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 crossproducts rule to determine whether two fractions are equivalent. They solve rate problems by writing proportions and using cross multiplication. Playing Fraction/Whole Number TopIt 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 FollowUp Briefly go over answers. Have students share strategies for solving Problem. Teaching the Lesson Math Message FollowUp (Math Journal, p. ) WHOLECLASS 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 WHOLECLASS 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 crossproducts method because they can solve many of the problems in this lesson more quickly using other methods. Explain that the advantage of the crossproducts 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
4 Solving Proportions with Cross Products continued For Problems, set up a proportion and solve it using cross multiplication. Show how the units cancel. Then write the answer. Example: Jessie swam lengths of the pool in minutes. At this rate, how many lengths will she swim in minutes? lengths Proportion: minutes Solution: _ n_ n n Answer: Jessie will swim n lengths minutes minutes lengths minutes n lengths minutes lengths minutes n lengths minutes lengths minutes n lengths lengths n lengths lengths in minutes.. Belle bought yards of ribbon for $. Solution: How many yards could she buy for $? yards n yards $ $ Answer: Belle could buy yards of ribbon for $. $ yards $ n yards $ yards $ n yards _ $ yards $ n yards yards n yards Guide students in solving a few more proportions using Steps. Example: n Cross multiply: n Solve: n Replace the variable: n _ n Check: Suggestions: _ x _ z_ r _ x r z. Math Journal, p. EMCS_S_G_MJ_U_.indd // : AM Adjusting the Activity Use a quick common denominator (QCD) or multiplicative inverses to explain why cross multiplication works: Find the QCD: Multiply both sides by. º º º º º º cross products Multiplicative inverses: Rewrite the proportion as _ _ and multiply both sides by the multiplicative inverses of _ and _. ( _ ) ( _ ) cross products 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. Before going to France, Maurice Solution: exchanged $ for euros. At that exchange rate, how many euros could he get for $? $ $ x euros _ $, euros $ euros Solving Proportions with Cross Products continued x euros Answer: Maurice could get euros for $. x euros $ euros $ x euros $ $, euros x euros euros Solving Problems Using Cross Multiplication (Math Journal, pp.,, A, and B) PARTNER PROBLEM SOLVING. One gloomy day, inches of rain Solution: fell in hours. At this rate, how many inches of rain had fallen after hours? inches p inches hours hours Answer:. _ inches of rain had fallen in hours.. Adelio s apartment building has Solution: flights of stairs. To climb to the top floor, he must go up steps. How many steps must he go up to climb flights? steps flights flights flights steps s steps Answer: Adelio must climb hours inches hours p inches hours inches hours p inches hours inches hours p inches. _ inches p inches s steps flights steps flights s steps flights steps flights s steps flights s steps steps steps. Algebraic Thinking Assign journal page. When most students have completed the problems, bring the class together and go over the answers. Ongoing Assessment: Recognizing Student Achievement Journal Page Problems Use journal page, Problems to assess students ability to use cross products to write an open number sentence. Students are making adequate progress if they are able to write open number sentences for Problems. Some students may be able to solve mentally for missing variables. [Operations and Computation Goal ] Math Journal, p. A A_B_EMCS_S_G_MJ_U_.indd A // : AM Unit Rates and Ratios
5 Work through the example at the top of journal page with the class. Show students how the units in the problem function much like numbers when they are included in the computation. Just as a number divided by itself is equal to, a unit divided by itself is also equal to. It is sometimes said that the units cancel and they can simply be crossed out as shown below. minutes lengths n lengths minutes NOTE This is a simple example of a strategy called dimensional analysis. Students will use dimensional analysis in future mathematics and science courses. It is not necessary to introduce the term at this time. Have students complete journal pages and A, showing how the units cancel. After most students have finished these problems, ask them why keeping track of the units is a useful strategy. Sample answers: It helps ensure that I have set up the proportion correctly. It helps me see the correct unit to use in my answer. Tell students that it is not necessary for them to include units every time they solve a proportion with cross multiplication, but it is a good strategy to use to check their work or help them on more difficult problems. Have students solve the problems on journal page B. It is not necessary for them to include units in their work on these problems, but they may do so if they wish. Solving Proportions with Cross Products continued Set up a proportion for each problem and solve it using cross multiplication.. Sarah uses scoops of coffee beans to brew Solution: cups of coffee. How many scoops of beans does Sarah use per cup? scoops s scoops cups cup _ Answer: Sarah uses beans per cup of coffee. scoop(s) of. Jeremiah ran _ miles in minutes. At this Solution: pace, how long would it take him to run miles? _ miles miles minutes m minutes Answer: It would take Jeremiah to run miles. minutes. It took Zach days to read a book that was Solution: pages long. If he read the same amount each day, how many pages did he read in one week? days days pages p pages Answer: Zach read. pages in one week.. At sea level, sound travels. mile in seconds. Solution: What is the speed of sound in miles per hour? (Hint: First find the number of seconds in hour.). mile d miles sec, sec Answer: Sound travels at the rate of A_B_EMCS_S_G_MJ_U_.indd B Math Journal, p. B miles per hour. s s _ s _ m _ m m m p p, p, _ p.,. d, d _, _ d d // : AM Links to the Future Students will apply their knowledge of cross products in future algebra and science courses. It is important that they be able to use cross products to write open number sentences. Ongoing Learning & Practice Math Boxes Playing Fraction/Whole Number TopIt (Student Reference Book, pp. and ) PARTNER Distribute four each of number cards (from the Everything Math Deck, if available) to each partnership. Students use cards to form whole numbers and fractions. They then find and compare the products. Math Boxes (Math Journal, p. ) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson . The skills in Problems and preview Unit content.. Which rate is equivalent to km in hr min? Fill in the circle next to the best answer. A. km in min B., m in min C. km in hr min D., m in min. A boat traveled kilometers in hours. Fill in the rate table. At this rate, how far did the boat travel in hours minutes? km. A bag contains red counter, blue counters, and white counter. You pick counter at random. Then you pick a second counter without replacing the first counter. a. Draw a tree diagram to show all possible counter combinations. R b. What is the probability of picking red counter and white counter (in either order)?. Add or subtract. a.  + () b.  c.  + d.  () W B W B B R B B R W B R W B _, or _   Math Journal, p. distance (km) hours. Insert parentheses to make each number sentence true. ( ) ( ) ( ) ( ) ( ) a.. + /. b. _  / c. _ / + B d. + EMCS_S_G_U_.indd // : PM Lesson
6 Study Link Master Name STUDY LINK Calculating Rates If necessary, draw a picture, find a perunit rate, make a rate table, or use a proportion to help you solve these problems. Study Link (Math Masters, pp. and ) INDEPENDENT. A can of worms for fishing costs $.. There are worms in a can. a. What is the cost per worm? b. At this rate, how much would worms cost?. An ounce bag of chips costs $.. a. What is the cost per ounce, rounded to the nearest cent? b. What is the cost per pound, rounded to the nearest cent?. Just gram of venom from a king cobra snake can kill people. At this rate, about how many people would kilogram kill?, people. A milking cow can produce nearly, quarts of milk each year. At this rate, about how many gallons of milk could a cow produce in months? gallons. A dogwalking service costs $, for months. What is the cost for months? $. per worm $. For years? $. per oz $. $ $, Home Connection Students solve rate problems on Math Masters, page. If you haven t already done so, review the instructions for Math Masters, page with the class. Students may postpone completing Parts B and C of the table until after they have completed Lesson . If a grocery store posts a unit price, ask students to check that the price is accurate. Try This py g g p _EMCS_B_G_MM_U_.indd. A pound bag of candy containing pieces costs. cents per ounce. What is the cost of piece of candy? Circle the best answer.. cents. cents. cent _ cent. Mr. Rainier s car uses about. fluid ounces of gas per minute when the engine is idling. One night, he parked his car but forgot to turn off the motor. He had just filled his tank. His tank holds gallons. About how many hours will it take before his car runs out of gas? Explain what you did to find the answer. Sample answer: oz gal; gal, oz; _. oz per min min; _ min hours, oz min per hour Sources: Fascinating Facts; Everything Has Its Price Math Masters, p. Teaching Master 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 perunit 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 cupsoffilling 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. hours 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 // : PM Differentiation Options READINESS Solving Equations (ax b) SMALLGROUP Min To provide experience solving equations of the form ax b, have students review and practice solving equations using the method of their choice. Suggestions: g g m m y y. k () k t t j j. x x f.() f () p p _ w _ w. ENRICHMENT Using Double Number Lines (Math Masters, pp. A and B) PARTNER Min Students explore an alternative way to solve rate problems by using double number lines. Tell students that a double number line is a number line that has two scales: one above the line and one below the line. Have students look at the double number line at the top of Math Masters, page A. Point out the two scales: dozens of tamales above, and cups of filling below. Have students read the top of Math Masters, page A with a partner. Ask them to locate the perunit rate on the number line and discuss how it can help to determine the scales on each side of the number line. Then have partnerships solve the problems on Math Masters, pages A and B. Math Masters, p. A AB_EMCS_B_MM_G_U_.indd A // : PM Unit Rates and Ratios
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 SMALLGROUP 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 dogwashing 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 AB_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 perunit 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 cupsoffilling 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/McGrawHill 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 dogwashing 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/McGrawHill 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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