# MA.5.GR.1.1

Classify triangles or quadrilaterals into different categories based on shared defining attributes. Explain why a triangle or quadrilateral would or would not belong to a category.

### Clarifications

Clarification 1: Triangles include scalene, isosceles, equilateral, acute, obtuse and right; quadrilaterals include parallelograms, rhombi, rectangles, squares and trapezoids.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

• There are no direct connections outside of this standard; however, teachers are encouraged to find possible indirect connections.

### Terms from the K-12 Glossary

• Acute Triangle
• Equilateral Triangle
• Isosceles Triangle
• Obtuse Triangle
• Parallelograms
• Rectangle
• Rhombus
• Right Triangle
• Scalene Triangle
• Square
• Trapezoid
• Triangle

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

The purpose of this benchmark is for students to understand that shapes can be classified by their attributes and these attributes may place them in multiple categories. In grade 3, students identified and drew quadrilaterals based on their attributes (MA.3.GR.1.2). In grade 4, students explored angle classifications and measures in two-dimensional figures (MA.4.GR.1.1). This past work built the understanding required for students to classify triangles and quadrilaterals in grade 5. Classification of geometric figures will return in high school geometry (MA.912.GR.3.2) using another grade 5 concept, the coordinate plane.
• The work in grade 5 will help students to understand that triangles can be defined by two different attributes that students can measure: the length of their sides (3 congruent sides, 2 congruent sides, or 0 congruent sides) and the size of their angle measures (3 acute angles, 2 acute angles and a right angle, or 2 acute angles and an obtuse angle).
• During instruction, it is important for students to have practice with classifying figures in multiple ways so they can better understand the relationship between attributes of the geometric figures. In addition, students should practice this concept by using graphic organizers such as, flow charts, T-charts and Venn diagrams (MTR.2.1).
• This benchmark requires a strong understanding and use of geometry vocabulary. Allow students to use math discourse throughout instruction to compare the attributes of geometric figures. o For example, pose questions such as, “Why is a square always a rhombus?” and “Why is a rhombus not always a square?” Lesson activities should require students to justify their thinking when making mathematical arguments about geometric figures (MTR.4.1).

### Common Misconceptions or Errors

• Students may think that when describing and classifying geometric shapes and placing them in subcategories, the last subcategory is the only classification that can be used.
• Students may think that a geometric figure can only be classified in one way.
• For example, a square (a shape with 4 congruent sides and 4 congruent angles) can also be a parallelogram because it contains 2 pairs of sides that are congruent and parallel.

### Strategies to Support Tiered Instruction

• Instruction includes providing a graphic organizer and having students place triangles and/or quadrilaterals into all the subcategories they belong to. Students then identify all the ways the figure could be classified.
• For example, students are provided with a graphic organizer like the one shown below to help them classify figures into subcategories. The name of the figure, an example, and the definition are provided. Students then identify which other categories the figure would also fit. For example, a parallelogram is a quadrilateral containing two pairs of parallel sides. A rectangle, rhombus, and square all also have two pairs of parallel sides so they would also fit in this subcategory. The teacher refers to the glossary, included with the standards, for several examples to provide students.
• Instruction includes providing a graphic organizer and having students use sticky notes with specific attributes on them to help them classify figures.
• For example, students are provided with a graphic organizer like the one shown below with an example of the figure filled in for them to refer to and yellow sticky notes that have “4 equal sides” written on them. Students determine which figures contain this attribute and place the sticky note under those figures (square and rhombus). The teacher then provides green sticky notes with “two pairs of parallel sides” written on them. Students place the sticky note under each figure that has that attribute (parallelogram, rhombus, rectangle, and square). Students would continue to add different color sticky notes with attributes that say, “One pair of parallel sides” and “four right angles”. Students are able to see that some figures have several sticky notes and which figures have the same sticky notes. Students will then name all the ways a figure can be classified based on the attributes they have.

• Part A. Roll a number cube twice and write a statement based on the key below.
Number Cube Key
1 – Equilateral
2 – Acute
3 – Right
4 – Obtuse
5 – Isosceles
6 – Scalene
• Part B. Write a statement that reads, “A(n) ___________ (roll 1) triangle is ______________ (always, sometimes or never) a(n) ____________ triangle (roll 2).” Complete your statement by determining whether the category of triangle from roll 1 is always, sometimes, or never the category of triangle from roll 2. Complete this process three more times for a total of four statements.
• Part C. Choose one of the statements that you said is sometimes true. Give an example of when the statement is true and when the statement is not true using picture models or words. If none of your statements are sometimes true, then create one to give an example.

### Instructional Items

Instructional Item 1

Choose all the shapes that can always be classified as parallelograms.
• a. Trapezoid
• b. Rectangle
• c. Rhombus
• d. Square Equilateral Triangle

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
5012070: Grade Five Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7712060: Access Mathematics Grade 5 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))
5012065: Grade 4 Accelerated Mathematics (Specifically in versions: 2019 - 2022, 2022 and beyond (current))
5012015: Foundational Skills in Mathematics 3-5 (Specifically in versions: 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.5.GR.1.AP.1a: Sort triangles into different categories based on the size of their angles. Triangles include acute, obtuse and right.
MA.5.GR.1.AP.1b: Sort quadrilaterals into different categories based on shared defining attributes. Explore why a quadrilateral would or would not belong to a category. Quadrilaterals include parallelograms, rhombi, rectangles, squares and trapezoids.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Classifying Shapes:

Type: Formative Assessment

Shape Clues:

Using shape attribute clues, the student will determine shapes that fit these clues and describe a category to which they both belong.

Type: Formative Assessment

Where Do They Belong:

Students are asked to sort shapes using a Venn diagram and to determine the label for each section of the diagram.

Type: Formative Assessment

Guess My Shape:

Using shape attribute clues, the student is asked to determine two shapes that fit these clues and describe a category to which both belong.

Type: Formative Assessment

Grouping Triangles:

Students are shown three triangles that fit a rule (each has a right angle) and are asked to determine which of three other triangles also fit the rule.

Type: Formative Assessment

Classifying Squares:

Students discuss the attributes of squares and rhombuses and consider how these two shapes are related.

Type: Formative Assessment

Students are given a diagram of quadrilaterals that have been sorted and are asked to determine how the shapes were sorted. Then, students are given two additional quadrilaterals and asked to place them into the appropriate region on the diagram.

Type: Formative Assessment

What Do You Know About Rectangles?:

Students are asked to describe the defining attributes of parallelograms and to consider the relationship between rectangles and parallelograms.

Type: Formative Assessment

Trapezoids:

Students are asked to consider how the hierarchy of quadrilaterals would change based on the two different definitions of trapezoids.

Type: Formative Assessment

Sketching Triangles:

Students are asked to use shape descriptions to sketch shapes and explain why some cannot be sketched.

Type: Formative Assessment

## Image/Photograph

Clipart: Geometric Shapes:

In this lesson, you will find clip art and various illustrations of polygons, circles, ellipses, star polygons, and inscribed shapes.

Type: Image/Photograph

## Lesson Plans

Polygon Park:

In this lesson students will use standards-based quadrilaterals and triangles to design a roller coaster tower. Students will use the Engineering Design Process to work through the processes in this lesson.

Type: Lesson Plan

Go TRI a Kite!:

In this lesson, students will identify and classify triangles. In addition, students will demonstrate how to solve problems requiring attention to approximation, selection of appropriate measuring tools, and precision of measurement using triangles.

Type: Lesson Plan

Where in the Venn are the Quadrilaterals?:

"Where in the Venn are the Quadrilaterals?" is an activity that helps the student to develop a better understanding of classifying two-dimensional figures in a hierarchy based on properties.

Type: Lesson Plan

A Closer Look at Quadrilaterals - The Parallelogram Connection:

Students work together as a class and in small groups to classify quadrilaterals.

Type: Lesson Plan

Quadrilaterals using Unit Origami - Sonobe Cube:

This lesson can be used as an introduction - unit attention grabber- or as a final review on quadrilaterals. As the class forms a Sonobe cube, the different quadrilaterals are formed with each new fold. Included is a Power Point introduction with instructions, a video I made demonstrating how to fold the unit origami design, a video of my introduction lesson  to quadrilaterals and 2 worksheets which accompany the lesson. I enjoy doing this activity each year and am amazed at all the prior knowledge the students have retained as the lesson proceeds.

Type: Lesson Plan

Triangles are Plane Easy:

The student will be engaged in a paper plane making activity while discovering the attributes of different triangles. The students will learned the similarities and differences of the following triangles: scalene, isosceles, equilateral, right, obtuse, and acute.

Type: Lesson Plan

## Original Student Tutorials

Action ‘Figures’ Unite! - Part 8:

Apply your understanding of the defining attributes of all 2-dimensional figures covered in this series to classify their relationships using Euler and Venn Diagrams.

This part 8 in a 8-part series. Click below to explore the other tutorials in the series.

Part 1: "Figuring Out" 2D Figures

Part 2: Exploring Relationships with Venn & Euler Diagrams

Part 3: Classifying Triangles by Angles Using Euler Diagrams

Part 4: Classifying Triangles by Sides & Angles Using Venn and Euler Diagrams

Part 6: Trapezoids

Type: Original Student Tutorial

By the end of this tutorial you should be able to identify examples of quadrilaterals and their defining attributes to classify them using diagrams. We will focus on kites and other quadrilaterals in this tutorial.

This part 7 in a 7-part series. Click below to explore the other tutorials in the series.

Type: Original Student Tutorial

Trapezoids Part 6:

Explore the defining attributes of trapezoids--a special type of quadrilateral--and classify them using diagrams in this interactive tutorial. You'll also learn how two different definitions for a trapezoid can change affect classifications of quadrilaterals.

This part 6 in a 6-part series. Click below to explore the other tutorials in the series.

Part 1: "Figuring Out" 2D Figures

Part 2: Exploring Relationships with Venn & Euler Diagrams

Part 3: Classifying Triangles by Angles Using Euler Diagrams

Part 4: Classifying Triangles by Sides & Angles Using Venn and Euler Diagrams

Type: Original Student Tutorial

Learn how to classify quadrilaterals--including parallelograms, rectangles, rhombi, and squares--based on their defining attributes using diagrams in this interactive tutorial.

This is part 5 in a 6-part series. Click below to explore the other tutorials in the series.

Type: Original Student Tutorial

Exploring Relationships with Venn & Euler Diagrams – Part 2:

Learn how to show relationships represented in Venn & Euler Diagrams as you complete this interactive geometry tutorial.

This is part two of four. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Classifying Triangles by Sides & Angles Using Venn and Euler Diagrams – Part 4:

Learn how triangles can be sorted and classified using side lengths and angle measures in this interactive tutorial.

This is the final tutorial in a four-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Classifying Triangles by Angles using Euler Diagrams – Part 3:

Learn to classify triangles and use Euler diagrams to show relationships, in this interactive tutorial.

This is part-three of four. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

"Figuring Out" 2D Figures - Part 1:

Explore 2D (two-dimensional) figures and see how every 2D figure possesses unique attributes in this interactive tutorial.

This is part one of four. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

The Nightmare of Pouring Polygons: Quadrilateral Classification:

Overcome the nightmare of quadrilateral classification based on the presence of parallel, perpendicular, and congruent sides as you complete this interactive tutorial. Learn about parallelogram, rectangles, rhombi and squares and how they are related.

Type: Original Student Tutorial

## Perspectives Video: Teaching Ideas

Estimating Angles:

Unlock an effective teaching strategy for teaching students to estimate angle measurements in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

Double Bubble Diagram to Compare Math Concepts:

Unlock an effective teaching strategy for helping students make connections in mathematics using the Double Bubble Diagram in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

Lines of symmetry for triangles:

This activity provides students an opportunity to recognize these distinguishing features of the different types of triangles before the technical language has been introduced. For finding the lines of symmetry, cut-out models of the four triangles would be helpful so that the students can fold them to find the lines.

This task provides students a chance to experiment with reflections of the plane and their impact on specific types of quadrilaterals. It is both interesting and important that these types of quadrilaterals can be distinguished by their lines of symmetry.

Are these right?:

The purpose of this task is for students to measure angles and decide whether the triangles are right or not. Students should already understand concepts of angle measurement  and know how to measure angles using a protractor before working on this task.

## Teaching Idea

This lesson is designed to introduce students to quadrilaterals and the terms and properties associated with quadrilaterals. This lesson provides links to discussions and activities related to quadrilaterals as well as suggested ways to integrate them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

Type: Teaching Idea

## Tutorials

Categorizing Geometric Shapes: Practice Examples:

This Khan Academy tutorial video presents examples and explanations for categorizations of perpendicular sides and right, obtuse, and acute triangles.

Type: Tutorial

Examples of Triangle Categories:

In this Khan Academy tutorial video triangles are categorized by angles or side lengths of a specified size.

Type: Tutorial

Type: Tutorial

## MFAS Formative Assessments

Students are given a diagram of quadrilaterals that have been sorted and are asked to determine how the shapes were sorted. Then, students are given two additional quadrilaterals and asked to place them into the appropriate region on the diagram.

Classifying Shapes:

Classifying Squares:

Students discuss the attributes of squares and rhombuses and consider how these two shapes are related.

Grouping Triangles:

Students are shown three triangles that fit a rule (each has a right angle) and are asked to determine which of three other triangles also fit the rule.

Guess My Shape:

Using shape attribute clues, the student is asked to determine two shapes that fit these clues and describe a category to which both belong.

Shape Clues:

Using shape attribute clues, the student will determine shapes that fit these clues and describe a category to which they both belong.

Sketching Triangles:

Students are asked to use shape descriptions to sketch shapes and explain why some cannot be sketched.

Trapezoids:

Students are asked to consider how the hierarchy of quadrilaterals would change based on the two different definitions of trapezoids.

What Do You Know About Rectangles?:

Students are asked to describe the defining attributes of parallelograms and to consider the relationship between rectangles and parallelograms.

Where Do They Belong:

Students are asked to sort shapes using a Venn diagram and to determine the label for each section of the diagram.

## Original Student Tutorials Mathematics - Grades K-5

"Figuring Out" 2D Figures - Part 1:

Explore 2D (two-dimensional) figures and see how every 2D figure possesses unique attributes in this interactive tutorial.

This is part one of four. Click below to open the other tutorials in the series.

Action ‘Figures’ Unite! - Part 8:

Apply your understanding of the defining attributes of all 2-dimensional figures covered in this series to classify their relationships using Euler and Venn Diagrams.

This part 8 in a 8-part series. Click below to explore the other tutorials in the series.

Part 1: "Figuring Out" 2D Figures

Part 2: Exploring Relationships with Venn & Euler Diagrams

Part 3: Classifying Triangles by Angles Using Euler Diagrams

Part 4: Classifying Triangles by Sides & Angles Using Venn and Euler Diagrams

Part 6: Trapezoids

Classifying Triangles by Angles using Euler Diagrams – Part 3:

Learn to classify triangles and use Euler diagrams to show relationships, in this interactive tutorial.

This is part-three of four. Click below to open the other tutorials in the series.

Classifying Triangles by Sides & Angles Using Venn and Euler Diagrams – Part 4:

Learn how triangles can be sorted and classified using side lengths and angle measures in this interactive tutorial.

This is the final tutorial in a four-part series. Click below to open the other tutorials in the series.

Exploring Relationships with Venn & Euler Diagrams – Part 2:

Learn how to show relationships represented in Venn & Euler Diagrams as you complete this interactive geometry tutorial.

This is part two of four. Click below to open the other tutorials in the series.

By the end of this tutorial you should be able to identify examples of quadrilaterals and their defining attributes to classify them using diagrams. We will focus on kites and other quadrilaterals in this tutorial.

This part 7 in a 7-part series. Click below to explore the other tutorials in the series.

Learn how to classify quadrilaterals--including parallelograms, rectangles, rhombi, and squares--based on their defining attributes using diagrams in this interactive tutorial.

This is part 5 in a 6-part series. Click below to explore the other tutorials in the series.

The Nightmare of Pouring Polygons: Quadrilateral Classification:

Overcome the nightmare of quadrilateral classification based on the presence of parallel, perpendicular, and congruent sides as you complete this interactive tutorial. Learn about parallelogram, rectangles, rhombi and squares and how they are related.

Trapezoids Part 6:

Explore the defining attributes of trapezoids--a special type of quadrilateral--and classify them using diagrams in this interactive tutorial. You'll also learn how two different definitions for a trapezoid can change affect classifications of quadrilaterals.

This part 6 in a 6-part series. Click below to explore the other tutorials in the series.

Part 1: "Figuring Out" 2D Figures

Part 2: Exploring Relationships with Venn & Euler Diagrams

Part 3: Classifying Triangles by Angles Using Euler Diagrams

Part 4: Classifying Triangles by Sides & Angles Using Venn and Euler Diagrams

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorials

Action ‘Figures’ Unite! - Part 8:

Apply your understanding of the defining attributes of all 2-dimensional figures covered in this series to classify their relationships using Euler and Venn Diagrams.

This part 8 in a 8-part series. Click below to explore the other tutorials in the series.

Part 1: "Figuring Out" 2D Figures

Part 2: Exploring Relationships with Venn & Euler Diagrams

Part 3: Classifying Triangles by Angles Using Euler Diagrams

Part 4: Classifying Triangles by Sides & Angles Using Venn and Euler Diagrams

Part 6: Trapezoids

Type: Original Student Tutorial

By the end of this tutorial you should be able to identify examples of quadrilaterals and their defining attributes to classify them using diagrams. We will focus on kites and other quadrilaterals in this tutorial.

This part 7 in a 7-part series. Click below to explore the other tutorials in the series.

Type: Original Student Tutorial

Trapezoids Part 6:

Explore the defining attributes of trapezoids--a special type of quadrilateral--and classify them using diagrams in this interactive tutorial. You'll also learn how two different definitions for a trapezoid can change affect classifications of quadrilaterals.

This part 6 in a 6-part series. Click below to explore the other tutorials in the series.

Part 1: "Figuring Out" 2D Figures

Part 2: Exploring Relationships with Venn & Euler Diagrams

Part 3: Classifying Triangles by Angles Using Euler Diagrams

Part 4: Classifying Triangles by Sides & Angles Using Venn and Euler Diagrams

Type: Original Student Tutorial

Learn how to classify quadrilaterals--including parallelograms, rectangles, rhombi, and squares--based on their defining attributes using diagrams in this interactive tutorial.

This is part 5 in a 6-part series. Click below to explore the other tutorials in the series.

Type: Original Student Tutorial

Exploring Relationships with Venn & Euler Diagrams – Part 2:

Learn how to show relationships represented in Venn & Euler Diagrams as you complete this interactive geometry tutorial.

This is part two of four. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Classifying Triangles by Sides & Angles Using Venn and Euler Diagrams – Part 4:

Learn how triangles can be sorted and classified using side lengths and angle measures in this interactive tutorial.

This is the final tutorial in a four-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Classifying Triangles by Angles using Euler Diagrams – Part 3:

Learn to classify triangles and use Euler diagrams to show relationships, in this interactive tutorial.

This is part-three of four. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

"Figuring Out" 2D Figures - Part 1:

Explore 2D (two-dimensional) figures and see how every 2D figure possesses unique attributes in this interactive tutorial.

This is part one of four. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

The Nightmare of Pouring Polygons: Quadrilateral Classification:

Overcome the nightmare of quadrilateral classification based on the presence of parallel, perpendicular, and congruent sides as you complete this interactive tutorial. Learn about parallelogram, rectangles, rhombi and squares and how they are related.

Type: Original Student Tutorial

Lines of symmetry for triangles:

This activity provides students an opportunity to recognize these distinguishing features of the different types of triangles before the technical language has been introduced. For finding the lines of symmetry, cut-out models of the four triangles would be helpful so that the students can fold them to find the lines.

This task provides students a chance to experiment with reflections of the plane and their impact on specific types of quadrilaterals. It is both interesting and important that these types of quadrilaterals can be distinguished by their lines of symmetry.

Are these right?:

The purpose of this task is for students to measure angles and decide whether the triangles are right or not. Students should already understand concepts of angle measurement  and know how to measure angles using a protractor before working on this task.

## Tutorials

Categorizing Geometric Shapes: Practice Examples:

This Khan Academy tutorial video presents examples and explanations for categorizations of perpendicular sides and right, obtuse, and acute triangles.

Type: Tutorial

Examples of Triangle Categories:

In this Khan Academy tutorial video triangles are categorized by angles or side lengths of a specified size.

Type: Tutorial

Type: Tutorial

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

## Image/Photograph

Clipart: Geometric Shapes:

In this lesson, you will find clip art and various illustrations of polygons, circles, ellipses, star polygons, and inscribed shapes.

Type: Image/Photograph

Lines of symmetry for triangles:

This activity provides students an opportunity to recognize these distinguishing features of the different types of triangles before the technical language has been introduced. For finding the lines of symmetry, cut-out models of the four triangles would be helpful so that the students can fold them to find the lines.