Table of Contents Multiplication and Division of Fractions and Decimal Fractions

Transcription

1 GRADE 5 UNIT 4 Table of Contents Multiplication and Division of Fractions and Decimal Fractions Lessons Topic 1: Line Plots of Fraction Measurements 1 Lesson 1: Measure and compare pencil lengths to the nearest ½, ¼, and 1/8 of an inch, and analyze the data through line plots. Topic 2: Fractions as Division 2-5 Lesson 2 & 3: Interpret a fraction as division. Lesson 4: Use tape diagrams to model fractions as division. Lesson 5: Solve word problems involving the division of whole numbers with answers in the form of fractions or whole numbers. Topic 3: Multiplication of a Whole Number by a Fraction 6-9 Lesson 6: Relate fractions as division to fraction of a set. Lesson 7: Multiply any whole number by a fraction using tape diagrams. Lesson 8: Relate a fraction of a set to the repeated addition interpretation of fraction multiplication. Lesson 9: Find a fraction of a measurement, and solve word problems. Topic 4: Fraction Expressions and Word Problems Lesson 10: Compare and evaluate expressions with parentheses. Lessons 11 & 12: Solve and create fraction word problems involving addition, subtraction, and multiplication.

2 Topic 5: Multiplication of a Fraction by a Fraction Lesson 13: Multiply unit fractions by unit fractions. Lesson 14: Multiply unit fractions by non-unit fractions. Lesson 15: Multiply non-unit fractions by non-unit fractions. Lesson 16: Solve word problems using tape diagrams and fraction-by- fraction multiplication. Lessons 17 & 18: Relate decimal and fraction multiplication. Lesson 19: Convert measures involving whole numbers, and solve multi- step word problems. Lesson 20: Convert mixed unit measurements, and solve multi-step word problems. Topic 6: Multiplication with Fractions and Decimals as Scaling and Word Problems Lessons 21: Explain the size of the product, and relate fraction and decimal equivalence to multiplying a fraction by 1. Lessons 22 & 23: Compare the size of the product to the size of the factors. Lesson 24: Solve word problems using fraction and decimal multiplication. Topic 7: Division of Fractions and Decimal Fractions Lesson 25: Divide a whole number by a unit fraction. Lesson 26: Divide a unit fraction by a whole number. Lesson 27: Solve problems involving fraction division. Lesson 28: Write equations and word problems corresponding to tape and number line diagrams. Lesson 29: Connect division by a unit fraction to division by 1 tenth and 1 hundredth. Lessons 30 & 31: Divide decimal dividends by non-unit decimal divisors. Topic 8: Interpretation of Numerical Expressions Lesson 32: Interpret and evaluate numerical expressions including the language of scaling and fraction division. Lesson 33: Create story contexts for numerical expressions and tape diagrams, and solve word problems.

3 Vocabulary Familiar Terms and Symbols Grade 5, Math Unit 4 for Parents and Students Array an arrangement or display of a number in equal rows and columns Conversion factor - a multiplier for converting a quantity expressed in one unit into an equivalent expressed in another unit. Commutative property the order of the numbers added or multiplied can be changed and the answer will remain the same. (i.e. 4 ½ = ½ 4) Decimal fraction - A decimal fraction is a fraction where the denominator (the bottom number) is a power of ten (such as tenths, hundredths, thousandths, etc). i.e. 43/100 is a decimal fraction and can be written as 0.43 Denominator the number on the bottom (names the fractional unit: fifths in 3 fifths, which is abbreviated to the 5 in 3/5) Distribute - with reference to the distributive property, (i.e. in 1 2/5 15 = (1 15) + (2/5 15)) Divide/division - partitioning a total into equal groups to show how many units in a whole, (i.e. 5 1/5 = 25) Equation a statement that two expressions are equal, will always have an equal sign (i.e. 3 4 = 6 2) Equivalent fraction fractions that name the same amount or part using different units (i.e. 3/5 = 6/10) Equivalent fractions - represent the same amount of area of a rectangle, the same point on the number line. Evaluate to find the value of an expression Expression - a group of numbers and symbols that shows a mathematical relationship (i.e. ½ + ¼ + ¾) Factors - numbers that are multiplied to obtain a product) Fraction a number that names a part of a whole or part of a group (i.e. 3 fifths or 3/5) Fraction greater than or equal to 1 can be written as an improper fraction where the numerator is greater than a the denominator or as a mixed number with a whole and a fraction part (i.e. 7/2 = 3 ½, an abbreviation for 3 + ½ ) Fractional unit (e.g., the fifth unit in 3 fifths denoted by the denominator 5 in 3/5) Hundredth - (1/100 or 0.01) Line plot - data shown on a number line using x s to show the number of times a particular fraction is used in the data (the frequency). (Examples located in Lesson 1) Mixed number names a whole plus a fractional part (i.e. 3 ½, an abbreviation for 3 + ½ ) Numerator - the number on the top (names the count of fractional units: 3 in 3 fifths or 3 in 3/5) Parentheses - symbols ( ) used around a fact or numbers within an equation or expression) Product the answer to a multiplication problem Quotient - the answer when one number is divided by another Tape diagram visual method for modeling problems, the whole is labeled on the top and the broken into the known and unknown parts. Example of Tape Diagram: Two hundred seventy-three vehicles were parked in a parking lot. One-third of the vehicles were trucks. How many trucks were in the parking lot? 3 units (sections) = unit (section) = = 91 trucks tape is 273 vehicles

4 Tenth - (1/10 or 0.1) Unit fraction a fraction with a numerator of 1 Units of measurement - feet, mile, yard, inch, gallon, quart, pint, cup, pound, ounce, hour, minute, second Whole unit (e.g., any unit that is partitioned into smaller, equally sized fractional pieces called units ) New Terms for 5 th Grade Unit 4 Decimal divisor - the number that divides the whole (dividend) and has units of tenths, hundredths, thousandths Frequency the number of times something occurs Simplify (using the largest fractional unit possible to express an equivalent fraction) Scaling may increase or decrease the size or quantity of something

5 Lesson by Lesson Suggestions Lesson 1: Line Plots and Fraction Measurements Students will read and create line plots using fractional measurements. In this lesson it is important that students be able to read a customary ruler with increments of halves, fourths, and eighths. Students use their knowledge of fraction operations to answer questions about the data such as, What is the total length of the five longest pencils in our class? It helps for students to interpret fractions as division in this lesson. To measure to the quarter inch, one inch must be divided into 4 equal parts, or 1 4. Example of a Line Plot The line plot below shows the growth of 10 sunflowers plants. The cross marks (x s) above each fraction represents the height of each plant after one month of planting. There are 10 x s because there were 10 plants measured. To summarize the data: 2 plants grew to 1 ¾ ft, 2 plants grew to 2 feet, 2 plants grew to 2 1/8 ft, 2 plants grew to 2 ½ ft, 1 plant grew to 2 5/8 ft and 1 plant grew to 2 7/8 ft. Example of a Line Plot 2 Joseph recorded the lengths of his classmates pencil erasers in the chart to the right. Examples of questions that may be asked: 1. How many erasers have a length of at least 1 ½ inch? 9 erasers 2. How many erasers measure less than a half inch? 2 erasers *3. What is the total length of all the erasers? inches 4. What is the difference between the shortest and longest erase lengths? 1 ¾ inches Students Length Student inch Student 2 1 inch Student 3 2 inches Student inch Student inches Student inches Student 7 2 inches Student 8 2inches Student inches Student inches Student inches Student 12 2 inches Student inches Student inches Student inches Student 16 1 inch 5. Which measurement appears most frequently? 2 inches *6. How many ¼-inch erasers would it take to equal the length of a 2-inch eraser? 8 one-fourth inch erasers *explanations on the next page* Explanations and work to be shown for questions 3 & 6.

6 *3. What is the total length of all the erasers? inches 6. How many ¼-inch erasers would it take to equal the length of a 2-inch eraser? To solve this problem you can use different strategies. One strategy is to take two whole rectangles and divide the rectangles into fourths. It would take 8 one-fourth inch erasers to equal the length of a 2-inch eraser.

7 Lessons 2-5: Fractions as Division Students will practice interpreting fractions as division. Students will solve problems using equal sharing with area models & tape diagrams to understand the division of whole numbers with answers in the form of fractions or mixed numbers (e.g., seven brownies shared by three girls, three pizzas shared by four people). Students will also interpret remainders as fractions. Students will solve real world problems using models and equations while reasoning about their results (e.g., between what two whole numbers does the answer lie?). Example 1: Regg has 7 crackers that he wants to share between his friend Gabe and himself equally. Method 1: If there are 7 crackers, you could give each boy 3 crackers. Then take the last cracker and split it in half and give each boy one of the halves. Method 2: Split all the crackers in half first, and then share. How many halves do we have to share in all? 14 halves Share them equally with each boy. How many crackers did each boy get? Each boy would get 7 halves. Although the crackers were shared in units of one-half, what is the total amount of crackers each boy receives? 3 whole crackers and 1 2 of another cracker. Each boy would get crackers.

8 Example 2: Using a picture, show how friends Sally, Adam, and Mandy could share two candy bars. Write an equation, solve, and check. Strategy: Draw two tape diagrams since there are 2 candy bars. Divide each candy bar into 3 equal parts and then share among the three friends. Unit Form: 6 thirds 3 = 2 thirds Example 3: Mark ran a total of 5 miles in 3 days. If Mark runs the same distance every day, how many miles does he run each day? To solve this problem use a tape diagram. We know that 3 units are equal to 5 miles. We want to know what 1 unit is equal to. Example 4: American Cookie Company uses 6 cups of chocolate chips to make 8 batches of mini chocolate chip cookies. If each batch uses the same amount of chocolate chips, how many cups of chocolate chips are used? (Solve using drawing, a standard algorithm, and check your answer.) 6 cups shared equally in 8 batches of cookies

9 Lessons 6-9: Multiplication of a Whole Number by a Fraction Students will use arrays and tape diagrams to find the fraction of a set. Students will then use the fraction of a set thinking to multiply a fraction times a whole number understanding that the word of is a signal to multiply. Students will also link division of a whole number to multiplication of a fraction, i.e. dividing by 2 is the same as multiplying by ½. Students also use the commutative property to relate fraction of a set to the Grade 4 repeated addition interpretation of multiplication by a fraction. This offers opportunities for students to reason about various strategies for multiplying fractions and whole numbers. Students apply their knowledge of fraction of a set and previous conversion experiences to find a fraction of a measurement, including converting a larger unit to an equivalent smaller unit (i.e. 1/3 min = 20 seconds and 2 1/3 feet = 27 inches). Lesson 6: Relate fractions as division to fraction of a set. Example 1: Tommy bought a dozen cookies, ¼ of the cookies were oatmeal. How many cookies were oatmeal? To find ¼ of 12, make an array with 12 circles. Use lines to divide the array into 4 equal groups. Write a division sentence to represent what was done = 3 or 12/4 = 3 Each group is ¼ of all the circles. So ¼ of 12 = 3 or Example 2: In a class of 15 students, 4/5 are boys. How many students are boys?

10 Lesson 7: Multiply any whole number by a fraction using tape diagrams. There are 42 students going on a field trip. Three-sevenths are girls. How many are boys? How many are girls? Solve using a tape diagram. The tape diagram shows that three sevenths of the 42 students are girls so the remaining pieces are boys which are 4 pieces or four sevenths. Each unit is equal to 6 students. The girls are 3 of the 7 units. To find how many girls are on the field trip we multiply 3 units by 6. 3 units = 6 x 3 = 18 students There is a total of 18 girls on the field trip. Boys are 4 of the 7 units. To find how many boys are on the field trip we multiply 4 units by 6. 4 units = 6 x 4 = 24 students There is a total of 24 boys on the field trip. Check: 18 girls + 24 boys = 42 total students Lesson 8: Using multiple ways to solve a multiplication of a fraction and a whole number. Ways to interpret the expression:

11 Lesson 9: Find a fraction of a measurement, and solve word problems. Example 1: Mrs. Collins baked 3 dozen cookies. Two-thirds of them were chocolate chip. How many chocolate chip cookies did she bake? 1 dozen is 12 cookies, so 3 dozen is 36 cookies (12 x 3) 2/3 of 36 cookies = chocolate chip cookies Using a Tape Diagram Using a Numerical Procedure Example 2: Tape Diagram lb pound oz ounce (16 oz is equal to 1 lb) Equation Example 3: Amanda measured the length of one of her books. It was ¾ of a foot. How long is her book in inches? ¾ of 1 foot = inches ft foot in inches Tape Diagram Equation

12 Lesson 10-12: Fraction Expressions and Word Problems In this topic students will write and evaluate expressions with parentheses, interpret numerical expressions, and solve and create fraction word problems. Lesson 10: Compare and evaluate expressions with parentheses. Write an expression to match a tape diagram. Then evaluate. Example 1: Example 2: Write and evaluate an expression from word form.

13 Lesson 11 & 12: Solve and create fraction word problems involving addition, subtraction, and multiplication. Example 1: Crissy and Crystal share a 16 ounce box of cereal. By the end of the week, Crissy has eaten 3/8 of the box and Crystal has eaten ¼ of the box of cereal. What fraction of the box is left? Example 2: 3/8 of the box of cereal is left. Create a story problem about a fish tank for the tape diagram below. Your story must include a fraction. There are 12 mollies in the fish tank.

14 Lesson 13-20: Multiplication of a Fraction by a Fraction Topic 5 introduces students to multiplication of fractions by fractions both in fraction and decimal form. The topic starts with multiplying a unit fraction by a unit fraction, and progresses to multiplying two non-unit fractions. Students use area models, rectangular arrays, and tape diagrams to model the multiplication. Students will also find fractional parts of customary measurements and calculate measurement conversion. Students will convert smaller units to fractions of a larger unit (i.e. 6 inches = ½ feet). Example 1: Solve. Draw a model to explain your thinking. Joseph has ¼ of a pound of strawberries. He gave his teacher 1/5 of the strawberries. What fraction of strawberries did Joseph give to his teacher? Think: We need to find 1/5 of ¼ strawberries. Step 1: Draw a rectangle and cut it vertically into 4 equal parts. Shade 1 part and label it ¼. Step 2: We need to find 1/5 of ¼. Split the whole rectangle into 5 equal parts by drawing horizontal lines. Now, shade 1 of the 5 parts (that are already shaded) and label it 1/5. Example 2: Of the students on Nia's track team, 3/5 participate in running events. Of the students who participate in running events, 2/3 are in the relay race. What fraction of the students on the track team ran in the relay race? Think: We need to find 2/3 of 3/5. Step 1: Draw a rectangle and cut it vertically into 5 equal parts. Shade 3 parts and label it 3/5. Step 2: Split the rectangle into 3 equal parts by drawing horizontal lines. Now shade 2 of the 3 parts (that are already shaded) and label it 2/3. How many units make our whole? 15 What s the name of these units? Fifteenths

15 In lesson 15 students will begin to recognize numerical strategies to multiplying fractions. Method 1: Students will eventually see a pattern and multiply numerator times numerator and denominator times denominator. Method 2: Students divide by common factors prior to multiplying. (*See video resources*) A common factor of 2 and 12 is 2. A common factor of 10 and 5 is 5. Students will also solve word problems using a tape diagram. Example: Dell has 14 blue marbles. His blue marbles make up 2/3 of his total number of marbles. How many marbles does Dell have? Dell has 35 marbles. Relate decimal and fraction multiplication Convert mixed unit measurements 2 ¼ ft = in 9 inches = ft Rename1 foot as 12 inches. Problem: A container can hold 4 ½ pints of water. How many cups can 2 containers hold? ( 1 pint = 2 cups)

16 Lesson 21-24: Multiplication with Fractions and Decimals as Scaling and Word Problems Students will extend their understanding of multiplication to include scaling. Students compare the product to the size of one factor, given the size of the other factor without calculation (i.e , is twice as large as 243 1, because 486 = 2 243). Students will begin to reason about the size of products when quantities are multiplied by 1, by numbers larger than 1, and numbers smaller than 1. Multiplying a number times a number equal to 1, results in the original number. ***These examples prove the statement that multiplying a number times a number equal to 1, does result in the original number. Therefore, if the scaling factor is equal to 1, the original number does not change. Multiplying a number times a number less than 1 results in a product less than the original number. ***These examples prove the statement that multiplying a number times a number less than 1, does result in a product less than the original number. Therefore, if the scaling factor is less than 1, the product will be less than the original number. Multiplying a number times a number greater than 1, results in a product greater than the original number. ***These examples prove the statement that multiplying a number times a number greater than 1, does result in a product greater than the original number. Therefore, if the scaling factor is greater than 1, the product will be greater than the original number.

17 A common misconception students have is the belief that multiplication always makes a quantity bigger. That is not always true. Suppose there are 6 students standing in line and ½ are wearing red shirts. How many students are wearing red shirts? ½ x 6 = 3 students. The product is smaller than the original number. Practice Problem: Without doing any calculating, choose a fraction to make the number sentence true. Explain how you know. Application Problem: At the book fair, Van spent all of his money on new books. Paul spent 2/3 as much as Van. Elliot spent 4/3 as much as Van. Who spent the most money? Who spent the least? Paul and Elliot are being compared to Van. Van spent all his money which is considered 1 whole in this problem. Using what we learned about scaling factor, 2/3 is less than 1 so Paul spent less than Van. 4/3 is greater than 1, so Elliot spent more than Van. Scaling with Decimals Whether you are working with fractions or decimals, the scaling factor statements still apply. Problem: Without calculating, fill in the blank using one of the scaling factors to make each number sentence true. Explain how you know. a x < 4.72 (4.72 x < 4.72) Since is less than 1, then the product will be less than b. x 4.72 > 4.72 ( x 4.72 > 4.72 ) Since is greater than 1, then the product will be greater than c x = 4.72 (4.72 x 1.00 = 4.72) Since 1.00 is equal to 1, then 4.72 does not change.

18 Lesson 25-31: Division of Fractions and Decimal Fractions Topic 7 begins the work of division with both fractions and decimal fractions. Students use tape diagrams and number lines to reason about the division of a whole number by a unit fraction and a unit fraction by a whole number. Using the same thinking developed in Unit 2 to divide whole numbers, students reason about how many fourths are in 5 when considering such cases as 5 ¼. They also reason about the size of the unit when ¼ is partitioned into 5 equal parts: ¼ 5. Using this thinking as a backdrop, students are introduced to decimal fraction divisors and use equivalent fraction and place value thinking to reason about the size of quotients, calculate quotients, and sensibly place the decimal in quotients. Divide a whole number by a unit fraction. Example: Garret is running a 5-K race. There are water stops every ½ kilometer, including at the finish line. How many water stops will there be? Number Sentence: 5 ½ Step 1: Draw a tape diagram to model the problem. Step 2: Since water stops are every ½ kilometer, each unit of the tape diagram is divided into 2 equal parts. The tape diagram is partitioned into 5 equal units. Each unit represents 1 kilometer of the race. When you count the number of halves in the tape diagram, you will determine that there Step 3: Draw a number line under the tape are a total of 10. diagram to show that there are 10 halves in 5 wholes. Therefore, there will be 10 water stops during the 5-K race. Misconception: Students may believe that the quotient in division is always smaller than the dividend (whole) and the divisor. It is about asking how many groups there are of a certain size. For example, what happens to the number of pieces if we cut a carrot into 6 equal pieces? (There are more pieces of carrot.) This is the meaning of dividing a whole by a unit fraction.

19 Practice Problem: Francois picked 2 pounds of blackberries. If he wants to separate the blackberries into pound bags, how many bags can he make? Number Sentence: 2 ¼ = 8 One whole has 4 fourths and 2 wholes has 8 fourths. Francois can make 8 bags with ¼ pound of blackberries in each. Divide a unit fraction by a whole number Randy and 2 of his friends will share a pizza equally. What fraction/portion of the pizza will each get? Number Sentence: 1 3 Now suppose there is only ½ of a pizza that is shared equally among Randy and his 2 friends. What fraction/portion of the pizza does each person get? Number Sentence: ½ 3 ½ 3 = 1/6 3 sixths 3 = 1 sixth (The unshaded part is showing 3 sixths.) Each person will receive 1/6 of the pizza. Practice Problem: 1 If Bridget poured ½ liter of lemonade equally into 4 bottles, how many liters of lemonade are in each bottle? Number Sentence: ½ 4 = 1/8 There is 1/8 liter in each bottle.

20 Divide by decimal divisors Example 1: Step 1: Rewrite the division expression as a fraction. Step 2: Rename the divisor/denominator as a whole number by multiplying a fraction equal to 1. Step 3: Divide. Example 2: Step 1: Step 2: Step 3: Application Problem: 18 2 = 9 Explain why it is true that and have the same quotient. They all have the same quotient because I can rename each fraction without changing their value by multiplying each by a fraction that equals 1. In the first fraction since both the numerator and denominator are in tenths, multiplying by 10/10 resulted in both the numerator and denominator being whole numbers. In the second fraction both numerator and denominator are in hundredths. When I multiply each by 100/100, it resulted in both numerator and denominator being whole numbers. Each fraction resulted in Application Problem: Mrs. Morgan has 21.6 pounds of peaches to pack for shipment. She plans to pack 2.4 lb of peaches in each box. How many boxes are required to ship all the peaches? Mrs. Morgan needs 9 boxes to ship all the peaches.

21 Lesson 32-33: Interpretation of Numerical Expressions In the last topic of this unit numerical expressions involving fraction-by-fraction multiplication are interpreted and evaluated. Students create and solve word problems involving both multiplication and division of fractions and decimal fractions. Write word form expressions numerically Example 1: Half the sum of 3/5 and 1 ½ Possible Responses: Example 2: 3 times as much as the quotient of 1.2 and 0.4 Possible Responses: 3 x ( ) or ( ) x 3 Practice Problem: Which expression is equivalent to the sum of 5 and 3 divided by ½? Correct answer: C Some will pick A but this expression represents the sum of 5 and 3 divided by 4. Application Problem: Susie picked 12 cucumbers from her garden. She cut up 2 of them for a salad and then gave 2/5 to her neighbor. Write an expression that tells how many cucumbers she gave to her neighbor. Expression: 2/5 x (12 2) Write a numerical expression in word form Example 1: (1/ ) 1/2 The sum of 1/4 and 1.25 divided by ½ Example2: 5/6 (1/5 x 0.2) The difference between 5/6 and the product of 1/5 and 0.2 Evaluate the following expressions: Students should recognize that when evaluating expressions that contain grouping symbols, any operation inside grouping symbols should be performed before operations outside of grouping symbols.

22 Extra Example Problems Problem: Without evaluating, compare the first expression to the second expression. Explain your reasoning. ( /4) x 3/2 2/3 x ( /4) In both expressions you are finding the sum of the same two numbers. In the first expression the sum is being multiplied by a fraction greater than 1, which would result in an answer greater than the sum of the two numbers. In the second expression the same sum is being multiplied by a fraction less than 1 which would result in an answer less than the sum of the two numbers. Therefore first expression will be greater than the second expression. Problem Solving: ( /4) x 3/2 > 2/3 x ( /4) Luke has 3.5 hours left in his workday as a car mechanic. He needs ½ of an hour to complete one oil change. a. How many oil changes can Luke complete during the rest of his workday? Luke can complete 7 oil changes during the 3.5 hours. b. Luke can complete two car inspections in the same amount of time it takes him to complete one oil change. How long does it take him to complete one car inspection? Luke can complete one car inspection in ¼ hour. c. If he only completes car inspections in the rest of his workday, how many can he complete? Since Luke can complete 2 car inspections in the same amount 7 x 2 = 14 of time it takes him to complete one oil change, he can complete 14 inspections (twice as many as 7) in 3.5 hours.

23 Recommended Resources IXL skills covered in this unit: S.10 Interpret line plots S.11 Create line plots S.12 Create and interpret line plots with fractions N.1 Multiply unit fractions by whole numbers using number lines N.2 Multiply unit fractions by whole numbers using models N.3 Multiples of fractions N.4 Multiply unit fractions and whole numbers: sorting N.5 Multiply fractions by whole numbers I N.6 Multiply fractions by whole numbers using number lines N.7 Multiply fractions by whole numbers using models N.8 Multiply fractions by whole numbers II N.9 Multiply fractions and whole numbers: sorting N.10 Multiply fractions by whole numbers: word problems N.11 Multiply fractions by whole numbers: input/output tables N.12 Multiply two unit fractions using models N.13 Multiply two fractions using models: fill in the missing factor N.14 Multiply two fractions using models N.15 Multiply two fractions N.16 Multiply two fractions: word problems N.17 Scaling whole numbers by fractions N.18 Scaling fractions by fractions N.21 Complete the fraction multiplication sentence **This site has a video providing guidance for every homework page.** (Click on 5 th Grade Select the Module 4 Select the lesson) Videos Using a ruler & line plots: Reading a ruler Or use quick code: LZ2683 Multiply fractions by whole numbers using repeat addition: Or use quick code: LZ210 Multiply fractions by whole numbers using tape diagrams/bar models: Or use quick code: LZ212 Multiply fractions by fractions using area models: Or use quick code: LZ213 Scaling with multiplication: Simplify fraction multiplication problems: cancelling common factors (lesson 15 method 2 ) Or use quick code: LZ373