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: Analyze proportional relationships and use them to solve real-world and mathematical problems.

Understand ratio concepts and use ratio reasoning to solve problems.
problem_activity: Activity
The purpose of this activity is to help students understand that quantities measured using the same two units of measure form a set of equivalent ratios. All of the strategies and representations they have for reasoning about equivalent ratios can be used for reasoning about converting from one unit of measure to another. Any ratio
\(a:b\)
has two associated unit rates:
\(\frac{a}{b}\)
and
\(\frac{b}{a}\)
, with a particular meaning in the context. For example, since there are 8 kilometers in approximately 5 miles, there are
\(\frac85\)
kilometers in 1 mile, and there are
\(\frac58\)
of a mile in one kilometer. We want students to notice that finding “how much per 1” and reasoning with these unit rates is efficient, but to make sense of these efficient strategies by using familiar representations like double number lines and tables. In a constant speed context, students are explicitly asked to compute each unit rate, and then they are asked to solve a problem where either unit rate can be used. For the second problem, monitor for one student who uses each strategy to solve the problem:
Creating a double number line or a table to represent the association between miles and the equivalent distance in kilometers as a set of equivalent ratios. (If both of these representations are used, it is fine to include both.)
Converting 80 kilometers into 50 miles by evaluating
\(80 \boldcdot \frac58\)
(in order to compare the resulting 50 miles per hour with 75 miles per hour)
Converting 75 miles into 120 kilometers by evaluating
\(75 \boldcdot \frac85\)
(in order to compare the resulting 120 kilometers per hour with 80 kilometers per hour)
Continuing to draw connections between representations of equivalent ratios and more efficient methods will help students make sense of the more efficient methods.
Launch
Display the image from the task statement for all to see, tell students it is a traffic sign you might see while driving, and ask students to explain what it means. They will likely guess it is a speed limit sign and assume it means 80 miles per hour. If no one brings it up, tell students that this is a sign you might see while driving in Canada or another country that uses the metric system for more things than we use it for in the United States.
Give students 2 minutes of quiet work time and ask them to pause after the first question. Ensure that everyone has correct answers for the first question before proceeding with the second question. Follow with whole-class discussion.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
Elena and her mom are on a road trip outside the United States. Elena sees this road sign.
Elena’s mom is driving 75 miles per hour when she gets pulled over for speeding.
[IMG]
The police officer explains that 8 kilometers is approximately 5 miles.
How many kilometers are in 1 mile?
How many miles are in 1 kilometer?
If the speed limit is 80 kilometers per hour, and Elena’s mom was driving 75 miles per hour, was she speeding? By how much?
Student Response
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Anticipated Misconceptions
It is acceptable to express the answers to the first question in either fraction or decimal form. If students express uncertainty about carrying out the division of
\(5 \div 8\)
or
\(8 \div 5\)
, encourage them to express the quotient in fraction form.
Activity Synthesis
Focus discussion on different approaches to the second question. If any students with less-efficient methods were selected, have them go first in the sequence, or present one of these representations yourself. As students are presenting their work, encourage them to explain the meaning of any numbers used and the reason they decided to use particular operations. For example, if a student multiplies 80 by
\(\frac58\)
, ask them to explain what
\(\frac58\)
means in this context and why they decided to multiply 80 by it. It can be handy to have representations like double number lines or tables displayed to facilitate these explanations.
Representing, Listening, Speaking: MLR7 Compare and Connect.
Ask students to display their approaches to determine whether or not Elena’s mom was speeding. As students share their work, encourage them to explain the meaning of each quantity they use. For example, if they convert 80 miles per hour into kilometers per hour, where 80 is multiplied by
\(\frac58\)
, ask what
\(\frac58\)
means in this context and why they decide to multiply it by 80. If students used a table or a double number line, ask how these representations connect with other strategies. This will help students make sense of the various approaches to reason about equivalent ratios which can be used for reasoning about converting one unit of measure to another.
Design Principle(s): Cultivate conversation; Maximize meta-awareness


correct_option_index: 1