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: 
Recognize and represent proportional relationships between quantities.

Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

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.
|                                                          | **           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?


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_neg_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_pos_0

8
standard_description: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.)
problem_activity: Narrative
The purpose of this activity is for students to analyze pairs of fractions to determine if they are equivalent. Students may use any representation that makes sense to them. Students will create a visual display and have a gallery walk to consider the different ways of looking for equivalence. Highlight representations such as diagrams and number lines, which will support students as they learn to compare fractions with the same numerator or denominator in this section.
This activity uses
MLR7 Compare and Connect. Advances: representing, conversing
Engagement: Develop Effort and Persistence.
Invite students to generate a list of shared expectations for group work. Record responses on a display and keep visible during the activity.
Supports accessibility for: Social-Emotional Functioning
Required Materials
Materials to Gather
Materials for creating a visual display
Launch
Groups of 2
“Decide if these pairs of fractions are equivalent and show your thinking for each one. You can use any representation that makes sense to you.”
Activity
3–5 minutes: independent work time
“Share your ideas with your partner.”
2–3 minutes: partner discussion
MLR7 Compare and Connect
“Work with your partner to create a visual display that shows your thinking about
\(\frac{4}{6}\)
and
\(\frac{5}{6}\)
. You may want to include details such as notes, diagrams, drawings, and so on, to help others understand your thinking.”
Give students materials for creating a visual display.
2–5 minutes: partner work time
5–7 minutes: gallery walk
Student Facing
Are these fractions equivalent? Show your thinking using diagrams, symbols, or other representations.
\(\frac{1}{2}\)
and
\(\frac{1}{3}\)
\(\frac{4}{6}\)
and
\(\frac{5}{6}\)
\(\frac{3}{4}\)
and
\(\frac{6}{8}\)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
If students don’t determine whether or not fractions are equivalent, consider asking:
“What do you know about these fractions?”
“How could you use your fraction strips or diagrams to decide if they are equivalent?”
Activity Synthesis
“What are the different representations we can use to decide if
\(\frac{4}{6}\)
and
\(\frac{5}{6}\)
are equivalent?” (diagrams, number lines)
“How did each representation show that
\(\frac{4}{6}\)
and
\(\frac{5}{6}\)
are not equivalent?” (In the diagram, you can see that
\(\frac{5}{6}\)
has more space shaded. On the number line, they are not at the same location.)


aligns: no