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.


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.

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

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


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

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


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

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, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: im_task_001477_pos_0

K

fl_problem_001097_pos_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
problem_activity: Narrative
The purpose of this activity is for students to continue to solve problems about dividing a unit fraction by a whole number. The unit fraction in both problems is
\(\frac{1}{2}\)
so that students will consider the relationship between the number of people sharing the macaroni and cheese and the size of the serving each person gets. When students connect the quantities in the story problem to an equation and a diagram representing the story, they reason abstractly and quantitatively (MP2).
Launch
Groups of 2
Activity
1–2 minutes: independent think time
5–8 minutes: partner work time
Monitor for students who :
draw diagrams like the ones in the student responses
describe the amount each person gets as a fraction of the whole pan
describe a relationship between the number of people sharing and the size of the serving each person gets
Student Facing
4 people equally share
\(\frac {1}{2}\)
a pan of macaroni and cheese.
Draw a diagram to represent the situation.
Explain how your diagram represents
\(\frac {1}{2} \div 4\)
.
How much of the whole pan of macaroni and cheese did each person get? Be prepared to explain your reasoning.
5 people equally share
\(\frac {1}{2}\)
a pan of macaroni and cheese.
Draw a diagram to represent the situation.
Explain how your diagram represents
\(\frac {1}{2} \div 5\)
.
How much of the whole pan of macaroni and cheese did each person get? Be prepared to explain your reasoning.
How are the problems the same? How are they different?
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Advancing Student Thinking
If students do not identify or explain how much macaroni and cheese each person gets, consider asking, “How much of the whole pan of macaroni and cheese did each person get?”
Activity Synthesis
“How are the situations the same? How are they different?” (Both situations are about
\(\frac {1}{2}\)
a pan of macaroni and cheese, but this one has more people sharing it, so each person gets less macaroni and cheese. When 4 people share, each person gets
\(\frac {1}{8}\)
of the whole pan. When 5 people share, each person gets
\(\frac {1}{10}\)
of the whole pan.)


aligns: no