Math 6 Unit 6

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In Section 6.1, students extend their knowledge about operations with portions to include division with fractions. Students focus on making sense of the operation of division, relying on diagrams and reasoning, before moving to an algorithm. They begin by distributing some number of whole units among some smaller number of people, generating expressions that include division of fractions. Students then focus on the connection between the operation of division (as they divide units among people) and fractions (the amounts that each person ends up with).

At the beginning of Section 6.2, students are provided distinctly different rules for finding the area of a trapezoid and are challenged to visualize how the trapezoid might have been decomposed and recomposed based on the rule. Students apply the Order of Operations when they are challenged to use the different rules to calculate the area of a trapezoid. They use the knowledge that a trapezoid with defined dimensions has a specific area in order to verify that they have correctly evaluated different expressions for the area.

For the remainder of Section 6.2, students use a concrete manipulative (called “algebra tiles”) to build shapes that have an unknown dimension. That specific but unknown dimension will be represented with a variable, most often x. Students write expressions to represent the perimeter and area of these shapes. Because students will “see” the shapes and build their expressions differently, they will generate multiple expressions, creating the need to decide whether expressions are equivalent. This motivates the use and practice of combining like terms in an expression. Students will also learn to substitute a given value for a variable and to evaluate an expression.

In Section 6.1, making sense of problems involving division is the main goal. In Section 6.2, students will begin using appropriate tools, in this case, algebra tiles, to focus on Order of Operations and writing algebraic expressions. Throughout the chapter, it is recommended that you focus very clearly on attention to precision, both as students communicate with others and when they use tools, units, etc

6.NS.1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?

6.NS.2. Fluently divide multi-digit numbers using the standard algorithm.

6.EE.1. Write and evaluate numerical expressions involving whole-number exponents.

6.EE.2b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

6.EE.2c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s³ and A = 6s² to find the volume and surface area of a cube with sides of length s = 1/2.

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.

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.

6.EE.6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

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