few_shots: 
Example 1:
Concept/skill: 
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.

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.


Answer: 
Yes. This task involves representing addition with objects such as number cards and dice.

Example 2:
Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

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
$$-\frac{1}{4}$$


Answer: 
Yes. 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).

Example 3:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Answer: 
Yes. 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.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: im_task_001477_pos_0

K

fl_problem_001097_pos_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))^(3) = 5(^(1/3))^(3) to hold, so (5^(1/3))^(3) must equal 5.
problem_activity: Activity
The purpose of this activity is for students to extend what they learned about square roots and
\(\frac12\)
exponents to cube roots and
\(\frac13\)
exponents.
Launch
Tell students that they will use the same reasoning as in the previous activity to make sense of numbers to the
\(\frac12\)
power or
\(\frac13\)
power. Give a few minutes of quiet work time followed by a brief, whole-class discussion.
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. Check that students are finding the exact values and remind them of how they could use the exponent rules to check their answers if needed. Invite students to show their thinking by rewriting the fractional exponents.
Supports accessibility for: Memory; Organization
Student Facing
Use the exponent rules and your understanding of roots to find the exact value of:
\(25^{\frac12}\)
\(15^{\frac12}\)
\(8^{\frac13}\)
\(2^{\frac13}\)
Student Response
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Activity Synthesis
The key takeaway is that extending exponent rules to these fractions means that
\(b^{\frac12}=\sqrt{b}\)
and
\(b^{\frac13}=\sqrt[3]{b}\)
. Select students to share their responses for
\(15^{\frac12}\)
and
\(2^{\frac13}\)
.
Conclude the discussion by reminding students that these expressions which involve the
\(\sqrt{}\)
symbol are often called
radical
expressions because the technical name for the symbol is the
radix
(which is also where the word root comes from). Expressions like
\(\sqrt{15}\)
and
\(\sqrt[3]{2}\)
are examples of radical expressions.
Speaking: MLR8 Discussion Supports.
As students share their responses for
\(15^{\frac{1}{2}}\)
and
\(2^{\frac{1}{3}}\)
, press for details by asking how they know that
\(15^{\frac{1}{2}}\)
is equal to
\(\sqrt{15}\)
and
\(2^{\frac{1}{3}}\)
is equal to
\(\sqrt[3]{2}\)
. Also ask students which properties of exponents they used to show that the exponential and radical expressions are equivalent. Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. This will help students justify why
\(b^{\frac{1}{2}} = \sqrt{b}\)
and
\(b^{\frac{1}{3}} = \sqrt[3]{b}\)
.
Design Principle(s): Support sense-making; Optimize output (for justification)


aligns: yes