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Without using any fractions or decimals in your workings, find the marked lengths. How would your students do it? Diagrams are not drawn to scale!

3

29

10

Source: Don Steward

2

20

10

15

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Area in Depth

Jo Morgan

resourceaholic.com

@mathsjem

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Planning lessons in any topic: �what to think about...

Topic progression: KS1 – 5 curriculum, prerequisites, skills, links & next steps

Subject knowledge: definitions, history, etymology, methods

Examples and explanation

Assessment

Misconceptions

Tasks and resources

Enrichment and narrative

Depth and challenge

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The need to measure land areas was one of the ancient problems which led to the development of geometry.

Both the early Egyptians and Babylonians had formulas for the areas of rectangles, triangles, and trapeziums.

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Know Your Maths, 1957

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Curriculum�and Assessment

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Year 4

Year 5

Year 6

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2019 KS2 SATS Reasoning Paper

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KS3

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KS3 SATs 2001

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KS3 SATs 2002

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KS4

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OCR GCSE

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Edexcel GCSE

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AQA GCSE

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A level

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Coordinate Geometry at A level

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D&T 2019

Science 2019

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‘Life skills’

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1961 edition (first published 1933)

Curriculum development

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Japanese Textbook

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First chapter on area in Book 4B

Concept of area
Area of rectangles
Area of L shapes
Units for large areas (m², hectares, km²)

Book 5B

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Depth and Challenge

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Reasoning with area and fractional thinking

Addressing triangle misconceptions

Today… we only have time to look at two aspects of teaching area

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1. Reasoning with area and fractional thinking

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“Polygon Areas”

from SMILE

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‘Area and Volume’, Heylings, 1983

Boxing-In

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“Area by Subtraction” from Don Steward

“Area of Polygons” from SMILE

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Checkpoint 1: Bigger?

Which of these shapes do you think is the biggest? Why?
Does including a square grid help?
In what way/s is the shape you chose bigger than the others? In what way/s is it the same?

Can you create shapes with this same area? How many possibilities are there?

Answers

a) There is no single correct answer as it will depend on the students' criteria for 'biggest’. The task is designed to be open so that students can suggest their own ideas. If they suggest area and perimeter as measures (or you choose to prompt them to) then:

Area: all of the shapes have the same area (five square units).

Perimeter (left to right from top left and then bottom left): blue 12 units; purple 10 units; pink 12 units; red 12 units; grey 12 units; green 12 units; orange 12 units.

b) The grid may make it easier for students to identify the areas and perimeters listed in the answer to part a.

c) This is designed as an open-ended question, so students can make their own comparisons and suggest their own justifications. An example response might be that the orange rectangle in the bottom right position is bigger because it's the tallest, but it's the same because it has the same area as all of the other shapes, and the same perimeter as most of them. As with part a, students may or may not automatically link to area and perimeter. It is for teachers’ own judgment whether to prompt them.

? Any arrangement of five squares into a shape. These are examples of pentominoes: seven of the twelve possible pentominoes are shown here. Examples of the twelve pentominoes are readily available online.

Suggested questioning

What is the same/different about ___?

I think __ is the biggest because ___, do you agree?

Things to think about

In Checkpoint 1, the first question (“which of these shapes is the biggest?”) is intended to give a context for students to talk about what they know about measures and properties of shapes. The teacher's role in this assessment task is to listen to the ways in which students talk about these properties but, as a part of this, you may like to prompt students to expand on definitions, or to compare one property or measure with another shape

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Solve My Maths

MathsPad

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OCR GCSE

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‘Dissecting a Square’ from Standards Unit

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Enrichment: Pick’s Theorem (1899)

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2. Addressing triangle misconceptions

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“You do not halve this sum as the triangle is made of two right angled triangles”

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mrseteachesmath.com

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Japanese primary textbook

Heylings

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ponderingplanning.wordpress.com

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Activity I: Base and height

Mark on a pair of base and perpendicular height for triangles 1 to 3.
Can you mark on more than one different pair?

1

2

3

4

How about for triangle 4?

Printable copies of the images in this task are available in ‘Printable resources’, slide 58. An animated version of the solution is available on slide 50.

This slide is animated. Each question will appear separately so you can choose the pace at which to work with your class.

Answers

The answers for this activity are given as animations on the next slide.

Suggested questioning

Which side do you most naturally ‘see’ as the base?

Could any side be the base?

How do you know the height is perpendicular?

Things to think about

Additional activity I supports understanding of the area of a triangle. At Key Stage 2, students will have had experience of calculating the area of triangles, but this may have been with the support of a formula or grid. Students are also likely to have mainly experienced examples where the base and its related perpendicular height are relatively obvious. This activity might be used as a further assessment of students’ understanding of the information needed to find the area of a triangle, or as a next step to support teaching of area at Key Stage 3. The area of a triangle is deliberately not referenced here in order to ensure students’ focus is on their definitions of base and perpendicular height.

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MathsPad

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Checkpoint 10: Area or perimeter?

For each triangle, do you have enough information to calculate the area, the perimeter, both or neither?

Explain how you know.

Drawn to scale.

For the ones that are not possible, write the minimum additional information you need. Is there more than one way to do this?

A

B

C

D

E

F

This slide is animated. Each question will appear separately so you can choose the pace at which to work with your class.

Answers

A: can calculate area but not perimeter

B: can calculate perimeter but not area

C: can calculate area but not perimeter*

D: can calculate area and perimeter

E: cannot calculate area or perimeter

F: can calculate area but not perimeter

*teachers may wish to mention that, once students have learned Pythagoras’ Theorem, they could calculate both!

? A: the remaining side

B: any perpendicular line from one vertex to a side (or an extension of that side)

C: the remaining side

E: both of the remaining sides (just the base would allow you to calculate area, as would a perpendicular line from the 4cm side to the opposite vertex)

F: the remaining side

Suggested questioning

What's the same and what's different in the labelling of triangle A and triangle E? Triangle C and triangle F? Triangle C and triangle D?

Things to think about

You might like to think about the use of language when discussing Checkpoint 10. Do students specify the perpendicular height for area and understand what this means? Although students may not have enough information to calculate the perimeter or area at this stage, you may like to clarify with them that it is possible to calculate precisely using techniques that they will learn later in KS3. For example, they cannot find the perimeter of triangle C using the mathematics they know now but, by using Pythagoras' Theorem, the problem will become straightforward.

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Checkpoint 11: From one vertex

Drawn to scale.

Carefully draw a different parallelogram with a base of 10 cm and a perpendicular height of 6 cm. Measure the other base and perpendicular height. What is the product of these values? Why?

This slide is animated. Each question will appear separately so you can choose the pace at which to work with your class. A printable version of the image is provided on slide 54.

Answers

Both Gaynor and Ollie are correct. They have used different lengths as the base, but as they are multiplying by the appropriate perpendicular height each time, they will both find the area of the parallelogram: 10 × 6 = 7.5 × 8 = 60

? The two values would also have a product of 60. Students may find that, due to the accuracy of their drawing, this is not exact!

Suggested questioning

What information is needed to calculate the area of a parallelogram?

What strategies could we use to test Gaynor and Ollie's methods?

Things to think about

The dimensions of the parallelogram have been chosen so that it can be drawn accurately with both vertical heights within the parallelogram and with 'friendly' numbers. What are the benefits of this? Are there disadvantages? You might like to explore the range of possible values that fit within these two constraints.

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Don Steward

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Don’t say this:

Area of parallelogram equals base times height

Always say this:

Area of parallelogram equals base times perpendicular height

Oracy

Always correct students if they omit the word perpendicular.

Ensure students both say and hear this formula multiple times.

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ponderingplanning.wordpress.com

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Enrichment: Heron’s Formula (c60 AD)

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What next?
Follow me on Twitter	@mathsjem
Like me on Facebook!	facebook.com/resourceaholic
Buy me a drink!	buymeacoffee.com/mathsjem
Check out my blog	resourceaholic.com
Watch my CPD	bit.ly/2FyLJcg
Buy my book	A Compendium of Mathematical Methods

In summary:

We’ve just scratched the surface of area here. There’s a lot to explore with your students to deepen their understanding, enrich their knowledge, and provide opportunities to develop their reasoning skills.