This week we will develop and practice the
algorithms for dividing fractions and long division. An algorithm is a set of rules for solving a math problem which, if done properly, will give a correct answer each time. Algorithms generally involve repeating a series of steps over and over, as in the borrowing and carrying algorithms and in the long multiplication and division algorithms. Children gain valuable confidence and insight when permitted to explore algorithms of their own invention. A given child may be more comfortable with this way or that. A given approach may be more useful for this problem or that one. Although you probably learned only one or two algorithms for each kind of arithmetic, it is important that you support your child’s use of many. In fact, if you closely observe your own computations in a variety of real-life settings — counting change, making estimates, balancing your checkbook, etc. — you will probably find that you use different algorithms at different times, and some of them are probably your own inventions. As we progress this week and explore NS.1 and NS.2, students will view the following instructional videos: http://gabelweb.org/six/6%20Grade%20Unit%20One%20Standards/2-Dividing%20Fractions%20with%20the%20Algorithm.pdf (Dividing Fractions 6:04)EQ: Why does the process of invert and multiply work when dividing fractions? http://gabelweb.org/six/6%20Grade%20Unit%20One%20Standards/3-Divide%20Multi-Digit%20Numbers.pdf (Long Division 7:58)How do we use an algorithm to divide whole numbers? Sample test question: MCC6.NS.1The length of a rectangular parking lot at the airport is 2/3 mile. If the area is 1/2 square mile, what is the width of the parking lot? A. 1/3 mile B. 3/4 mile C. 1 1/6 miles D. 1 1/3 miles Commentary: This item measures 6.NS.1 because it requires the student to interpret and solve a word problem involving division of a fraction by a fraction.Extended RationaleAnswer Choice A: “1/3 mile” This response is incorrect and may occur when a student selects an incorrect operation based on the question in the word problem. The student uses multiplication (2/3 × 1/2 = 2/6 = 1/3 ) rather than division to solve the word problem. Answer Choice B: “3/4 mile” The student has correctly interpreted the word problem and applied the area for a rectangle ( A = lw) to find the width of the parking lot. The student divided the total area by the given length in order to find the width of the parking lot:1/2 ÷ 2/3 = 1/2 × 3/2 = 3/4 Answer Choice C: “1 1/6 miles” This response is incorrect and may occur when a student selects an incorrect operation based on the question in the word problem. The student may have used addition (1/2 + 2/3 = 3/6 + 4/6 = 7/6 = 1 1/ 6 ) rather than division to solve the word problem. Answer Choice D: “1 1/3 miles” This response is incorrect and may occur when a student confuses the divisor with the dividend. A student who selects this response may have some understanding of computing quotients of fractions. However, there may be a lack of conceptual understanding of how to interpret word problems involving the division of a fraction by a fraction. The student also may have applied the formula for the area of a rectangle incorrectly either in the creation of an equation or the evaluation and solving of that equation. Answer options A, C, and D are plausible but incorrect. They are based on a conceptual misunderstanding of how to interpret and compute fractions, and solve a word problem involving division of a fraction by a fraction. CORRECT ANSWE
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