standard_description: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.)
problem_activity: Problem 1
a.   Here are two tables with expressions. Find the value of each expression. Use a model if it's helpful. Leave the last two rows of each table blank for now.
{"table": [{"_attributes": {"align": "center", "class": ["table"], "style": "width: 100%;align: center"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align: center;"}, "strong": [{"_value": "Set A"}]}, {"_attributes": {"style": "text-align: center;"}, "strong": [{"_value": "Set B"}]}]}, {"td": [{"_attributes": {"style": "text-align: center;"}, "div": [{"_attributes": {"class": ["table-responsive"]}, "table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "width: 60%;"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "strong": [{"_value": "expression"}]}, {"_attributes": {"scope": "col"}, "strong": [{"_value": "value"}]}]}]}], "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align: center;"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$1\\times\\frac{1}{8}$$"}]}, {}]}, {"td": [{"_attributes": {"style": "text-align: center;"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$2\\times\\frac{1}{8}$$"}]}, {}]}, {"td": [{"_attributes": {"style": "text-align: center;"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$3\\times\\frac{1}{8}$$"}]}, {}]}, {"td": [{"_attributes": {"style": "text-align: center;"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$4\\times\\frac{1}{8}$$"}]}, {}]}, {"td": [{"_attributes": {"style": "text-align: center;"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$5\\times\\frac{1}{8}$$"}]}, {}]}, {"td": [{"_attributes": {"style": "text-align: center;"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$6\\times\\frac{1}{8}$$"}]}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}]}]}]}]}, {"_attributes": {"style": "text-align: center;"}, "div": [{"_attributes": {"class": ["table-responsive"]}, "table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "width: 60%;"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "strong": [{"_value": "expression"}]}, {"_attributes": {"scope": "col"}, "strong": [{"_value": "value"}]}]}]}], "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align: center;"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$2\\times \\frac{1}{3}$$"}]}, {"_attributes": {"style": "text-align: center;"}}]}, {"td": [{"_attributes": {"style": "text-align: center;"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$2\\times \\frac{1}{4}$$"}]}, {"_attributes": {"style": "text-align: center;"}}]}, {"td": [{"_attributes": {"style": "text-align: center;"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$2\\times \\frac{1}{5}$$"}]}, {"_attributes": {"style": "text-align: center;"}}]}, {"td": [{"_attributes": {"style": "text-align: center;"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$2\\times \\frac{1}{6}$$"}]}, {"_attributes": {"style": "text-align: center;"}}]}, {"td": [{"_attributes": {"style": "text-align: center;"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$2\\times \\frac{1}{7}$$"}]}, {"_attributes": {"style": "text-align: center;"}}]}, {"td": [{"_attributes": {"style": "text-align: center;"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$2\\times \\frac{1}{8}$$"}]}, {"_attributes": {"style": "text-align: center;"}}]}, {"td": [{"_attributes": {"style": "text-align: center;"}}, {"_attributes": {"style": "text-align: center;"}}]}, {"td": [{"_attributes": {"style": "text-align: center;"}}, {"_attributes": {"style": "text-align: center;"}}]}]}]}]}]}]}]}]}]}
b.   Study your completed tables. What patterns do you see in how the expressions and values are related?
c.   In the last two rows of the table of Set A, write
$$\frac{11}{8}$$
and
$$\frac{13}{8}$$
in the "value" column. Write the expressions with that value.
d.   In the last two rows of the table of Set B, write
$$\frac{2}{12}$$
and
$$\frac{2}{15}$$
in the "value" column. Write the expressions with that value.
Problem 2
a.
Draw a model to represent
$$\frac{2}{3}$$
.
Show how you would use or adjust the model above to represent
$$4\times \frac{2}{3}$$
.
Based on your model in Part (ii), what is the value of
$$4\times \frac{2}{3}$$
?
b.   Find the value of each expression below. Draw a model if it would be helpful.
$$2\times \frac{1}{5}$$
$$2\times \frac{3}{5}$$
$$2\times \frac{4}{5}$$
$$6\times \frac{4}{5}$$
c.   Willa said that to multiply a whole number by a fraction, she’d multiply the whole number and the numerator and use the same denominator. Do you agree with Willa? Explain your reasoning.
Problem 3
a.   Find the number that makes each equation true. Draw a model if it is helpful.
{"table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "width: 50%;"}, "tbody": [{"tr": [{"td": [{"span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$\\frac{12}{5} = 12\\space \\times $$"}], "_value": "_____"}, {"p": [{"span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$\\frac{12}{5} = 3\\space \\times $$"}], "_value": "_____"}]}]}, {"td": [{"p": [{"span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$\\frac{12}{5} = 6\\space \\times $$"}], "_value": "_____"}]}, {"p": [{"span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$\\frac{12}{5} = 2\\space \\times $$"}], "_value": "_____"}]}]}, {"td": [{"p": [{"span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$\\frac{12}{5} = 4\\space \\times $$"}], "_value": "_____"}]}, {"p": [{"span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$\\frac{12}{5} = 1\\space \\times $$"}], "_value": "_____"}]}]}]}]}]}
b.   Here are two sets of numbers:
{"table": [{"_attributes": {"class": ["table"], "style": "width: 100%;"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align: center;"}, "strong": [{"_value": "Set A"}]}, {"_attributes": {"style": "text-align: center;"}, "strong": [{"_value": "Set B"}]}]}, {"td": [{"_attributes": {"style": "text-align: center;"}, "_value": "1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11"}, {"_attributes": {"style": "text-align: center;"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$\\frac17, \\frac27,\\frac37,\\frac47,\\frac57,\\frac67,\\frac77$$"}]}]}]}]}]}
Choose a number from Set A and a number from Set B to complete this equation and make it true:
$$\frac67 =$$
_____
$$\times$$
_____
Choose a different number from Set A and a number from Set B to complete the equation to make it true:
$$\frac67 =$$
_____
$$\times$$
_____
c.   Explain or show how you know that the two equations you wrote are both true.


aligns: no