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Question

How can you figure out the size of a part of a part without having to draw a diagram? Work with your team or your class to explore this question as you consider the example of $\frac{2}{3} \cdot \frac{4}{5}.$ a. Describe how you could draw a diagram to make this calculation. b. If you completed the diagram, how many parts would there be in all? How do you know? c. How many of the parts would be counted for the numerator of your result? Again, describe how you know. d. How can you know what the numerator and denominator of a product will be without having to draw or envision a diagram each time? Discuss this with your team and be prepared to explain your ideas to the class.

Solution

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$\textbf{a. }$A rectangle is divided into $5$ parts and $4$ parts are shaded. Those $4$ parts will then be divided into $3$ parts and $2$ parts of those divided $3$ parts will be shaded.

$\textbf{b. }$Total parts in the diagram would be

$5 \times 3=15$

since each part (divided into $5$ parts) is further divided into $3$ parts.

$\textbf{c. }$Parts in the numerator of the result would be

$4 \times 2=8$

since only $4$ parts are divided into $3$ parts and out of those $3$ parts, $2$ parts are shaded.

$\textbf{d. }$So, the resulting fraction is

$\frac{4}{5} \times \frac{2}{3}=\frac{8}{15}$

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