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: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in Grade 8.)
problem_activity: Activity
In this activity, students find intercepts and slopes for pairs of equations. They should notice that the equations in each pair are multiples of one another and therefore are equivalent equations. Students look for and make use of structure (MP7) when they notice the relationships between the pairs of equations.
Launch
Ask students what they remember about
\(x\)
- and
\(y\)
-intercepts. One useful thing to remember is that the
\(x\)
-intercept of a graph occurs where the
\(y\)
-coordinate is zero. (Demonstrate using a graph, if needed.) So, to find the
\(x\)
-intercept of a graph representing an equation, you can replace
\(y\)
with 0 and solve the equation for
\(x\)
. (And vice versa to find the
\(y\)
-intercept.)
Ask students how they can use two points like
\((1,0)\)
and
\((0,8)\)
to find the slope of a line. (Using a formula like
\(m = \frac{y_2 - y_1}{x_2 - x_1}\)
, or a less-formal version like “subtract the y’s, subtract the x’s, then divide.
”
For this example the slope is -8 since
\(\frac{8 - 0}{0-1} = \text{-}8\)
.)
Student Facing
Here are some equations in pairs. For each equation:
Find the
\(x\)
-intercept and
\(y\)
-intercept of a graph of the equation.
Find the slope of a graph of the equation.
\(x + y = 6\)
and
\(2x + 2y = 12\)
\(3y - 15x = \text{-}33\)
and
\(y - 5x = \text{-}11\)
\(5x + 20y = 100\)
and
\(4x + 16y = 80\)
\(3x - 2y = 10\)
and
\(4y - 6x = \text{-}20\)
What do you notice about the pairs of equations?
Choose one pair of equations and rewrite them into slope-intercept form (
\(y = mx + b\)
). What do you notice about the equations in this form?
Student Response
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Activity Synthesis
The purpose of the discussion is to show that equations that are multiples of one another will have graphs with the same intercepts and slope. Select students to share their solutions and methods for finding the intercepts and slope as well as things they notice. While students may notice many things, ensure it is mentioned that in each pair of equations, one is a multiple of the other.
Some questions for discussion:
"How would the graphs of each pair of equations look?" (Each graph would look like a single line since the two equations are equivalent. Time permitting, demonstrate this using graphing technology.)
"Select a pair of equations. Find another equation whose graph would have the same slope and intercepts as the pair already given." (Another multiple of either equation will work.)


correct_option_index: 0

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