Use visual models and patterns to develop a general rule to divide with fractions.
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The number $$3$$ is divided by unit fractions $${\frac{1}{2}}$$, $$\frac{1}{3}$$, $${\frac{1}{4}}$$, and $${\frac{1}{5}}$$. For each division problem, draw a visual model to represent the problem and to find the solution. Then complete the rest of the chart and answer the questions that follow.
Division Problem | Visual Model | Quotient | Multiplication Problem |
$$3\div \frac{1}{2}$$ | |||
$$3\div \frac{1}{3}$$ | |||
$$3\div \frac{1}{4}$$ | |||
$$3\div \frac{1}{5}$$ |
What pattern do you notice? What generalization can you make? Explain your reasoning.
The number 3 is now divided by fractions $${\frac {1}{4}}$$, $${\frac{2}{4}}$$, $${\frac{3}{4}}$$, and $${\frac{4}{4}}$$. For each division problem, draw a visual model to represent the problem and to find the solution. Then complete the rest of the chart and answer the questions that follow.
Division Problem | Visual Model | Quotient | Multiplication Problem |
$$3\div \frac{1}{4}$$ | |||
$$3\div \frac{2}{4}$$ | |||
$$3\div \frac{3}{4}$$ | |||
$$3\div \frac{4}{4}$$ |
What pattern do you notice? What generalization can you make?
For each problem, draw a diagram and write a division problem. Find the solution using both the diagram and by calculating, and check that your answers are the same by each method.
a. How many fives are in 15?
b. How many halves are in 3?
c. How many sixths are in 4?
d. How many two-thirds are in 2?
e. How many three-fourths are in 2?
f. How many $${{\frac{1}{6}}}$$s are in $${\frac{1}{3}}$$?
g. How many $${{\frac{1}{6}}}$$s are in $${{\frac{2}{3}}}$$?
h. How many $${\frac{1}{4}}$$s are in $${{\frac{2}{3}}}$$?
i. How many $${\frac{5}{12}}$$s are in $${\frac{1}{2}}$$?
How Many ___ Are in ... ?, accessed on Sept. 28, 2017, 12:55 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.
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Draw a visual model to represent the solution to the division problem $${6 \div \frac{2}{3}}$$.
Then, use your model to explain why this can also be solved with the multiplication problem $${6 \times \frac{3}{2}}$$.
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