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.


Topic options: 
A. Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
B. Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.
C. Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
D. Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
E. Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.
F. Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.
G. Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
H. Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.
I. Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
J. Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.
K. Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.


Thought: This task involves learning addition, which is a mathematical operation, using objects.
Answer: F

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Topic options: 
A. Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.
B. Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
C. Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
D. Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
E. Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
F. Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.
G. Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
H. Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.
I. Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
J. Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.
K. Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.


Thought: This problem teaches students different ways of writing numbers that express the same value. This helps students better understand the complex number system, which includes rational numbers.
Answer: I

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Topic options: 
A. Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.
B. Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
C. Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
D. Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.
E. Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
F. Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.
G. Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
H. Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.
I. Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.
J. Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
K. Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.


Thought: This problem pertains to functions because it asks students to find the general solution to a trigonometric function.
Answer: B

Now, assign the following problem or activity to one or more relevant topics it teaches.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
problem_activity: Activity
This activity connects the ideas of roots, rational exponents, graphs of exponential functions, and decimal approximations. Students graph
\(y=2^x\)
for a few integer values of
\(x\)
, approximate a smooth curve passing through those points, and use that curve to estimate the value of
\(2^x\)
for various rational inputs. Students then test these estimates using their understanding of the connection between rational exponents and roots.
Launch
Arrange students in groups of 2–3. Encourage them to compare their answers with their group after each question.
Since the first problem in this activity was designed to be completed without technology, ask students to put away any devices until after they complete it.
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to prepare students for the whole class discussion about the graph of
\(y=2^x\)
. After students complete the table and sketch the graph of
\(y=2^x\)
, invite them to create a visual display of their work. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their tables and graphs. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast the estimates for
\(2^x\)
for various values of
\(x\)
between 0 and 2.
Design Principle(s): Cultivate conversation
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer collaboration. When students share their work with a partner, display sentence frames to support conversation such as: “First, I _____ because. . .”, “How did you get . . .?”, “That could/couldn’t be true because . . . .”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Complete the table as much as you can without using a calculator. (You should be able to fill in three spaces.)
|                        \(x\)                         |   |   |   |   |   |   |
|------------------------------------------------------|---|---|---|---|---|---|
|    \(2^x\)                      (using exponents)    |   |   |   |   |   |   |
| \(2^x\)                      (decimal approximation) |   |   |   |   |   |   |
Plot the points that you filled in.
[IMG]
Connect the points as smoothly as you can.
Use this graph of
\(y=2^x\)
to estimate the value of the other powers in the table, and write your estimates in the table.
Let’s investigate
\(2^{\frac13}\)
:
Write
\(2^{\frac13}\)
using radical notation.
What is
\(\left( 2^{\frac13}\right)^3\)
?
Raise your estimate from the table of
\(2^{\frac13}\)
to the third power. What should it be? How close did you get?
Let’s investigate
\(2^{\frac23}\)
:
Write
\(2^{\frac23}\)
using radical notation.
What is the value of
\(\left( 2^{\frac23}\right)^3\)
?
Raise your estimate from the table of
\(2^{\frac23}\)
to the third power. What should it be? How close did you get?
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?
Answer these questions using the fact that
\((1.26)^3 = 2.000376\)
.
Explain why
\(\sqrt[3]{2}\)
is very close to
\(1.26\)
. Is it larger or smaller than
\(1.26\)
?
Is it possible to write
\(\sqrt[3]{2}\)
exactly with a finite decimal expansion? Explain how you know.
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 several student graphs of
\(y=2^x\)
for all to see and invite students to share how close their estimates for the cubes of
\(2^{\frac13}\)
and
\(2^{\frac23}\)
were. The important takeaway from this activity is that whether we write the number as
\(2^{\frac23}\)
or
\(\sqrt[3]{2^2}\)
, the expression is just a number and its value can be approximated.


correct_option_index: 4