few_shots: 
Example 1:
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.

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?


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:
Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

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
$$-\frac{1}{4}$$


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:
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.

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?


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, assess whether the following problem or activity is 'aligned' with the given Common Core standard in Mathematics.

shot_info: fl_problem_000972_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
problem_activity: Activity
The purpose of this activity is for students to see that to find what percentage one number is of another, divide them and then multiply by 100. First they find what percentage of 20 various numbers are, then they organize everything into a table. Using the table, students then describe the relationship they see between dividing the numbers by 20 and finding the percentage (MP8).
Launch
Depending on students' strengths in grade 5 work, it may be beneficial to offer some strategies for writing a fraction in an equivalent decimal form before students start working. Display the fraction
\(\frac{15}{25}\)
and ask students how they might think about writing it in decimal form. Possible strategies are writing the equivalent fraction
\(\frac{60}{100}\)
and knowing that "60 hundredths" is written 0.60 or 0.6. Another possibility is to write the equivalent fraction
\(\frac35\)
, which is equal to
\(\frac{6}{10}\)
and knowing that "6 tenths" can be written as 0.6.
Give students quiet think time to complete the activity and then time to share their explanations with a partner.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Speaking: MLR8 Discussion Supports.
To amplify mathematical uses of language to communicate about equivalent fractions and percentages (e.g., 15 minutes is 75% of 20 minutes, 75% of 20 minutes is equivalent to
\(\frac{15}{20}\)
), invite students to use these phrases when stating their ideas, revoicing and rephrasing as necessary.
Design Principle(s): Support sense-making, Optimize output (for explanation)
Student Facing
A school held a jump-roping contest. Diego jumped rope for 20 minutes.
Jada jumped rope for 15 minutes. What percentage of Diego’s time is that?
Lin jumped rope for 24 minutes. What percentage of Diego’s time is that?
Noah jumped rope for 9 minutes. What percentage of Diego’s time is that?
Record your answers in this table. Write the quotients in the last column as decimals.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "time (minutes)"}, {"_attributes": {"scope": "col"}, "_value": "percentage"}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "39226"}, "_value": "\\(\\text{time} \\div 20\\)"}]}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "Diego"}], "td": [{"_value": "20"}, {"_value": "100"}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "failed-conversion"}, "_value": "\\(\\frac{20}{20} = 1\\text{ \u00a0 }\\)"}]}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "Jada"}], "td": [{"_value": "15"}, {}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "failed-conversion"}, "_value": "\\(\\frac{15}{20} = \\text{ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0}\\)"}]}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "Lin"}], "td": [{"_value": "24"}, {}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "failed-conversion"}, "_value": "\\(\\frac{24}{20} = \\text{ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0}\\)"}]}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "Noah"}], "td": [{"_value": "9"}, {}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "failed-conversion"}, "_value": "\\(\\frac{9}{20} = \\text{ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0}\\)"}]}]}]}]}]}
What do you notice about the numbers in the last two columns of the table?
Student Response
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Activity Synthesis
Have students share their strategies for the first three problems. Then ask what they noticed about the last two columns. There are many ways to formulate the relationship. Make sure everyone sees that to find what percentage a number is of 20, divide it by 20 and then multiply by 100. A table showing the percentage for 1 minute might also help:
{"table": [{"_attributes": {"border": "1", "cellpadding": "1", "cellspacing": "1", "style": "width:500px"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "time (minutes)"}, {"_attributes": {"scope": "col"}, "_value": "percentage"}]}]}], "tbody": [{"tr": [{"th": [{"_value": "Diego"}], "td": [{"_value": "20"}, {"_value": "100"}]}, {"th": [{"_value": "unit rate"}], "td": [{"_value": "1"}, {"_value": "5"}]}, {"th": [{"_value": "Jada"}], "td": [{"_value": "15"}, {"_value": "75"}]}, {"th": [{"_value": "Lin"}], "td": [{"_value": "24"}, {"_value": "120"}]}, {"th": [{"_value": "Noah"}], "td": [{"_value": "9"}, {"_value": "45"}]}]}]}]}
Note that to find the percentage for 1 minute, we divide 100 by 20, and to find the percentage for any number of minutes, we multiply the result by that number of minutes. That is the same as dividing the number of minutes by 20 and multiplying by 100.


aligns: no