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: Activity
This is the final task in a series that leads students toward a general procedure for dividing fractions. Students verify previous observations about the steps for dividing non-unit fractions (namely, multiplying by the denominator and dividing by the numerator) and contrast the results with those found using diagrams. They then generalize these steps as an algorithm and apply it to answer other division questions.
As students discuss in their groups, listen to their observations and explanations. Select students with clear explanations to share later.
Digital
Launch
Arrange students in groups of 2. Give students 5–7 minutes of quiet think time and 2–3 minutes to share their response with their partner. Provide access to colored pencils. Some students may find it helpful to identify whole groups and partial groups on a tape diagram by coloring.
Students using the digital materials can use an applet to investigate division of fractions.
Representation: Internalize Comprehension.
Provide additional examples and counterexamples to illustrate concepts. Students may benefit from additional examples of division problems that they can “test” to reinforce understanding of their conclusion about how to divide a number by any fraction.
Supports accessibility for: Visual-spatial processing; Organization
Conversing, Representing: MLR8 Discussion Supports.
Provide the following sentence frames for students to use when they share their response with their partner: “I know there are ___
\(\frac18\)
s in
\(\frac34\)
because….”, “I drew the diagram like this ___ because.…”, and “First, I ___ because….”. This will help students produce statements that describe how to divide a number by any fraction.
Design Principle(s): Cultivate conversation; Support sense-making
Student Facing
Work with a partner. One person works on the questions labeled “Partner A” and the other person works on those labeled “Partner B.”
Partner A:
Find the value of each expression by completing the diagram.
\(\frac 34 \div \frac 18\)
How many
\(\frac 18\)
s in
\(\frac 34\)
?
[IMG]
\(\frac {9}{10} \div \frac 35\)
How many
\(\frac 35\)
s in
\(\frac{9}{10}\)
?
[IMG]
Use the applet to confirm your answers and explore your own examples.
Partner B:
Elena said: “If I want to divide 4 by
\(\frac 25\)
, I can multiply 4 by 5 and then divide it by 2 or multiply it by
\(\frac 12\)
.”
Find the value of each expression using the strategy Elena described.
\(\frac 34 \div \frac 18\)
\(\frac{9}{10} \div \frac35\)
What do you notice about the diagrams and expressions? Discuss with your partner.
Complete this sentence based on what you noticed:
To divide a number
\(n\)
by a fraction
\(\frac {a}{b}\)
, we can multiply
\(n\)
by ________ and then divide the product by ________.
Select
all
equations that represent the statement you completed.
\(n \div \frac {a}{b} = n \boldcdot b \div a\)
\(n \div \frac {a}{b}= n \boldcdot a \div b\)
\(n \div \frac {a}{b} = n \boldcdot \frac {a}{b}\)
\(n \div \frac {a}{b} = n \boldcdot \frac {b}{a}\)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 2. Give students 5–7 minutes of quiet think time and 2–3 minutes to share their response with their partner. Provide access to colored pencils. Some students may find it helpful to identify whole groups and partial groups on a tape diagram by coloring.
Students using the digital materials can use an applet to investigate division of fractions.
Representation: Internalize Comprehension.
Provide additional examples and counterexamples to illustrate concepts. Students may benefit from additional examples of division problems that they can “test” to reinforce understanding of their conclusion about how to divide a number by any fraction.
Supports accessibility for: Visual-spatial processing; Organization
Conversing, Representing: MLR8 Discussion Supports.
Provide the following sentence frames for students to use when they share their response with their partner: “I know there are ___
\(\frac18\)
s in
\(\frac34\)
because….”, “I drew the diagram like this ___ because.…”, and “First, I ___ because….”. This will help students produce statements that describe how to divide a number by any fraction.
Design Principle(s): Cultivate conversation; Support sense-making
Student Facing
Work with a partner. One person works on the questions labeled “Partner A” and the other person works on those labeled “Partner B.”
Partner A: Find the value of each expression by completing the diagram.
\(\frac 34 \div \frac 18\)
How many
\(\frac 18\)
s in
\(\frac 34\)
?
[IMG]
\(\frac {9}{10} \div \frac 35\)
How many
\(\frac 35\)
s in
\(\frac{9}{10}\)
?
[IMG]
Partner B:
Elena said, “If I want to divide 4 by
\(\frac 25\)
, I can multiply 4 by 5 and then divide it by 2 or multiply it by
\(\frac 12\)
.”
Find the value of each expression using the strategy Elena described.
\(\frac 34 \div \frac 18\)
\(\frac{9}{10} \div \frac35\)
What do you notice about the diagrams and expressions? Discuss with your partner.
Complete this sentence based on what you noticed:
To divide a number
\(n\)
by a fraction
\(\frac {a}{b}\)
, we can multiply
\(n\)
by ________ and then divide the product by ________.
Select
all
the equations that represent the sentence you completed.
\(n \div \frac {a}{b} = n \boldcdot b \div a\)
\(n \div \frac {a}{b}= n \boldcdot a \div b\)
\(n \div \frac {a}{b} = n \boldcdot \frac {a}{b}\)
\(n \div \frac {a}{b} = n \boldcdot \frac {b}{a}\)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Invite a couple of students to share their conclusion about how to divide a number by any fraction. Then, review the sequence of reasoning that led us to this conclusion using both numerical examples and algebraic statements throughout. Remind students that in the past few activities, we learned that:
Dividing by a whole number
\(n\)
is the same as multiplying by a unit fraction
\(\frac{1}{n}\)
(e.g., dividing by 5 is the same as multiplying by
\(\frac15\)
).
Dividing by a unit fraction
\(\frac{1}{n}\)
is the same as multiplying by a whole number
\(n\)
(e.g., dividing by
\(\frac 17\)
is the same as multiplying by 7).
Dividing by a fraction
\(\frac{a}{b}\)
is the same as multiplying by a unit fraction
\(\frac{1}{a}\)
and multiplying by a whole number
\(b\)
, which is the same as multiplying by
\(\frac ba\)
(e.g., dividing by
\(\frac57\)
is the same as multiplying by
\(\frac 15\)
, and then by 7. Performing these two steps gives the same result as multiplying by
\(\frac75\)
).
Finish the discussion by trying out the generalized method with other fractions such a
\(18\div\frac97\)
, or
\(\frac{15}{14}\div\frac{5}{2}\)
. Explain that although we now have a reliable and efficient method to divide any number by any fraction, sometimes it is still easier and more natural to think of the quotient in terms of a multiplication problem with a missing factor and to use diagrams to find the missing factor.


correct_option_index: 0