Math Misconceptions 6.NS.2 6.NS.3 6.NS.4
3 + 4 = 34 Look closely at errors in students’ work (formative assessment) to help you reflect and make instructional decisions to suit all students’ needs.
The traditional long-division algorithm is difficult for many students. Often it is because the process of dividing contains so many steps and each step needs an exact answer in the quotient. Another reason is because the algorithm treats the dividend as a set of digits rather than a complete number. During the process, students must make a guess, and then using another operation check their guess. The guess must be as large as possible but cannot go over. When students routinely work through the steps they lose all sense of place value and the complete number. Continuing the development of number sense through understanding of the algorithm is important.
MISCONCEPTION: This student probably said to himself, “Eight can’t go into 3, but it goes into 32 four times.” “Eight doesn’t go into zero, but it does go into 8 one time.” “Eight goes into 3208 Forty-one times.”
WHAT TO DO: The above student is looking at the digits rather than the complete number. Ask him questions like, “How many eights are in 3200?” to help him realize the answer is at least 400. Why? Because there are 400 eights in 3200 but the dividend is really 3208, which is greater than 3200.
Students may encounter difficulty trying to make sense of the division algorithm. In the Number of Groups Unknown model, we know the size of each group and must find the number of groups, however, in the Group Size Unknown model of division we know the number of groups to be formed and must determine the size of each group. When trying to interpret the algorithm students sometimes get confused as they try to explain what is going on using place value notions. EX.
Number of Groups Unknown Mr. Smith has 32 flowers for his garden. He wants to divide them into groups with 8 flowers in each group. How many groups will he have?
MISCONCEPTION:
WHAT TO DO:
Size of Group Unknown Mr. Smith has 32 flowers for his garden. He wants to divide them into 8 groups. How many flowers will be in each group?
For students to be fluent with decimal computation, all they need to do is remember a few rules for placing the decimal point and then perform operations as if they are whole numbers. “Multiply just like whole numbers, count the digits behind the decimal point in the factors, and count the same number of decimal places in the product� is a quick way to teach the algorithm but may be counterproductive in the long run. Students should be taught to use estimation as a strategy to determine the reasonableness of an answer so that when they apply the algorithm they can self-check and self-correct.
MISCONCEPTION:
WHAT TO DO: Questioning students and leading rich classroom discussions are valuable tools. These tools can help reveal misconceptions that have occurred and get students to start questioning their own conjectures. The following questions are a few examples that can be used in a classroom discussion. • • • • • • • • •
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What does .08 mean? What does .8 mean? What does .80 mean? 127 x 1 = 127. Is .08 close to one? Should the answer be close to 127? Is it reasonable to say the answer is almost 102? The math is correct so what is incorrect? What does .08 look like in fraction form? What patterns do you notice in the following problems? o 0.08 ÷ 0.04 = 2 o 0.8 ÷ 0.4 = 2 o 8÷4 = 2 o 80 ÷ 40 = 2 Even though we multiplied the dividend and divisor by 10 each time what happened to the quotient? Let’s try 16 ÷ 0.5 . How many halves fit into 16? 2(16 ÷ 0.5) 32 ÷1 32 Is it reasonable to say 0.08 or 8 hundredths fits into 101 only127 times?
Students in middle school routinely divide with whole numbers but when asked to divide decimals we casually add the “move the decimal point” step to the algorithm. The problem is we forget to say why. If students can relate the moving of the decimal point to equivalent fractions they will be more likely to remember what to move, when to move it, and why. 0.15 0.15 ÷ 0.5 Consider writing the problem as a fraction. 0.5 When know how to divide by whole numbers so how can we change the divisor, or denominator, into a whole number? 0.15 10 1.5 Divide by a power of 10. We can do this because it does not × = 0.5 10 5 10 change the value of the fraction because remember = 1 . We have moved 10 the decimal point one place to the right in both the numerator and the denominator or the dividend and the divisor. Now we have 1.5 ÷ 5 .
Greatest Common Factor and Least Common Multiple are taught in close proximity of each other. When taught in isolation without any correlation to other concepts, students may confuse factor with multiple or they may look at the vocabulary “greatest” and think it must be a big number or “least” and think it must be a small number. When students think about GCF in terms of simplifying fractions or decomposing numbers, now called factoring, and LCM in terms of finding common denominators their confusion may be diminished. Students should also realize that both could be used to solve real life questions. For example, “I have 16 yellow balloons and 28 purple balloons. I want to use all of them to create balloon bouquets that are the same size to decorate the school gym. How many bouquets can I make?”
MISCONCEPTION:
WHAT TO DO: