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: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

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_neg_0

HS_0
standard_description: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
problem_activity: Activity
The motion of a falling object is commonly modeled with a quadratic function. This activity prompts students to build a very simple quadratic model using given time-distance data of a free-falling rock. By reasoning repeatedly about the values in the data, students notice regularity in the relationship between time and the vertical distance the object travels, which they then generalize as an expression with a squared variable (MP8). The work here prepares students to make sense of more complex quadratic functions later (that is, to model the motion of an object that is launched up and then returns to the ground).
Launch
Display the image of the falling object for all to see. Students will recognize the numbers from the warm-up. Invite students to make some other observations about the information. Ask questions such as:
“What do you think the numbers tell us?”
“Does the object fall the same distance every successive second? How do you know?”
Arrange students in groups of 2. Tell students to think quietly about the first question and share their thinking with a partner. Afterward, consider pausing for a brief discussion before proceeding to the second question.
Speaking, Reading: MLR5 Co-Craft Questions.
Begin the launch by displaying only the context and the diagram of the building. Give students 1–2 minutes to write their own mathematical questions about the situation before inviting 3–4 students to share their questions with the whole-class. Listen for and amplify any questions involving the relationship between elapsed time and the distance that a falling object travels.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Student Facing
A rock is dropped from the top floor of a 500-foot tall building. A camera captures the distance the rock traveled, in feet, after each second.
[IMG]
How far will the rock have fallen after 6 seconds? Show your reasoning.
Jada noticed that the distances fallen are all multiples of 16.
She wrote down:
\(\displaystyle \begin {align}16 &= 16 \boldcdot 1\\64 &= 16 \boldcdot 4\\144 &= 16 \boldcdot 9\\256 &= 16 \boldcdot 16\\400 &=16 \boldcdot 25 \end {align}\)
Then, she noticed that 1, 4, 9, 16, and 25 are
\(1^2, 2^2, 3^2, 4^2\)
and
\(5^2\)
.
Use Jada’s observations to predict the distance fallen after 7 seconds. (Assume the building is tall enough that an object dropped from the top of it will continue falling for at least 7 seconds.) Show your reasoning.
Write an equation for the function, with
\(d\)
representing the distance dropped after
\(t\)
seconds.
Student Response
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Anticipated Misconceptions
Some students may question why the distances are positive when the rock is falling. In earlier grades, negative numbers represented on a vertical number line may have been associated with an arrow pointing down. Emphasize that the values shown in the picture measure how far the rock fell and not the direction it is falling.
Activity Synthesis
Discuss the equation students wrote for the last question. If not already mentioned by students, point out that the
\(t^2\)
suggests a quadratic relationship between elapsed time and the distance that a falling object travels. Ask students:
“How do you know that the equation
\(d=16t^2\)
represents a function?” (For every input of time, there is a particular output.)
“Suppose we want to know if the rock will travel 600 feet before 6 seconds have elapsed. How can we find out?” (Find the value of
\(d\)
when
\(t\)
is 6, which is
\(16 \boldcdot 6^2\)
or 576 feet.)
Explain to students that we only have a few data points to go by in this case, and the quadratic expression
\(16t^2\)
is a simplified model, but quadratic functions are generally used to model the movement of falling objects. We will see this expression appearing in some other contexts where gravity affects the quantities being studied.


aligns: no