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# 193 Results

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Algebraic Reasoning

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Add, subtract, multiply, and divide rational numbers.
Evaluate expressions for a value of a variable.
Use the distributive property to generate equivalent expressions including combining like terms.
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?
Write and solve equations of the form x+p=q and px=q for cases in which p, q, and x are non-negative rational numbers.
Understand and graph solutions to inequalities x<c or x>c.
Use equations, tables, and graphs to represent the relationship between two variables.
Relate fractions, decimals, and percents.
Solve percent problems included those involving percent of increase or percent of decrease.

Lesson Flow

This unit covers all of the Common Core State Standards for Expressions and Equations in Grade 7. Students extend what they learned in Grade 6 about evaluating expressions and using properties to write equivalent expressions. They write, evaluate, and simplify expressions that now contain both positive and negative rational numbers. They write algebraic expressions for problem situations and discuss how different equivalent expressions can be used to represent different ways of solving the same problem. They make connections between various forms of rational numbers. Students apply what they learned in Grade 6 about solving equations such as x+2=6 or 3x=12 to solving equations such as 3x+6=12 and 3(x−2)=12. Students solve these equations using formal algebraic methods. The numbers in these equations can now be rational numbers. They use estimation and mental math to estimate solutions. They learn how solving linear inequalities differs from solving linear equations and then they solve and graph linear inequalities such as −3x+4<12. Students use inequalities to solve real-world problems, solving the problem first by arithmetic and then by writing and solving an inequality. They see that the solution of the algebraic inequality may differ from the solution to the problem.

Subject:
Algebra
Mathematics
Provider:
Pearson
Conditional Remix & Share Permitted
CC BY-NC
Rating
0.0 stars

Subject:
Mathematics
Material Type:
Lesson Plan
Author:
Pearson
11/02/2020
Conditional Remix & Share Permitted
CC BY-NC
Rating
0.0 stars

Constructions and Angles

Unit Overview

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Use a protractor and ruler.
Identify different types of triangles and quadrilaterals and their characteristics.

Lesson Flow

After an initial exploratory lesson involving a paper folding activity that gets students thinking in general about angles and figures in a context, the unit is divided into two concept development sections. The first section focuses on types of angles—adjacent, supplementary, complementary, and vertical—and how they are manifested in quadrilaterals. The second section looks at triangles and their properties, including the angle sum, and how this affects other figures.

In the first set of conceptual lessons, students explore different types of angles and where the types of angles appear in quadrilaterals. Students fold paper and observe the angles formed, draw given angles, and explore interactive sketches that test many cases. Students use a protractor and ruler to draw parallelograms with given properties. They explore sketches of parallelograms with specific properties, such as perpendicular diagonals. After concluding the investigation of the angle types, students move on to the next set.

In the second set of conceptual development lessons, students focus on triangles. Students again fold paper to create figures and certain angles, such as complementary angles.

Students draw, using a protractor and ruler, other triangles with given properties. Students then explore triangles with certain known and unknown elements, such as the number of given sides and angles. This process starts with paper folding and drawing and continues with exploration of interactive sketches. Students draw conclusions about which cases allow 0, 1, 2, or an infinite number of triangles. In the course of the exploration, students discover that the sum of the measure of the interior angles of a triangle is 180°. They also learn that the sum of the measures of the interior angles of a quadrilateral is 360°. They explore other polygons to find their angle sum and determine if there is a relationship to angle sum of triangles. The exploration concludes with finding the measure of the interior angles of regular polygons and speculating about how this relates to a circle.

Lastly, students solve equations to find unknown angle measures. Using their previous experience, students find the remaining angle measures in a parallelogram when only one angle measure is given. Students also play a game similar to 20 Questions to identify types of triangles and quadrilaterals. Having completed the remaining lessons, students have a four-day Gallery to explore a variety of problems.

The unit ends with a unit assessment.

Subject:
Geometry
Mathematics
Provider:
Pearson
Conditional Remix & Share Permitted
CC BY-NC
Rating
0.0 stars

Students critique and revise their work from the Self Check after receiving feedback. Students then take a quiz to review the goals of the unit.Key ConceptsStudents reflect on their work and apply what they've learned about the characteristics of geometric figures.Goals and Learning ObjectivesCritique and revise work on the Self Check.Apply skills learned in the unit.Understand the relationship of angles:Created by intersecting lines.Found in quadrilaterals, triangles, and polygons.

Subject:
Geometry
Material Type:
Lesson Plan
Author:
Pearson
11/02/2020
Conditional Remix & Share Permitted
CC BY-NC
Rating
0.0 stars

Samples and ProbabilityType of Unit: ConceptualPrior KnowledgeStudents should be able to:Understand the concept of a ratio.Write ratios as percents.Describe data using measures of center.Display and interpret data in dot plots, histograms, and box plots.Lesson FlowStudents begin to think about probability by considering the relative likelihood of familiar events on the continuum between impossible and certain. Students begin to formalize this understanding of probability. They are introduced to the concept of probability as a measure of likelihood, and how to calculate probability of equally likely events using a ratio. The terms (impossible, certain, etc.) are given numerical values. Next, students compare expected results to actual results by calculating the probability of an event and conducting an experiment. Students explore the probability of outcomes that are not equally likely. They collect data to estimate the experimental probabilities. They use ratio and proportion to predict results for a large number of trials. Students learn about compound events. They use tree diagrams, tables, and systematic lists as tools to find the sample space. They determine the theoretical probability of first independent, and then dependent events. In Lesson 10 students identify a question to investigate for a unit project and submit a proposal. They then complete a Self Check. In Lesson 11, students review the results of the Self Check, solve a related problem, and take a Quiz.Students are introduced to the concept of sampling as a method of determining characteristics of a population. They consider how a sample can be random or biased, and think about methods for randomly sampling a population to ensure that it is representative. In Lesson 13, students collect and analyze data for their unit project. Students begin to apply their knowledge of statistics learned in sixth grade. They determine the typical class score from a sample of the population, and reason about the representativeness of the sample. Then, students begin to develop intuition about appropriate sample size by conducting an experiment. They compare different sample sizes, and decide whether increasing the sample size improves the results. In Lesson 16 and Lesson 17, students compare two data sets using any tools they wish. Students will be reminded of Mean Average Deviation (MAD), which will be a useful tool in this situation. Students complete another Self Check, review the results of their Self Check, and solve additional problems. The unit ends with three days for students to work on Gallery problems, possibly using one of the days to complete their project or get help on their project if needed, two days for students to present their unit projects to the class, and one day for the End of Unit Assessment.

Subject:
Mathematics
Statistics and Probability
Provider:
Pearson
Conditional Remix & Share Permitted
CC BY-NC
Rating
0.0 stars

Lesson OverviewGroups will begin presentations for their unit project. Students will provide constructive feedback on others' presentations.Key ConceptsStudents should demonstrate their understanding of the unit concepts.Goals and Learning ObjectivesPresent projects and demonstrate an understanding of the unit concepts.Provide feedback for others' presentations.Clarify any misconceptions or areas of difficulty.Review the concepts from the unit.

Subject:
Mathematics
Material Type:
Lesson Plan
Author:
Pearson
11/02/2020
Conditional Remix & Share Permitted
CC BY-NC
Rating
0.0 stars

Remaining groups present their unit projects and students discuss teacher and peer feedback.Key ConceptsStudents should demonstrate their understanding of the unit concepts.Goals and Learning ObjectivesPresent projects and demonstrate an understanding of the unit concepts.Provide feedback for others' presentations.Review the concepts from the unit.Review presentation feedback and reflect.

Subject:
Mathematics
Material Type:
Lesson Plan
Author:
Pearson
11/02/2020
Rating
0.0 stars

For this assessment, students measure the length of their desk using a ruler and write to tell how long their desk is in inches. Measuring the desk a second time, students write how long their desk is in centimeters. Lastly, students write to explain whether there were more inches or more centimeters and why.

Subject:
Mathematics
Material Type:
Assessment
Provider:
Hawaii State Department of Education
02/26/2019
Rating
0.0 stars

For this assessment, students measure the length of a chart paper using a ruler and write to tell how long the chart paper is in inches. Measuring the chart paper a second time, students write how long he paper is in feet. Lastly, students write to explain why the measurements are different.

Subject:
Mathematics
Material Type:
Assessment
Provider:
Hawaii State Department of Education
02/26/2019
Rating
0.0 stars

For this assessment, students are told that Justing just picked a variety of oranges from different trees. They must decide how many oranges Justing picked altogether.

Subject:
Mathematics
Material Type:
Assessment
Provider:
Hawaii State Department of Education
02/26/2019
Rating
0.0 stars

In this lesson from The New York Times Learning Network, students consider the value of a photograph and the circumstances under which photographs are taken, research the importance of a particular photograph, and create original photographs that illustrate the emotions, events or ideas portrayed in photographs of international news.

Subject:
Arts Education
Visual Arts
Material Type:
Lesson Plan
Provider:
New York Times
Author:
Michelle Sale and Javaid Khan
02/26/2019
Rating
0.0 stars

This is the news article attached to the New York Times lesson activity of the same name.

Subject:
Arts Education
Visual Arts
Material Type:
Lesson Plan
Provider:
New York Times
Author:
Michelle Sale and Javaid Khan
02/26/2019
Conditional Remix & Share Permitted
CC BY-SA
Rating
0.0 stars

This article discusses research findings about classroom questioning and provides tips for teachers and online and print resources.

Subject:
Science
Material Type:
Lesson Plan
Provider:
Ohio State University College of Education and Human Ecology
Provider Set:
Beyond Penguins and Polar Bears: An Online Magazine for K-5 Teachers
Author:
Jessica Fries-Gaither
07/30/2019
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars

This resource is part of Tools4NCTeachers.

This task calls for students to solve an addition problem using a variety of equations. The task may be used for instruction or assessment purposes.

Subject:
Mathematics
Material Type:
Assessment
Formative Assessment
07/10/2019
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars

This assessment may be used for instructional or assessment purposes. A scoring rubric is included.

Subject:
Mathematics
Material Type:
Assessment
Formative Assessment
11/07/2019
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars

This resource is part of Tools4NCTeachers.

Students are asked to solve a real world problem, show their work using pictures, numbers or words, and record a matching equation. A scoring rubric is included. The task may be used for instructional or assessment purposes.

Subject:
Mathematics
Material Type:
Assessment
Formative Assessment
07/11/2019
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars

This resource is part of Tools4NCTeachers.

Students are asked to solve a real world problem, show their work using pictures, numbers or words, and record a matching equation. A scoring rubric is included. The task may be used for instructional or assessment purposes.

Subject:
Mathematics
Material Type:
Assessment
Formative Assessment
07/10/2019
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars

This resource is part of Tools4NCTeachers.

In this task, students sort counting cubes by color, and count each group. The task may be used for instruction or as formative assessment. A rubric is provided in order to determine the students current level of understanding.

Subject:
Mathematics
Material Type:
Activity/Lab
Formative Assessment
07/05/2019
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars

This resource is part of Tools4NCTeachers.

In this task, students sort a set of pattern blocks into categories, and count the amount in each group. The task may be used for instruction or as formative assessment. A rubric is provided in order to determine the students current level of understanding.

Subject:
Mathematics
Material Type:
Activity/Lab
Formative Assessment
The text passage provides an overview of a unit on algebraic reasoning, focusing on the skills and concepts students should be able to apply, such as evaluating expressions, writing equivalent expressions, and solving equations and inequalities. It also mentions the use of various tools like protractors, rulers, and interactive sketches to explore these concepts. However, it does not contain a specific mathematical question or provide an answer to one.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Updating search results...

# 193 Results

View
Selected filters:
• assessment
Conditional Remix & Share Permitted
CC BY-NC
Rating
0.0 stars

Algebraic Reasoning

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Add, subtract, multiply, and divide rational numbers.
Evaluate expressions for a value of a variable.
Use the distributive property to generate equivalent expressions including combining like terms.
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?
Write and solve equations of the form x+p=q and px=q for cases in which p, q, and x are non-negative rational numbers.
Understand and graph solutions to inequalities x<c or x>c.
Use equations, tables, and graphs to represent the relationship between two variables.
Relate fractions, decimals, and percents.
Solve percent problems included those involving percent of increase or percent of decrease.

Lesson Flow

This unit covers all of the Common Core State Standards for Expressions and Equations in Grade 7. Students extend what they learned in Grade 6 about evaluating expressions and using properties to write equivalent expressions. They write, evaluate, and simplify expressions that now contain both positive and negative rational numbers. They write algebraic expressions for problem situations and discuss how different equivalent expressions can be used to represent different ways of solving the same problem. They make connections between various forms of rational numbers. Students apply what they learned in Grade 6 about solving equations such as x+2=6 or 3x=12 to solving equations such as 3x+6=12 and 3(x−2)=12. Students solve these equations using formal algebraic methods. The numbers in these equations can now be rational numbers. They use estimation and mental math to estimate solutions. They learn how solving linear inequalities differs from solving linear equations and then they solve and graph linear inequalities such as −3x+4<12. Students use inequalities to solve real-world problems, solving the problem first by arithmetic and then by writing and solving an inequality. They see that the solution of the algebraic inequality may differ from the solution to the problem.

Subject:
Algebra
Mathematics
Provider:
Pearson
Conditional Remix & Share Permitted
CC BY-NC
Rating
0.0 stars

Subject:
Mathematics
Material Type:
Lesson Plan
Author:
Pearson
11/02/2020
Conditional Remix & Share Permitted
CC BY-NC
Rating
0.0 stars

Constructions and Angles

Unit Overview

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Use a protractor and ruler.
Identify different types of triangles and quadrilaterals and their characteristics.

Lesson Flow

After an initial exploratory lesson involving a paper folding activity that gets students thinking in general about angles and figures in a context, the unit is divided into two concept development sections. The first section focuses on types of angles—adjacent, supplementary, complementary, and vertical—and how they are manifested in quadrilaterals. The second section looks at triangles and their properties, including the angle sum, and how this affects other figures.

In the first set of conceptual lessons, students explore different types of angles and where the types of angles appear in quadrilaterals. Students fold paper and observe the angles formed, draw given angles, and explore interactive sketches that test many cases. Students use a protractor and ruler to draw parallelograms with given properties. They explore sketches of parallelograms with specific properties, such as perpendicular diagonals. After concluding the investigation of the angle types, students move on to the next set.

In the second set of conceptual development lessons, students focus on triangles. Students again fold paper to create figures and certain angles, such as complementary angles.

Students draw, using a protractor and ruler, other triangles with given properties. Students then explore triangles with certain known and unknown elements, such as the number of given sides and angles. This process starts with paper folding and drawing and continues with exploration of interactive sketches. Students draw conclusions about which cases allow 0, 1, 2, or an infinite number of triangles. In the course of the exploration, students discover that the sum of the measure of the interior angles of a triangle is 180°. They also learn that the sum of the measures of the interior angles of a quadrilateral is 360°. They explore other polygons to find their angle sum and determine if there is a relationship to angle sum of triangles. The exploration concludes with finding the measure of the interior angles of regular polygons and speculating about how this relates to a circle.

Lastly, students solve equations to find unknown angle measures. Using their previous experience, students find the remaining angle measures in a parallelogram when only one angle measure is given. Students also play a game similar to 20 Questions to identify types of triangles and quadrilaterals. Having completed the remaining lessons, students have a four-day Gallery to explore a variety of problems.

The unit ends with a unit assessment.

Subject:
Geometry
Mathematics
Provider:
Pearson
Conditional Remix & Share Permitted
CC BY-NC
Rating
0.0 stars

Students critique and revise their work from the Self Check after receiving feedback. Students then take a quiz to review the goals of the unit.Key ConceptsStudents reflect on their work and apply what they've learned about the characteristics of geometric figures.Goals and Learning ObjectivesCritique and revise work on the Self Check.Apply skills learned in the unit.Understand the relationship of angles:Created by intersecting lines.Found in quadrilaterals, triangles, and polygons.

Subject:
Geometry
Material Type:
Lesson Plan
Author:
Pearson
11/02/2020
Conditional Remix & Share Permitted
CC BY-NC
Rating
0.0 stars

Samples and ProbabilityType of Unit: ConceptualPrior KnowledgeStudents should be able to:Understand the concept of a ratio.Write ratios as percents.Describe data using measures of center.Display and interpret data in dot plots, histograms, and box plots.Lesson FlowStudents begin to think about probability by considering the relative likelihood of familiar events on the continuum between impossible and certain. Students begin to formalize this understanding of probability. They are introduced to the concept of probability as a measure of likelihood, and how to calculate probability of equally likely events using a ratio. The terms (impossible, certain, etc.) are given numerical values. Next, students compare expected results to actual results by calculating the probability of an event and conducting an experiment. Students explore the probability of outcomes that are not equally likely. They collect data to estimate the experimental probabilities. They use ratio and proportion to predict results for a large number of trials. Students learn about compound events. They use tree diagrams, tables, and systematic lists as tools to find the sample space. They determine the theoretical probability of first independent, and then dependent events. In Lesson 10 students identify a question to investigate for a unit project and submit a proposal. They then complete a Self Check. In Lesson 11, students review the results of the Self Check, solve a related problem, and take a Quiz.Students are introduced to the concept of sampling as a method of determining characteristics of a population. They consider how a sample can be random or biased, and think about methods for randomly sampling a population to ensure that it is representative. In Lesson 13, students collect and analyze data for their unit project. Students begin to apply their knowledge of statistics learned in sixth grade. They determine the typical class score from a sample of the population, and reason about the representativeness of the sample. Then, students begin to develop intuition about appropriate sample size by conducting an experiment. They compare different sample sizes, and decide whether increasing the sample size improves the results. In Lesson 16 and Lesson 17, students compare two data sets using any tools they wish. Students will be reminded of Mean Average Deviation (MAD), which will be a useful tool in this situation. Students complete another Self Check, review the results of their Self Check, and solve additional problems. The unit ends with three days for students to work on Gallery problems, possibly using one of the days to complete their project or get help on their project if needed, two days for students to present their unit projects to the class, and one day for the End of Unit Assessment.

Subject:
Mathematics
Statistics and Probability
Provider:
Pearson
Conditional Remix & Share Permitted
CC BY-NC
Rating
0.0 stars

Lesson OverviewGroups will begin presentations for their unit project. Students will provide constructive feedback on others' presentations.Key ConceptsStudents should demonstrate their understanding of the unit concepts.Goals and Learning ObjectivesPresent projects and demonstrate an understanding of the unit concepts.Provide feedback for others' presentations.Clarify any misconceptions or areas of difficulty.Review the concepts from the unit.

Subject:
Mathematics
Material Type:
Lesson Plan
Author:
Pearson
11/02/2020
Conditional Remix & Share Permitted
CC BY-NC
Rating
0.0 stars

Remaining groups present their unit projects and students discuss teacher and peer feedback.Key ConceptsStudents should demonstrate their understanding of the unit concepts.Goals and Learning ObjectivesPresent projects and demonstrate an understanding of the unit concepts.Provide feedback for others' presentations.Review the concepts from the unit.Review presentation feedback and reflect.

Subject:
Mathematics
Material Type:
Lesson Plan
Author:
Pearson
11/02/2020
Rating
0.0 stars

For this assessment, students measure the length of their desk using a ruler and write to tell how long their desk is in inches. Measuring the desk a second time, students write how long their desk is in centimeters. Lastly, students write to explain whether there were more inches or more centimeters and why.

Subject:
Mathematics
Material Type:
Assessment
Provider:
Hawaii State Department of Education
02/26/2019
Rating
0.0 stars

For this assessment, students measure the length of a chart paper using a ruler and write to tell how long the chart paper is in inches. Measuring the chart paper a second time, students write how long he paper is in feet. Lastly, students write to explain why the measurements are different.

Subject:
Mathematics
Material Type:
Assessment
Provider:
Hawaii State Department of Education
02/26/2019
Rating
0.0 stars

For this assessment, students are told that Justing just picked a variety of oranges from different trees. They must decide how many oranges Justing picked altogether.

Subject:
Mathematics
Material Type:
Assessment
Provider:
Hawaii State Department of Education
02/26/2019
Rating
0.0 stars

In this lesson from The New York Times Learning Network, students consider the value of a photograph and the circumstances under which photographs are taken, research the importance of a particular photograph, and create original photographs that illustrate the emotions, events or ideas portrayed in photographs of international news.

Subject:
Arts Education
Visual Arts
Material Type:
Lesson Plan
Provider:
New York Times
Author:
Michelle Sale and Javaid Khan
02/26/2019
Rating
0.0 stars

This is the news article attached to the New York Times lesson activity of the same name.

Subject:
Arts Education
Visual Arts
Material Type:
Lesson Plan
Provider:
New York Times
Author:
Michelle Sale and Javaid Khan
02/26/2019
Conditional Remix & Share Permitted
CC BY-SA
Rating
0.0 stars

This article discusses research findings about classroom questioning and provides tips for teachers and online and print resources.

Subject:
Science
Material Type:
Lesson Plan
Provider:
Ohio State University College of Education and Human Ecology
Provider Set:
Beyond Penguins and Polar Bears: An Online Magazine for K-5 Teachers
Author:
Jessica Fries-Gaither
07/30/2019
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars

This resource is part of Tools4NCTeachers.

This task calls for students to solve an addition problem using a variety of equations. The task may be used for instruction or assessment purposes.

Subject:
Mathematics
Material Type:
Assessment
Formative Assessment
07/10/2019
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars

This assessment may be used for instructional or assessment purposes. A scoring rubric is included.

Subject:
Mathematics
Material Type:
Assessment
Formative Assessment
11/07/2019
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars

This resource is part of Tools4NCTeachers.

Students are asked to solve a real world problem, show their work using pictures, numbers or words, and record a matching equation. A scoring rubric is included. The task may be used for instructional or assessment purposes.

Subject:
Mathematics
Material Type:
Assessment
Formative Assessment
07/11/2019
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars

This resource is part of Tools4NCTeachers.

Students are asked to solve a real world problem, show their work using pictures, numbers or words, and record a matching equation. A scoring rubric is included. The task may be used for instructional or assessment purposes.

Subject:
Mathematics
Material Type:
Assessment
Formative Assessment
07/10/2019
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars

This resource is part of Tools4NCTeachers.

In this task, students sort counting cubes by color, and count each group. The task may be used for instruction or as formative assessment. A rubric is provided in order to determine the students current level of understanding.

Subject:
Mathematics
Material Type:
Activity/Lab
Formative Assessment
07/05/2019
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars

This resource is part of Tools4NCTeachers.

In this task, students sort a set of pattern blocks into categories, and count the amount in each group. The task may be used for instruction or as formative assessment. A rubric is provided in order to determine the students current level of understanding.

Subject:
Mathematics
Material Type:
Activity/Lab
Formative Assessment
user
The text passage provides an overview of a unit on algebraic reasoning, focusing on the skills and concepts students should be able to apply, such as evaluating expressions, writing equivalent expressions, and solving equations and inequalities. It also mentions the use of various tools like protractors, rulers, and interactive sketches to explore these concepts. However, it does not contain a specific mathematical question or provide an answer to one.

TYPE: NO_QUESTION