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.


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.


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.


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: Represent complex numbers and their operations on the complex plane.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Use complex numbers in polynomial identities and equations.

Perform arithmetic operations with complex numbers.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Reason quantitatively and use units to solve problems.

Perform operations on matrices and use matrices in applications.

Apply and extend previous understandings of numbers to the system of rational numbers.

Represent and model with vector quantities.

Extend the properties of exponents to rational exponents.

Use properties of rational and irrational numbers.

Compute fluently with multi-digit numbers and find common factors and multiples.

Perform operations on vectors.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
problem_activity: Activity
In this activity, students are building skills that will help them in mathematical modeling (MP4). Students formulate a model by designing a ramp that they think will be accessible. Then they validate their results by comparing them to the Americans with Disabilities Act (ADA) guidelines. Students then use measurement and computation to determine if their ramp is acceptable and redesign it if necessary.
Monitor for the methods students choose to validate their design:
Use the angle to check their ramps by measuring the angle with a protractor.
Use the slope information by measuring to make sure that for every inch of vertical length, there are 12 inches of horizontal length.
Construct one or more slope triangles along their ramp to demonstrate that it meets the guidelines.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
You may want to do an image search for “bad wheelchair ramp” to show some examples that are not safe to help students come up with good characteristics.
Arrange students in groups. After students make their design, distribute one copy of the ADA guidelines, cut from the blackline master, per group.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “How did you get. . .?”
Supports accessibility for: Language; Social-emotional skills
Student Facing
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Some buildings offer ramps in addition to stairs so people in wheelchairs have access to the building. What characteristics make a ramp safe?
A school has 4 steps to the front door. Each step is 7 inches tall. Design a ramp for the school.
Your teacher will give you the Americans with Disabilities Act (ADA) guidelines. Does your design follow the rules of this law? If not, draw a new design that does.
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?
[IMG]
A ramp with a length to height ratio of
\(12:1\)
forms a right triangle with a 4.8 degree angle.
What is the angle measurement if the base is only 6 units long and the height is 1 unit tall?
When the length is half as long does that make the angle half as big?
What is the angle measurement if the base is 6 units long and the height is increased to 2 units tall?
When the height is twice as tall does that make the angle twice as big?
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 be struggling to design a ramp. Ask them what shape is a good model for the side view of a ramp. (A right triangle.)
Activity Synthesis
The goal of this discussion is to see that having a 4.8 degree angle and a ratio of vertical to horizontal length of
\(1:12\)
are equivalent.
Display previously selected drawings of student ramp designs. Tell students, “Simplified diagrams of right triangles are a mathematical model of the cross section of a ramp.” This previews the work with cross sections that students will do in a subsequent unit but there’s no need to define cross section now.
Invite students who used the ratio of
\(1:12\)
to check their ramps to share their process. Make sure that all students understand how to use the ratio to generate the horizontal length of the ramp given the vertical length, as students will need to do this in the cool-down.
Next, invite students who used the angle of 4.8 degrees to share.
Ask students to confirm that ramps in which students checked for a ratio of
\(1:12\)
also have an angle of 4.8, and vice versa. Students should end convinced that having a 4.8 degree angle and a ratio of vertical to horizontal length of
\(1:12\)
are equivalent. In the lesson synthesis, they will connect this concept to triangle similarity.
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to prepare students for the whole-class discussion. At the appropriate time, invite students to create a visual display that includes their ramp design and a justification for whether it meets ADA guidelines. Invite students to quietly circulate and read at least 2 other displays in the room. Next, ask students to find a new partner to discuss what is the same and what is different about the different methods groups chose to validate their designs. Listen for and amplify observations that include mathematical language and reasoning about angles and ratios of side lengths.
Design Principle(s): Cultivate conversation; Optimize output (for justification)


correct_option_index: 5