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DIVISION 6.NS.1 Dividing by Fractions and Mixed Numbers Purpose: To divide by fractions and mixed numbers and to approximate quotients by rounding Materials: Fraction Bars and water-base pens TEACHER MODELING/STUDENT COMMUNICATION Activity 1 Dividing fractions whose quotient is not a whole number Fraction Bars water-base pens 1. Show students and have them find the red bar for 5/6 and the yellow bar for 1/3. How many times greater is the shaded amount of the 5/6 bar than the shaded amount of the 1/3 bar? (2 and ½ times greater) Illustrate by drawing lines on bars or transparent bars with water-base pens. 5/6 ÷ 1/3 = 2 ½ pencils and paper Compute 5/6 ÷ 1/3 using the invert and multiply rule and write the division equation. (5/6 ÷ 1/3 = 5/6 × 3 = 15/6 = 2 ½) Discuss the fact that his answer makes sense because in the illustration with the bars, the remaining shaded amount of the red bar is ½ the size of the shaded part of the yellow bar. 2. Show students and have them find bars for 7/12 and 1/4. What is the whole number of times the shaded amount of the ¼ bar "fits into" the shaded amount of the bar for 7/12? (2 times) 7/12 ÷ 1/4 = 2 1/3 Compute 7/12 ÷ 1/4 using invert and multiply and write the division equation. Explain why the bars show that this answer makes sense. (The shaded amount of the orange bar that is left over is 1/3 the size of the shaded amount of the blue bar.) Activity 2 Quotients less than one 1. Show students and have them find the red bar for 1/6 and the green bar for 1/2. Fraction Bars pencils and paper How many times will the shaded amount of the bar for 1/2 "fit into" the shaded amount of the bar for 1/6? ( 0 times) Compute 1/6 ÷ 1/2 and write the division equation. 1/6 ÷ 1/2 = 1/6 × 2/1 = 1/3 Explain why the bars help to show that the answer 1/3 makes sense. (The shaded amount of the 1/6 bar is 1/3 the size of the shaded amount of the 1/2 bar.) 2. Write the following quotients on the board or overhead 1 2 ÷ 1 3 1 4 ÷ 2 3 7 10 ÷ 1 3 3 4 ÷ 1 5 2 5 ÷ 7 8 Without computing, determine whether each quotient is less than 1 or greater than 1 and explain your reasoning. (1/2 ÷ 1/3 is greater than 1 because 1/3 is less than 1/2; etc. Discuss with students that if a number is divided into a larger number, the quotient will be larger than 1; and if a number is divided into a smaller number, the quotient will be less than 1.) Activity 3 Division involving mixed numbers and fractions Fraction Bars 1. Have students find the red bars for 3/6 and 5/6 and place their shaded amounts end to end. What is the total shaded amount? (1 2/6 bars) water-base pens Find the yellow bar for 1/3 and determine how many times its shaded amount fits into the total shaded amount of the two red bars. (4 times) Show with lines. Use the invert and multiply rule to compute 1 2/6 ÷ 1/3. 1 2 6 ÷ 1 8 1 8 3 24 = ÷ = × = = 4 3 6 3 6 1 6 2. Have students compute the following quotients by first replacing the mixed numbers by improper fractions. Then ask them to approximate each quotient by rounding the mixed numbers to whole numbers and dividing. (The rounded results are in brackets.) 2 7 8 ÷1 1 3 =2 [3] 4 10 1 1 4 ÷1 1 3 =1 [1] 10 22 4 1 2 ÷2 1 13 =1 [2 ½] 3 14 Note: NCTM's Standards 2000 (page 152) notes that there is a common, but incorrect, generalization that "division always makes things smaller." Ask students to compute the following quotients and write a statement about what happens as a given number is divided by smaller and smaller numbers. 1 ÷ 1/2; 1 ÷ 1/3; 1 ÷ 1/4; 1 ÷ 1/5; 1 ÷ 1/6 INDEPENDENT PRACTICE and ASSESSMENT Worksheets 6.NS.1 #3 and #4 fractionbars.com Set 1 Equations Game (Fractions from the bars or cards are placed into 10 blanks to form equations involving the four operations. It is often challenging, but is always possible. By clicking GO, the computer will check the equations that have been completed.)