standard_description: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
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