Math Misconceptions 6.NS.1
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.
A common misconception is that division always makes the quotient smaller. This misconception can occur in grades 3-5 when students divide whole numbers by whole numbers. In 5th grade students start exploring division of 1 whole numbers by unit fractions, i.e. 4 á , which can start to dispel this 5 misconception. Students will now divide fractions by fractions, which will further show that the quotient can get bigger or be in-between the dividend and divisor. Students should experience solving division problems using models to gain a concrete understanding before being introduced to the algorithm.
MISCONCEPTION:
WHAT TO DO: In the first example, the divisor (1/3) is smaller than the dividend (2/3). 1/3 can fit into 2/3 two times. In this example, the quotient is bigger than the dividend or divisor. The second example also shows the quotient being bigger than the dividend or divisor. The last example shows a quotient that is between the dividend and divisor. These last two models may be more difficult to draw because the pieces in the models don’t line up exactly. However, these models can lead to powerful classroom discussions.
Students may try to change the value of the whole as they draw models. In the misconception example below, the whole has been changed when divided by fourths.
MISCONCEPTION:
To begin, one whole is broken into 2 pieces and shaded to represent ½. Next, the ½ was broken into fourths. This is where the misconception occurs. The whole should have been broken into fourths rather than the ½. This means the shaded part represents 4/8s not 4/4s. In this example, the whole was really broken into eighths, which is equal to 2 wholes.
WHAT TO DO:
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2
The value of the whole should remain the same. The fractional pieces are the only things that change. First, divide the whole into half. Then divide the whole into fourths and compare where they overlap.