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 a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x^(2 )+ y^(2) = 3.

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Represent a system of linear equations as a single matrix equation in a vector variable.

Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

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: Activity
In this activity, students solve a system of equations by graphing then include a third line through the solution point that is either vertical or horizontal. In the associated Algebra lesson, students will graph equations in a system and additional equations that arise from the elimination method to show that they all intersect at the solution.
To focus on the work of the activity, it is recommended that students use the digital version of the activity or have access to Desmos or another graphing tool to more quickly and accurately find the solutions to the system of equations through graphing. If students need practice graphing lines representing equations by hand or if digital access is not available, the print version provides a graph given one of the lines and asks students to graph the second line by hand.
Launch
Display the instructions for all to see. Ask students,
"What do you know about equations that are represented by vertical lines?" (They are of the form
\(x = a\)
for a number
\(a\)
.)
"What do you know about equations that are represented by horizontal lines?" (They are of the form
\(y = a\)
for a number
\(a\)
.)
Student Facing
For each system of equations:
Solve the system of equations by graphing. Write the solution as an ordered pair.
Write an equation that would represented by a vertical or horizontal line that also passes through the solution of the system of equations.
Graph your new equation along with the system.
\(\begin {cases} \begin {align}y = 3x+5\\ y=\text{-}x+1\end{align} \end {cases}\)
The line representing
\(y = 3x+5\)
is shown
[IMG]
\(\begin {cases} \begin {align}y = \frac{1}{3}x-2\\ y=x-6\end{align} \end {cases}\)
The line representing
\(y = \frac{1}{3}x-2\)
is shown
[IMG]
\(\begin {cases} \begin {align} 2x+3y=10\\ x+y=3\end{align} \end {cases}\)
The line representing
\(2x+3y=10\)
is shown
[IMG]
Student Response
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Activity Synthesis
The purpose of the discussion is for students to recognize that, for a system of equations represented by graphs that intersect at
\((a,b)\)
, the lines
\(x = a\)
and
\(y = b\)
also go through the solution. Select students to share their solutions including students who found vertical as well as horizontal lines through the intersection point.
Some questions for discussion:
"How could you use your answers to these questions to write another system of equations that have the same solution as the original system of equations?" (The system
\(\begin {cases} \begin {align} x = \text{-}1\\ y=2\end{align} \end {cases}\)
has the same solution as the original equation
\(\begin {cases} \begin {align}y = 3x+5\\ y=\text{-}x+1\end{align} \end {cases}\)
since all 4 lines go through the point
\((\text{-}1,2)\)
.)
"Two lines intersect at the point
\((3,5)\)
. How could you know whether the equation
\(3x - 2y = \text{-}1\)
is represented by a line that goes through that same point?" (Substitute the value 3 for
\(x\)
and 5 for
\(y\)
. If the equation is still true, then the associated line will go through the same point. If the equation is false, the associated line will not go through the same point.)


correct_option_index: 1