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: Extend the properties of exponents to rational exponents.

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

Represent complex numbers and their operations on the complex plane.

Perform arithmetic operations with complex numbers.

Perform operations on vectors.

Use properties of rational and irrational numbers.

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

Use complex numbers in polynomial identities and equations.

Reason quantitatively and use units to solve problems.

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

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Perform operations on matrices and use matrices in applications.

Represent and model with vector quantities.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
problem_activity: Activity
Students ended the previous activity wondering whether solutions that look like
\(7 \pm \sqrt2\)
and
\(4\sqrt5\)
are rational or irrational. In this activity, they pursue that question and experiment with adding and multiplying different types of numbers. The goal here is to make conjectures about the sums and products by noticing regularity in repeated reasoning with concrete numbers (MP8). In an upcoming lesson, students will reason logically and abstractly about what the sums and products of rational and irrational numbers must be.
Launch
Tell students that they will now further investigate what happens when different types of numbers are combined by addition and multiplication. Are the results rational or irrational? Can we come up with general rules about what types of numbers the sums and products will be?
Consider keeping students in groups of 2. Provide access to calculators for numerical calculations, in case requested. Students who choose to use technology to help them analyze patterns practice choosing a tool strategically (MP5).
Engagement: Internalize Self Regulation.
Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to begin by sorting and labeling the initial numbers into rational or irrational. Invite each partner group to choose and respond to either the sum or the product statements. Provide sentence frames for students to use in their discussion. For example, “I agree/disagree with _____ because . . . .” and “Here is an example that shows . . . .” Then, have partners switch or select students to share with the whole class, so that each student has examples and conclusions for all truth statements.
Supports accessibility for: Organization; Attention
Student Facing
Here is a list of numbers:
\(\displaystyle 2\)
\(\displaystyle 3\)
\(\displaystyle \frac13\)
\(\displaystyle 0\)
\(\displaystyle \sqrt2\)
\(\displaystyle \sqrt3\)
\(\displaystyle \text-\sqrt3\)
\(\displaystyle \frac{1}{\sqrt3}\)
Here are some statements about the sums and products of numbers. For each statement, decide whether it is
always
true, true for
some
numbers but not others, or
never
true.
Sums:
The sum of two rational numbers is rational.
The sum of a rational number and an irrational number is irrational.
The sum of two irrational numbers is irrational.
Products:
The product of two rational numbers is rational.
The product of a rational number and an irrational number is irrational.
The product of two irrational numbers is irrational.
Experiment with sums and products of two numbers in the given list to help you decide.
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?
It can be quite difficult to show that a number is irrational. To do so, we have to explain why the number is impossible to write as a ratio of two integers. It took mathematicians thousands of years before they were finally able to show that
\(\pi\)
is irrational, and they still don’t know whether or not
\(\pi^{\pi}\)
is irrational.
Here is a way we could show that
\(\sqrt{2}\)
can’t be rational, and is therefore irrational.
Let's assume that
\(\sqrt{2}\)
were rational and could be written as a fraction
\(\frac{a}{b}\)
, where
\(a\)
and
\(b\)
are non-zero integers.
Let’s also assume that
\(a\)
and
\(b\)
are integers that no longer have any common factors. For example, to express 0.4 as
\(\frac {a}{b}\)
, we write
\(\frac25\)
instead of
\(\frac{4}{10}\)
or
\(\frac {200}{500}\)
. That is, we assume that
\(a\)
and
\(b\)
are 2 and 5, rather than 4 and 10, or 200 and 500.
If
\(\sqrt{2}=\frac{a}{b}\)
, then
\(2 = \frac {\boxed{\phantom{300}}}{\boxed{\phantom{300}}}\)
.
Explain why
\(a^2\)
must be an even number.
Explain why if
\(a^2\)
is an even number, then
\(a\)
itself is also an even number. (If you get stuck, consider squaring a few different integers.)
Because
\(a\)
is an even number, then
\(a\)
is 2 times another integer, say,
\(k\)
. We can write
\(a=2k\)
. Substitute
\(2k\)
for
\(a\)
in the equation you wrote in the first question. Then, solve for
\(b^2\)
.
Explain why the resulting equation shows that
\(b^2\)
, and therefore
\(b\)
, are also even numbers.
We just arrived at the conclusion that
\(a\)
and
\(b\)
are even numbers, but given our assumption about
\(a\)
and
\(b\)
, it is impossible for this to be true. Explain why this is.
If
\(a\)
and
\(b\)
cannot both be even,
\(\sqrt{2}\)
must be equal to some number other than
\(\frac{a}{b}\)
.
Because our original assumption that we could write
\(\sqrt{2}\)
as a fraction
\(\frac{a}{b}\)
led to a false conclusion, that assumption must be wrong. In other words, we must not be able to write
\(\sqrt{2}\)
as a fraction. This means
\(\sqrt{2}\)
is irrational!
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
Display the six statements for all to see. Invite students to share their responses. After each student shares, ask others whether they agree or disagree. Ask students who disagree for an explanation or a counterexample. Develop a consensus on what the class thinks is true when we combine numbers by addition and multiplication. Record and display the consensus for all to see, for instance:
\(\text {rational} + \text {rational} = \)
\(\text {rational} + \text {irrational} =\)
\(\text {irrational} + \text {irrational} =\)
\(\text {rational} \boldcdot \text {rational} =\)
\(\text {rational} \boldcdot \text {irrational} =\)
\(\text {irrational} \boldcdot \text {irrational} =\)
Explain to students that in an upcoming lesson, they will have a chance to test their conjectures.
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: "I agree because . . .” or "I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language, and invite students to chorally repeat phrases that include relevant vocabulary in context.
Design Principle(s): Support sense-making


correct_option_index: 5