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: 
Understand a fraction as a number on the number line; represent fractions on a number line diagram.
 Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.

Problem/activity:
Task
 Arrange the fractions in order from least to greatest. Explain your answer with a picture.
 $\frac{1}{5}, \frac{1}{7}, \frac{1}{3} $
 $\frac{2}{5}, \frac{2}{7}, \frac{2}{3} $
 $\frac{5}{6}, \frac{3}{6}, \frac{1}{6} $
 $\frac{5}{12}, \frac{8}{12}, \frac{4}{12} $

Answer: 
No. Completing this task does not involve representing fractions on a number line diagram.

Example 3:
Concept/skill: 
Fluently multiply multi-digit whole numbers using the standard algorithm.

Problem/activity:
Problem 1
Solve using the standard algorithm. If you get stuck, use an area model and/or the partial products algorithm to help. Then assess the reasonableness of your answer.
a.
$$5,803\times46$$
b.
$$63\times2,116$$
Problem 2
Dunkin' introduced a croissant doughnut in 2014. In the first three months of its debut, they sold 7,870 dozen of them per day. How many croissant doughnuts were sold in the first three months?


Answer: 
Yes. This set of problems involves multi-digit whole numbers because the numbers in each problem are four and two digit integers, and the mathematical calculations involved in both problems focus on multiplication. The first problem explicitly asks the student to solve its subproblems using the standard algorithm.

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

im_task_000157_neg_0

3

fl_problem_000674_pos_0

5
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: Optional activity
This activity is optional.  This activity is similar to the previous (optional) one from a function point of view, but now students investigate the volume of a cylinder instead of a rectangular prism. Students continue working with functions to investigate what happens to the volume of a cylinder when you halve the height. The exploration and representations resemble what was done in the previous activity, and students continue to identify the effect of the changing dimension on the graph and the equation of this function. As groups work on the task, identify those who make the connection between the graph and equation representations, and encourage groups to look for similarities and differences between what they see in this activity and what they saw in the previous activity. Invite these groups to share during the whole-class discussion.
Digital
Launch
Keep students in the same groups. Tell students that this activity is similar to the previous one, but they will work with a cylinder instead of a rectangular prism. Give students 2–3 minutes of quiet work time followed by time to discuss the last question with their partner. Follow with a whole-class discussion.
Students using the digital activity can graph their equations using the applet.
Student Facing
There are many cylinders with radius 5 units. Let
\(h\)
represent the height and
\(V\)
represent the volume of these cylinders.
Write an equation that represents the relationship between
\(V\)
and
\(h\)
. Use 3.14 as an approximation of
\(\pi\)
.
Graph this equation.
What happens to the volume if you halve the height,
\(h\)
? Where can you see this in the graph? How can you see it algebraically?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Keep students in the same groups. Tell students that this activity is similar to the previous one, but they will work with a cylinder instead of a rectangular prism. Give students 2–3 minutes of quiet work time followed by time to discuss the last question with their partner. Follow with a whole-class discussion.
Students using the digital activity can graph their equations using the applet.
Student Facing
There are many cylinders with radius 5 units. Let
\(h\)
represent the height and
\(V\)
represent the volume of these cylinders.
Write an equation that represents the relationship between
\(V\)
and
\(h\)
. Use 3.14 as an approximation of
\(\pi\)
.
Graph this equation and label the axes.
[IMG]
What happens to the volume if you halve the height,
\(h\)
? Where can you see this in the graph? How can you see it algebraically?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Suppose we have a rectangular prism with dimensions 2 units by 3 units by 6 units, and we would like to make a rectangular prism of volume 216 cubic units by stretching
one
of the three dimensions.
What are the three ways of doing this? Of these, which gives the prism with the smallest surface area?
Repeat this process for a starting rectangular prism with dimensions 2 units by 6 units by 6 units.
Can you give some general tips to someone who wants to make a box with a certain volume, but wants to save cost on material by having as small a surface area as possible?
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
Ask previously identified groups to share their graphs and equations.
If students completed the optional activity, display student graphs from both activities and ask:
“Compare the graph in this activity to the graph in the last activity. How are they alike? How are they different?”
“Compare the equation in this activity to the equation in the last activity. How are they alike? How are they different?”
“How can you tell that this is a linear relationship?”
For the last question, make sure students understand that “looks like a line” is insufficient evidence for saying a relationship is linear since the scale of the axes can make non-linear graphs look linear. It is important students connect that the equation is linear (of the form y=mx+b) or relate back to the relationship between the height and volume: that the volume is
\(25\pi\)
, or about 78.5, times the height.
If students did not complete the optional activity, ask:
“Which of your variables is the independent? The dependent?” (The height,
\(h\)
, is the independent variable, and volume, V, is dependent variable.)
“Which variable is a function of which?” (Volume is a function of height.)
“How can you tell that this is a linear relationship?”
See notes above on the last question in this list. If it has not been brought up in students’ explanations, ask what the volume equation looks like when we double the height
\(h\)
. Display volume equation
\(V=78.5(2h)\)
for all to see. Ask how we can write this equation to show that the volume doubled when
\(h\)
doubled (using algebra, we can rewrite
\(V=78.5(2h)\)
as
\(V=2(78.5h)\)
. Since the volume for
\(h\)
was
\(78.5h\)
, this shows that the volume for
\(2h\)
is twice the volume for
\(h\)
).
Representing, Listening, Speaking: MLR7 Compare and Connect.
Invite students to create a visual representation that shows what is the same and what is different between this activity and the previous one. If students did not complete the optional activity, students can create multiple representations that illustrate why volume is a function of height. Give students time to do a gallery walk of the displays. Look for opportunities to highlight approaches that compare the multiple representations of the function of volume. Lead a whole-class discussion that connects features on the displays, such as comparing the equations and graphs. This will help students communicate about the multiple representations of volume functions while using mathematical language.
Design Principle(s): Optimize output; Maximize meta-awareness


aligns: yes