areas of composite figures

Lesson 2

Learning outcomes

By the end of this lesson, students will be able to

recognise and calculate the area of a composite figure as the sum of the areas of the dissected shapes (triangles and quadrilaterals)

Content indicators

calculate the areas of composite figures by dissection into triangles, special quadrilaterals, quadrants, semicircles and sectors

WM indicators

identify different possible dissections for a given composite figure and select an appropriate dissection to facilitate calculation of the area (Problem Solving)

Prior knowledge and understanding

Students are assumed to be able to:

dissect irregular composite shapes into three or more triangles and special quadrilaterals
recall and use area formulas of triangles and special quadrilaterals (including rectangles, squares, rectangles and squares, parallelograms, rhombuses, trapeziums and kites)
find the area of simple composite shapes that are comprised of two plane shapes such as rectangles, squares, parallelograms and triangles

Lesson plan

Lesson length: 60 mins

Materials/Resources

GeoGebra applet link: https://www.geogebra.org/m/dyd4srak
GeoGebra template link: https://drive.google.com/file/d/1z7daJvwmBFjaZQO6Y-a2aBWo6v-TQ_Kz/view
Grid paper
Tangram kits
Exit cards
Square pieces of paper (for homework)

10:00 Introduction

Collect students' homework as students settle in class (AFL)

Check if completed
Assess if students understand what a composite shape is from their dissections

(10 mins, ICT)

Review of how to dissect simple composite figures into at least three triangles and special quadrilaterals
Recap area formulas of triangles and special quadrilaterals.

Demonstrate how to find the area of a composite figure by dissecting it into basic triangles and quadrilaterals shapes. (ICT)

Download template beforehand (link)
Open GeoGebra application (link) and load template
Supply students with grid paper to copy the diagram and for working out.

Ask students to first find the area by counting the squares of the grid of the basic template (Total area = 64 cm²)
On screen, dissect the composite figure into two basic shapes using the segment tool in the same manner students did in the introduction of Lesson 1. Identify that it is now separated into two shapes that have area formulas and label them A1 and A2. Have students copy the diagram with labels down onto grid paper.
Ask students to identify the shapes (parallelograms) and recall the formula to find the area of one shape (A = bh). Ask students what the breath and height of one parallelogram is and to label them on their papers.
Ask how to find the area of A1. Ask students to show working in their book. Roam around the class to check their dissections and working out (formative assessment). Check that everyone calculated A1 to be 32 cm². Have students find the area of A2 in the same manner.
Show that since both A1 and A2 make up the original composite figure, in order to find the total area, we must sum the areas of A1 and A2 together. Have students calculate this in their book and recognise this gives the same area that students counted at the beginning (Total area = 64 cm²).

5. Total area of the composite figure

1. Basic template

2. Dissected and labelled figure

3. Dissected and labelled figure with dimensions

4. Calculated areas of separated components

10:10 Body activity 1

(15 mins, ICT): Calculating areas of composite figures by dissecting into at least three basic shapes I

Students download the basic template and open it in the GeoGebra app as above. Using their dissections from Activity 1 of Lesson 1, they recreate dissections into three or more basic shapes in the app using the segment tool. Following the steps in the introduction, they calculate areas of the separate components, and sum them to find the total area.

Have student compare their solutions with others at their table. Ask (AFL):

How did they cut up their composite figure? Is it different to your dissection?
Did you end up with the same area for the original shape?

Students should recognise that the area of the composite shape will be the same irrespective of the manner of dissection.

10:25 Body activity 2

(20 mins, LIT): Calculating areas of composite figures by dissecting into at least three basic shapes II

(5 mins) Individually, students create a composite figure using at least two tangram pieces and copy or take a photo of this. Example figures and dissections are provided below.

Students find the areas of each component to find the total area of their created figure. The student writes their working out and final total area and record this in their workbook. They then hand copy their own design on a separate piece of paper.

Resource adapted from: https://www.teacherspayteachers.com/Product/Tangram-Composite-Figures-2433559

Figure 1.

Figure 2.

Figure 3.

(15 mins) in pairs, students swap shapes with partners to and must find:

An appropriate dissection that can be created using the tangram shapes provided.
The area calculations for each of their components and for the total figure.

When students are working, ask questions such as (AFL):

How many shapes are you cutting the figure into?
How do your two dissections compare?
What is the area formula this shape? Can you show me the sides/dimensions you need to calculate the area?
What is the total area of your figure? How did you calculate it?
How does it compare to the area of the figure after you have dissected it another way.
If you didn't get the same area, why do you think that happened?

Once completed, students compare the dissections for the same shape and check that their final total area for this shape are the same. If pairs students have successfully completed the task, they should create another composite figure together and find an appropriate dissection that will be used to find its total area.

NB: The manner of creating the dissections may differ between students:

Low-ability students should use the tangram pieces as an aid for their dissections and to further practice dissecting composite figures into basic shapes.
Higher-ability students could use a ruler to create shapes that aren't present in the tangram kits.

10:45 - Conclusion

(5 mins) Class discussion: Ask students:

What did we cover today?
How can we calculate the areas of composite figures?
- Reinforce that the way to calculate the areas of composite shapes is to dissect them into two or more basic shapes that we can find the areas of, then sum.
Does changing the manner of dissection have an effect on the total area of the original composite shape?

(5 mins) Exit card: Provide students with the following exit card. Collect cards before the end of class and look over to gauge students' understanding of the dissection of composite figures. (AFL)

Composite figure 1 image retrieved from: https://www.onlinemath4all.com/exploring-areas-of-composite-figures.htmlComposite figure 2 image retrieved from: https://www.mathpractice101.com/DisplayPage/QuestionsAnswers?page_id=3110

(5 mins) Homework and pack up

Students select one of the following flat origami figures. Following the instructions in the links, they create the figurine. Afterwards copy the outline and visible folds lines of the finished origami in their workbook.

On one outline, students dissect the composite shapes into basic shapes that they can use to calculate its total area. They may use the existing fold lines to guide them and must clearly show the dissected pieces. They must also provide the full working out. Afterwards, they paste their original origami figure in their workbook.

Students repeat this for at least one other origami figure.

Tadpole

Origami instructions: https://origami.me/tadpole/

Angelfish

Origami instructions: https://origami.me/angelfish/

Penguin

Origami instructions: https://origami.me/penguin-baby/

Ice cream

Origami instructions: https://origami.me/ice-cream/

Evaluation

Did students have the relevant prior knowledge?
- Did they recall how to dissect a composite figure from the previous lesson?
- Do they remember their area formulas?
- Review feedback from homework
Were my instructions clear?
Were all students engaged in the activities?
Were my questions clear/encouraged students to think about the concept?
Do students understand that multiple possible dissections that all lead to the same answer?
- For students whose area didn't match up for different dissections, was it because of a calculation error or because they have misconceptions?
Can students dissect a composite shape into smaller, basic shapes and calculate its area?
- Review feedback from exit card
What do I need to modify in my upcoming lessons to support the needs of all students?

Additional resources:

Homework activity is interchangeable with tangram puzzle activity: https://sites.google.com/view/areasurfacearea-s51/additional-resources#h.p_CrPgzeZRREJ2

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Unless otherwise indicated, all diagrams on this page were produced myself, using the GeoGeobra application, MS Word and MS Paint.

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