Multiply and Divide Fractions

Transcription

1 5NS.4 Understand the concept of multiplication and division of fractions. 5NS.5 Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems. GetTING started LearNING objectives In this lesson, the student will: Multiply fractions. Divide fractions. prerequisites In order to complete this lesson, the student is expected to: Know multiplication facts. Understand reciprocals. Convert between mixed numbers and improper fractions. tap students prior knowledge Briefly review with students the concept of reciprocals as inverse fractions where the numerator and denominator flip, or switch places. Then remind students that improper fractions are easier to work with in fraction computations. Reciprocals Explain to students that a reciprocal is the number multiplied by a fraction to get a product of. Model for students how and are reciprocals since can be written as the fraction and 5. Ask students to give and verify the reciprocals for 7,, and : Improper Fractions and Mixed Numbers Explain that improper fractions are converted to mixed numbers by dividing the numerator by the denominator and using the remainder as a fraction. Division finds the number of wholes in the numerator and the remainder is the part leftover. Mixed numbers are converted to improper fractions by the inverse process: multiply the denominator by the whole number (rather than divide) and add to the numerator of the fraction (rather than subtracting the remainder). The process adds all the parts in the wholes to the fraction. Ask students to convert 7 and 9 to mixed numbers: 6 and Ask students to convert and 9 to improper fractions: 4 0 and Using the Interactive Whiteboard You can project each lesson page on the IWB to enhance instruction. When discussing the examples in the Introduction and Modeled Instruction pages, walk through the solutions step-by-step with the class, using the IWB to illustrate each step. You can record equations, draw diagrams, and show related problems. Or, zoom in on a difficult step and shade the rest. Have students, individually or in pairs, demonstrate their solutions to the Try It and Guided Practice problems. Encourage them to use the pen, the highlighter, the eraser, and other tools. Be sure to save your work and students work.

2 Introduction At a Glance Read and discuss the Introduction on page with the class to help students understand the concept of rounding numbers to large or small place values. INSTRUCTIONAL SUPPorT Use the following to explain each part of the Introduction in greater depth. Convert Mixed Numbers Multiplication and division of fractions tends to come easier to students since the numerators and denominators get the same treatment, but mixed numbers are much harder to work. Improper fractions are quite easy and make multiplication and division relatively simple. Rewriting Division as Multiplication The reciprocal of the divisor is used because it effectively cancels out the division by changing the divisor to a. However, since the divisor is multiplied by the reciprocal, the dividend must also be multiplied to maintain an equivalent fraction. Real-World Connection Multiplication and division of fractions are often used in baking, construction, crafts, and purchases of quantities of materials. Multiply and Divide Fractions Introduction You do not need common denominators to multiply or divide fractions. In order to multiply or divide with a mixed number, you will need to convert the mixed number to an improper fraction first. An improper fraction is a fraction in which the numerator is greater than the denominator. Multiply by Convert to an improper fraction. Multiply the 5 denominator by the whole number, and then add the numerator. 5. Multiply the numerators. Multiply the denominators Write as a mixed number. Divide 9 by 8 and use the remainder as the fraction R To divide fractions, first convert the problem to a product. Multiply the dividend by the reciprocal of the divisor. In the reciprocal of a fraction, the numerator and denominator switch places. Divide by Rewrite the divisor as its reciprocal by flipping the to numerator and denominator.. Rewrite the problem as a product of the dividend 5 and reciprocal. 6. Multiply the fractions Simplify the fraction to simplest form Rewrite the improper fraction as a mixed number Ns.4 Understand the concept of multiplication and division of fractions. 5Ns.5 Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems.

3 Modeled Instruction AT A GLANce Read and discuss each Example problem on pages and with the class. Model the steps used to solve each example. Then have students solve the Try It problems that follow the example. Modeled Instruction STEP BY STEP Guide the class through each Example problem. Read aloud each question and discuss how to solve the problem. To ensure understanding, demonstrate each step beneath the problem. The steps model the thinking designed to lead students to a correct solution. Pause to allow students to fill in the missing information for each step. Then discuss each response. After discussing each example, direct students to solve the related Try It problems. Read each question with students. Then have students, individually or in pairs, solve the problem and write the solution. Discuss their solutions as a class. Page : ; 8. Page : 6; TeacHer TIPS Use the following to discuss the example problems in more detail. Multiplying Fractions (Example, Page ) Students often get confused by the simplicity of multiplication of fractions after adding and subtracting fractions with unlike denominators. Explain to students that the denominators can be different because fractions are division problems and multiplication has the same priority in the order of operations as division, while addition and subtract are lower priority. EXAMPLE Jay has used 5 of the space on his camera card. of the pictures are from his 8 4 summer vacation. What fraction of the space on Jay s camera card are pictures from his summer vacation? Follow these steps to solve the problem. Step Determine what operation is needed to solve the problem. The problem asks for a fraction of the total space used for vacation pictures. of can be written as Tip: What operation times does of represent? Step Write the multiplication problem Step Multiply the numerators and the denominators SOLUTION: Jay has used 5 of the space on his camera card for pictures from his summer vacation. TRY IT Use what you know to solve these problems Melissa lives miles from her school. The park is the 4 distance from her house than the school is. How many miles is the park from Melissa s house? 8

4 teacher tips (continued) Dividing Fractions (Example, Page ) Teach students the phrase Keep It; Change It; Flip It as a memory tool for division of fractions. Keep the dividend, change the divide to multiply, and flip the divisor. Modeled Instruction EXAMPLE Modeled Instruction Sanju has quarts of smoothie. She splits it equally among three glasses. How 7 8 much smoothie is in one glass? Follow these steps to solve the problem. Using the Interactive Whiteboard Use a pen tool or a new page to show students the reasoning behind changing the division to multiplication by the reciprocal: Write 7 4 as a vertical quotient (fraction). Then multiply both the 8 numerator and denominator by, the reciprocal of. Show students how the denominator is cancelled out leaving the multiplication problem of 7. 8 Step Determine what operation is needed to solve the problem. Splitting equally is the same as the operation of division. Step Write the division problem. 7 4 Tip: A whole number can be written as 8 a fraction with a denominator of. Step Rewrite the problem as a multiplication problem by using the reciprocal of the divisor. 7 8 Step 4 Multiply the numerators and denominators SOLUTION: One glass has 7 quarts of smoothie. 4 TRY IT Use what you know to solve these problems Nathan spends 4 dollars for pounds of shrimp. What is the price per pound of the shrimp? 5 4

5 Guided Practice AT A GLANce Have each student complete short-answer problems on page 4. STEP BY STEP Guided Practice Before students complete short-answer problems, tell students that the Hints provide clues for solving the problems. Make sure students understand that they must write a solution and an explanation for each problem. Have pairs of students share and discuss solutions and explanations. Follow up the Pair/Share activity with a whole-class discussion of their work. Hints Remember to change the problem to multiplication after flipping the divisor. Solve each problem. Use the Hints to help you. Then explain how you found your solution. Solution: 0 4 5? Explanation: 5 Responses will vary. SoLUTIons and SaMPLe ExPLanaTIons For Discussion Solution: 5 Explanation: I changed the problem to, multiplied the 0 numerators and denominators, and then simplified. Solution: 7 Explanation: I multiplied and reduced to simplest form Solution: Explanation: I converted 4 to 4. Then I found of 4 : , which simplifies to. To assign students more practice on this topic, please visit the Queue tab in i-ready. SImplify the product to simplest form. Convert the mixed number to an improper fraction. With your partner, share and discuss your solutions and explanations. In a parking lot, of the vehicles are white. of the white vehicles in the parking lot are cars. What fraction of the vehicles in the parking lot are white cars? Solution: 7 Explanation: Responses will vary. Karen walks 4 miles in a week. What is the average number of miles she walks per day? Solution: Explanation: Responses will vary. 4 5

6 Mini-Lesson: Finding Equivalent Fractions Tell students that they are going to practice finding equivalent fractions. Write the fraction and the addition problem on the board.. To solve problems, you often need to rewrite fractions either to find like denominators or to simplify. This means you need to find another way to write a fraction with a different denominator, but without changing the value of the fraction.. Look at the fraction. Is that simplified? If there is a way to 6 rewrite it with a smaller denominator, then it is not simplified. Since both the numerator and denominator can be divided by, the fraction can be simplified.. Divide the numerator and denominator by : 4 5. simplifies to Look at the fraction 7. It is simplified, but what if you want to 0 add it to? To do that, you need to change the denominator so 0 that it is 0. How do you get from 0 to 0? (multiply by ) 5. Multiply the numerator and denominator of 7 by : can be rewritten as 0 4. You can then add 0 4 to to get and, and and 0 4 are equivalent fractions. They use different numbers but have the same value. 6