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: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
problem_activity: Activity
The mathematical purpose for this activity is to allow students to practice solving for a variable in an equation with two variables. In the associated Algebra lesson, expressing one variable in terms of another is necessary before substituting an expression for a variable in another equation.
Launch
Demonstrate how to solve for a variable by showing students the equation
\(a + 2b = 5\)
. Solving for
\(a\)
results in
\(a = 5 - 2b\)
or
\(a = \text{-}2b + 5\)
. Solving for
\(b\)
results in
\(b = \frac{5-a}{2}\)
or
\(b = \frac{5}{2} - \frac{a}{2}\)
or another equivalent expression.
Student Facing
Solve for the indicated variable.
Solve for
\(k\)
.
\(2t+k=6\)
Solve for
\(n\)
.
\(10n=2p\)
Solve for
\(c\)
.
\(12-6d=3c\)
Solve for
\(g\)
.
\(h=8g+4\)
Solve for
\(x\)
.
\(4x+3y=12\)
Solve for
\(y\)
.
\(4x+3y=12\)
Student Response
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Activity Synthesis
The purpose of the discussion is to share strategies for solving for one variable in an equation including more than one variable. Select students to share their answers and strategies. As students share, consider asking another student why each step makes sense to do.
Some questions for discussion:
"How do you know when you are done solving for a variable?" (When the variable I want to solve for is the only thing on one side of the equation.)
"How is solving for
\(x\)
in an equation like
\(4x+3y=12\)
similar or different from solving for
\(x\)
in an equation like
\(4x + 6 = 12\)
?" (The steps are very similar, but when there is only one variable, we can know the value of
\(x\)
that makes the equation true.)
"If you know the value for
\(y\)
is 2 and need to know the value for
\(x\)
, would you rather begin with the equation
\(4x + 3y = 12\)
or the equivalent equation
\(x = 3 - \frac{3}{4}y\)
? Explain your reasoning." (I would rather use the second equation since I only need to substitute in the value for
\(y\)
and evaluate the expression rather than solve an equation.)


correct_option_index: 2