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: 
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_pos_0

7

fl_problem_001392_pos_0

8
standard_description: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., p^(2)). For example, by truncating the decimal expansion of Ö2, show that Ö2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
problem_activity: Optional activity
If students investigated the answers to their questions in the previous activity, they need access to their findings. They can either work with the questions generated by their group, or everyone can work from a master list of questions compiled by the teacher.
If students have access to a spreadsheet program, they can use it to find the results (MP5). Students need to communicate their findings clearly in graphical displays (MP6). They can choose to illustrate all or some of the questions in their graphical displays. Encourage the students to be creative with their displays. For example, if students are examining the question “How many people would live on each continent?” they might sketch the continents and place stick figures on the continents to represent the class. They might also draw a bar graph showing the same information in a different way.
Launch
Students work in same groups of 2–4 from previous activity. Provide access to four-function calculators. If students are doing their own research, provide access to internet-enabled devices. If conducting a gallery walk at the end, provide access to tools for making a visual display.
To give students a sense of the magnitude of the world versus the classroom, consider zooming in from Earth to North America, the United States, your state, your city, and finally your school.
Invite students to work on the question or questions that are of most interest to them either from their own group work on from the compiled list.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check to make sure students are beginning to choose characteristics about the world’s population that are available to find, and that they can calculate percentages for the class for those characteristics.
Supports accessibility for: Memory; Organization
Student Facing
Suppose your class represents all the people in the world.
Choose several characteristics about the world’s population that you have investigated. Find the number of students in
your
class that would have the same characteristics.
Create a visual display that includes a diagram that represents this information. Give your display the title “If Our Class Were the World.”
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may struggle with the difficult numbers for the percent of the world population with different characteristics. For example, there are about 1.2 billion people living in Africa. Ask them  to calculate
\(1.2 \div 7.4 \approx 0.162\)
. "What percent of the world population this is?" (16%). "If our class were the world, then about 16% of the class would live in Africa.  So how many students in our class would live in Africa?" (Since 16% of 25 is 4, that means that 4 students in our class would live in Africa.)
In some cases, there might be two acceptable answers to how many students in our class have a certain characteristic. About 10% of the world population is in Europe. For our class, we could have 2 students (8%) or 3 students (12%) that are in Europe. Either one is an acceptable answer. Make sure that students understand that 2.5 (10% of 25) is not a reasonable answer.
Activity Synthesis
Invite groups to share their displays. Consider doing a gallery walk and asking:
Were you surprised by any of your findings?
In what ways is the class actually representative of the world? (possibly gender distribution, religious adherence, etc.)
In what ways is it not? (age, place of residence, access to clean water, etc.)
Make sure to highlight the fact that having a small class limits the percentages that are possible. This can make deciding how many students in the class have a particular characteristic difficult. Consider asking:
What percentages are possible for students in the class? (0, 4, 8, . . . multiples of 4 up to 100)
There are 1 billion people living in the Americas. That’s a little more than 13.5% of the world population. If our class was the world, how many students in our class would be from the Americas? (3 students would be 12%, and 4 would be 16%. So 3 students is as close as we can get.)


aligns: no