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.
shot_info: im_task_001477

K

im_task_000157

3

fl_problem_000674

5
options: Make geometric constructions.

Prove geometric theorems.

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

Understand similarity in terms of similarity transformations.

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

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

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

Prove theorems involving similarity.

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

Apply geometric concepts in modeling situations.

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

Understand and apply theorems about circles.

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

Find arc lengths and areas of sectors of circles.

Define trigonometric ratios and solve problems involving right triangles.

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

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

Reason with shapes and their attributes.

Analyze, compare, create, and compose shapes.

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

Experiment with transformations in the plane.

Understand congruence in terms of rigid motions.

Understand and apply the Pythagorean Theorem.

Use coordinates to prove simple geometric theorems algebraically.

Apply trigonometry to general triangles.

Explain volume formulas and use them to solve problems.

Classify two-dimensional figures into categories based on their properties.
problem_activity: Optional activity
In this activity, students transfer what they learned with the pattern blocks to calculate the area of other scaled shapes (MP8). In groups of 2, students draw scaled copies of either a parallelogram or a triangle and calculate the areas. Then, each group compares their results with those of a group that worked on the other shape. They find that the scaled areas of two shapes are the same (even though the starting shapes are different and have different measurements) and attribute this to the fact that the two shapes had the same original area and were scaled using the same scale factors.
[IMG]
While students are not asked to reason about scaled areas by tiling (as they had done in the previous activities), each scaled copy can be tiled to illustrate how length measurements have scaled and how the original area has changed. Some students may choose to draw scaled copies and think about scaled areas this way.
[IMG]
As students find the areas of copies with scale factors 5 and
\(\frac35\)
without drawing (for the last question), monitor for these methods, depending on their understanding of or comfort with the
\(\text{(scale factor)}^2\)
pattern:
Scaling the original base and height and then multiplying to find the area
Multiplying the original area by the square of the scale factor
Select students using each approach. Invite them to share their reasoning, sequenced in this order, during the discussion.
You will need the Area of Scaled Parallelograms and Triangles blackline master for this activity.
Launch
Arrange students in groups of 2. Provide access to geometry toolkits.
Distribute slips showing the parallelogram to half the groups and the triangle to the others. Give students 1 minute of quiet work time for the first question, and then time to complete the rest of the task with their partner.
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, draw students attention to the warm-up and remind them how to build scaled copies of the rhombus with scale factors of 2, 3, and 4. Ask students how they can use this technique to draw scaled copies of the parallelogram or triangle with scale factors of 2, 3, and 5.
Supports accessibility for: Social-emotional skills; Conceptual processing
Student Facing
Your teacher will give you a figure with measurements in centimeters. What is the
area
of your figure? How do you know?
Work with your partner to draw scaled copies of your figure, using each scale factor in the table. Complete the table with the measurements of your scaled copies.
|  scale factor   | base (cm) | height (cm) | area (cm                       2                      ) |
|-----------------|-----------|-------------|---------------------------------------------------------|
|        1        |           |             |                                                         |
|        2        |           |             |                                                         |
|        3        |           |             |                                                         |
| \(\frac{1}{2}\) |           |             |                                                         |
| \(\frac{1}{3}\) |           |             |                                                         |
Compare your results with a group that worked with a different figure. What is the same about your answers? What is different?
If you drew scaled copies of your figure with the following scale factors, what would their areas be? Discuss your thinking. If you disagree, work to reach an agreement. Be prepared to explain your reasoning.
|  scale factor   | area (cm                       2                      ) |
|-----------------|---------------------------------------------------------|
|        5        |                                                         |
| \(\frac{3}{5}\) |                                                         |
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 not remember how to calculate the area of parallelograms and triangles. Make sure that they have the correct area of 10 square units for their original shape before they calculate the area of their scaled copies.
When drawing their scaled copies, some students might not focus on making corresponding angles equal. As long as they scale the base and height of their polygon correctly, this will not impact their area calculations. If time permits, however, prompt them to check their angles using tracing paper or a protractor.
Some students might focus unnecessarily on measuring other side lengths of their polygon, instead of attending only to base and height. If time is limited, encourage them to scale the base and height carefully and check or measure the angles instead.
Activity Synthesis
Invite selected students to share their solutions. Then focus class discussion on two themes: how the values in the tables for the two shapes compare, and how students determined the scaled areas for the scale factors 5 and
\(\frac35\)
. Ask questions such as:
What did you notice when you compared your answerswithanother group that worked with the other figure? (When the scale factors are the same, the scaled areas are the same, though the bases and heights are different.)
How did you find the scaled areas for scale factors of 5 and
\(\frac35\)
? (By scaling the original base and height and multiplying the scaled measurements; by multiplying the original area by
\(\text{(scale factor)}^2\)
.)
How is the process for finding scaled area here the same as and different than that in the previous activities with pattern blocks? (The area units are different; the pattern of squaring the scale factor is the same.)
Highlight the connection between the two ways of finding scaled areas. Point out that when we multiply the base and height each by the scale factor and then multiply the results, we are essentially multiplying the original lengths by the scale factor two times. The effect of this process is the same as multiplying the original area by
\(\text{(scale factor)}^2\)
.
Representing, Speaking, and Listening: MLR7 Compare and Connect.
Invite students to prepare a display that shows their approach to finding the areas for scale factors of 5 and
\(\frac35\)
. Ask students to research how other students approached the problem, in search of a method that is different from their own. Challenge students to describe why the different approaches result in the same answers. During the whole-class discussion, emphasize the language used to explain the different strategies, especially phrases related to “squaring” and “multiplying a number by itself.” This will strengthen students' mathematical language use and reasoning based on the relationship between scale factors and area.
Design Principle(s): Maximize meta-awareness


correct_option_index: 8