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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options: Understand and apply the Pythagorean Theorem.

Define trigonometric ratios and solve problems involving right triangles.

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

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

Analyze, compare, create, and compose shapes.

Understand congruence in terms of rigid motions.

Reason with shapes and their attributes.

Prove geometric theorems.

Apply trigonometry to general triangles.

Apply geometric concepts in modeling situations.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

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

Prove theorems involving similarity.

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

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

Find arc lengths and areas of sectors of circles.

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

Make geometric constructions.

Use coordinates to prove simple geometric theorems algebraically.

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

Understand and apply theorems about circles.

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

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

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

Experiment with transformations in the plane.

Explain volume formulas and use them to solve problems.

Understand similarity in terms of similarity transformations.
problem_activity: Optional activity
In the previous lesson, students built trundle wheels using three different wheel sizes, and they tested the functionality of their wheels in the classroom. In this activity, they use their trundle wheels to measure a longer path of about 50–100 meters. This is the same path that they measured during a previous lesson with a different method. If several groups are sharing a trundle wheel, then they each measure the given path once and share their data with each other.
After students measure, they spend the remainder of the lesson on computations and sharing results. Students get a chance to connect the mathematical formulas and computations with the aspects of the hands-on experience they had in making and using the wheels. They attend to precision when they are deciding on how to report their results and when they are comparing results with other groups (MP6).
Digital
Launch
Keep students in the same groups from the previous lesson. Remind students of the path they should measure. Instruct them to come back to the classroom to finish their calculations as soon as they have recorded their measurements.
Give students 10–20 minutes to take turns measuring and 10 minutes of group work time to finish their calculations, followed by whole-class discussion.
There is also a digital version of a trundle wheel available. Students can be assigned a path, numbered from 1 to 5, and collect data on different-sized trundle wheels. The applet is programmed to stop automatically at one of the five distances. The students must keep track of the number of rotations the wheel makes before it stops.
Action and Expression: Provide Access for Physical Action.
Support access to tools and assistive technologies. Monifor for students who may need an additional demonstration, or assistance using their trundle wheel to measure a longer path. Consider modeling a systematic way to record the number of clicks to calculate an example distance as students complete this task.
Supports accessibility for: Visual-spatial processing; Fine-motor skills
Student Facing
Earlier you made trundle wheels so that you can measure long distances.
If you do not have a trundle wheel, you can use an applet to simulate measuring. Your teacher will tell you a path whose distance you will measure.
Measure the path with your trundle wheel three times and calculate the distance. Record your results in the table.
|         | diameter | number of rotations | computation | distance |
|---------|----------|---------------------|-------------|----------|
| trial 1 |          |                     |             |          |
| trial 2 |          |                     |             |          |
| trial 3 |          |                     |             |          |
Decide what distance you will report to the class. Be prepared to explain your reasoning.
Compare this distance with the distance you measured the other day for this same path.
Compare your results with the results of two other groups. Express the differences between the measurements in terms of percentages.
To use the applet:
Choose a diameter for the wheel, and enter a number 1 -- 5 for the path you will measure.
Watch carefully to keep track of the number of rotations that the wheel makes before it stops.
Try three different diameters for the same path, and compare your results.
Answer the questions.
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 from the previous lesson. Remind students of the path they should measure. Instruct them to come back to the classroom to finish their calculations as soon as they have recorded their measurements.
Give students 10–20 minutes to take turns measuring and 10 minutes of group work time to finish their calculations, followed by whole-class discussion.
There is also a digital version of a trundle wheel available. Students can be assigned a path, numbered from 1 to 5, and collect data on different-sized trundle wheels. The applet is programmed to stop automatically at one of the five distances. The students must keep track of the number of rotations the wheel makes before it stops.
Action and Expression: Provide Access for Physical Action.
Support access to tools and assistive technologies. Monitor for students who may need an additional demonstration, or assistance using their trundle wheel to measure a longer path. Consider modeling a systematic way to record the number of clicks to calculate an example distance as students complete this task.
Supports accessibility for: Visual-spatial processing; Fine-motor skills
Student Facing
Earlier you made trundle wheels so that you can measure long distances. Your teacher will show you a path to measure.
Measure the path with your trundle wheel three times and calculate the distance. Record your results in the table.
| trial number | number of clicks | computation | distance |
|--------------|------------------|-------------|----------|
|      1       |                  |             |          |
|      2       |                  |             |          |
|      3       |                  |             |          |
Decide what distance you will report to the class. Be prepared to explain your reasoning.
Compare this distance with the distance you measured the other day for this same path.
Compare your results with the results of two other groups. Express the differences between the measurements in terms of percentages.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may lose count when counting the number of rotations along the path. Encourage them find a systematic way to record the number of clicks while their group member is walking along the path.
Activity Synthesis
Ask each group to report their measurement for the length of the path and record their answers for all to see. Guide students to compare these answers by asking questions like these:
“Do all of these answers seem reasonable? Do any of these answers seem unreasonable? Why?”
“Why are these answers not all exactly the same? What are some sources of error?” (Not going in a straight line, the wheel wobbles, the ground is uneven, only counting number of clicks but not parts of rotations etc.)
“What units did you use? What units would be most convenient for designing the course of a 5K walk-a-thon?” (Metric, since we are designing a 5 km course.)
“What degree of precision is appropriate to report?" (To the closest 1 meter at most. Reporting cm or mm on such long distances with a tool like a trundle wheel would be implying a degree of precision that would not be appropriate.)
If time permits, consider asking “If you could choose your own diameter for a trundle wheel, what would it be?” (A diameter that creates a circumference of 1 m would be convenient, i.e. about 32 cm.)
Collect and store students’ trundle wheels so they will have access to them again in the next lesson.
Speaking: MLR8 Discussion Supports.
To support students’ discussion of their measurements comparisons, invite them to use a sentence frame such as: “Our measurement is _____% (higher/lower) than group _____’s measurement because . . . ." Call on another student or group to restate what they heard using mathematical language. Encourage students to press each other for clear explanations and precise use of language.
Design Principle(s): Optimize output (for comparison)


correct_option_index: 3