few_shots: 
Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Concept/skill: 
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Answer: 
No. Though this task involves addition, it does not involve any word problems.

Example 2:
Problem/activity:
Task
 Arrange the fractions in order from least to greatest. Explain your answer with a picture.
 $\frac{1}{5}, \frac{1}{7}, \frac{1}{3} $
 $\frac{2}{5}, \frac{2}{7}, \frac{2}{3} $
 $\frac{5}{6}, \frac{3}{6}, \frac{1}{6} $
 $\frac{5}{12}, \frac{8}{12}, \frac{4}{12} $

Concept/skill: 
Understand a fraction as a number on the number line; represent fractions on a number line diagram.
 Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.

Answer: 
No. Completing this task does not involve representing fractions on a number line diagram.

Example 3:
Problem/activity:
Problem 1
Solve using the standard algorithm. If you get stuck, use an area model and/or the partial products algorithm to help. Then assess the reasonableness of your answer.
a.
$$5,803\times46$$
b.
$$63\times2,116$$
Problem 2
Dunkin' introduced a croissant doughnut in 2014. In the first three months of its debut, they sold 7,870 dozen of them per day. How many croissant doughnuts were sold in the first three months?


Concept/skill: 
Fluently multiply multi-digit whole numbers using the standard algorithm.

Answer: 
Yes. This set of problems involves multi-digit whole numbers because the numbers in each problem are four and two digit integers, and the mathematical calculations involved in both problems focus on multiplication. The first problem explicitly asks the student to solve its subproblems using the standard algorithm.

Now, does the following problem or activity enable students to learn the full intent of the following concept or skill?

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im_task_000157_neg_0

3

fl_problem_000674_pos_0

5
standard_description: 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.


aligns: yes