From Rockets to Polygons! A Geometry Experience for Grades K-2

Mathematician Marcus Du Sautoy once said, “The power of mathematics is often to change one thing into another, to change geometry into language.” His quote reminds me of one of my favorite math experiences that I first learned about from my colleague Chris Confer (Math by All Means: Geometry, Math Solutions, 1994) and that we adapted for our book, Supporting English Language Learners, K-2 (Bresser, Melanese, & Sphar, Math Solutions, 2009).

When geometry gets its time in the limelight, students are often asked to identify shapes and their attributes from a picture in a textbook. Shapes’ names and attributes are important, but what we want is for students to take these shapes off the textbook page and do something with them. From Rockets to Polygons does just that! The experience literally brings Sautoy’s quote to life as students take a square and transform it into other shapes and then describe what they’ve made; they change one thing into another and then change geometry into language.

Launching the Lesson with Literature
Michael Hall’s book, Perfect Square is the perfect book to launch this lesson. The story is about a square that is transformed into different things: fountains, a garden, a park, bridge, river, and so on. It’s the ideal story to set the stage for the geometry activity, and the language in the book and its illustrations are exactly the kind of creativity I look for when I vet children’s books to use in math class.

Introducing the Activity

Materials Needed:

• 4½-inch colored construction paper square, 1 per student
• Pair of scissors, 1 per student
• Glue or paste
• White copier paper, 1 per student

To begin, I hold up a 4½-inch paper square (of any color) and ask students what they notice and what they would call the shape. Most everyone knows it’s a square, but some students are aware of other names like rhombus, quadrilateral, and parallelogram. Most know about sides, some mention corners or angles, and even a few identify vertices. It’s a good time to assess prior knowledge and introduce some new math terms.

Folding and Cutting the Square
Next, I distribute a paper square and scissors to each student, and step-by-step I model and help them fold and cut to create different shapes.

For kindergartners and maybe some first graders, I recommend folding and cutting in a small group since many young children are still developing their fine motor skills. Once they are finished, they should have these four shapes to work with: 2 small triangles, a large triangle, and a rectangle (see step 5 above).

Exploring
When students have their four shapes cut out and assembled in front of them, I ask them what they notice. Starting with an open question serves several different purposes. It lowers the floor and raises the ceiling, allowing for a variety of responses, accepting the students from where they are in their experience. I find that kids in grades K-2 generally know a lot about shapes, since they’ve been encountering them for much of their short lives. We talk about the shapes’ names and attributes: rectangle, square, triangles, edges, corners.. We talk about size: big and small. I throw in my own thinking and mention that the two small triangles are congruent – the same size and shape. Students share where they’ve seen these shapes in the world and where they see them in the classroom. As children share, I write the key geometry terms on chart paper so students can access them later.

Several students are already trying to put the pieces back together into the original square. I follow their lead and give the class time to just explore and make different shapes from other shapes, an important idea in geometry.

Levels of Geometric Thinking
Learners go through different levels of geometric thinking on their learning paths (see the levels below from the Common Core Progressions, 2011 and Van Hiele, “A Child’s Thought and Geometry”, NSF, 1984)). Children in grades K-2 generally work at the Visual and Descriptive levels. The levels aren’t set in stone, and students move back and forth between them. Research tells us that instructional practices that develop students’ geometric thinking include opportunities to explore, discuss, and get their hands on real shapes, not just pictures of shapes in textbooks (Teaching Children Mathematics, NCTM., May 2010).

Visual
Students recognize shapes, e.g., a rectangle “looks like a door.”
Descriptive
Students perceive properties of shapes, e.g., a rectangle has four sides, all its sides are straight, opposite sides have equal length.
Analytic
Students characterize shapes by their properties, e.g., a rectangle has opposite sides of equal length and four right angles.
Abstract
Students generalize about shapes (e.g. they understand that a rectangle is a parallelogram because it has all the properties of parallelograms (prepares students for higher-level geometry).

The ‘Make a Rocket Challenge’
For kindergarteners and first graders, I wait to pose the make a rocket challenge until after they’ve created polygons from the cut-up shapes. For second graders, I begin with the rocket challenge by showing them a rocket shape I’ve made from using all the shapes (see below). The challenge for them is to see if they can recreate the rocket from looking at my model.

The rocket challenge can be difficult for some second graders, but I think it’s worth it because it gives them a chance to build perseverance and develop their spatial skills. They rotate, flip, and rearrange the shapes until they make the rocket. Most relish the challenge.

From Rockets to Polygons
After students have made the rocket, I propose three rules for creating polygons from the shapes they made from the original square. Showing them examples and counterexamples helps.

1. The sides that touch must be the same length.
2. You cannot put a shape on top of another shape.
3. You can use two, three, or four shapes to make a new shape.

After modeling the rules, I explain to students that they are to choose some or all their shapes and put them together to make a new shape, or polygon. Once they find a polygon they like, they should glue or paste it on paper and describe the new shape.

Supporting Language Learners
Geometry involves a lot of language! There’s content vocabulary including the names of polygons, and there’s also words that describe the shapes such as line, side, angle, and vertex. But language is more than vocabulary. When describing shapes, children use complete sentences using words and phrases that connect the key vocabulary.

From Rockets to Polygons provides children with many opportunities to talk and write about the shapes they’ve made. Some need language support, especially those children learning English as a second or third language. Therefore, before setting them off to create polygons, I introduce and have the children practice saying the following frames and stems that will help them describe their polygons once they’ve made them and glued/pasted them to their paper.

It has _____________________.
My shape has _______, ________, and __________.
I built my shape with __________ and ____________.
It looks like a _______________.

Examples from the Classroom
My friend and colleague Lila Murphy tried the lesson in her second-grade class. Following are some student examples.

Here’s Adam’s rocket shape. He wrote, “My shape has 6 vertices/angles and 6 sides. I built my shape with 1 big triangle, 2 small triangles, and 1 rectangle. It looks like a rocket ship.”

Saron created the following shape. He wrote, “I have vertices, sides, angles, rectangles. My shape has 3 triangles and one rectangle.

Aeris definitely thought outside the box! Her writing provides a window into what she knows about shapes and some hints about what she still needs to learn. She wrote, “My shape has thirteen vertices and 12 sides. I built my shape with 2 rectangles and a long square. Then a big rectangle on top of it. Then I separated them. They are barely close to each other.”

 

Changing One Thing to Another
Composition and decomposition of shapes is an important activity in helping children develop their spatial knowledge and language. In fact, children who acquire a solid understanding of space and spatial language tend to demonstrate higher math achievement (Stanford University, DREME Network, 2024). From Rockets to Polygons is the perfect way to engage children in changing one shape into another and then changing geometry into language. The activity is deceptively simple and straightforward and requires little preparation. More importantly, it allows children to experience success with geometry and poses a challenge for everyone. We hope you’ll try it!