few_shots: 
Example:
Problem/activity:
Task
Cruz and Erica were both getting ready for soccer.
Cruz ran 1 lap around the school.
Erica ran 3 laps around the playground.
Erica said,
I ran more laps, so I ran farther.
Cruz said,
4 laps around the school is 1 mile, but it takes 12 laps around the playground to go 1 mile. My laps are much longer, so I ran farther.
Who is right? Draw a picture to help you explain your answer.


Concept/skill: 
Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Answer: 
Yes. This task asks students to compare two fractions that are equal. The students need to explain why the two fractions are equal by drawing a picture, which involves using visual fraction models.

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_000212_pos_0

4
standard_description: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
problem_activity: Activity
In this task, students use the constant of proportionality they estimated in the previous task to calculate circumferences of circles given the diameter and vice versa. The purpose is to reinforce the proportional relationship between circumference
\(c\)
and diameter
\(d\)
and use the formula
\(c=kd\)
, where
\(k\)
is about 3 or 3.1 (the value agreed upon in the previous task).
If there is not sufficient time to allow students to work through all of the calculations, consider dividing up the work, assigning one circle to each student or group of students. It is important to save enough time for the discussion about the meaning and value of
\(\pi\)
at the end of this activity.
Before students begin the task, a digital applet found at
http://ggbm.at/H8UuD96V
can be used to reinforce the constant of proportionality.
Launch
Ask students to label these measurements given in the table on the picture of the circles. Instruct students to use the constant of proportionality the class estimated in the previous activity. Give students quiet work time followed by whole-class discussion.
Student Facing
Here are five circles. One measurement for each circle is given in the table.
[IMG]
Use the constant of proportionality estimated in the previous activity to complete the table.
|          | diameter (cm) | circumference (cm) |
|----------|---------------|--------------------|
| circle A |               |                    |
| circle B |               |                    |
| circle C |               |                    |
| circle D |               |                    |
| circle E |               |                    |
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?
The circumference of Earth is approximately 40,000 km. If you made a circle of wire around the globe, that is only 10 meters (0.01 km) longer than the circumference of the globe, could a flea, a mouse, or even a person creep under it?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students may multiply the circumference by the constant of proportionality instead of dividing by it. Prompt them to consider whether the diameter can be longer than the circumference of a circle.
Some students may struggle to divide by 3.1 if that is the constant of proportionality decided on in the previous lesson. Ask these students if they could use an easier number as their constant, and allow them to divide by 3 instead. Then ask them how their answer would have changed if they divided by 3.1.
Activity Synthesis
Display the table from the task statement for all to see. Ask students to share their answers and write these in the table, resolving any discrepancies.
Ask students if they noticed any connections between the tables and the graphs from the previous activity. Students will have used the equation
\(C=kd\)
, possibly without realizing it, to fill in the missing information so it is important for them to make this connection. To reinforce this connection, add the points from the table they just completed to the graph from the previous activity. Point out that the constant of proportionality appears in the table as the unit rate in the row with diameter 1 cm.
Ask if students have heard of the number
pi
or seen the symbol
\(\pi\)
. Define
\(\pi\)
as the exact value of the constant of proportionality for this relationship. Explain that the exact value of
\(\pi\)
is a decimal with infinitely many digits and no repeating pattern, so an approximation is often used. Frequently used approximations for
\(\pi\)
include
\(\frac{22} 7\)
, 3.14, and 3.14159, but none of these are exactly equal to
\(\pi\)
.


aligns: yes