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March 13, 2017

# The Greedy Triangle: Geometry for Every Grade

Picture books have long been a favorite way of introducing mathematical concepts in my class. What’s even better is when you can take one book and apply it across grade levels. Mathematical concepts grow throughout the grades, and perhaps nowhere is this more evident than in geometry, when the concept of shape starts before students even enter school. Marilyn Burns, mathematician and author extraordinaire, captures a simple idea of adding sides to a triangle in The Greedy Triangle but this one picture book can support learning throughout the grades.

Extended Learning | Printable Manipulatives | Great Geometry Resources

## Kindergarten

Standards

K.G.A.2 Correctly name shapes regardless of their orientations or overall size.

K.G.B.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.

Learning Objective

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).  Note: 3-D shapes are not in The Greedy Triangle.

Activity

Read The Greedy Triangle aloud. Pause to let students guess what the triangle will become each time he adds a side. “Quadrilateral” is not a word kindergarteners need to know, but if they can identify that a quadrilateral is a square or rectangle, that is supporting future math vocabulary.

After reading, have students become their favorite shape and tell what they would be, the same way The Greedy Triangle enjoys being a book page as a quadrilateral, or a beehive as a hexagon. Supply students shapes to cut out and create their image. Download a sentence stem starter, or let students construct their own.

Standards

1.G.A.2 Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Learning Objective

Reason with shapes and their attributes

Activity

Read The Greedy Triangle and pause to let students guess the next shape. Discuss the attributes of each shape. Ask students if it matters that the triangle is a certain color or orientation throughout the book.

Go on a “shape hunt” in the room or school. Look for items that are various shapes. Students should be able to tell the difference between a square and rectangle and also watch for the misconception that non-equilateral triangles are not actually triangles – they are!

Provide students with geoblocks to trace or paper cutouts of shapes. Allow them to construct a new shape using smaller shapes. Then have them write what their shape became. Early finishers can try and make shapes such as hexagons in all the various ways possible using smaller shapes. For added craftiness, have students use their shapes in a picture with the stem “My shape became a…”

Standards

2.GA.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Learning Objective

Reason with shapes and their attributes

Activity

Read The Greedy Triangle and discuss the attributes of each shape. Create an anchor chart of the different shapes and their attributes. Create some categories, such as quadrilaterals, and show how multiple shapes fit into the category. While reading, look for those various shapes on the page.

After reading, play “Guess My Rule.” Using a variety of 2-D blocks or cutouts, sort your shapes into groups using a specific attribute. Have students guess what “rule” you used to sort your shapes. They can ask yes/no questions to help decide the rule. Create trickier rules with more groups, multiple rules at once, or more complex shapes. Give students a variety of geoblocks to work with and allow them to play in partners, taking turns creating their own rules and having the partner guess the rule. To record and display work, use cutout shapes and have students glue and label their work.

Standards

3.GA.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

Learning Objective

Reason with shapes and their attributes

Activity

Read The Greedy Triangle and pause when the shape changes. Decide what other shapes could fit in the category, such as squares, rhombuses, and rectangles all belonging to the quadrilateral page. Create an anchor chart to display student thinking.

After reading, challenge groups to create a poster that shows the relationship of different shapes, particularly in the quadrilaterals category. For example, students could show that a square fits in the category of rectangles and quadrilaterals. Have students create non-examples as well and then allow time for groups to present their posters.

Standards

G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

Learning Objective

Classify shapes by properties of their lines and angles

Activity

While reading, ask students to describe what shape will come next using number of sides, vertices, types of angles, parallel and non-parallel sides, etc. Students may be able to describe more than one shape because more than one type of quadrilateral exists. Engage students in a discussion of regular versus irregular polygons.

After reading, have students draw a large equilateral triangle, like the triangle in the story. Check for understanding of what defines an equilateral triangle. Then have students create another shape, even if it is irregular. Have students write the characteristics of the shape they made including types of sides, angles, vertices, and any other defining characteristic. Then, have students create a short story about how the triangle became their new shape. If time allows, students can illustrate their shapes to go with their stories.

Standards

G.B.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
G.B.4 Classify two-dimensional figures in a hierarchy based on properties.

Learning Objective
Classify two-dimensional figures into categories based on their properties

Activity
While reading, ask students to identify all the different types of shapes that fit into the polygon category that the triangle becomes. Students can use geoboards or drawings to create different quadrilaterals, pentagons, hexagons, and more. When the triangle starts rapidly going through shape options, slow down and allow time for students to make each shape. Before the triangle changes into a circle, ask students to create a regular polygon with as many sides as they can using their geoboard. Ask students what would happen as the sides increase. Look for an understanding that the angles become more and more obtuse and eventually will look like a circle. Discuss if a circle can be a polygon (it can’t) and why.

Pass out various shape cards to each student. Hold up a category card, such as “hexagon” and have every student holding a hexagon stand with their card. Repeat this with various categories. Be sure to include sub-categories such as “rectangle” to determine if students realize the hierarchy of squares. Then, use the cards to create a chart with a hierarchy of polygons.

## Extended Learning

As students get older, they will apply angle measurements to polygons, understanding the sum of the angle measurements of a triangle is always 180 degrees. While this is typically taught in middle school and beyond, it is accessible for younger students who understand measuring angles or can be done as a whole-class activity.

Learning Objective

Determine the pattern of angle measures in polygons

Activities

While reading, have students measure the angles in an equilateral triangle, like The Greedy Triangle, and begin recording a chart for each type of polygon and the sum of the angles in that polygon. As the triangle changes shape, discuss the attributes of the shape, specifically focusing on how many angles the shape has. Measure and record the sum of the angles in the various polygons as the triangle changes.

After reading, challenge the students to see a pattern in the change in angle measurements when a polygon adds one side. To find why this is 180 degrees every time, draw triangles inside the polygon and notice that additional sides create additional triangles that can be drawn.

If you want to extend further, have students measure the exterior angles of the polygons and see that they always total 360 degrees. Older students may quickly realize this is the same as the degrees in a circle!

To make it cross-curricular, have students write a new version of The Greedy Triangle using changing angle measurements instead of changing sides.

## Print Manipulatives

Need some great shape manipulatives? Try printing from these great sites:

## Geometry Resources

My fellow Scholastic bloggers have great geometry resources to keep your learning going. Try some of these great posts:

·       “Five Ways to Make Geometry Memorable

·      “Three Great Geometry Games

Picture books are a great way to generate interest and learning, no matter the grade level. One simple story like The Greedy Triangle can open a world of possible in the math classroom at any age. Click here to purchase a copy of The Greedy Triangle for your classroom.

What books do you use in your math lessons?

Picture books have long been a favorite way of introducing mathematical concepts in my class. What’s even better is when you can take one book and apply it across grade levels. Mathematical concepts grow throughout the grades, and perhaps nowhere is this more evident than in geometry, when the concept of shape starts before students even enter school. Marilyn Burns, mathematician and author extraordinaire, captures a simple idea of adding sides to a triangle in The Greedy Triangle but this one picture book can support learning throughout the grades.

Extended Learning | Printable Manipulatives | Great Geometry Resources

## Kindergarten

Standards

K.G.A.2 Correctly name shapes regardless of their orientations or overall size.

K.G.B.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.

Learning Objective

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).  Note: 3-D shapes are not in The Greedy Triangle.

Activity

Read The Greedy Triangle aloud. Pause to let students guess what the triangle will become each time he adds a side. “Quadrilateral” is not a word kindergarteners need to know, but if they can identify that a quadrilateral is a square or rectangle, that is supporting future math vocabulary.

After reading, have students become their favorite shape and tell what they would be, the same way The Greedy Triangle enjoys being a book page as a quadrilateral, or a beehive as a hexagon. Supply students shapes to cut out and create their image. Download a sentence stem starter, or let students construct their own.

Standards

1.G.A.2 Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Learning Objective

Reason with shapes and their attributes

Activity

Read The Greedy Triangle and pause to let students guess the next shape. Discuss the attributes of each shape. Ask students if it matters that the triangle is a certain color or orientation throughout the book.

Go on a “shape hunt” in the room or school. Look for items that are various shapes. Students should be able to tell the difference between a square and rectangle and also watch for the misconception that non-equilateral triangles are not actually triangles – they are!

Provide students with geoblocks to trace or paper cutouts of shapes. Allow them to construct a new shape using smaller shapes. Then have them write what their shape became. Early finishers can try and make shapes such as hexagons in all the various ways possible using smaller shapes. For added craftiness, have students use their shapes in a picture with the stem “My shape became a…”

Standards

2.GA.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Learning Objective

Reason with shapes and their attributes

Activity

Read The Greedy Triangle and discuss the attributes of each shape. Create an anchor chart of the different shapes and their attributes. Create some categories, such as quadrilaterals, and show how multiple shapes fit into the category. While reading, look for those various shapes on the page.

After reading, play “Guess My Rule.” Using a variety of 2-D blocks or cutouts, sort your shapes into groups using a specific attribute. Have students guess what “rule” you used to sort your shapes. They can ask yes/no questions to help decide the rule. Create trickier rules with more groups, multiple rules at once, or more complex shapes. Give students a variety of geoblocks to work with and allow them to play in partners, taking turns creating their own rules and having the partner guess the rule. To record and display work, use cutout shapes and have students glue and label their work.

Standards

3.GA.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

Learning Objective

Reason with shapes and their attributes

Activity

Read The Greedy Triangle and pause when the shape changes. Decide what other shapes could fit in the category, such as squares, rhombuses, and rectangles all belonging to the quadrilateral page. Create an anchor chart to display student thinking.

After reading, challenge groups to create a poster that shows the relationship of different shapes, particularly in the quadrilaterals category. For example, students could show that a square fits in the category of rectangles and quadrilaterals. Have students create non-examples as well and then allow time for groups to present their posters.

Standards

G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

Learning Objective

Classify shapes by properties of their lines and angles

Activity

While reading, ask students to describe what shape will come next using number of sides, vertices, types of angles, parallel and non-parallel sides, etc. Students may be able to describe more than one shape because more than one type of quadrilateral exists. Engage students in a discussion of regular versus irregular polygons.

After reading, have students draw a large equilateral triangle, like the triangle in the story. Check for understanding of what defines an equilateral triangle. Then have students create another shape, even if it is irregular. Have students write the characteristics of the shape they made including types of sides, angles, vertices, and any other defining characteristic. Then, have students create a short story about how the triangle became their new shape. If time allows, students can illustrate their shapes to go with their stories.

Standards

G.B.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
G.B.4 Classify two-dimensional figures in a hierarchy based on properties.

Learning Objective
Classify two-dimensional figures into categories based on their properties

Activity
While reading, ask students to identify all the different types of shapes that fit into the polygon category that the triangle becomes. Students can use geoboards or drawings to create different quadrilaterals, pentagons, hexagons, and more. When the triangle starts rapidly going through shape options, slow down and allow time for students to make each shape. Before the triangle changes into a circle, ask students to create a regular polygon with as many sides as they can using their geoboard. Ask students what would happen as the sides increase. Look for an understanding that the angles become more and more obtuse and eventually will look like a circle. Discuss if a circle can be a polygon (it can’t) and why.

Pass out various shape cards to each student. Hold up a category card, such as “hexagon” and have every student holding a hexagon stand with their card. Repeat this with various categories. Be sure to include sub-categories such as “rectangle” to determine if students realize the hierarchy of squares. Then, use the cards to create a chart with a hierarchy of polygons.

## Extended Learning

As students get older, they will apply angle measurements to polygons, understanding the sum of the angle measurements of a triangle is always 180 degrees. While this is typically taught in middle school and beyond, it is accessible for younger students who understand measuring angles or can be done as a whole-class activity.

Learning Objective

Determine the pattern of angle measures in polygons

Activities

While reading, have students measure the angles in an equilateral triangle, like The Greedy Triangle, and begin recording a chart for each type of polygon and the sum of the angles in that polygon. As the triangle changes shape, discuss the attributes of the shape, specifically focusing on how many angles the shape has. Measure and record the sum of the angles in the various polygons as the triangle changes.

After reading, challenge the students to see a pattern in the change in angle measurements when a polygon adds one side. To find why this is 180 degrees every time, draw triangles inside the polygon and notice that additional sides create additional triangles that can be drawn.

If you want to extend further, have students measure the exterior angles of the polygons and see that they always total 360 degrees. Older students may quickly realize this is the same as the degrees in a circle!

To make it cross-curricular, have students write a new version of The Greedy Triangle using changing angle measurements instead of changing sides.

## Print Manipulatives

Need some great shape manipulatives? Try printing from these great sites:

## Geometry Resources

My fellow Scholastic bloggers have great geometry resources to keep your learning going. Try some of these great posts:

·       “Five Ways to Make Geometry Memorable

·      “Three Great Geometry Games

Picture books are a great way to generate interest and learning, no matter the grade level. One simple story like The Greedy Triangle can open a world of possible in the math classroom at any age. Click here to purchase a copy of The Greedy Triangle for your classroom.

What books do you use in your math lessons?

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