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: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
problem_activity: Activity
The purpose of this activity is for students to interact with contexts in which the direction of inequality is the opposite of what they might expect if they try to solve like they would with an equation. For example, in the second problem, the original inequality is
\(9(7-x) \leq 36\)
, but the solution to the inequality is
\(x \geq 3\)
.
Some students might solve the associated equation and then test values
of
\(x\)
to determine the direction of inequality. That method will be introduced in more generality in the next lesson. This activity emphasizes thinking about the context in deciding the direction of inequality.
Launch
Keep students in the same groups. Give 5–10 minutes of quiet work time and partner discussion followed by a whole-class discussion.
Student Facing
Kiran has
$
100 saved in a bank account. (The account doesn’t earn interest.) He asked Clare to help him figure out how much he could take out each month if he needs to have at least
$
25 in the account a year from now.
Clare wrote the inequality
\(\text-12x + 100 \geq 25\)
, where
\(x\)
represents the amount Kiran takes out each month. What does
\(\text-12x\)
represent?
Find some values of
\(x\)
that would work for Kiran.
We could express
all
the values that would work using either
\(x \leq \text{__ or } x \geq \text{__}\)
. Which one should we use?
Write the answer to Kiran’s question using mathematical notation.
A teacher wants to buy 9 boxes of granola bars for a school trip. Each box usually costs
$
7, but many grocery stores are having a sale on granola bars this week. Different stores are selling boxes of granola bars at different discounts.
If
\(x\)
represents the dollar amount of the discount, then the amount the teacher will pay can be expressed as
\(9(7-x)\)
. In this expression, what does the quantity
\(7-x\)
represent?
The teacher has
$
36 to spend on the granola bars. The equation
\(9(7-x)=36\)
represents a situation where she spends all
$
36. Solve this equation.
What does the solution mean in this situation?
The teacher does not have to spend all
$
36. Write an inequality relating 36 and
\(9(7-x)\)
representing this situation.
The solution to this inequality must either look like
\(x \geq 3\ \text{or } x \leq 3\)
. Which do you think it is? Explain your reasoning.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Student Facing
Are you ready for more?
Jada and Diego baked a large batch of cookies.
They selected
\(\frac14\)
of the cookies to give to their teachers.
Next, they threw away one burnt cookie.
They delivered
\(\frac25\)
of the remaining cookies to a local nursing home.
Next, they gave 3 cookies to some neighborhood kids.
They wrapped up
\(\frac23\)
of the remaining cookies to save for their friends.
After all this, they had 15 cookies left. How many cookies did they bake?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The purpose of the discussion is to let students voice their reasoning about the direction of the inequality by reasoning about the context. Ask students to share their reasons for choosing the direction of inequality in their solutions. Some students may notice that the algebra in both problems involves multiplying or dividing by a negative number. Honor this observation, but again, the goal is not to turn this observation into a rule for students to memorize and follow. Interpreting the meaning of the solution in the context should remain at the forefront.
As students model real-world situations, questions about the interpretation of the mathematical solution should continue to come up in the conversation. For instance, the amount of the granola bar discount cannot be
$
3.5923, even though this is a solution to the inequality
\(x \geq 3\)
. The value -10 is a solution to Kiran’s inequality, even though he can’t withdraw -10 dollars. Students can argue that negative values for
\(x\)
simply don’t make sense in this context. Some may argue that we should interpret
\(x=\text-10\)
to mean that Kiran deposits
$
10 every month.
Engagement: Develop Effort and Persistence
. Break the class into small discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Language; Social-emotional skills; Attention
Representing, Writing, Conversing: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. After a student shares their reasons for the direction of the inequality in their solution, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. This will help students understand how context determines the direction of inequality.
Design Principle(s): Support sense-making


correct_option_index: 0