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 to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

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

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fl_problem_001097_neg_0

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fl_problem_001894_pos_0

HS_0
standard_description: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
problem_activity: Warm-up
The goal of this warm-up is to review the meaning of a function presented graphically. While students do not need to use function notation here, interpreting the graph in terms of the context will prepare them for their work with functions in the rest of the unit.
Launch
Tell students to close their books or devices. Display the graph for all to see. Ask students to observe the graph and be prepared to share one thing they notice and one thing they wonder. Select students to briefly share one thing they noticed or wondered to help ensure all students understand the information conveyed in the graph. Ask students to open their books or devices and answer the questions about the graph. Follow with a whole-class discussion.
Student Facing
Here is a graph of the accumulated rainfall in Las Vegas, Nevada, in the first 60 days of 2017.
[IMG]
Use the graph to support your answers to the following questions.
Is the accumulated amount of rainfall a function of time?
Is time a function of accumulated rainfall?
Student Response
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Anticipated Misconceptions
If students struggle to see from the graph how the accumulated rainfall is a function of time but time is not a function of accumulated rainfall, consider displaying the data in a table. Shown here is the data for the first 20 days of 2017. Help students see that for every value of
\(t\)
, the time in days, there is one value of
\(r\)
, the accumulated rainfall in inches, but this is not true the other way around.
{"table": [{"_attributes": {"border": "1", "class": ["c-table-compressed"]}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "166"}, "_value": "\\(t\\)"}], "_value": "(days)"}], "td": [{"_value": "1"}, {"_value": "2"}, {"_value": "3"}, {"_value": "4"}, {"_value": "5"}, {"_value": "6"}, {"_value": "7"}, {"_value": "8"}, {"_value": "9"}, {"_value": "10"}, {"_value": "11"}, {"_value": "12"}, {"_value": "13"}, {"_value": "14"}, {"_value": "15"}, {"_value": "16"}, {"_value": "17"}, {"_value": "18"}, {"_value": "19"}, {"_value": "20"}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "832"}, "_value": "\\(r\\)"}], "_value": "(inches)"}], "td": [{"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0.03"}, {"_value": "0.1"}, {"_value": "0.1"}, {"_value": "0.1"}, {"_value": "0.1"}, {"_value": "0.1"}, {"_value": "0.1"}, {"_value": "0.11"}, {"_value": "0.38"}]}]}]}]}
Activity Synthesis
Make sure students understand why the accumulated rain is a function of time but not the other way around.
Then, recall the notation for writing, using function notation, accumulated rain fall as a function of time. If
\(r\)
represents the amount of rainfall in inches and
\(t\)
is time in days, then
\(r(t)\)
is the amount of rain that has fallen in the first
\(t\)
days of 2017. For example,
\(r(2) = 0\)
tells us that the accumulated rainfall after in the first two days of the year was 0 inches.
\(r(48)=1\)
means that there was 1 inch of accumulated rain in the first 48 days of the year. Ask students to write and explain the meaning of a few other statements using function notation.


aligns: yes