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.)


Your response:
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.


Your response:
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.


Your response:
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 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.

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.

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 ordering and absolute value of rational numbers.
problem_activity: Activity
The purpose of this activity is for students to extend the vertical and horizontal axes to include 4 quadrants just as they extended the number line to include negative numbers. Students are introduced to the term
quadrant.
Students plot and label coordinates using ordered pairs and identify their quadrants.
Digital
Launch
Introduce the concept of the 4-region coordinate plane by explaining that, just like we extended the number line include negative numbers, we can extend both the number lines of the coordinate plane (the axes) to include negative coordinates. Use the word
quadrant
to describe the four regions of the coordinate plane. It may be helpful to explain that the prefix "quad-" means 4 and give other examples from English and other languages that use the prefix (quadriceps, quadrilateral, cuatro). Give students 10 minutes to work followed by whole-class discussion.
Classes using the digital version have an interactive applet to use. Instead of naming the coordinates, students enter them into the Input Bar. If correct, the new points will hit the targets and turn them black.
Representation: Develop Language and Symbols.
Activate or supply background knowledge. During the launch, take time to review terms that students will need to access for this activity. Invite students to suggest language or diagrams to include that will support their understanding of: vertical number line, horizontal number line.
Supports accessibility for: Conceptual processing; Language
Student Facing
[IMG]
The colored points on the coordinate plane are like targets. Hit each point by entering its coordinates as an ordered pair in the Input Bar, like this:
What do you notice about the locations and ordered pairs of
\(B\)
,
\(C\)
, and
\(D\)
? How are they different from those for point
\(A\)
?
Plot a point at
\((\text-2, 5)\)
. Label it
\(E\)
. Plot another point at
\((3, \text-4.5)\)
. Label it
\(F\)
.
The coordinate plane is divided into four quadrants, I, II, III, and IV, as shown here.
[IMG]
\(G = (5, 2)\)
\(H=(\text-1, \text-5)\)
\(I=(7,\text-4)\)
In which quadrant is
\(G\)
located?
\(H\)
?
\(I\)
?
A point has a positive
\(y\)
-coordinate. In which quadrant could it be?
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
Introduce the concept of the 4-region coordinate plane by explaining that, just like we extended the number line include negative numbers, we can extend both the number lines of the coordinate plane (the axes) to include negative coordinates. Use the word
quadrant
to describe the four regions of the coordinate plane. It may be helpful to explain that the prefix "quad-" means 4 and give other examples from English and other languages that use the prefix (quadriceps, quadrilateral, cuatro). Give students 10 minutes to work followed by whole-class discussion.
Classes using the digital version have an interactive applet to use. Instead of naming the coordinates, students enter them into the Input Bar. If correct, the new points will hit the targets and turn them black.
Representation: Develop Language and Symbols.
Activate or supply background knowledge. During the launch, take time to review terms that students will need to access for this activity. Invite students to suggest language or diagrams to include that will support their understanding of: vertical number line, horizontal number line.
Supports accessibility for: Conceptual processing; Language
Student Facing
Label each point on the coordinate plane with an ordered pair.
[IMG]
What do you notice about the locations and ordered pairs of
\(B\)
,
\(C\)
, and
\(D\)
? How are they different from those for point
\(A\)
?
Plot a point at
\((\text-2, 5)\)
. Label it
\(E\)
. Plot another point at
\((3, \text-4.5)\)
. Label it
\(F\)
.
The coordinate plane is divided into four
quadrants
, I, II, III, and IV, as shown here.
[IMG]
\(G = (5, 2)\)
\(H=(\text-1, \text-5)\)
\(I=(7,\text-4)\)
In which quadrant is point
\(G\)
located? Point
\(H\)
? Point
\(I\)
?
A point has a positive
\(y\)
-coordinate. In which quadrant could it be?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may confuse the order of the coordinate pairs when plotting points. Direct students to look at their work with vertical and horizontal lines in the warm up to remind them which coordinate is vertical.
Students may locate -4.5 between -3 and -4, rather than between -4 and -5. Direct them to the number lines used in earlier lessons to remind them of locating non integer numbers on number lines.
Activity Synthesis
The most important idea for students to understand is that by extending the two number lines that form the coordinate axes for the first quadrant, we now have 4 quadrants. We describe points in these quadrants using negative and positive numbers as the
\(x\)
and
\(y\)
coordinates. Focus on responses from question 4 for discussion. Invite students to share their reasoning about how to identify the quadrants for the points
\(G\)
,
\(H\)
, and
\(I\)
. As time allows, consider asking the following questions:
“If a point has a negative
\(x\)
-coordinate, what quadrant could it be in?”
“If a point has a negative
\(y\)
-coordinate, what quadrant could it be in?”
“If a point has a positive
\(x\)
-coordinate, what quadrant could it be in?”
To involve more students in the conversation, consider asking:
“Do you agree or disagree? Why?”
“Who can restate ___’s reasoning in a different way?”
“Does anyone want to add on to _____’s reasoning?”
Speaking: MLR8 Discussion Supports.
To support students in producing statements about features of the quadrants that have negative and/or positive numbers provide sentence frames for students to use such as: “Point _____ is in quadrant _____ because _____.” This will support class discussion in providing low entry points for students to start speaking about their reasoning.
Design Principle(s): Support sense-making


correct_option_index: 1