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Cut A Rug: Areas of Basic Shapes and Composite Figures

Context for this Lesson

Teaching Strategies: 

TOPIC: Area of right triangles, Squares, Rectangles and Composite Figures

GRADE LEVEL: 4th grade

GOAL: Reinforcing and reviewing characteristics of right triangles, rectangles and squares and offering a “real world” situation involving irregular shapes.


  • How do we find area for the different kinds of shapes?
  • How can we use squares, right triangles and rectangles to solve irregular shapes?

MATERIALS: 1 Big paper example room to work through together Big paper equations for areas of right triangle, square, rectangle 3 problem rooms



§111.6. Math, 4th grade

(b) Knowledge and skills.

  • (1) Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
    • (A) apply mathematics to problems arising in everyday life, society, and the workplace;
    • (B) use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;
  • (5) Algebraic Reasoning
    • (C) use models to determine the formulas for the perimeter of a rectangle (l + w + l + w or 2l + 2w), including the special form for perimeter of a square (4s) and the area of a rectangle (l x w); and
    • (D) solve problems related to perimeter and area of rectangles where dimensions are whole numbers.
  • (8) Geometry and Measurement
    • (C) solve problems that deal with measurements of length, intervals of time, liquid volumes, mass

Common Core State Standards:

Grade 4 Overview

Operations and Algebraic Thinking: 

  • Use the four operations with whole numbers to solve problems
    • (2) Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison

Measurement and Data: 

  • Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
    • (3) Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.


Objective: practice/prior knowledge/anticipatory set

Assemble the group in a large open space. Explain that you have a challenge for the group which they must do by working together. They may not talk or make any noise. Ask how they might communicate nonverbally? After taking a few suggestions, explain that when you say “Go,” participants should create one large triangle that involves every person without making any sounds. The challenge is to see whether this task can be completed in one minute with absolutely no sound. Go.


  • Look at triangle that was created. If each person is a unit of measurement, how long is each side? Knowing the lengths of the sides, what kind of triangle is this?
  • Next let’s make a right triangle (what are the attributes of a right triangle?).
  • How might we find the area of our right triangle (using a person as a unit of measurement) (A=1/2 bh)
  • Next we need to make a shape as group, this time it’s a square. Let’s see how quickly we can make a square using a person as unit of measurement. How long is each of our length?
  • How might we find the area of our square? A=B x h Transition: (Have students put down yarn and return to seats.)


I’d like to continue thinking about shapes, specifically squares, right triangles, and rectangles and how they are used in our everyday lives. Let’s take a look at this object I have here.


 Objective: Setting up /hooking into story Random piece of flooring material.


  • What do you see?
  • What do you notice about this object?
  • What shape is it?


  • Based on those observations, what do you think this could be?


  • How do we use this item?
  • What different shapes do you think it comes in?


Great, now we are going to explore how we might be able to use what we know about these shapes and calculating the areas of these shapes to help a special guest solve a problem.



Explain to participants that we will now explore a practical problem by going into role. When the leader puts on a costume piece, participants will know she is going into role. They will be acting as experts to help with a particular problem, which they will learn more about from the character. Count down from 3, 2, 1, and go into role as a carpet installer by putting on the costume piece.

Rachel: "Hi everyone, I’m Rachel Brown from Cut a Rug Carpet Design and Installation. I’ve got a big issue and I’m hoping you can help me. See, Mark Zuckerberg, founder of Facebook, just hired my company to install carpet in their new company headquarters. But my boss just gave me these drawings of the rooms in this building and they are CRAZY, I mean I’ve never seen rooms shaped like this. In my training they barely taught me to find the area of any shapes. And look at these rooms! Will you let me show you the plans for one of the rooms? Maybe you might have some ideas…(put up big copy of one room) This room is just so oddly shaped. What do I do? Staying in role, have students coach Leader through the steps of finding the area for each shape. Once they have walked through the steps and found the total area of the room, say: wow! Thanks for helping me.

(get’s a phone call) Yes? Uh huh? Oh yeah, I totally got it boss no problem. (listens) What? He wants what on the floor? But, we’re a carpet company and…right, lots and lots of money. Yes. Okay, no problem. (hangs up phone) Okay, so now I need to find the area of each room AND I need to propose a type of floor covering that I think will best suit the THEME of the room. Money is no object. Here, the theme of this room is the beach room. A room for folks to relax and get some sun even when they are working late at night. What do you think? What could we put on the floor here? (takes suggestions) Listen. You all did a great job helping me with the first room. What if we divide and conquer for the rest. Each group will get a floor plan. Your job is to solve for the area of the floor that we need to cover, using our formulas, then based on the room, offer a suggestion about what you think might best cover the floor to support the theme. Any questions? Great let’s get started."

Pass out the floor plan worksheets to groups of 3-5 students. Students have 15 minutes to work.

Share each design as they talk through how they found the area.


Rachel: "Now we need you to tell us in detail how you solved this problem so I can take that information back to my colleagues at Cut a Rug Carpet Design. We are very good at laying carpet, but not so good at figuring out how to find the area."

Come together as whole group to brainstorm the steps we took to solve the problem.  Decide on the steps, order 4-5 steps Divide into 2 groups, and each group take 2 steps, makes a frozen image for those steps.

Facilitators rotate, give feedback and an opportunity to revise images and they write a caption along with their image.

Share out Quick DAR each image: What do you see? What tells you that? What important step does this image illustrate?

Rachel: "Thanks for all your hard work! Thanks so much for creating this video so I can take it to my staff. It will greatly help them figure out how to do this in the future!"

Teacher takes off hat. And, thanks for your willingness to think and play with me today in math class. Let’s talk a little bit about our work together.




  • What did we do today?


  • How did you figure out how to help Rachel with the carpet installation? What was your favorite potential floor cover offered?


  • What other jobs/situations might require someone to know how to deal with irregular shapes like this?