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 rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”

Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
problem_activity: Activity
This activity is an opportunity for students to practice rewriting expressions using the distributive property. It is a step up from the same type of work in grade 6 because arithmetic with signed numbers is required.
The row with
\(6a-2b\)
is designed to allow students to figure out how to factor by reasoning based on structure they already understand, instead of learning how to factor based on a procedure that a teacher demonstrates first. The rows with
\(k(4 - 17)\)
and
\(10a-13a\)
are designed to prepare students for combining like terms in the next lesson.
Launch
Draw students’ attention to the organizers that appear above the table, and tell them that these correspond to the first three rows in the table. Let them know that they are encouraged to draw more organizers like this for other rows as needed.
Arrange students in groups of 2. Instruct them to take turns writing an equivalent expression for each row. One partner writes the equivalent expression and explains their reasoning while the other listens. If the partner disagrees, they work to resolve the discrepancy before moving to the next row.
Action and Expression: Internalize Executive Functions.
Provide students with blank diagrams to organize their work for the factored and expanded expressions. Include example diagrams labeled “expanded” and “factored.”
Supports accessibility for: Language; Organization
Student Facing
In each row, write the equivalent expression. If you get stuck, use a diagram to organize your work. The first row is provided as an example. Diagrams are provided for the first three rows.
[IMG]
|       factored        |      expanded      |
|-----------------------|--------------------|
| \(\text- 3(5 - 2y)\)  | \(\text- 15 + 6y\) |
|      \(5(a-6)\)       |                    |
|                       |     \(6a-2b\)      |
|  \(\text- 4(2w-5z)\)  |                    |
|  \(\text- (2x-3y)\)   |                    |
|                       |  \(20x-10y+15z\)   |
|      \(k(4-17)\)      |                    |
|                       |    \(10a-13a\)     |
|  \(\text- 2x(3y-z)\)  |                    |
|                       |   \(ab-bc-3bd\)    |
| \(\text- x(3y-z+4w)\) |                    |
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
Expand to create an equivalent expression that uses the fewest number of terms:
\(\left(\left(\left(\left(x\strut+1\right)\frac12\right)+1\right)\frac12\right)+1\)
. If we wrote a new expression following the same pattern so that there were 20 sets of parentheses, how could it be expanded into an equivalent expression that uses the fewest number of terms?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students are unsure how to proceed, remind them of tools and understandings they have seen recently that would be helpful. For example, “Draw an organizer and think about how the organizer represents terms in the expression.” Also, “Rewrite subtraction as adding the opposite.”
Activity Synthesis
Much of the discussion will take place in small groups. Display the correct equivalent expressions and work to resolve any discrepancies. Expanding the term
\(\text-(2x-3y)\)
may require particular care. One way to interpret it is to rewrite as
\(\text-1 \boldcdot (2x-3y).\)
If any confusion about handling subtraction arises, encourage students to employ the strategy of rewriting subtraction as adding the opposite.
To wrap up the activity, ask:
“Which rows did you and your partner disagree about? How did you resolve the disagreement?”
“Which rows are you the most unsure about?”
“Describe a process or procedure for taking a factored expression and writing its corresponding expanded expression.”
“Describe a process or procedure for taking an expanded expression and writing its corresponding factored expression.”
Writing: MLR3 Clarify, Critique, Correct.
Present an incorrect response for one of the expanded expressions in the table. For example, “An equivalent expression for
\(6a − 2b\)
is
\(4(a−b)\)
because
\(6−2\)
is 4 and
\((a−b)\)
is left on its own.” Prompt students to clarify any of the language and reasoning in the incorrect response and then to identify the error(s). Invite students to work with a partner to write a correct response. This helps students evaluate and improve on the written mathematical arguments of others.
Design Principle(s): Maximize meta-awareness


correct_option_index: 1