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

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fl_problem_000674

5
options: Use polynomial identities to solve problems.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Write expressions in equivalent forms to solve problems.

Gain familiarity with factors and multiples.

Perform arithmetic operations on polynomials.

Use properties of operations to generate equivalent expressions.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Create equations that describe numbers or relationships

Reason about and solve one-variable equations and inequalities.

Analyze patterns and relationships.

Add and subtract within 20.

Interpret the structure of expressions.

Solve systems of equations.

Work with radicals and integer exponents.

Rewrite rational expressions.

Represent and analyze quantitative relationships between dependent and independent variables.

Generate and analyze patterns.

Use the four operations with whole numbers to solve problems.

Understand properties of multiplication and the relationship between multiplication and division.

Represent and solve problems involving addition and subtraction.

Represent and solve problems involving multiplication and division.

Multiply and divide within 100.

Work with addition and subtraction equations.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Understand and apply properties of operations and the relationship between addition and subtraction.

Represent and solve equations and inequalities graphically.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Write and interpret numerical expressions.

Understand solving equations as a process of reasoning and explain the reasoning.

Work with equal groups of objects to gain foundations for multiplication.

Understand the connections between proportional relationships, lines, and linear equations.

Understand the relationship between zeros and factors of polynomials.

Solve equations and inequalities in one variable.
problem_activity: Activity
This activity requires students to make sense of negative powers of 10 as repeated multiplication by
\(\frac{1}{10}\)
in order to distinguish between equivalent exponential expressions. If students have time, instruct them to write the other expressions in each table as a power of 10 with a single exponent as well. Look for students who have productive debate regarding the interpretation of, for example,
\((10^2)^{\text-3}\)
versus
\((10^{\text-2})^3\)
and ask them to share their reasoning later (MP3).
Launch
Arrange students in groups of 2. Give students 15 minutes of partner work time followed by a whole-class discussion. Ask students to explain their reasoning to their partner as they work. If there is disagreement, tell students to work to reach an agreement.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time.
Supports accessibility for: Organization; Attention
Student Facing
Match each exponential expression with an equivalent multiplication expression:
\(\left(10^2\right)^3\)
\(\left(10^2\right)^{\text -3}\)
\(\left(10^{\text -2}\right)^3\)
\(\left(10^{\text -2}\right)^{\text-3}\)
|                              \(\frac{1}{(10 \boldcdot 10)} \boldcdot \frac{1}{(10 \boldcdot 10)} \boldcdot \frac{1}{(10 \boldcdot 10)}\)                               |
|------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
|          \(\left(\frac{1}{10} \boldcdot \frac{1}{10}\right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right)\)          |
| \(\frac{1}{ \frac{1}{10} \boldcdot \frac{1}{10} }\boldcdot \frac{1}{ \frac{1}{10} \boldcdot \frac{1}{10} } \boldcdot \frac{1}{ \frac{1}{10} \boldcdot \frac{1}{10} }\) |
|                                                        \((10 \boldcdot 10)(10 \boldcdot 10)(10 \boldcdot 10)\)                                                         |
Write
\((10^2)^{\text-3}\)
as a power of 10 with a single exponent. Be prepared to explain your reasoning.
Match each exponential expression with an equivalent multiplication expression:
\(\frac{10^2}{10^5}\)
\(\frac{10^2}{10^{\text -5}}\)
\(\frac{10^{\text -2}}{10^5}\)
\(\frac{10^{\text -2}}{10^{\text -5}}\)
| \(\frac{ \frac{1}{10} \boldcdot \frac{1}{10} }{ \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\boldcdot \frac{1}{10}\boldcdot \frac{1}{10} }\) |
|------------------------------------------------------------------------------------------------------------------------------------------------------------|
|                                     \(\frac{10 \boldcdot 10}{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10}\)                                     |
|                          \(\frac{ \frac{1}{10} \boldcdot \frac{1}{10} }{ 10 \boldcdot 10\boldcdot 10\boldcdot 10\boldcdot 10 }\)                           |
|           \(\frac{ 10 \boldcdot 10 }{ \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\boldcdot \frac{1}{10}\boldcdot \frac{1}{10}}\)            |
Write
\(\frac{10^{\text -2}}{10^{\text -5}}\)
as a power of 10 with a single exponent. Be prepared to explain your reasoning.
Match each exponential expression with an equivalent multiplication expression:
\(10^4 \boldcdot 10^3\)
\(10^4 \boldcdot 10^{\text -3}\)
\(10^{\text -4} \boldcdot 10^3\)
\(10^{\text -4} \boldcdot 10^{\text -3}\)
|                                \((10 \boldcdot 10 \boldcdot 10 \boldcdot 10) \boldcdot ( \frac{1}{10} \boldcdot \frac{1}{10}\boldcdot \frac{1}{10})\)                                 |
|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| \(\left(\frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\right) \boldcdot \left( \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\right)\) |
|                 \(\left(\frac{1}{10}\boldcdot \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\right) \boldcdot \left(10 \boldcdot 10 \boldcdot 10\right)\)                 |
|                                               \((10 \boldcdot 10 \boldcdot 10 \boldcdot 10) \boldcdot (10 \boldcdot 10 \boldcdot 10)\)                                                |
Write
\(10^{\text-4} \boldcdot 10^3\)
as a power of 10 with a single exponent. Be prepared to explain your reasoning.
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
Priya, Jada, Han, and Diego stand in a circle and take turns playing a game.
Priya says, SAFE. Jada, standing to Priya's left, says, OUT and leaves the circle. Han is next: he says, SAFE. Then Diego says, OUT and leaves the circle. At this point, only Priya and Han are left. They continue to alternate. Priya says, SAFE. Han says, OUT and leaves the circle. Priya is the only person left, so she is the winner.
Priya says, “I knew I’d be the only one left, since I went first.”
Record this game on paper a few times with different numbers of players. Does the person who starts always win?
Try to find as many numbers as you can where the person who starts always wins. What patterns do you notice?
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 struggle to distinguish between
\(\left(10^{\text-2}\right)^3\)
and
\(\left(10^2\right)^{\text-3}\)
. Breaking down each expression into parts that emphasize repeated multiplication will help to illustrate the difference. For the first expression,
\(10^{\text-2} = \frac{1}{10} \boldcdot \frac{1}{10}\)
. The outer exponent of 3 means that
\(\frac{1}{10} \boldcdot \frac{1}{10}\)
is multiplied repeatedly 3 times. Similarly for the second expression,
\(10^2 = 10 \boldcdot 10\)
and the outer exponent of
\(\text- 3\)
means that the
reciprocal
of
\(10 \boldcdot 10\)
is multiplied repeatedly 3 times. In the end, both expressions are equal to
\(10^{\text-6}\)
.
Activity Synthesis
Select students who had disagreements during the activity to share what they disagreed about and how they came to an agreement. Consider asking:
“How is
\((10^{\text-2})^3\)
different from
\((10^2)^{\text-3}\)
? How are they the same?”
“Do you agree or disagree? Why?”
“Could you restate __'s reasoning in a different way?”
It is important for students to understand that the exponent rules work even with negative exponents. To see why, the whole class discussion must make a clear connection between the exponent rules and the process of multiplying repeated factors that are 10 and
\(\frac{1}{10}\)
. Contrast the expanded version of
\(\left(10^{\text-2}\right)^3\)
and
\(\left(10^2\right)^{\text-3}\)
. For
\((10^{\text-2})^3\)
, there are 3 factors that are
\(10^{\text-2}\)
, where
\(10^{\text-2}\)
is two factors that are
\(\frac{1}{10}\)
, so
\(\displaystyle \left(10^{\text-2}\right)^3 = \left( \frac{1}{10} \boldcdot \frac{1}{10} \right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right) = \frac{1}{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10} = 10^{\text-6}.\)
For
\((10^2)^{\text-3}\)
, there are 3 factors that are
\(\frac{1}{10^2}\)
, so
\(\displaystyle (10^2)^{\text-3} = \left( \frac{1}{10 \boldcdot 10} \right)\left(\frac{1}{10 \boldcdot 10}\right)\left(\frac{1}{10 \boldcdot 10}\right) = \frac{1}{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10} = 10^{\text-6}.\)
Both
\(\left(10^{\text-2}\right)^3\)
and
\(\left(10^2\right)^{\text-3}\)
are equal to
\(10^{\text-6}\)
in the same way that
\(\text-2 \boldcdot 3 = 2 \boldcdot \text-3 = \text-6\)
.
Listening, Conversing: MLR8 Discussion Supports.
As students share the disagreements they had with their partner, press for details in students’ statements by requesting that students elaborate on an idea or give an example of their process of reconciliation. For example, ask students, “How did you explain your reasoning to your partner in a way that convinced them?” In this discussion, it may be useful to revoice student ideas to model mathematical language use by restating a statement in order to clarify, apply appropriate language, or involve more students.
Design Principle(s): Maximize meta-awareness


correct_option_index: 13