few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Standard options: 
A. Represent addition and subtraction with objects, fingers, mental images, drawings (Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
B. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
C. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
D. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
E. Fluently add and subtract within 5.


Your response, in json format:
{
  "explanation": "This task involves representing addition with objects such as number cards and dice.",
  "answer": "A"
}

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, in json format:
{
  "explanation": "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:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Your response, in json format:
{
  "explanation": "An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation. Solving this problem does not require use of the Pythagorean identity.",
  "answer": "B"
}

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Represent a system of linear equations as a single matrix equation in a vector variable.

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.

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.

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).
problem_activity: Activity
Earlier, students saw that adding or subtracting the equations in a system creates a third equation that can help us solve the system. In this activity, students reconnect the idea of the solution to a system to the intersection of the graphs of the equations. They graph each original pair of equations and the equation that results from adding or subtracting them. They then observe that the graph of the third equation intersects the other two graphs at the exact same point—at the intersection of the first two.
At this point, students simply get a graphical confirmation that adding or subtracting equations can help them find the solution to a system. They are not yet expected to be able to articulate why this is the case. That understanding will be developed over a few upcoming lessons.
Launch
Remind students that earlier they added or subtracted pairs of equations to form new equations. Explain that they will now graph each pair of equations in the systems given earlier, as well as the third equation that came from adding or subtracting those equations, and then make some observations about them.
Arrange students into groups of 3 and provide access to graphing technology. Assign one system for each group member to graph. Ask students to discuss their observations after graphing.
If possible, make available graphing technology that allows users to enter linear equations in standard form, such as Desmos (available under Math Tools). Otherwise, give students time to rearrange the equations into a form that can be used with the technology and to check their equivalent equations. If time is limited, provide these equivalent equations:
System A
\(\begin {cases}\begin {align}y &= \text-\frac43x+\frac{10}{3}\\ y&=\hspace{2mm}\frac45x + \frac65 \end{align} \end{cases}\)
System B
\(\begin {cases}\begin {align}y &= \text-2x+4\\ y&=\hspace{3mm}x-11 \end{align} \end{cases}\)
System C
\(\begin {cases}\begin {align}y &= \text-\frac{8}{11}x+\frac{37}{11}\\ y&=\hspace{3.7mm}\text-8x + 7 \end{align} \end{cases}\)
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Look for students who are able to find the sum or difference of the equations with ease. If students struggle, direct students to use parentheses around the entirety of the equation being added or subtracted to enhance clarity and reduce errors.
Supports accessibility for: Memory; Organization
Student Facing
Here are three systems of equations you saw earlier.
System A
\(\begin {cases}\begin {align}4x + 3y &= 10\\ \text-4x + 5y &= \hspace{2mm}6 \end{align} \end{cases}\)
System B
\(\begin {cases} \begin {align}2x + y &= 4\\ x - y &= 11 \end {align} \end {cases} \)
System C
\(\begin {cases} \begin{align} 8x + 11y &= 37\\ 8x + \hspace{4mm} y &= \hspace{2mm} 7 \end{align} \end{cases}\)
For each system:
Use graphing technology to graph the original two equations in the system. Then, identify the coordinates of the solution.
Find the sum or difference of the two original equations that would enable the system to be solved.
Graph the third equation on the same coordinate plane. Make an observation about the graph.
Student Response
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Student Facing
Are you ready for more?
Mai wonders what would happen if we multiply equations. That is, we multiply the expressions on the left side of the two equations and set them equal to the expressions on the right side of the two equations.
In system B write out an equation that you would get if you multiply the two equations in this manner.
Does your original solution still work in this new equation?
Use graphing technology to graph this new equation on the same coordinate plane. Why is this approach not particularly helpful?
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
When solving system B, some students may not notice that the
\(y\)
-variable in one equation has a positive coefficient and the other has a negative coefficient, and consequently decide to subtract the second equation from the first, rather than to add the two equations. They may struggle to figure out why the solution pair they find doesn't match what is on the graph. Suggest that they express the second equation in terms of addition,
\(x+ (\text-y) = 11\)
, and try eliminating one variable again.
Activity Synthesis
Display the graphs that students generated for all to see and ask students to share their observations. Highlight that the graph of the new equation intersects the graphs of the equations in the original system at the same point.
Time permitting, ask students to subtract the equations they previously added (or add the equations they previously subtracted) and then to graph the resulting equation on the same coordinate plane. Ask them to comment on the graphs. Students are likely to see that the graphs of the new equations are no longer horizontal or vertical lines, but they still intersect at the same point as the original graphs.
Invite students to make some conjectures as to why the graph of the new equation intersects the other two graphs at the same point. Without confirming or correcting their conjectures, tell students that they will investigate this question in the coming activities.


correct_option_index: 3