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: Work with numbers 11–19 to gain foundations for place value.

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.

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.

Understand place value.

Extend the counting sequence.

Use place value understanding and properties of operations to perform multi-digit arithmetic. (A range of algorithms may be used.)

Understand the place value system.

Use place value understanding and properties of operations to add and subtract.

Perform operations with multi-digit whole numbers and with decimals to hundredths.
problem_activity: Narrative
This optional activity gives students additional opportunities to compare multi-digit numbers by reasoning about the value of the digits in different places. It also prompts students to generalize the relative size of two numbers based on their understanding of place value. Students practice constructing logical arguments (MP3) as they explain whether it is possible for
\(4 \underline{\phantom{5}},\!300\)
to be less than
\(3 \underline{\phantom{5}},\!400\)
, or for
\( \underline{\phantom{5}}4,\!300\)
to be less than
\( \underline{\phantom{5}}3,\!400\)
.
Required Materials
Materials to Gather
Materials from a previous activity
Required Preparation
Each group of 2 needs a set of cards from the previous activity.
Launch
Groups of 2
Give each group a card with a digit between 0 and 9 for each group
Activity
“Use the digit on the card to complete the comparison statements in the first problem and decide if they are true.”
4–5 minutes: group work time
Pause to collect responses from all the groups.
“Is the first comparison statement true for all digits?” (Yes, because if the digits are the same, we can just compare 999 to 500.)
Repeat the question with all statements. Record responses in a chart such as shown:
|                                statement                                 |    true    |                 false                 |
|--------------------------------------------------------------------------|------------|---------------------------------------|
| \( \underline{\phantom{00}} \ , 999 > \underline{\phantom{00}} \ , 500\) | all digits |                                       |
|   \(15,2 \underline{\phantom{00}}0 > 15, \underline{\phantom{00}}02\)    |  0, 1, 2   | 3, 4, 5, 6,                   7, 8, 9 |
|  \(4 \underline{\phantom{00}},700 < 7 \underline{\phantom{00}} ,400 \)   | all digits |                                       |
|  \(1 \underline{\phantom{00}}5,000 > 5 \underline{\phantom{00}}1,000\)   |            |              all digits               |
“Why are the statements in parts a and c true no matter what digit is used?” (Sample response:
In part a, both numbers are missing the first digit, in the thousands place. Since the missing digit is the same, we’re comparing the hundreds, and 999 is always greater than 500.
In part c, the missing digit is in the thousands place of both numbers, but the digits in the ten-thousands place need to be compared first, and 4 ten-thousand is always less than 7 ten-thousand.)
“Why is the statement in part d false no matter what digit is used?” (Ten-thousand is never greater than 50 thousand.)
“Take a few quiet minutes to work on the last two problems independently.”
5 minutes: independent work time
Student Facing
Each of the following pairs of numbers is missing the same digit but in different places.
Your teacher will assign a digit to you. Use it as the missing digit and decide if each comparison statement is true.
\(\large \boxed{\phantom{0}} \ , \boxed{9} \ \boxed{9} \ \boxed{9} > \boxed{\phantom{0}} \ , \boxed{5} \ \boxed{0} \ \boxed{0}\)
\(\large \boxed{1} \ \boxed{5} \ , \boxed{2} \ \boxed{\phantom{0}} \ \boxed{0} > \boxed{1} \ \boxed{5} \ , \boxed{\phantom{0}} \ \boxed{0} \ \boxed{2}\)
\(\large \boxed{4} \ \boxed{\phantom{0}} \ , \boxed{7} \ \boxed{0} \ \boxed{0} < \boxed{7} \ \boxed{\phantom{0}} \ , \boxed{4} \ \boxed{0} \ \boxed{0}\)
\(\large \boxed{1} \ \boxed{\phantom{0}} \ \boxed{5} \ , \boxed{0} \ \boxed{0} \ \boxed{0} > \boxed{5} \ \boxed{\phantom{0}} \ \boxed{1} \ , \boxed{0} \ \boxed{0} \ \boxed{0}\)
Here are two numbers, each with the same missing digit.
\(\large \boxed{4} \ \boxed{\phantom{0}} \ , \boxed{3} \ \boxed{0} \ \boxed{0} \qquad \qquad \boxed{3} \ \boxed{\phantom{0}} \ , \boxed{4} \ \boxed{0} \ \boxed{0}\)
Choose a digit to complete the numbers and show where they would be on the number line.
[IMG]
Is it possible to fill in the two blanks with the same digit to make each statement true? If you think so, give at least one example of what the digits could be. If not, explain why it is not possible.
\(\large \boxed{4} \ \boxed{\phantom{0}} \ , \boxed{3} \ \boxed{0} \ \boxed{0}\)
is less than
\(\large {\boxed{3}} \ \boxed{\phantom{0}} \ , \boxed{4} \ \boxed{0} \ \boxed{0}\)
.
\(\large \boxed{\phantom{0}} \ \boxed{4} \ , \boxed{3} \ \boxed{0} \ \boxed{0}\)
is less than
\(\large \boxed{\phantom{0}} \ \boxed{3} \ , \boxed{4} \ \boxed{0} \ \boxed{0}\)
.
Student Response
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Activity Synthesis
Select students to share their responses to the last two problems.
Highlight explanations that make it clear that:
\(4 \underline{\phantom{5}},\!300\)
will always be greater than
\(3 \underline{\phantom{5}},\!400\)
because 4 ten-thousand is always greater than 3 ten-thousands.
\( \underline{\phantom{5}}4,\!300\)
will always be greater than
\( \underline{\phantom{5}}3,\!400\)
because, given the first digit is the same, the digits to compare are the thousands, and 4 thousands is always greater than 3 thousands.


correct_option_index: 1