Embedded Files
Chapter Overview
Chapter Overview
This chapter extends the concept of ratios as comparisons that began in Chapter 4. Students will use situations involving fundraising, running, and triathlons to solve rate problems. They will use ratios, tables, and graphs to compare rates with both like and unlike units. They will identify that steeper lines on the same graph indicate faster rates. This provides a strong foundation for later work with linear functions.
In Section 7.2, students extend their learning about operations with portions to include methods for efficiently dividing with fractions, mixed numbers, and decimals. Students build on the work that they did in Chapter 6 where they relied on diagrams and reasoning to move to an algorithm.
In Section 7.3, students analyze mathematical “magic tricks” in which any number can be chosen to start, but after a series of operations, the same result always occurs. These number tricks motivate simplifying expressions and working with the Distributive Property. Students will represent the tricks with algebra tiles and then variables in order to understand how the trick is operating. From the visual representation with algebra tiles, students will transition to writing algebraic expressions to represent the steps of the trick. Students will draw connections between the words of the trick, the concrete representation with algebra tiles, and the algebraic expressions. Students will also build expressions with algebra tiles on an expression mat and simplify them. Students identify inverse operations as they determine why specific results occur and as they construct their own magic tricks. They will also briefly look at one-variable inequalities, comparing them to equations and learning how to graph them on number lines.
Standards for Mathematical Practice
Standards for Mathematical Practice
Students will focus on many mathematical structures in Chapter 7. In developing these structures, give special attention to their developing skills in reasoning and constructing viable arguments. Most of all, students should make sense of these problems.
As always, it is recommended that you focus very clearly on attention to precision, both as students communicate with others and as they use tools, units, etc.
CA Content Standards
CA Content Standards
6.RP.3a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
6.RP.2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”(non complex fractions)
6.RP.3b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
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.3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
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.2a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y.
6.EE.5. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
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.
6.EE.7. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
6.EE.8. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
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