Grade Level: Unit:



Time Frame: Approximately 2-3 weeks

Connections to Previous Learning:

In 6th grade, students read, write and evaluate numerical expressions involving variables and whole number exponents. They apply properties of operations using the appropriate order of operations to generate equivalent expressions.

Focus of this unit:

Students simplify general linear expressions that involve rational coefficients and distribute negative numbers to solve real world and mathematical problems.

Connections to Subsequent Learning:

A more complete understanding of order of operations and their properties will lay the foundation for the extensive study of functions next year.

From the 6-8, Expressions and Equations Progression Document pp. 8-9:

Use properties of operations to generate equivalent expressions In Grade 7 students start to simplify general linear expressions with rational coefficients. Building on work in Grade 6, where students used conventions about the order of operations to parse, and properties of operations to transform, simple expressions such as 2(3 + 8x) or 10 – 2p, students now encounter linear expressions wit more operations an whose transformation may require an understanding of the rules for multiplying negative numbers, such as 7 – 2(3 – 8x).

In simplifying this expression students might come up with answers such as:

• 5(3 – 8x), mistakenly detaching the 2 from the indicated multiplication.

• 7 – 2(-5x), through a determination to perform the computation in parentheses first, even though no simplification is possible.

• 7 – 6 – 16x, through an imperfect understanding of the way the distributive law works or of the rules for multiplying negative numbers.

In contrast with the simple linear expressions they see in Grade 6, the more complex expressions student seen in Grade 7 afford shifts of perspective, particularly because of their experience with negative numbers: for example, students might see 7 – 2(3 – 8x) as 7 – (2(3 – 8x)) or as

7 + (-2)(3 + (-8)x) (MP7).

As students gain experience with multiple ways of writing an expression, they also learn that different ways of writing expressions can serve different purposes and provide different ways of seeing a problem. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” In the example on the right, the connection between the expressions and the figure emphasize that they all represent the same number, and the connection between the structure of each expression and a method of calculation emphasize the fact that expressions are built up from operations on numbers.

|Desired Outcomes |

|Standard(s): |

|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. |

|7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a +0.05a = 1.05a means that “increase by 5% |

|is the same as multiply by 1.05.” |

|Supporting Standards: |

|7.NS1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. |

|Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. |

|Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive |

|inverses). Interpret sums of rational numbers by describing real‐world contexts. |

|Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply |

|this principle in real‐world contexts. |

|Apply properties of operations as strategies to add and subtract rational numbers. |

|7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. |

|Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products |

|such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real‐world contexts. |

|Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non‐zero divisor) is a rational number. If p and q are integers then – (p/q) = (-p)/q = p/(-q). |

|Interpret quotients of rational numbers by describing real‐world contexts. |

|Apply properties of operations as strategies to multiply and divide rational numbers. |

|Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats |

|WIDA Standard: (English Language Learners) |

|English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. |

|English language learners benefit from: |

|explicit vocabulary instruction with regard to the components of an algebraic expression. |

|Understandings: Students will understand that … |

|Variables can be used to represent numbers in any type of mathematical problem. |

|Understand the difference between an expression and an equation. |

|Expressions you simplify and equations you solve for the variable’s value. |

|Write and solve multi-step equations including all rational numbers. |

|Some equations may have more than one solution and understand inequalities. |

|Properties of operations allow us to add, subtract, factor, and expand linear expressions. |

|Essential Questions: |

|When and how are expressions, equations, inequalities and graphs applied to real world situations? |

|How can the order of operations be applied to evaluating expressions, and solving from one-step to multi-step equations? |

|Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.) |

|1. Make sense of problems and persevere in solving them. |

|*2. Reason abstractly and quantitatively. Students demonstrate quantitative reasoning by representing and solving real world situations using visuals, numbers, and symbols. They demonstrate abstract reasoning by |

|translating expressions, equations, inequalities and linear relationships into real world situations. |

|*3. Construct viable arguments and critique the reasoning of others. Students will discuss the differences among expressions, equations and inequalities using appropriate terminology and tools/visuals. Students |

|will apply their knowledge of equations and inequalities to support their arguments and critique the reasoning of others while supporting their own position. |

|*4. Model with mathematics. Students will model an understanding of expressions, equations, inequalities, and graphs using tools such as algebra tiles/blocks, counters, protractors, compasses, and visuals to |

|represent real world situations. |

|*5. Use appropriate tools strategically. Students demonstrate their ability to select and use the most appropriate tool (pencil/paper, manipulatives, calculators, protractors, etc.) while |

|simplifying/evaluating/analyzing expressions, solving equations and representing and analyzing linear relationships. |

|*6. Attend to precision. Students demonstrate precision by correctly using numbers, variables and symbols to represent expressions, equations and linear relationships, and correctly label units. Students use |

|precision in calculation by checking the reasonableness of their answers and making adjustments accordingly. Students will use appropriate geometric language to describe and label figures. |

|7. Look for and make use of structure. to solve for the variable. Students will also examine the relationship between the structure of a circle and the formulas for area and circumference. |

|8. Look for and express regularity in repeated reasoning. |

|Prerequisite Skills/Concepts: |Advanced Skills/Concepts: |

|Students should already be able to: |Some students may be ready to: |

|Apply and extend previous understandings of arithmetic to algebraic expressions. (6.EE.1-4) |Solve real-life and mathematical problems using numerical and algebraic expressions and equations. (7.EE.3-4) |

|Add, subtract multiply and divide positive and negative numbers. (7.NS.1-2) | |

|Knowledge: Students will know… |Skills: Students will be able to do… |

| |Use Commutative, Associative, Distributive, Identity, and Inverse Properties to add and subtract linear expressions |

|All standards in this unit go beyond the knowledge level. |with rational coefficients. (7.EE.1) |

| |Use Commutative, Associative, Distributive, Identity, and Inverse Properties to factor and expand linear expressions |

| |with rational coefficients. (7.EE.1) |

| |Rewrite an expression in a different form. (7.EE.2) |

| |Choose the form of an expression that works best to solve a problem. (7.EE.2) |

| |Explain your reasoning for the choice of expression used to solve a problem. (7.EE.2) |

|Academic Vocabulary: |

| | | |

|Critical Terms: |Supplemental Terms: | |

| | | |

|Distributive Property |Algebra | |

|Commutative Property |Property | |

|Associative Property |Order of operations | |

|Multiplicative Property of Zero |Evaluate | |

|Variable |Simplest form | |

|Numerical expression | | |

|Algebraic expression | | |

|Term | | |

|Coefficient | | |

|Constant | | |

|Equation | | |

|Inequality | | |

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