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: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

Understand ordering and absolute value of rational numbers.

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
problem_activity: Activity
Students reason abstractly and quantitatively about temperatures over time graphed on coordinate axes (MP2). The goal of this activity is for students to use inequalities to describe the location of points on a coordinate grid in one direction. This activity also introduces the idea of vertical difference on the coordinate plane using a familiar context. Students may use previous strategies such as counting squares, but are not expected to explicitly add or subtract using negative numbers.
Launch
Arrange students in groups of 2. Allow students 3–4 minutes of quiet work time and 1–2 minutes to check results with their partner. Follow with a whole-class discussion.
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, remind students about the sea levels and elevation number line representations.
Supports accessibility for: Social-emotional skills; Conceptual processing
Student Facing
The coordinate plane shows the high and low temperatures in Nome, Alaska over a period of 8 days. The axis labeled
\(T\)
represents temperatures in degrees Fahrenheit. The axis labeled
\(d\)
represents the day.
[IMG]
What was the warmest high temperature?
Write an inequality to describe the high temperatures,
\(H\)
, over the 8-day period.
What was the coldest low temperature?
Write an inequality to describe the low temperatures,
\(L\)
, over the 8-day period.
On which day(s) did the
largest
difference between the high and low temperatures occur? Write down this difference.
On which day(s) did the
smallest
difference between the high and low temperatures occur? Write down this difference.
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
Before doing this problem, do the problem about taxicab distance in an earlier lesson.
The point
\((0,3)\)
is 4 taxicab units away from
\((\text{-}4,3)\)
and 4 taxicab units away from
\((2,1)\)
.
Find as many other points as you can that are 4 taxicab units away from
both
\((\text{-}4,3)\)
and
\((2,1)\)
.
Are there any points that are 3 taxicab units away from both points?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Because the high temperature on day 7 was positive and the low temperature was negative, some students may not notice that the difference was 10 degrees on this day as well. Consider prompting them to use tracing paper to compare the temperature differences on days 6 and 7.
Activity Synthesis
This discussion should lead to two key takeaways. First, students express the range of values for the low and high temperatures using inequalities. Second, students share strategies for finding a difference between two values on the coordinate plane.
Ask students to share their inequalities for
\(H\)
and
\(L\)
. It is expected that students have inequalities that describe the maximum high temperature for
\(H\)
and the minimum low temperature for
\(L\)
, but the discussion should bring out that each variable has 4 statements that capture its possible values:
\(L>\text-3\)
,
\(L=\text-3\)
,
\(L < 26\)
, and
\(L=26\)
for the variable
\(L\)
and
\(H>2\)
,
\(H=2\)
,
\(H < 28\)
, and
\(H=28\)
for the variable
\(H\)
.
Ask students to share their strategies for finding the vertical distance between points. Push them to explain how they took the scale of the vertical axis into account. Invite students to explain how they used the context to make sense of their answers.
Listening, Speaking: MLR8 Discussion Supports.
To support students in producing statements about details in the inequalities, provide sentence frames for students to use when they explain their reasoning to their peers, such as: “This inequality should/should not also be equal to because _____”, “A situation uses an equal sign when _____”, “I know that the inequality _____ is correct because _____.”
Design Principle(s): Support sense-making


correct_option_index: 0

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