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: 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.

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

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.

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.

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 in Base Ten: students learn to work with the base-ten system and build place value understanding.

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

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
problem_activity: Activity
The purpose of this activity is for students to make connections between the polynomial division reasoning they did in the previous lesson and polynomial long division. The remainders for division in this activity continue to be zero since the focus is on dividing by known linear factors. In future lessons, students will focus on what a non-zero remainder means for both polynomials and when rewriting expressions of rational functions.
Identify students who make errors while working out the long division (for example, adding a term they meant to subtract) who are willing to share what they did during the discussion.
Launch
After 5 minutes of work time, pause the class and display the long division and the incomplete diagram in the first question for all to see. Select students to explain how they completed the diagram and any connections they see between terms in the long division and the diagram. Remind students that since they are dividing by known linear factors, the results of any long division should have a remainder of 0.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation, such as: “First, I _____ because . . . .”, “Next, I . . . .”, “I tried _____ , and what happened was . . . .”, and “How did you get . . . ?” Encourage students to become error detectives and positively reinforce when students are able to explain their mistakes.
Supports accessibility for: Language; Social-emotional skills
Student Facing
Diego used the long division shown here to figure out that
\(6x^2-7x-5 = (2x+1)(3x-5)\)
. Show what it would look like if he had used a diagram.
\(\displaystyle  \displaystyle \require{enclose} \begin{array}{r}   3x-5 \\   2x+1 \enclose{longdiv}{6x^2 - 7x - 5} \\      \underline{\text-6x^2 - 3x} \phantom{-55} \\    \text-10x-5 \\  \underline{10x+5} \\    0  \end{array} \)
|        |   |   |
|--------|---|---|
| \(2x\) |   |   |
|   1    |   |   |
Pause here for a whole-class discussion.
\((x-2)\)
is a factor of
\(2x^3 - 7x^2 + x + 10\)
, which means there is some other factor
\(A\)
where
\(2x^3 - 7x^2 + x + 10=(x-2)(A)\)
. Finish the division started here to find the value of
\(A\)
.
\(\displaystyle \displaystyle \require{enclose} \begin{array}{r}   2x^2 \phantom{+x+100} \\   x-2 \enclose{longdiv}{2x^3-7x^2+x+10} \\      \underline{\text-2x^3+4x^2} \phantom{+x+100} \end{array}\)
Jada used the diagram shown here to figure out that
\(2x^3 + 13x^2 + 16x + 5 = (2x+1)(x^2+6x+5)\)
. Show what it would look like if she had used long division.
|        | \(x^2\) | \(6x\) | 5 |
|--------|---------|--------|---|
| \(2x\) |         |        |   |
|   1    |         |        |   |
\(\displaystyle \displaystyle \require{enclose} \begin{array}{r}    2x+1 \enclose{longdiv}{2x^3+13x^2+16x+5} \end{array}\)
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
What is
\((x^4-1) \div (x-1)\)
?
Use your response to predict what
\((x^7-1) \div (x-1)\)
is, and then use division to check your prediction.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students may forget that they are subtracting the terms in each new row of long division. Prompt these students to consider the signs in front of the terms they write carefully.
Activity Synthesis
Since polynomial long division can prove challenging due to errors in arithmetic, such as adding instead of subtracting terms, the focus of this discussion is on errors. Invite students previously identified to share the error they made and how they revised their work.
Students have now seen two ways of representing division: using a diagram to work backwards to determine what to multiply the divisor by, and using long division to find the quotient directly. This activity asks students to translate between the two methods so that they see how they are related. Students should understand that these are two strategies for doing the same thing, and that they each have advantages.
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.
Use this routine to give students the opportunity to discuss a common error that arises when doing long division of polynomials. For example, “In Jada’s problem, I placed the term “
\(x\)
2
” under “
\(13x\)
2
” because I multiplied and got the answer.” Ask students to identify the error, critique the reasoning, and write an improved explanation. As students discuss with a partner, listen for students who identify and clarify the incorrect and ambiguous language in the statement. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to explain why subtracting is the appropriate operation when combining these terms. This helps students evaluate, and improve on, the written mathematical arguments of others, as they discuss common arithmetic errors in long division.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness


correct_option_index: 1