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 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_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
problem_activity: Problem 1
a.   Fill in the following blanks to make true statements. Use base ten blocks to help you.
1 thousand is the same as _______ hundreds.
1 hundred is the same as _______ tens.
1 ten is the same as _______ ones.
1 one is the same as _______ tenths.
b.   Below is an area model that represents 1 one. It is partitioned into tenths. Partition it to show what you would expect the next smallest unit to look like so that it continues the pattern in Part (a).
[IMG]
c.   Use what you notice in Part (a) and (b) to write a similar sentence that starts “1 tenth is the same as…”
Problem 2
The shaded part of this diagram represents 1 hundredth. We can write this as the fraction
$$\frac{1}{100}$$
or as the decimal
0.01.
[IMG]
Fill out the following table to represent each given number.
{"table": [{"_attributes": {"class": ["table", "table-bordered"], "style": "width: 100%;"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align: center;"}, "strong": [{"_value": "Diagram"}]}, {"_attributes": {"style": "text-align: center;"}, "strong": [{"_value": "Unit Form"}]}, {"_attributes": {"style": "text-align: center;"}, "strong": [{"_value": "Fraction Form"}]}, {"_attributes": {"style": "text-align: center;"}, "strong": [{"_value": "Decimal Form"}]}]}, {"td": [{"_attributes": {"style": "text-align: center;"}, "p": [{"img": [{"_attributes": {"src": "https://d21royfkw9g4l6.cloudfront.net/UXbX2I0Zc5A1kLpwT6BAA2_C9OM62.png"}}]}]}, {"_attributes": {"style": "text-align: center;"}, "_value": "7 hundredths"}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align: center;"}, "p": [{"img": [{"_attributes": {"src": "https://d21royfkw9g4l6.cloudfront.net/UXbX2I0Zc5A1kLpwT6BAA2_C9OM62.png"}}]}]}, {}, {"_attributes": {"style": "text-align: center;"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$\\frac{32}{100}$$"}]}, {}]}, {"td": [{"_attributes": {"style": "text-align: center;"}, "p": [{"img": [{"_attributes": {"src": "https://d21royfkw9g4l6.cloudfront.net/UXbX2I0Zc5A1kLpwT6BAA2_C9OM62.png"}}]}]}, {}, {}, {"_attributes": {"style": "text-align: center;"}, "_value": "0.85"}]}, {"td": [{"_attributes": {"style": "text-align: center;"}, "p": [{"img": [{"_attributes": {"src": "https://d21royfkw9g4l6.cloudfront.net/xJTfKSJObLTngvzpCrjGcc_LY71VX.png"}}]}]}, {}, {}, {}]}]}]}]}
Problem 3
a.   Explain why
$${{1\over10}={10\over100}}$$
. Draw a picture to illustrate your explanation.
b.   Explain why
$${0.20 = 0.2}$$
. Draw a picture to illustrate your explanation.
Problem 4
Jani and Kai are representing a decimal number. Here are the area models they drew:
{"table": [{"_attributes": {"align": "center", "class": ["table"], "style": "width:50%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align: center;"}, "_value": "Jani"}, {"_attributes": {"style": "text-align: center;"}, "_value": "Kai"}]}, {"td": [{"p": [{"_attributes": {"style": "text-align: center;"}, "img": [{"_attributes": {"src": "https://d21royfkw9g4l6.cloudfront.net/pCjOWHyq93ljOqoarBpFRi_XAG8G1.png"}}]}]}, {"p": [{"_attributes": {"style": "text-align: center;"}, "img": [{"_attributes": {"src": "https://d21royfkw9g4l6.cloudfront.net/IX0bwHJs13FGRL7TldfH6i_9P5SGX.png"}}]}]}]}]}]}]}
a.   What value do you think Jani and Kai are trying to model? How do you know?
b.   Draw pictures to show what Kai might have drawn for each area model in Anchor Task 2.


aligns: no