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: Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.
problem_activity: Narrative
The purpose of this activity is for students to practice division within 100 by playing a game called Race to 1. The goal of the game is to repeatedly divide numbers until they reach one.
Engagement: Develop Effort and Persistence.
Check in and provide each group with feedback that encourages collaboration and community.
Supports accessibility for: Social-Emotional Functioning, Organization
Required Materials
Materials to Gather
Number cubes
Launch
Groups of 2
“Take a minute to read the directions for Race to 1.”
1 minute: quiet think time
“Let’s look at a sample game. Jada rolled a 3 on her first turn, then rolled 2 a few times afterwards. Talk with your partner about what her next move should be if she rolls 2 on her next turn.” (She should divide 4 or 6 by 2 because those moves get her really close to one.)
1 minute: partner discussion
Share responses.
Give each group a number cube.
Activity
“Play Race to 1 with your partner.”
10–15 minutes: partner work time
If students finish early, have them play again. Students can fill in the same numbers from the first game at the top of the columns of the second game or use the numbers 24, 36, 48, and 72 for a challenge.
Student Facing
Directions for Race to 1
On your first turn:
Roll a number cube. The number you roll is your divisor. (If you roll a 5, roll again.)
Choose one number from the top row of your table. Divide it by your divisor.
Write a division expression and record the quotient in the table. Your turn is done.
On all your other turns:
Roll a number cube. The number you roll is your divisor. (If you roll a 5, roll again.)
Choose either a quotient from one of your past turns or a new number from the top row of the table. Divide it by your divisor.
Write a division expression and record the quotient in the table.
The first player to get a quotient of 1 wins.
Sample Game
Jada rolled 3 on her first turn, then rolled 2 a few times afterwards.
{"table": [{"_attributes": {"border": "1", "style": "width:100%"}, "thead": [{"tr": [{"th": [{"p": [{"_attributes": {"style": "text-align:center"}, "_value": "start number"}]}, {"_attributes": {"scope": "col", "style": "text-align:center; width:18%"}, "_value": "12"}, {"_attributes": {"scope": "col", "style": "text-align:center; width:18%"}, "_value": "16"}, {"_attributes": {"scope": "col", "style": "text-align:center; width:18%"}, "_value": "18"}, {"_attributes": {"scope": "col", "style": "text-align:center; width:18%"}, "_value": "24"}]}]}], "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math"]}, "_value": "\\(12\\div3\\)"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math"]}, "_value": "\\(16\\div2\\)"}]}, {}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math"]}, "_value": "\\(24\\div2\\)"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {"_attributes": {"style": "text-align:center"}, "_value": "4"}, {"_attributes": {"style": "text-align:center"}, "_value": "8"}, {}, {"_attributes": {"style": "text-align:center"}, "_value": "12"}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math"]}, "_value": "\\(12\\div2\\)"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {"_attributes": {"style": "text-align:center"}, "_value": "6"}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}]}]}]}
Game 1
{"table": [{"_attributes": {"border": "1", "style": "width:100%"}, "thead": [{"tr": [{"th": [{"p": [{"_attributes": {"style": "text-align:center"}, "_value": "start number"}]}, {"_attributes": {"scope": "col", "style": "text-align:center; width:18%"}, "_value": "12"}, {"_attributes": {"scope": "col", "style": "text-align:center; width:18%"}, "_value": "16"}, {"_attributes": {"scope": "col", "style": "text-align:center; width:18%"}, "_value": "18"}, {"_attributes": {"scope": "col", "style": "text-align:center; width:18%"}, "_value": "24"}]}]}], "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}]}]}]}
Game 2
{"table": [{"_attributes": {"border": "1", "style": "width:100%"}, "thead": [{"tr": [{"th": [{"p": [{"_attributes": {"style": "text-align:center"}, "_value": "start number"}]}, {"_attributes": {"scope": "col", "style": "text-align:center; width:18%"}}, {"_attributes": {"scope": "col", "style": "text-align:center; width:18%"}}, {"_attributes": {"scope": "col", "style": "text-align:center; width:18%"}}, {"_attributes": {"scope": "col", "style": "text-align:center; width:18%"}}]}]}], "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "division expression"}, {}, {}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "quotient"}, {}, {}, {}, {}]}]}]}]}
Activity Synthesis
“How was having an efficient way to divide helpful as you played Race to 1?” (I had to decide which number I was going to divide, so it was helpful to be able to do it quickly. I had to divide a few times on each turn.)
“What was a helpful strategy as you played Race to 1?” (I had to think about which number I could divide by the number I rolled. I divided the smallest number by the largest divisor to try to get to the quotient to be as small as possible so I could get to 1 first.)


aligns: no