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.


Standard options: 
A. 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.
B. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
C. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
D. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
E. Fluently add and subtract within 5.


Your response:
Thought: This task involves representing addition with objects such as number cards and dice.
Answer: A

Example 2:
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} $

Standard options: 
A. 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.
B. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
C. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.


Your response:
Thought: By arranging the fractions in this problem, students learn to compare fractions by reasoning about their size.
Answer: B

Example 3:
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?


Standard options: 
A. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
B. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
C. Fluently multiply multi-digit whole numbers using the standard algorithm.


Your response:
Thought: 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.
Answer: C

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_task_001477

K

im_task_000157

3

fl_problem_000674

5
options: 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.

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
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
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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
\(\pi\)
.


correct_option_index: 0