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.


Thought: 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.


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:
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.


Thought: 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: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
problem_activity: Optional activity
This activity is optional because it goes beyond the depth of understanding required by the standards.
The purpose of this activity is to connect graphs of quadratic functions with solutions to related quadratic equations to tell how many solutions an equation has, and whether those solutions are real or non-real. Students have already solved two of the equations in the previous activity, and the equation they haven’t solved involves the perfect square that connects all three of the equations. Graphically,
\(y=f(x)\)
intersects the
\(x\)
-axis twice, which corresponds to the two real solutions to
\(x^2 - 8x + 13 = 0\)
. The graph
\(y=g(x)\)
intersects the
\(x\)
-axis once, which corresponds to the single real solution to
\(x^2 - 8x + 16 = 0\)
. However, the graph
\(y=h(x)\)
doesn’t intersect the
\(x\)
-axis, which reflects the fact that the complex solutions to
\(x^2 - 8x + 19 = 0\)
can’t be described with a real number line.
Launch
Provide access to graphing technology.
Student Facing
How many real solutions does each equation have? How many non-real solutions?
\(x^2 - 8x + 13 = 0\)
\(x^2 - 8x + 16 = 0\)
\(x^2 - 8x + 19 = 0\)
How do the graphs of these functions help us answer the previous question?
\(f(x) = x^2 - 8x + 13\)
\(g(x) = x^2 - 8x + 16\)
\(h(x) = x^2 - 8x + 19\)
Student Response
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for free access to Student Response.
Anticipated Misconceptions
Some students may attempt to plot the complex solutions in the coordinate plane as if it were the complex plane. Remind these students that points in the coordinate plane are pairs of real numbers
\((x,y)\)
, whereas points in the complex plane are single complex numbers. In other words, the point
\((2,4)\)
in the coordinate plane is a different kind of object than the point
\(2+4i\)
in the complex plane.
Activity Synthesis
Select students to share their solutions and discuss the connections between the graphs and the equations. Discuss the idea that if a graph of a quadratic function doesn’t cross a given horizontal line (in this case,
\(x=0\)
) this doesn’t indicate that the related equation has no solutions, but rather that it has no
real
solutions that can be graphed with real number lines.
Speaking: MLR8 Discussion Supports.
To help students produce statements about the connection between graphs of quadratic functions and solutions to related quadratic equations, provide sentence frames such as: “If _____ then _____ because…” and  “On the graph, I see _____, so I know….” Show concepts multi-modally by displaying the graphs of
\(f(x)\)
,
\(g(x)\)
, and
\(h(x)\)
.
Design Principle(s): Support sense-making


correct_option_index: 1