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.


Your response:
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.


Your response:
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.


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

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

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.

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

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.

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

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.

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.

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.

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.
problem_activity: Activity
In this activity, students begin to build their own table of trigonometric ratios (ratios of side lengths in right triangles, by angle measure of one acute angle). For now, call this the right triangle table so students remember these ratios only apply to right triangles. Building these ratios from the similar triangles they are based on helps students build connections between similar triangles and trigonometry.
Measuring several examples of triangles with the same angle measurements is essential in both the digital and paper versions of this activity. In the paper version, it provides more data points to get a more accurate ratio for the given angle. In both versions, it helps students understand that all right triangles with their angle as one of the angles will be similar and thus have the same ratios.
In the next activity and the cool-down of this lesson, students will look for patterns in the right triangle table, and also apply the ratios in the right triangle table to find unknown angle measures given a ratio of side lengths. Do not use the word trigonometry or any of the names of the ratios yet.
Digital
Launch
Provide access to devices that can run GeoGebra. Students will be working with the applet at
ggbm.at/bfxeb6cr
.
Arrange students in groups of 2-4. Invite students to create a spreadsheet to organize their data and perform calculations. Students can access a GeoGebra spreadsheet in the math tool kit or at
geogebra.org/spreadsheet
. For students needing extra support, consider providing this spreadsheet for them to use:
ggbm.at/jvvqrngb
.
Assign each group a pair of complementary angles. If students did the optional lesson choose to skip 30 and 60 or to give them to a group that needs extra practice.
Explain to students that the computer measurement is precise to at least the ten-thousandths place, but the thousandths place is sufficient for our needs.
Representation: Internalize Comprehension.
Use color coding and annotations to highlight connections between representations in a problem. For example color code the side adjacent to angle
\(A\)
, the side opposite angle
\(A\)
, and the hypotenuse.
Supports accessibility for: Visual-spatial processing.
Student Facing
Your teacher will give you some angles.
Use the applet to build 4 different right triangles for each of your angles.
Record the side lengths of each of the triangles.
Compute these 3 quotients for the acute angles in each triangle:
The length of the leg adjacent to your angle divided by the length of the hypotenuse
The length of the leg opposite from your angle divided by the length of the hypotenuse
The length of the leg opposite from your angle divided by the length of the leg adjacent to your angle
Find the mean of each type of quotient.
What do you notice? What do you wonder?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 3-4. Give each group a copy of the blackline master to complete.
Assign each group a pair of complementary angles. If students did the optional lesson choose to skip 30 and 60 or to give them to a group that needs extra practice.
Discuss rounding with students before they begin. Ask students whether it makes sense to measure the short side of a triangle as about 5.2 cm and the hypotenuse as pretty close to 7.1 cm and then report that the ratio of the short side to the hypotenuse is exactly 0.73239437. (No, that's too many digits to report.)
Explain to students that the rule of thumb used by scientists is that if your measurements have two digits then your calculated answer should have two digits as well. For example, if your tools measure to the tenths place, say 5.2 cm, so that your measurements have one digit you’re confident in - the 5 in the one’s place, and one digit you may have had to estimate a bit - the 2 in the tenths place, then your answer should be rounded to two digits - 0.73. In the case where the number starts with a zero we count digits starting at the first non-zero digit, so in this case we will round to the hundredths place.
As students finish their measurements, have them find the mean of each column and record that on the class Right Triangle Table.
Check students’ mean data before they enter it to ensure that it is close to the actual values:
|   angle    | adjacent leg                           \(\div\)                          hypotenuse | opposite leg                           \(\div\)                          hypotenuse | opposite leg                           \(\div\)                          adjacent leg |
|------------|-------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------|
| 10 degrees |                                        0.985                                        |                                        0.174                                        |                                         0.176                                         |
| 20 degrees |                                        0.940                                        |                                        0.342                                        |                                         0.364                                         |
| 30 degrees |                                        0.866                                        |                                        0.500                                        |                                         0.577                                         |
| 40 degrees |                                        0.766                                        |                                        0.643                                        |                                         0.839                                         |
| 50 degrees |                                        0.643                                        |                                        0.766                                        |                                         1.192                                         |
| 60 degrees |                                        0.500                                        |                                        0.866                                        |                                         1.732                                         |
| 70 degrees |                                        0.342                                        |                                        0.940                                        |                                         2.747                                         |
| 80 degrees |                                        0.174                                        |                                        0.985                                        |                                         5.671                                         |
This table is needed for the next activity, so it should be recorded to display at that time.
Representation: Internalize Comprehension.
Use color coding and annotations to highlight connections between representations in a problem. For example color code the side adjacent to angle
\(A\)
, the side opposite angle
\(A\)
, and the hypotenuse.
Supports accessibility for: Visual-spatial processing.
Student Facing
Your teacher will give you some angles.
Draw several right triangles using the angles you receive.
Precisely measure the side lengths of the triangles.
Complete the tables by computing 3 quotients for the acute angles in each triangle:
The length of the leg adjacent to your angle divided by the length of the hypotenuse
The length of the leg opposite from your angle divided by the length of the hypotenuse
The length of the leg opposite from your angle divided by the length of the leg adjacent to your angle
Find the mean of each column in your table.
What do you notice about your table? What do you wonder about your table?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students might struggle to identify opposite and adjacent legs. Prompt students to highlight the angle in question to accurately identify sides. If you point to “your angle” then the leg you are touching is adjacent and the leg you are not touching is opposite. The hypotenuse is always the longest side.
[IMG]
Activity Synthesis
Invite groups to share what they noticed and wondered about their data. (Each ratio is about the same for the angle. The ratio of the opposite side to the hypotenuse for angle
\(A\)
is the same as the ratio of the adjacent side to the hypotenuse for angle
\(B\)
.)
Speaking: MLR8 Discussion Supports.
Give students additional time to make sure that everyone in their group can explain what they noticed and wondered about the relationships between the quantities represented in the table. Invite groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking, and will improve the quality of explanations shared during the whole-class discussion.
Design Principle(s): Support sense-making; Cultivate conversation


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