01-Math 7a-CC2 Ch7

In this chapter, students will extend the use of scale factors that they started in Chapter 4, specifically, using them to generally describe a multiplicative relationship between quantities and to find unknown quantities. In order to identify various equal ratios and scaling relationships, students are encouraged to organize information on both linear diagrams (like those used in Chapter 4) and on similar triangle diagrams to build connections between the numerical and geometric work they have done in the past. In this chapter, viewing multiplication as scaling supports reasoning around why enlarging a quantity by 5% can be achieved by multiplying by 1.05 and why reducing the quantity by 13% can be achieved by multiplying by 0.87. This way of thinking about multiplication lays a foundation for future work with exponential growth, as well as for work with other functions as they study Algebra and other advanced math courses.

To start their exploration, in Lesson 7.1.1, students discover the relationship between distance, rate and time. In each of these situations, time can be thought of as a scale factor. In the middle lessons of Section 7.1, students will also extend their equation-solving skills as they develop strategies for writing and solving equations with a coefficient that is a fraction. In this context, they will also develop reasoning around why dividing by a fraction is the same as multiplying by that fraction’s reciprocal. This work is connected to the geometric enlarging and reducing from Chapter 4, and starts revisiting the multiplicative inverse in the context of similar shapes. Therefore, instead of only memorizing a rule with no grounded understanding, students connect both division by a fraction and multiplication by the reciprocal to the act of “undoing” the original scaling.

At the end of the section, students are introduced to percent increase and decrease, and simple interest, which provide additional practice solving equations with non-integer coefficients as these applications are related to scale factors. Throughout these lessons, it is important to encourage students to identify the whole to which portions are being compared to help them identify the quantity on which the scale factor is acting as well as whether the scale factor is greater than or less than one.

In Section 7.2, students consider proportional relationships explicitly. Students have worked with proportional relationships in Chapter 1 (the million penny tower) and Chapter 4. Students will make sense of various methods to solve proportions.

Note: While it may be tempting prior to Lesson 7.2.2 to show students how to use cross products to solve equal ratio problems such as the ones they encounter in Section 7.1, resist the urge to do so! The introduction of this strategy is purposefully done at a point in this course where it will be one of many strategies available to students for solving these sorts of problems and when students have sufficient understanding of the concepts of ratios and proportions to apply it correctly.

Students will spend this chapter modeling with mathematics through the concepts of percent. Throughout this chapter, 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, we recommend that you focus very clearly on attention to precision, both in their communication with others and their use of units.

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.”

7.EE.3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

7.EE.4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

a) Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

7.NS.3. Solve real-world and mathematical problems involving the four operations with rational numbers.

7.RP.3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.