few_shots: 
Example 1:
Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Concept/skill: 
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Answer: 
Yes. This problem solves a real-world problem involving mixing ingredients for cookies. A student doing this problem would need to reason about equivalent ratios of butter amounts to sugar amounts.

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

Example 3:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
{"table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "table-layout:fixed; width:70%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "No solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Exactly 1 solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Infinite number of solutions"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9=4x+7}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y=3y+5}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y={x\\over2}}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${3x+6y-9=0}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${x=y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${1.25x=1.25y}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9.9=6x+8y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9x=2.5y-8.8}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=1.5}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=2.5}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {}]}]}]}]}
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Concept/skill: 
Analyze and solve pairs of simultaneous linear equations.

Answer: 
Yes. The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_000972_pos_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_pos_0

8
standard_description: Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
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