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: 
Recognize and represent proportional relationships between quantities.

Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

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 and division and of fractions to multiply and divide rational numbers.

Answer: 
Yes. This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).

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: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

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_neg_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
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