few_shots: 
Example 1:
Concept/skill: 
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.

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?


Answer: 
Yes. 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.

Example 2:
Concept/skill: 
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

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
$$-\frac{1}{4}$$


Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

Example 3:
Concept/skill: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

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?


Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_000972_pos_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
problem_activity: Narrative
This activity prompts students to compare multiple fractions and put them in order by size. The work gives students opportunities to look for and make use of structure (MP7) in each set of fractions and make comparisons strategically. For instance, rather than comparing two fractions at a time and in the order they are listed, students could first classify the given fractions as greater or less than
\(\frac{1}{2}\)
or 1, look for fractions with a common numerator or denominator, and so on.
If time is limited, consider asking students to choose two sets of fractions to compare and order.
Representation: Internalize Comprehension.
Synthesis: Invite students to identify which details were most useful when putting fractions in order. Display the sentence frame, “The next time I put fractions in order, I will pay attention to . . . .”
Supports accessibility for: Memory
Launch
Groups of 2
Activity
“Work independently on two sets. Then, discuss your work with your partner and complete the rest together.”
10 minutes: independent work time
Monitor for students who look for and make use of structure. Ask them to share during lesson synthesis.
3–4 minutes: partner discussion
Student Facing
Put each set of fractions in order, from least to greatest. Be prepared to explain your reasoning.
\(\frac{3}{12} \qquad \frac{2}{4} \qquad \frac{2}{3} \qquad \frac{1}{8}\)
\(\frac{8}{5} \qquad \frac{5}{6} \qquad \frac{11}{12} \qquad \frac{11}{10}\)
\(\frac{21}{20} \qquad \frac{9}{10} \qquad \frac{6}{5} \qquad \frac{101}{100}\)
\(\frac{5}{8} \qquad \frac{2}{5} \qquad \frac{3}{7} \qquad \frac{3}{6}\)
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Advancing Student Thinking
Some students may try to write equivalent fractions with a common denominator for all four fractions in each set before comparing them but may be unable to do so. Encourage them to try reasoning about two fractions at a time, and to use what they know about the fractions to determine how they compare (to one another or to familiar benchmarks).
Activity Synthesis
See lesson synthesis.


aligns: no