive Property
2x + 2x – 24 Associative Property
2. Draw a model to show that 2(y + 6) is equal to 2y + 12.
3. Write an equivalent expression for 4(b+2) + 3(d+1).
4b + 11 + 3d traditional answer
4(2 + b) + 3(1 + d) non- traditional answer using associative and commutative properties
2(2 +b) + 2(2 + b) + 3( 1 + d) using distributive property
4. Write two expressions that are equivalent to 5( g + 4) – 3
5g + 5 and 2(g + 2) + 3g + 4 – 3 Answers may vary
5. Write an equivalent expression for ½ ( 24x – 4)
12x – 2
2(6x – 1)
IL Classrooms in Action - Formative Assessment
PBS Podcast: Commutative and Associative Properties