What do you notice about the diagrams and equations? Discuss with your partner.
Complete this sentence based on what you noticed:[br]Dividing by a whole number a produces the same result as multiplying by ________.
To find the value of [math]6\div\frac{1}{2}[/math], Elena thought, “How many [math]\frac{1}{2}[/math]s are in 6?” and then she drew this tape diagram. It shows 6 ones, with each one partitioned into 2 equal pieces.[br][math]6\div\frac{1}{2}[/math] [br][img]data:image/png;base64,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[/img][br][br]
Examine the expressions and answers more closely. Look for a pattern. How could you find how many halves, thirds, fourths, or sixths were in 6 without counting all of them? Explain your reasoning.
[math]6\div\frac{1}{8}[/math]
[math]6\div\frac{1}{10}[/math]
[math]6\div\frac{1}{25}[/math]
[math]6\div\frac{1}{b}[/math]
[list][*][math]8\div\frac{1}{4}[/math][br][/*][*][math]12\div\frac{1}{5}[/math][br][/*][*][math]a\div\frac{1}{2}[/math][br][/*][*][math]a\div\frac{1}{b}[/math][/*][/list]
Elena says, “To find [math]6\div\frac{2}{3}[/math], I can just take the value of [math]6\div\frac{1}{3}[/math] and then either multiply it by [math]\frac{1}{2}[/math] or divide it by 2.” Do you agree with her? Explain your reasoning.
Elena examined her diagrams and noticed that she always took the same two steps to show division by a fraction on a tape diagram. She said:[br][br]“My first step was to divide each 1 whole into as many parts as the number in the denominator. So if the expression is [math]6:\frac{3}{4}[/math], I would break each 1 whole into 4 parts. Now I have 4 times as many parts.[br][br]My second step was to put a certain number of those parts into one group, and that number is the numerator of the divisor. So if the fraction is [math]\frac{3}{4}[/math], I would put 3 of the [math]\frac{1}{4}[/math]s into one group. Then I could tell how many [math]\frac{3}{4}[/math]s are in 6.”[br][br]Which expression represents how many [math]\frac{3}{4}[/math]s Elena would have after these two steps? Be prepared to explain your reasoning.
[list][*][math]6\div\frac{2}{7}[/math][br][/*][*][math]6\div\frac{3}{10}[/math][br][/*][*][math]6\div\frac{6}{25}[/math][br][/*][/list]