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: Apply and extend previous understandings of arithmetic to algebraic expressions.

Create equations that describe numbers or relationships

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

Reason about and solve one-variable equations and inequalities.

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

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

Solve systems of equations.

Analyze patterns and relationships.

Write and interpret numerical expressions.

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

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

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

Rewrite rational expressions.

Multiply and divide within 100.

Write expressions in equivalent forms to solve problems.

Use properties of operations to generate equivalent expressions.

Add and subtract within 20.

Represent and analyze quantitative relationships between dependent and independent variables.

Understand the relationship between zeros and factors of polynomials.

Use polynomial identities to solve problems.

Represent and solve equations and inequalities graphically.

Work with addition and subtraction equations.

Interpret the structure of expressions.

Represent and solve problems involving multiplication and division.

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

Use the four operations with whole numbers to solve problems.

Solve equations and inequalities in one variable.

Gain familiarity with factors and multiples.

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

Generate and analyze patterns.

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

Perform arithmetic operations on polynomials.

Work with radicals and integer exponents.

Represent and solve problems involving addition and subtraction.
problem_activity: Activity
The goal of this activity is to help students flexibly transition between different notations for powers of 10 and introduce the property of multiplication of values with the same base. Students observe that
\(10^n \boldcdot 10^m = 10^{n+m}\)
for values of
\(n\)
and
\(m\)
that are positive integers. The second question hints at the reasoning that will extend exponent rules to include zero exponents, but students will investigate that more deeply in a later lesson.
Notice students who need help writing the general rule in in terms of
\(n\)
and
\(m\)
. You might ask, “What patterns did you notice with the exponents in the table? So if the exponents are
\(n\)
and
\(m\)
, how do you write what you did with the exponents?”
Launch
Give students 1 minute of quiet think time to complete the first unfinished row in the table before asking 1–2 students to share and explain their answers. When it is clear that students know how to complete the table, explain to them that they can skip one entry in the table, but they have to be able to explain why they skipped it. Give students 7–8 minutes to work before a brief whole-class discussion.
Student Facing
Complete the table to explore patterns in the exponents when multiplying powers of 10. You may skip a single box in the table, but if you do, be prepared to explain why you skipped it.
|          expression           |                                          expanded                                          | single power of 10 |
|-------------------------------|--------------------------------------------------------------------------------------------|--------------------|
|    \(10^2 \boldcdot 10^3\)    |                     \((10 \boldcdot 10)(10\boldcdot 10 \boldcdot 10)\)                     |      \(10^5\)      |
|    \(10^4 \boldcdot 10^3\)    |                                                                                            |                    |
|    \(10^4 \boldcdot 10^4\)    |                                                                                            |                    |
|                               | \((10 \boldcdot 10 \boldcdot 10)(10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10)\) |                    |
| \(10^{18} \boldcdot 10^{23}\) |                                                                                            |                    |
If you chose to skip one entry in the table, which entry did you skip? Why?
Use the patterns you found in the table to rewrite
\(10^n \boldcdot 10^m\)
as an equivalent expression with a single exponent, like
\(10^{\boxed{\phantom{3}}}\)
.
Use your rule to write
\(10^4 \boldcdot 10^0\)
with a single exponent.  What does this tell you about the value of
\(10^0\)
?
The state of Georgia has roughly
\(10^7\)
human residents. Each human has roughly
\(10^{13}\)
bacteria cells in his or her digestive tract. How many bacteria cells are there in the digestive tracts of all the humans in Georgia?
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?
There are four ways to make
\(10^4\)
by multiplying powers of 10 with smaller, positive exponents.
\(\displaystyle 10^1 \boldcdot 10^1 \boldcdot 10^1 \boldcdot 10^1\)
\(\displaystyle 10^1 \boldcdot 10^1 \boldcdot 10^2\)
\(\displaystyle 10^1 \boldcdot 10^3\)
\(\displaystyle 10^2 \boldcdot 10^2\)
(This list is complete if you don't pay attention to the order you write them in. For example, we are only counting
\(10^1 \boldcdot 10^3\)
and
\(10^3 \boldcdot 10^1\)
once.)
How many ways are there to make
\(10^6\)
by multiplying smaller powers of 10 together?
How about
\(10^7\)
?
\(10^8\)
?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Create and post visual displays showing the exponent rules for reference throughout the unit, with one visual display for each rule. The visual display could include an example to illustrate how the rule works, along with visual aids and use of color. Here is a sample visual display:
[IMG]
Explain the visual display to students and display it for all to see throughout the unit.
Writing: MLR3 Clarify, Critique, Correct.
Present a hypothetical student statement that represents a misunderstanding of the exponent rule, such as “
\(10^2 ⋅ 10^3 = 10^6 \)
because
\(2⋅3 = 6\)
.” Prompt discussion by asking, “Do you agree with the statement? Why or why not?” Then, ask students to individually write an improved statement. Improved responses should include connections between the representations of a single expression, expanded form, and the single power of 10. This will help students evaluate, and improve on, the written mathematical arguments of others and highlight why
\(10^n\boldsymbol\cdot 10^m=10^{n+m}\)
.
Design Principle(s): Maximize meta-awareness.


correct_option_index: 32