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 activity gives students additional practice in using diagrams and equations to represent division situations involving whole numbers and fractions.
For each problem, many kinds of visual representations are possible, but creating a meaningful representation may be challenging nonetheless. Urge students to use the contexts to generate ideas for useful diagrams, and to start with a draft and modify it as needed. Students may also use the fraction strips in the warm-up as a starting point for drawing diagrams.
As students work and discuss, monitor for effective diagrams or those that can be generalized to different situations (e.g., rectangles, tape diagrams, and number lines). Assign one problem for each group to record on a visual display.
Launch
Arrange students in group of 3–4. Give students 8–10 minutes of quiet work time and a few minutes to share their responses with their group. Tell each group they will be asked to present their solution to one problem. Provide access to geometry toolkits and tools for creating a visual display.
During group discussion, ask students to exchange feedback on each other’s diagrams and to notice any that might be particularly effective, efficient, or easy to understand. Then, they should record the diagram and equations for their assigned problem on a visual display and be prepared to explain them.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to illustrate connections between representations. For example, use the same color to represent the 3 miles in the diagram and the equation,
\(3\div \frac32=2\)
, then label each as “dividend.”
Supports accessibility for: Visual-spatial processing
Student Facing
For each situation, draw a diagram for the relationship of the quantities to help you answer the question. Then write a multiplication equation or a division equation for the relationship. Be prepared to share your reasoning.
The distance around a park is
\(\frac32\)
miles. Noah rode his bicycle around the park for a total of 3 miles. How many times around the park did he ride?
You need
\(\frac34\)
yard of ribbon for one gift box. You have 3 yards of ribbon. How many gift boxes do you have ribbon for?
The water hose fills a bucket at
\(\frac13\)
gallon per minute. How many minutes does it take to fill a 2-gallon bucket?
Student Response
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Student Facing
Are you ready for more?
How many heaping teaspoons are in a heaping tablespoon? How would the answer depend on the shape of the spoons?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may mistake the divisor and the dividend in the problems. Ask students to discuss (in their groups) the number or quantity being divided, and the reasonableness of the different ways of setting up each problem given the context. Representing the situations with objects may also help.
Activity Synthesis
Invite each group to present the solution on their visual display. Ask the rest of the class to think about two things: whether the equations make sense, and how the presented diagram shows the number of groups, the size of each group, and a total amount. Doing so will help students see the structure of the problems in the equations and diagrams.
Make sure students understand how each situation can be expressed by multiplication and division equations. Students should recognize that a question such as “how many batches are in 4 cups if each batch requires
\(\frac23\)
cups?” can be written as both
\(? \boldcdot \frac 23 = 4\)
and
\(4 \div \frac 23 = \,?\)
. With repeated reasoning, they see that a division expression such as
\(5 \div \frac 38\)
can be interpreted as “how many
\(\frac38\)
s are in 5?”
Speaking, Representing: MLR8 Discussion Supports.
Give students additional time to make sure that everyone in their group can explain their visual display and the relationship between the quantities represented. Prompt groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking. This will help students improve their explanations of their group’s reasoning during the whole-class discussion.
Design Principle(s): Optimize output (for explanation)


correct_option_index: 0