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Clarifications
Clarification 1: Triangles include scalene, isosceles, equilateral, acute, obtuse and right; quadrilaterals include parallelograms, rhombi, rectangles, squares and trapezoids.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
- Quadrilateral
- 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.
- Instruction may include providing a variety of graphic organizers that allow students to build physical and cognitive connections
- For example, have students create or provide them with a template to complete a double-bubble graphic organizer; then allow them to compare and contrast assorted examples.
Instructional Tasks
Instructional Task 1 (MTR.3.1, MTR.4.1)
- Part A. Roll a number cube twice and write a statement based on the key below.
- 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
What type of triangle is shown?
- a. Equilateral Triangle
- b. Isosceles Triangle
- c. Obtuse Triangle
- d. Scalene Triangle
Instructional Item 3
Which shape is an obtuse triangle?
Instructional Item 4
Using the table, identify the angles of the triangles by putting an X in the corresponding column.
Instructional Item 5
The figure below has equal side lengths. Classify the figure, selecting all that apply.
- a. Quadrilateral
- b. Rectangle
- c. Rhombus
- d. Square
- e. Trapezoid
Instructional Item 6
Which shape is a rhombus?
Instructional Item 7
Using the table, identify the angles of the triangles by putting an X in the corresponding column.
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
Related Access Points
Related Resources
Formative Assessments
Image/Photograph
Lesson Plans
Original Student Tutorials
Perspectives Video: Teaching Ideas
Problem-Solving Tasks
Teaching Idea
Tutorials
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.
Students are asked to classify quadrilaterals and trapezoids by their properties.
Students discuss the attributes of squares and rhombuses and consider how these two shapes are related.
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.
Using shape attribute clues, the student is asked to determine two shapes that fit these clues and describe a category to which both belong.
Using shape attribute clues, the student will determine shapes that fit these clues and describe a category to which they both belong.
Students are asked to use shape descriptions to sketch shapes and explain why some cannot be sketched.
Students are asked to consider how the hierarchy of quadrilaterals would change based on the two different definitions of trapezoids.
Students are asked to describe the defining attributes of parallelograms and to consider the relationship between rectangles and parallelograms.
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
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.
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
Identify and classify quadrilaterals, including squares, rectangles, and rhombi, using their defining attributes in tables and Venn diagrams with this interactive tutorial.
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.
- Part 1: "Figuring Out" 2D Figures - Part 1
- Part 2 Exploring Relationships with Venn & Euler Diagrams
- Part 3: Classifying Triangles by Angles using Euler Diagrams
- Part 4: Classifying Triangles by Sides and Angles using Venn and Euler Diagrams
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.
- Part 1: "Figuring Out" 2D Figures - Part 1
- Part 2 Exploring Relationships with Venn & Euler Diagrams
- Part 3: Classifying Triangles by Angles using Euler Diagrams
- Part 4: Classifying Triangles by Sides and Angles using Venn and Euler Diagrams
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.
- Part 1: "Figuring Out" 2D Figures - Part 1
- Part 2 Exploring Relationships with Venn & Euler Diagrams
- Part 3: Classifying Triangles by Angles using Euler Diagrams
- Part 4: Classifying Triangles by Sides and Angles using Venn and Euler Diagrams
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.
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 5: Quadrilaterals
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.
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.
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
Original Student Tutorials
Identify and classify quadrilaterals, including squares, rectangles, and rhombi, using their defining attributes in tables and Venn diagrams with this interactive tutorial.
Type: Original Student Tutorial
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
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.
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 5: Quadrilaterals
Type: Original Student Tutorial
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.
- 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: (Coming Soon)
Type: Original Student Tutorial
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.
- Part 1: "Figuring Out" 2D Figures - Part 1
- Part 2 Exploring Relationships with Venn & Euler Diagrams
- Part 3: Classifying Triangles by Angles using Euler Diagrams
- Part 4: Classifying Triangles by Sides and Angles using Venn and Euler Diagrams
Type: Original Student Tutorial
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.
- Part 1: "Figuring Out" 2D Figures - Part 1
- Part 2 Exploring Relationships with Venn & Euler Diagrams
- Part 3: Classifying Triangles by Angles using Euler Diagrams
- Part 4: Classifying Triangles by Sides and Angles using Venn and Euler Diagrams
Type: Original Student Tutorial
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.
- Part 1: "Figuring Out" 2D Figures - Part 1
- Part 2 Exploring Relationships with Venn & Euler Diagrams
- Part 3: Classifying Triangles by Angles using Euler Diagrams
- Part 4: Classifying Triangles by Sides and Angles using Venn and Euler Diagrams
Type: Original Student Tutorial
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.
- Part 1: "Figuring Out" 2D Figures - Part 1
- Part 2 Exploring Relationships with Venn & Euler Diagrams
- Part 3: Classifying Triangles by Angles using Euler Diagrams
- Part 4: Classifying Triangles by Sides and Angles using Venn and Euler Diagrams
Type: Original Student Tutorial
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
Problem-Solving Tasks
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.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
Tutorials
This Khan Academy tutorial video presents examples and explanations for categorizations of perpendicular sides and right, obtuse, and acute triangles.
Type: Tutorial
In this Khan Academy tutorial video triangles are categorized by angles or side lengths of a specified size.
Type: Tutorial
This Khan Academy tutorial video introduces quadrilaterals. their categories, and subcategories.
Type: Tutorial
Parent Resources
Image/Photograph
In this lesson, you will find clip art and various illustrations of polygons, circles, ellipses, star polygons, and inscribed shapes.
Type: Image/Photograph
Problem-Solving Tasks
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.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
