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?


Standard options: 
A. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
B. 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.
C. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)


Thought: 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.
Answer: B

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


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Thought: 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).
Answer: B

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.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
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?


Standard options: 
A. Solve linear equations in one variable.
B. Analyze and solve pairs of simultaneous linear equations.


Thought: 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.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_000972

6

fl_problem_001097

7

fl_problem_001392

8
options: 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: Optional activity
This optional activity gives students another opportunity to use what they have learned about all operations to model and solve a problem in a baking context. Students need to make sense of the problem and persevere in solving it (MP1).
Students may approach the problem in different ways (by drawing diagrams, making computations, reasoning verbally, etc.). Students may also choose different operations to obtain the information they need. For instance, instead of dividing by a fraction, they may perform repeated subtraction. Notice the different methods students use and identify strategies or explanations that should be shared with the class.
Launch
Consider arranging students in new groups of 2. Give students 7–8 minutes of quiet work time and then 1–2 minutes to discuss their response with their partner. Ask students to be prepared to explain their reasoning.
Conversing, Writing: MLR5 Co-craft Questions.
Use this routine to increase students' awareness of the language of mathematical questions. Display the problem about baking cookies, but withhold the final section that includes the question. Give students 2–3 minutes to write down any mathematical questions they have about the situation presented, then to share their questions with a partner. Circulate and listen for questions that involve combining fractional amounts of ingredients. Reveal the actual question and ask students to compare their questions with the original.  Listen for and amplify any questions involving the relationships between the two quantities in this task (number of cups of flour and number of cups of butter).
Design Principle(s): Cultivate conversation; Support sense-making
Student Facing
Mai, Kiran, and Clare are baking cookies together. They need
\(\frac 34\)
cup of flour and
\(\frac 13\)
cup of butter to make a batch of cookies. They each brought the ingredients they had at home.
Mai brought 2 cups of flour and
\(\frac 14\)
cup of butter.
Kiran brought 1 cup of flour and
\(\frac 12\)
cup of butter.
Clare brought
\(1\frac 14\)
cups of flour and
\(\frac34\)
cup of butter.
If the students have plenty of the other ingredients they need (sugar, salt, baking soda, etc.), how many whole batches of cookies can they make? Explain your reasoning.
Student Response
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Activity Synthesis
Consider combining every 2–3 groups of 2 students and having students discuss their responses and methods in larger groups of 4–6.
If time permits, reconvene as a class to highlight a couple of strategies and reflect on the effectiveness and efficiency of students’ strategies. For example, if some students performed repeated addition instead of multiplying (or repeated subtraction instead of dividing), ask if repeated addition (or subtraction) is always as efficient as multiplication (or division), or under what circumstances one method might be preferred over the other.


correct_option_index: 0