Perimeter and Area of Composite Shapes: Teacher Notes

Perimeter and Area of Composite Shapes: Teacher Notes

Perimeter and Area of Composite Shapes: Teacher Notes

Overview

In this activity students determine the perimeter and area of composite shapes.

Important Mathematical Ideas

  • Composite shapes are made up of simpler shapes like rectangles, triangles and circles.
  • By decomposing a shape into simpler shapes you can calculate its perimeter and area.
  • Shapes can have the same perimeter but different areas.

Prior Knowledge

  • Applying area formulas to calculate the area of simple composite shapes.
  • Applying perimeter formulas to calculate the perimeter of simple composite shapes.

Common Misconceptions

  • Including lengths inside the shape that do not belong to its perimeter.
  • Confusing when to use area and when to use perimeter formulas.
  • Inability to modify area formulas to calculate the areas of associated partial shapes (e.g., semi-circle).
  • Inability to decompose a shape into simple shapes in order to calculate its area.
  • Inability to use subtraction to determine the area of a composite shape.
  • Mixing up the radius and diameter in formulas for the area and circumference of a circle.
  • Inability to use the properties of a given shape to determine a missing measurement.

Information to Support/ Enhance/ Extend Learning

  • Add definition of composite shapes to theWord Wall.
  • Consider a variety of formats as an alternative to journal entries (e.g., student note, pair/share, group discussion, exit card, poster, electronic posting).
  • Students may benefit from a review of formulas for area and perimeter of 2D shapes (e.g.,EQAO formula sheet).
  • Recognize this is not a one-day activity.

Minds On

Task 1: Creating Composite Shapes

  • Students will usePatch Toolto recreate given composite shapes
  • an exploratory tool that uses shapes and transformations
  • Patch Tool contains bulleted information relevant to each piece (i.e., activity sheet that needs to be downloaded, what kind of feedback is available, materials needed)
  • Students can use pattern blocks or tangrams to recreate their shape
  • A coach and be coached strategy can be used
  • Other resources worth considering:
  • Exploring Composite Figures
  • Area of a Triangle: interactive review

Journal Prompt and Sample Response

When you have completed a composite shape, capture a screenshot of it and paste this into your journal. Name each of the simpler shapes used to create your composite shape.

I see two triangles, a square,a parallelogram and a trapezoid.

Action

Task 2: Perimeter and Composite Shapes

  • May wish to review area of composite figures prior to the perimeter of composite figures.
  • Area of Composite Shapespractice
  • Students will:
  • investigate and answer questions regarding the perimeter of composite shapes inEverything You Wanted to Know about Perimeter and Area
  • an interactive website investigating perimeter and area; engaging activities; instructions; hints; full solutions
  • determine the perimeter of at least three composite shapes
  • solve questions involving area of composite shapes later in this activity

Journal Prompt and Sample Response

Capture a screenshot of one of the composite shapes. Show your calculations to find its perimeter in your journal.

P = 8 + 4 + 3 + 6 + 5 + 10

P = 36 cm

Task 3: Perimeter Problem Practice

Students practise answering perimeter problems inStep Perimeter ProblemandCross Perimeter.

  • the dimensions on the cross change each time the link is opened
  • hints are provided
  • consider usingaThink/Pair/Share and/or using manipulatives (e.g., pattern blocks)

Journal Prompts and Sample Responses

1)Record your solutions to theStep Perimeterand theCross Perimeter problems.

Step Perimeter

P = 10 + 10 + 12 + 12

P = 42 cm

Cross Perimeter

P = 24 + 24 + 26 + 26

P = 100 cm

2)Summarize the strategy that was shown by the hint on theStep Perimeterwebsite.

I noticed that the horizontal distances on the steps combine to be the total at the bottom of the stairs and the vertical distances on the steps combine to be the total distance on the left side of the stairs. I can calculate the distance around the step shape using two lengths and two widths. This is the same as finding the perimeter of a rectangle.

3)Describe how your solutions are the same or different than this strategy.

My solutions are the same because I used the same strategy.

Task 4: Assignment 1 Perimeter of Composite Shapes Discussion Prompts and Sample Responses

  • Students can complete this assignment in pairs and results shared with at least one other pair.
  • discussion prompts given can be used to guide their conversations

1)Make a prediction: How do the perimetersof these two shapes compare to each other? Are they the same? If different, which one is larger?

Some students might say: “I think Shape 1 has a larger perimeter because it is covering up more of the graph paper.” (A typical misconception of confusing perimeter and area)

2)Determine the perimeter of Shape 1.

To calculate the perimeter of Shape 1, addthe lengths of the three sides of the rectangle and the circumference of the semi-circle.

Circumference of semi-circle =2πr ÷ 2 (π = 3.14)

Perimeter of Shape 1

=9 + 6 + 9 + 2π(3) ÷ 2 ≈ 9 + 6 + 9 + 9.42 ≈ 32.42 units

3)Determine the perimeter of Shape 2.

To calculate the perimeter of Shape 2 add the length of the three sides of the rectangle and the circumference of the semi-circle.

Circumference of semi-circle =2πr ÷ 2 (π = 3.14)

Perimeter of Shape 2

=9 + 6 + 9 + 2π(3) ÷ 2 ≈ 9 + 6 + 9 + 9.42 ≈ 32.42 units

Discussion Notes

Solution should include:

  • a correct formula, Circumference of a semi-circle:C = πr or C = ½ πd
    orfull circumference divided by 2
  • substituted values
  • accurate calculation
  • appropriate units

Common Errors:

  • using four sides to calculate the perimeter of the rectangle
  • using the incorrect formula for perimeter of a semi-circle; might use area formula or forget to divide by 2
  • missing or incorrect units
  • subtracting the perimeter of a semi-circle for shape 2 instead of adding it

Task 5: Area of Composite Shapes

  • Students will investigate questions involving area of composite figures given inEverything you wanted to know about Perimeter and Area
  • an interactive website investigating area and perimeter with engaging activities, instructions, hints and full solutions
  • Students are to determine the area of at least three composite shapes

Journal Prompt and Sample Response

  • Capture a screenshot of one of the composite shapes. In your journal, record calculations to find its area.

A = A1 + A2 + A3

A = (5)(10) + (2)(20) + (5)(10)

A = 140 cm2

Task 6: Assignment 2 Area of Composite Shapes (See Task 4 Shapes)

  • Students can complete this assignment in pairs and results shared with at least one other pair.

1)How are Shape 1 and Shape 2 the same?

They have the same three side lengths and a semi-circular edge.

2)How are Shape 1 and Shape 2 different?

Shape 1 includes the semi-circle and Shape 2 has it cut out.

3)Determine the area of Shape 1. (see Task 4 for measurements)

Area of rectangle=l x w

=9 x 6

=54 units2

Area of semi-circle =πr2÷2

=π(3(32)÷2

≐14.1 units2

Total Area = Area of rectangle + Area of semi-circle

≐54 + 14.1

≐68 units2

4)Determine the area of Shape 2. (see Task 4 for measurements)

Area of rectangle =l x w

=9 x 6

=54 units2

Area of semi-circle =πr2÷2

=π(32)÷2

≐14.1 units2

Total Area = Area of rectangle - Area of semi-circle

≐54 – 14.1

≐39.9 units2

Discussion Prompt and Sample Response

  • Students can use the Discussion prompt given to guide their conversations.
  • What do you notice about the area of the two shapes? Explain.

The area of Shape 1 is larger than the area of Shape 2 because you add the area of the semi-circle for Shape 1 and subtract the area of the semi-circle for Shape 2.

Discussion Notes

Solution should include:

  • a correct formula, Area of a semi-circle:
    A =orarea of full circle divided by 2
  • substituted values
  • accurate calculation
  • appropriate units

Common Errors:

  • using the incorrect formula for area of a semi-circle; might use circumference formula or forget to divide by 2
  • missing or incorrect units

Consolidation

Task 7: Area of Composite ShapesPractice Problems

  • Students determine the area of two composite shapes of their choice for level 3, 4 and 5
  • students check their answers
  • students may share their solutions in aGallery Walk
  • Additional Practiceactivities

Task 8:Assignment 3 Composite Shape Problems

  • Posted with unit.
  • See sample solution in the Teacher Notes posted on the Virtual Learning Environment (vLE).

Task 9: Student Reflection

  • Students are asked to reflect on their understanding of this topic.
  • These reflections can be used as an assessment for learning to help determine next steps for individual students.

Grade 9 Applied Blended Learning: Unit 1 Activity 3 Page 1 of 6

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