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: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

Write and evaluate numerical expressions involving whole-number exponents.

Write, read, and evaluate expressions in which letters stand for numbers.

Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
problem_activity: Activity
In the previous activity students looked at 9 different equations. In this activity, each student focuses on just one of those equations to learn about the real-world situation it represents. They interpret the variables and values in terms of the situation (MP7). Adding to the table they saw in the previous activity gives students an opportunity to practice both evaluating expressions and solving equations.
This activity also asks students to create a graph of their assigned relationship on graph paper. In previous activities, students were given a grid with the axes already labeled and numbered whenever they were asked to create a graph of a relationship. Here they must decide for themselves how to scale each axis.
Launch
Decide whether you want to assign the equations or allow students to select which equation they will work with. Make sure there is at least one student working with each equation.
Give students 5–10 minutes of quiet work time followed by partner and whole-class discussion. If there is more than one student working with the same equation, consider letting them share their answers and reasoning with each other, after they have had some time to work on their own.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. Check in with students within the first 2-3 minutes of work time. Look for students who need additional support deciding how to scale each axis for table. Invite students to share strategies they have used so far, as well as any questions they have before continuing.
Supports accessibility for: Memory; Organization
Student Facing
The equations in the previous activity represent situations.
\(S - 2 = T\)
where
\(S\)
is the number of sides on a polygon and
\(T\)
is the number of triangles you can draw inside it (from one vertex to the others, without overlapping)
\(G = J + 13\)
where
\(G\)
is a day in the Gregorian calendar and
\(J\)
is the same day in the Julian calendar
\(P = I - 47.50\)
where
\(I\)
is the amount of income and
\(P\)
is the profit after
$
47.50 in expenses
\(C + 273.15 = K\)
where
\(C\)
is a temperature in degrees Celsius and
\(K\)
is the same temperature in Kelvin
\(e = 6s\)
where
\(e\)
is the total edge length of a regular tetrahedron and
\(s\)
is the length of one side
\(m = 8.96V\)
where
\(V\)
is the volume of a piece of copper and
\(m\)
is its mass
\(y = \frac{1}{12}x\)
where
\(x\)
is the number of eggs and
\(y\)
is how many dozens that makes
\(t = \frac{d}{2.5}\)
where
\(t\)
is the amount of time it takes in seconds to jog a distance of
\(d\)
meters at a constant speed of 2.5 meters per second
\(g = 28.35z\)
where
\(g\)
is the mass in grams and
\(z\)
is the same amount in ounces
Your teacher will assign you one of these equations to examine more closely.
Rewrite your equation using words. Use words like product, sum, difference, quotient, and term.
In the previous activity, you matched equations and tables. Copy the values from the table that matched your assigned equation into the first 3 rows of this table. Make sure to label what each column represents.
| independent variable:                             ________________________ | dependent variable:                             ________________________ |
|----------------------------------------------------------------------------|--------------------------------------------------------------------------|
|                                                                            |                                                                          |
|                                                                            |                                                                          |
|                                                                            |                                                                          |
|                                     60                                     |                                                                          |
|                                                                            |                                   300                                    |
Select one of the first 3 rows of the table and explain what those values mean in this situation.
Use your equation to find the values that complete the last 2 rows of the table. Explain your reasoning.
On graph paper, create a graph that represents this relationship. Make sure to label your axes.
Student Response
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Anticipated Misconceptions
Some students may struggle to understand what each step is asking them to do. Consider preparing an example to share with them, such as this for the equation
\(2.54i = c\)
, and point out which part of the example corresponds to the question they are on.
The number of centimeters is the product of the number of inches and 2.54.
| independent variable:                       length (inches) | dependent variable:                       length (centimeters) |
|-------------------------------------------------------------|----------------------------------------------------------------|
|                              5                              |                              12.7                              |
|                             36                              |                             91.44                              |
|                             75                              |                             190.5                              |
|                             60                              |                                                                |
|                                                             |                              300                               |
If something is 36 inches long, this is equivalent to 91.44 when measured in centimeters.
In the table above, fill in 152.4 in the right column, next to the 60, because
\(2.54 \boldcdot 60 = 152.4\)
. Fill in 118.11 in the left column, next to the 300, because
\(2.54 \boldcdot 118.11 \approx 300\)
A graph with “length (inches)” on the horizontal axis, “length (centimeters)” on the vertical axis, and points at
\((5, 12.7), (36, 91.44), (75, 190.5), (60, 152.4)\)
, and
\((118.11, 300)\)
Some students may struggle with setting up their graph from scratch on graph paper. Prompt them to think about the maximum value they want to represent on each axis and what number they could count by to get to that maximum value in the amount of space they have.
Activity Synthesis
Continue to the next activity. The discussion of these relationships occurs after students have created their visual displays.


correct_option_index: 2