# Solving Proportions by Cross Multiplication Objective To introduce and use cross multiplication to solve proportions.

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1 Solving Proportions by Cross Multiplication Objective To introduce and use cross multiplication to solve proportions. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Apply multiplication and division facts to find cross products. [Operations and Computation Goal ] Multiply whole or decimal numbers. [Operations and Computation Goal ] Use cross products to write open number sentences. [Operations and Computation Goal ] Describe rules for patterns and use them to solve problems. [Patterns, Functions, and Algebra Goal ] Use a method to solve equations. [Patterns, Functions, and Algebra Goal ] Key Activities Students use the cross-products rule to determine whether two fractions are equivalent. They solve rate problems by writing proportions and using cross multiplication. Playing Fraction/Whole Number Top-It Student Reference Book, pp. and per partnership: each of number cards (from the Everything Math Deck, if available), calculator (optional) Students practice calculating and comparing products of fractions and whole numbers. Math Boxes Math Journal, p. Students practice and maintain skills through Math Box problems. Study Link Math Masters, pp. and Students practice and maintain skills through Study Link activities. READINESS Solving Equations (ax b) Students practice solving simple equations. ENRICHMENT Using Double Number Lines Math Masters, pp. A and B Students use double number lines to solve rate problems. EXTRA PRACTICE Calculating Ingredient Amounts Math Masters, p. Students practice solving rate problems by calculating ingredient amounts for a recipe. ELL SUPPORT Illustrating Terms posterboard markers Students make posters illustrating how to use cross products to solve open proportions. Ongoing Assessment: Informing Instruction See page. Ongoing Assessment: Recognizing Student Achievement Use journal page. [Operations and Computation Goal ] Key Vocabulary cross products cross multiplication Materials Math Journal, pp., B Study Link calculator (optional) Advance Preparation Teacher s Reference Manual, Grades pp., Lesson

2 Getting Started Mathematical Practices SMP, SMP, SMP, SMP, SMP, SMP Content Standards.RP.,.RP.b,.RP.d,.EE.,.EE. Mental Math and Reflexes Students compare fractions using <, >, or. Suggestions: > _ > < < > _ If time permits, have students share the strategies they used to compare the fractions. Math Message Complete the problems on journal page. Study Link Follow-Up Briefly go over answers. Have students share strategies for solving Problem. Teaching the Lesson Math Message Follow-Up (Math Journal, p. ) WHOLE-CLASS DISCUSSION Algebraic Thinking Go over the answers to Problems. Review the following: Cross products are found by multiplying the numerator of each fraction by the denominator of the other fraction. Cross multiplication is the process of finding cross products. Adjusting the Activity ELL To support English language learners, demonstrate how an X is used to cross out a word or number. Relate this X to the terms cross products and cross multiplication. A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L _ Math Message For Part a of each problem, write or in the answer box. For Part b, calculate the cross products. _. a.. a. b. b. _. a.. a. b. b. Equivalent Fractions and Cross Products _. a.. a. b. b. _. a.. a. b. b.. a.. a. b. b. Sample answer: If the fractions are equivalent, the cross products are equal.. What pattern can you find in Parts a and b in the problems above? Math Journal, p. EMCS_S_G_U_.indd _ _ _ _ // : PM Discuss Problem. While there are several possible patterns, one stands out: If the fractions in Part a are equivalent, then the cross products in Part b are equal. If the fractions in Part a are not equivalent, then the cross products in Part b are not equal. Point out that this pattern provides a way to test whether two fractions are equivalent. Have students use this rule to test several pairs of fractions for equivalence. Suggestions: _? _ Cross products: ;. The cross products are equal; therefore, the fractions are equivalent. _? _ Cross products: ;. The cross products are not equal; therefore, the fractions are not equivalent. _? _ Not equivalent _? _ equivalent Pose additional problems as needed. Unit Rates and Ratios

3 Using Cross Products to Solve Proportions WHOLE-CLASS Algebraic Thinking Write the following proportion on the board: _ x_. Ask volunteers to explain how to solve the proportion using cross products. Students may suggest using the Identity Property of Multiplication, as was done in Lessons - and -. This is correct; however, remind students that they are supposed to find a solution using cross products. If no one is able to do so, demonstrate the following approach: Step Cross multiply. Note that the cross product of and x is written as x, or x. Adjusting the Activity Have students use pencil and paper or a calculator to calculate products as needed. AUDITORY KINESTHETIC TACTILE VISUAL x x or x Step Because we want the two fractions in the proportion to be equivalent, we also want the two cross products to be equal; that is, we want the product x to equal the product. Step Solve the equation from Step. x x _ x x Step Write in place of x in the proportion:. Use cross multiplication to check that the two fractions are equivalent. ; Ongoing Assessment: Informing Instruction Watch for students who doubt the need to apply and practice the cross-products method because they can solve many of the problems in this lesson more quickly using other methods. Explain that the advantage of the cross-products method is that it works for all proportions, not just those with convenient numbers. To prove your point, pose a problem such as the following: _. _. t... t...t. t Solving Proportions with Cross Products Use cross multiplication to solve these proportions. Example: _ p_ p p p p _ p p. _. _ y_ y. c c. m m. z z _ k k. _ t_ t. _ d_ d.. z z. _ r_ r. _ p _ p. h h. j j Math Journal, p. EMCS_S_G_U_.indd // : PM Lesson

7 Teaching Master EXTRA PRACTICE INDEPENDENT Calculating Ingredient Amounts (Math Masters, p. ) Min Students practice solving rate problems by calculating how much of each ingredient is needed to make pound and pounds of peanut butter fudge. ELL SUPPORT SMALL-GROUP Illustrating Terms Min To provide language support for solving proportions, have students create a poster that features the steps for using cross products to solve proportions. Their poster should include the terms cross products and cross multiplication. Planning Ahead Remind students to collect nutrition labels from containers of food, such as cans of soup, cups of yogurt, and cereal boxes. They will need to bring these labels to school for use in Lesson -. If you haven t already done so, provide students with a copy of Study Link - (Math Masters, page ) and remind them to collect data about the cost and weight of the listed items. They may postpone the calculations of unit price until after they have completed Lesson -. Name Double Number Lines continued For Problems, fill in the blanks on the double number lines and use them to help you solve the problem.. Jamie is ordering supplies for his dog-washing business. Last week, he washed dogs and used bottles of shampoo. Jamie uses the same amount of shampoo for each dog he washes. Dogs Bottles of shampoo a. How many dogs can he wash with one bottle of shampoo? dogs b. How many bottles of shampoo should he order if he expects to wash dogs this week? bottles. A craft store has skeins of yarn on special. They are selling skeins for \$. Skeins Cost \$ \$ \$. \$. \$ \$. \$ \$. a. What is the cost per skein of yarn? \$. b. Holly needs skeins of yarn to make an afghan. How much will the yarn cost? \$. Katie rode her bicycle to work today. The -mile ride took her minutes. Miles Minutes A-B_EMCS_B_MM_G_U_.indd B a. On average, how long does it take Katie to ride one mile? minutes b. At that rate, how long will it take her to ride miles to get from work to her sister s house? minutes Math Masters, p. B // : AM Teaching Master Name Ingredients for Peanut Butter Fudge. The list at the right shows the ingredients used to make peanut butter fudge but not how much of each ingredient is needed. Use the following clues to calculate the amount of each ingredient needed to make pound of peanut butter fudge. Record each amount in the ingredient list. Clues Use cups of sugar to make pounds of fudge. You need _ cups of milk to make pounds of fudge. You need cups of peanut butter to make pounds of fudge. (Hint: cup tablespoons) An -pound batch of fudge uses cup of corn syrup. Use teaspoons of vanilla for each pounds of fudge. Use _ teaspoon of salt for each pounds of fudge. Peanut Butter Fudge (makes pound) _ cups of sugar cup of milk tablespoons of peanut butter tablespoons of corn syrup teaspoons of vanilla teaspoon of salt. Suppose you wanted to make an -pound batch of fudge. Record how much of each ingredient you would need. Use the following equivalencies and your ingredient lists to complete each problem. teaspoons tablespoon tablespoons cup. cups of peanut butter are needed for pounds of fudge.. cups of corn syrup are needed for pounds of fudge. Ingredient List for Pounds of Peanut Butter Fudge cups of sugar cups of milk tablespoons of peanut butter tablespoons of corn syrup teaspoons of vanilla teaspoons of salt. tablespoons of vanilla are needed for pounds of fudge. Math Masters, p. -_EMCS_B_G_MM_U_.indd // : PM Lesson

8 Name Double Number Lines Howie is making tamales. He used cups of filling to make dozen tamales. How much filling does he need to make dozen tamales? The double number line below can be used to help solve this problem. Notice that the scale at the top of the number line is labeled in dozens of tamales. The scale at the bottom of the number line is labeled in cups of filling. Find the mark for cups of filling. Notice how it lines up with dozen tamales. This represents the information given in the problem. The per-unit rate is also shown on the number line: Howie uses cups of filling per dozen tamales, so the mark for dozen tamales lines up with the mark for cups of filling. This information was used to complete the double number line. Dozens of tamales Cups of filling The problem asks how much filling Howie needs to make dozen tamales. Find the mark for dozen tamales. Then find the number on the cups-of-filling scale that lines up with this mark. It s, so Howie needs cups of filling to make dozen tamales. Use double number lines to help you solve the problems.. A marine animal trainer noted that the aquarium s newest beluga whale ate pounds of food in days. The whale was fed the same amount of food each day. a. How many pounds of food does the whale eat per day? pounds b. Use your answer to Part a to fill in the blanks on the top scale of the double number line below. Pounds of food Days c. If he continues to eat at this rate, how many pounds of food will the whale eat in days? pounds Copyright Wright Group/McGraw-Hill A

9 Name Double Number Lines continued For Problems, fill in the blanks on the double number lines and use them to help you solve the problem.. Jamie is ordering supplies for his dog-washing business. Last week, he washed dogs and used bottles of shampoo. Jamie uses the same amount of shampoo for each dog he washes. Dogs Bottles of shampoo a. How many dogs can he wash with one bottle of shampoo? dogs b. How many bottles of shampoo should he order if he expects to wash dogs this week? bottles. A craft store has skeins of yarn on special. They are selling skeins for \$. Skeins Cost \$ \$ a. What is the cost per skein of yarn? Copyright Wright Group/McGraw-Hill b. Holly needs skeins of yarn to make an afghan. How much will the yarn cost?. Katie rode her bicycle to work today. The -mile ride took her minutes. Miles Minutes a. On average, how long does it take Katie to ride one mile? minutes b. At that rate, how long will it take her to ride miles to get from work to her sister s house? minutes B

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