1 Katrina Madden
CCLM^2 Project
Summer 2012
DRAFT DOCUMENT. This material was developed as part of the Leadership for the Common Core in Mathematics
project at the University of Wisconsin-Milwaukee.
Part 1. STANDARD 6.EE.4 Grade: 6 Domain: EE-­‐Expressions and Equations Cluster: Apply and extend previous understanding of arithmetic to algebraic expressions. Standard: 6.EE.4. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Part 2. EXPLANATION AND EXAMPLES OF THE STANDARD I chose to focus on 6.EE.4 (Identify when two expressions are equivalent-­‐ i.e. when the two expressions name the same number regardless of which value is substituted into them-­‐ For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for). This standard really stood out to me because it gets students’ to think algebraically and check their solution, along with making connections to finding equivalent expressions. This standard wants students to view 3y as y + y + y and understand that 3x + 4x = 7x. Having students understand that 3x represents 1x + 1x + 1x (or just x + x + x) and that 4x is the same as 1x + 1x + 1x + 1x can be an extremely difficult skill for students. From here students need to understand that when they combine all of their "x's" that is why they get the sum of 7x which can also be shown as x + x + x + x + x + x + x. It is also important that students think about combining like (common) terms. I can combine 3x and 4x to get a sum of 7x because those are like terms. For example 3x + 4x² is not equivalent to 7x because 3x and 4x² cannot be combined because they are not like terms; meaning they do not have identical variable parts (the exponents with x are not the same). Another example of a common misconception that students have is that 9x -­‐ 6x = 3. We know the correct solution to this subtraction problem is 3x. If students went back and substituted a value for x, they would see that the whole number 3 would not be a correct answer. Most of the examples above have shown breaking apart the expression and understanding what those parts in the expression really mean. 6.EE.4 also digs deeper and has students think about the distributive property and order of operations. Students must have a deep understanding of these topics as well to be able to prove expressions are equivalent and check their solutions. As written in the 6-­‐8 EE progression document on p. 6, "This can lead students to make mistakes such as 2 simplifying n -­‐ 2 + 5 as n -­‐ 7 (instead of the correct n + 3) because they add the 2 and the 5 before subtracting from n." 6.EE.4-­‐Something to think about: Using the commutative, associative and distributive properties simplify each expression into the same form. Are these expressions all equivalent? 4m + 8 4(m + 2) 3m + 8 + m 2 + 2m + m + 6 + m Part 3. SCHOOL MATHEMATICS TEXTBOOK PROGRAM
When I read 6.EE.4 it really stood out to be an important topic that students need to have a deep understanding of in order to fully understand and move forward to more complex problems/expressions in algebra. When going through a number of documents (my text, CCSSM, Progressions, etc.) there were many questions and ideas circling around in my head. One area in math that has always been important to me is vocabulary and modeling math strategies. I have been trying to improve my vocabulary around math year to year. This personal goal of mine specifically relates to SMP 6-­‐ Attend to precision. While reading various materials, I noticed the wording of "term and like terms" which was used to describe this standard. I was unsure if this language was appropriate or not. As I read through the CCSSM, I noticed it came up a number of times, mostly in high school, but was used in the progression document for 6th grade as well. It is also the wording used in my textbook. I was glad to know I could call this strategy being discussed, “combining like terms.” While going through my 6th grade MathThematics textbook I noticed pieces of this standard were touched, even more in-­‐depth in 7th grade but the overall concepts broken down were not taught in full until our 8th grade text. Our textbook does a nice job of introducing 6.EE.4 throughout grades 6-­‐8 but ALL of these ideas should be taught at once during 6th grade. Ideas are built around modeling, balancing (using inverse operations) solving, and checking expressions using the properties and order of operations from 6th-­‐8th grades but actually having the break down and understanding of 6.EE.4 is not covered until grade 8. In 6th grade-­‐Module 1, our text introduces order of operations, inverse operations and writing and modeling algebraic equations (x + 5 = 8) using algebraic tiles (shown on the next page). The distributive property is also used in Module 3 when discussing fractions. These ideas build in 7th grade-­‐Module 1 where using the order of operations has multiple steps, students solve one-­‐step equations/expressions with positive and negative integers (-­‐5 + 3 = -­‐2 and) within a context, and are introduced to zero pairs (1 + (-­‐1) = 0). In 7th grade-­‐Module 2 they touch more on the overall idea (not why or the breakdown) of equivalent expressions by modeling with algebraic tiles (x + 6 + (-­‐2) = x + 4) and introducing the distributive property. th
8 grade is when the distributive property is used more in depth and “like terms” is discussed. A like term is described in our book as having identical variable parts that can be combined by adding or subtracting coefficients. A coefficient is the number part of the term. For example 1x + 2x = 3x because 1x and 2x have identical variable parts so they can be combined to 3x resulting in like terms. The 8th grade text also had a table showing examples of like and unlike terms that I have drawn below. 3 Like Terms 1, 13 3rs, 4rs 2y, 8y ¼x, 3x Unlike Terms 4, 13x 3r, 4rs 2y, 8y² ¼, 3x 6.EE.4 is completely covered in 8th grade. Next year I will combine what is being taught throughout all three grade levels and combine it to be taught in 6th grade. I think students are ready to understand the why’s and how’s behind equivalent expressions and combining like terms so next year I will definitely add to the ideas covered in the 6th grade text book and bring in the 7th-­‐8th grade ideas so that student learning is aligned with the common core. *Hereare a few examplesof the useof algebratiles
Here are a few examples of the use of algebraic tiles Key:
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**Likeand UnlikeTermsTable
LikeTerms
UnlikeTerms