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.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Your response:
Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

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} $

Mathematical concept/skill options: 
A. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
B. Extend understanding of fraction equivalence and ordering.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
C. Use equivalent fractions as a strategy to add and subtract fractions.
D. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
E. Understand decimal notation for fractions, and compare decimal fractions.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
F. Develop Understanding of fractions as numbers.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.


Your response:
Thought: This problem involves understanding fractions as numbers by comparing their quantities.
Answer: F

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?


Mathematical concept/skill options: 
A. Use place value understanding and properties of operations to perform multi-digit arithmetic. (A range of algorithms may be used.)
B. Understand place value.
C. Understand the place value system.
D. Extend the counting sequence.
E. Work with numbers 11–19 to gain foundations for place value.
F. Use place value understanding and properties of operations to perform multi-digit arithmetic.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
G. Perform operations with multi-digit whole numbers and with decimals to hundredths.
H. Generalize place value understanding for multi-digit whole numbers.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
I. Use place value understanding and properties of operations to add and subtract.


Your response:
Thought: This set of problems asks students to perform multiplication, which is an operation, on multi-digit whole numbers.
Answer: G

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
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K

im_task_000157

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fl_problem_000674

5
options: Draw, construct, and describe geometrical figures and describe the relationships between them.

Understand and apply theorems about circles.

Make geometric constructions.

Explain volume formulas and use them to solve problems.

Apply geometric concepts in modeling situations.

Find arc lengths and areas of sectors of circles.

Reason with shapes and their attributes.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Experiment with transformations in the plane.

Understand and apply the Pythagorean Theorem.

Prove theorems involving similarity.

Analyze, compare, create, and compose shapes.

Solve real-world and mathematical problems involving area, surface area, and volume.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Use coordinates to prove simple geometric theorems algebraically.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Apply trigonometry to general triangles.

Define trigonometric ratios and solve problems involving right triangles.

Translate between the geometric description and the equation for a conic section.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Visualize relationships between two-dimensional and three-dimensional objects.

Classify two-dimensional figures into categories based on their properties.

Prove geometric theorems.

Understand similarity in terms of similarity transformations.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Understand congruence in terms of rigid motions.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
problem_activity: Optional activity
In the previous activity, students relate the distance a wheel travels to the number of rotations the wheel has made. This activity introduces a new quantity, the
speed
the wheel travels. As long as the rate the wheel rotates does not change, there is a nice relationship between the distance the wheel travels,
\(d\)
, and the amount of time,
\(t\)
. With appropriate units,
\(d = rt\)
: here
\(r\)
is the speed the wheel travels, which can be calculated in terms of the rate at which the wheel spins. The goal of this activity is to develop and explore this relationship.
As they work on this activity, students will observe repeatedly that the distance the wheel travels is determined by the amount of time elapsed and the rate at which the wheel spins.
Watch for students who use the calculations that they have made in earlier problems as a scaffold for answering later questions. For example, when the car wheels rotate once per second, the car travels about 3.7 mph. When the wheels rotate 5 times per second, that means that the car will travel 5 times as fast or about 18.5 miles per hour. Ask these students to share their work during the discussion.
Launch
Remind students to be careful with units as they work through the problems. Also remind them that there are 5,280 feet in a mile and 12 inches in a foot.
Student Facing
The circumference of a car wheel is about 65 inches.
If the car wheel rotates once per second, how far does the car travel in one minute?
If the car wheel rotates once per second, about how many miles does the car travel in one hour?
If the car wheel rotates 5 times per second, about how many miles does the car travel in one hour?
If the car is traveling 65 miles per hour, about how many times per second does the wheel rotate?
Student Response
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Anticipated Misconceptions
Some students may do the calculations in feet but not know how to convert their answers to miles. Remind them that there are 5,280 feet in 1 mile.
Activity Synthesis
Have students share their solutions to the first question and emphasize the different ratios and rates that come up when solving this problem. There are
65 inches of distance traveled
per
rotation of the wheels
1 rotation of the wheel
per
second
60 seconds
per
minute
Finally, there are 12 inches
per
foot if students convert the answer to feet. While this is not essential, it is useful in the next problem as feet are a good unit for expressing both inches and miles.
For the last problem, different solutions are possible including:
In the second question, students have found that the car travels 19,500 feet in an hour if the wheels rotate once per second. This is about 3.7 miles. The number of rotations per second is proportional to the speed. So one way to find how many times the wheels rotate at 65 miles per hour would be to calculate
\(65 \div 3.7\)
.
Students could similarly use their answer to the third question, checking what multiple of 18.5 miles (the distance traveled in an hour at 5 rotations per second) gives 65.
A calculation can be done from scratch, checking how many inches are in 65 miles, how many seconds are in an hour, and then finding how many rotations of the wheel per second result in traveling 65 miles.
The first of these approaches should be stressed in the discussion as it uses the previous calculations students have made in an efficient way.
Conversing: MLR8 Discussion Supports.
To help students to compare and justify their methods for answering the problem about the car traveling 65 mph, ask them to compare their responses with a partner. Listen for and amplify any language that describes different methods for calculations. Invite students to zero in on “why” there are more differences here. Provide students with question stems to help them compare and contrast, such as: “Do you have the same answer? If not, then why?” (Students may have estimated or converted differently or incorrectly.), “Did you use the same conversions for units?”, and “Why do your answers differ?”
Design Principle(s): Maximize meta-awareness; Cultivate conversation


correct_option_index: 24