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: 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.

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
problem_activity: Narrative
The purpose of this activity is for students to compare two numbers in context, to explain or show their reasoning, and record the results of the comparisons with the symbols >, =, or < (MP2). The numbers may be fractions with the same numerator or the same denominator, or a fraction and a whole number.
Students are likely to generate different comparison statements for the same situation. For example, they may write
\(\frac{5}{8} > \frac{3}{8}\)
or
\(\frac{3}{8}<\frac{5}{8}\)
to represent
\(\frac{5}{8}\)
being the greater fraction. During synthesis, discuss how both statements capture the comparison and are valid.
MLR8 Discussion Supports.
Synthesis: As students share the similarities and differences between the representations and comparison statements, use gestures to emphasize what is being described. For example, show with your fingers the partitions such as fourths or eighths that are the same in the representations that are being compared.
Advances: Listening, Representing
Engagement: Provide Access by Recruiting Interest.
Synthesis: Optimize meaning and value. Invite students to share a connection between activity content and their own lives. Ask “How can I use this in my own life?”
Supports accessibility for: Conceptual Processing
Launch
Groups of 2
“Let’s use what we learned about comparing fractions and recognizing equivalent fractions to solve problems about lengths.”
Activity
“Work independently to solve the problems. For each one, be sure to show your thinking and to write a comparison statement.”
6–8 minutes: independent work time
“Share your responses and reasoning with your partner.”
2–3 minutes: partner discussion
Monitor for:
different representations or reasoning strategies used for the same problem, such as diagrams, fraction strips, number lines, or explanations in words
different statements written for the same problem, such as
\(4=\frac{12}{3}\)
and
\(\frac{12}{3}=4\)
, or
\(\frac{2}{3} > \frac{2}{8}\)
and
\(\frac{2}{8} < \frac{2}{3}\)
Student Facing
For each problem:
Answer the question and explain or show your reasoning.
Represent your answer with a statement that uses the symbols >, <, or =.
A beetle crawled
\(\frac{2}{8}\)
of the length of a log. A caterpillar crawled
\(\frac{2}{3}\)
of the length of the same log. Which insect crawled farther?
A grasshopper is 4 centimeters long. A caterpillar is
\(\frac{12}{3}\)
centimeters long. Which insect is longer?
[IMG]
A ladybug crawled
\(\frac{3}{8}\)
the length of a branch. An ant crawled
\(\frac{5}{8}\)
the length of the same branch. Which insect crawled farther?
A grasshopper jumped
\(\frac{5}{8}\)
the width of the sidewalk. A frog jumped
\(\frac{5}{6}\)
the width of the same sidewalk. Which jumped a longer distance?
Student Response
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click here to register or sign in
for free access to Student Response.
Activity Synthesis
Select 2–3 students to share their reasoning strategies or representations for at least one of the situations.
“How did you use what you’ve learned in earlier lessons to compare fractions?” (If the fractions had the same numerator, I thought about the size of the denominators. If they had the same denominator, I compared the numerators.)
Select students who wrote different but equally valid comparison statements (for instance,
\(\frac{2}{8}<\frac{2}{3}\)
and
\(\frac{2}{3} > \frac{2}{8}\)
) to share.
Discuss how to read each statement and ask students whether both accurately represent the comparison.
Emphasize that we can write comparison statements in more than one way, but we need to check that the statements make sense given the numbers we write and the symbols we use.


correct_option_index: 1