Algebraic Summary

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Algebra-Linearequations1.pptx

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GCSE Mathematics – Algebra – Linear Equations

SLIDE NUMBER 1

April 2019

Elliott Wade

Session Objectives

The purpose of the session is to:

Differentiate between Expressions, Equations and Formula

Solve simple Linear Equations

Solve Linear Equations involving multi-step skills

Solve Linear Inequalities

SLIDE NUMBER 2

April 2019

Solve

Linear

Equation

Expression

Equation

Inequality

CONSIDER…

At this point you should consider the list of session objectives and ask yourself:

How many of the session objectives am I confident with

Could I explain these objectives in relation to teaching and learning

SLIDE NUMBER 3

April 2019

Session Objectives

The purpose of the session is to:

Differentiate between Expressions, Equations and Formula

Solve simple Linear equations

Solve Linear Equations involving multi-step skills

Solve Linear Inequalities

SLIDE NUMBER 4

April 2019

Solve

Linear

Equation

Expression

Equation

Inequality

Differentiate between Expressions, Equations and Formula

Definitions

Put into practice

Exam style questions

Linear Equations

SLIDE NUMBER 5

April 2019

Oracy

Linear

Equation

Expression

Equation

Inequality

Definitions

Equation – States that two things are equal.

For example: 6a = 12

Identity – An equation that is true for any given value substituted in.

For example: xy = yx

Formula – A mathematical rule usually containing an equals sign.

For example: Area of a circle

Expression – a group of numbers, letters and operations.

For example: 5a + 9

Linear Equations

SLIDE NUMBER 6

April 2019

B4L is key. Time to embed into practice. Students need to be comfortable with using all four key words under this topic. And so should teachers!

Formula

Expression

Identity

Equation

Equal

Linear Equations

SLIDE NUMBER 7

April 2019

Put into practice

Equation

Formula

Expression

x(x-4) = 12

x + x + x + x

V = u +at

C = d

2x +14

15t3 + 3v = 146

E=MC2

B4L is key. Time to embed into practice.

Resilience

Independence

Algebra

Equation

Deep understanding

Linear Equations

SLIDE NUMBER 8

April 2019

Exam style questions

A group of numbers, letters and operations

Dan: x

Harry : x + 5

Regan 2x

Mean = Total sum ÷ number of sellers

x + x + 5 + 2x

Expression = (4x + 5) ÷ 3

B4L is key. Time to embed into practice.

Resilience

Independence

Algebra

Equation

Deep understanding

CONSIDER…

Consider the following question:

SLIDE NUMBER 9

April 2019

Linear Equations

SLIDE NUMBER 10

April 2019

Consider task - Answer

T = 5x + 20y

B4L is key. Time to embed into practice.

Resilience

Independence

Algebra

Equation

Deep understanding

Session Objectives

The purpose of the session is to:

Differentiate between Expressions, Equations and Formula

Solve simple Linear equations

Solve linear equations involving multi-step skills

Solve linear inequalities

SLIDE NUMBER 11

April 2019

Solve simple Linear Equations

Methods

One step equations

Exam style questions

Linear Equations

SLIDE NUMBER 12

April 2019

Methods

Linear Equations

SLIDE NUMBER 13

April 2019

Balance method

a + 4 = 12

a = 8

Elimination method

a + 4 = 12

To eliminate +4 from the left we move it over the equals sign.

a + 4 = 12 – 4

Whenever moving across the equals sign, invert the operation

a = 8

-4

Weave solving equations into several topics.

One step equations

Linear Equations

SLIDE NUMBER 14

April 2019

÷10

b = 10

x = 7

÷10

10b = 100

x3 = 343

÷2

c = 16.5

÷2

33 = 2c

-12

v = -15

-12

12 + v = - 3

Weave solving equations into several topics.

www.mathsgenie.co.uk

Exam Style questions

Linear Equations

SLIDE NUMBER 15

April 2019

-5

y = 7

t = 11

Weave solving equations into several topics.

www.mathsgenie.co.uk

Exam Style questions

Linear Equations

SLIDE NUMBER 16

April 2019

÷2

y = 4

y = 18

Weave solving equations into several topics.

Task:

Complete the following one step equations:

4y = 10

10g = 37

a – 7 = -3

V + 2 = -6

= 2.7

5y = 24

SLIDE NUMBER 17

April 2019

CONSIDER…

Session Objectives

Consider Task Review

SLIDE NUMBER 18

April 2019

4y = 10

10g = 37

a – 7 = -3

V + 2 = -6

= 2.7

5y = 24

÷4

4y = 10

y = 2.5

÷10

10g = 37

g = 3.7

a – 7 = -3

a = 4

-2

V + 2 = -6

V = - 8

= 2.7

w = 10.8

÷5

5y = 24

a = 4.8

Session Objectives

The purpose of the session is to:

Differentiate between Expressions, Equations and Formula

Solve simple Linear equations

Solve linear equations involving multi-step skills

Solve linear inequalities

SLIDE NUMBER 19

April 2019

Solve linear equations involving multi-step skills

Multi-step examples

Exam Style Questions

Common misconception

Linear Equations

SLIDE NUMBER 20

April 2019

Multi-Step Examples – Two Step equations

Linear Equations

SLIDE NUMBER 21

April 2019

4x = 12

x = 3

-3

÷4

4x = 18

x = 4.5

https://variationtheory.com/category/algebra/equations/

Variation

Solve

Multi Step

Equal

Balance

Multi-Step Examples – Variables on both sides

Linear Equations

SLIDE NUMBER 22

April 2019

x = 2

x + 3 = 5

-x

-3

-x

-3

x = 1

2x + 3 = 5

÷2

2x = 2

https://variationtheory.com/category/algebra/equations/

Variation

Solve

Multi Step

Equal

Balance

Multi-Step Examples - Brackets

Linear Equations

SLIDE NUMBER 23

April 2019

-2

-x

Expand

2x + 2 = x +5

x + 2 = 5

x = 3

https://variationtheory.com/category/algebra/equations/

Variation

Solve

Multi Step

Equal

Balance

Exam Style questions

Linear Equations

SLIDE NUMBER 24

April 2019

Perimeter = 2x + 2x + 10

Perimeter = 4x + 10

34 = 4x + 10

24 = 4x

6 = x

-10

÷4

https://www.mathsgenie.co.uk/resources/64_forming-and-solving-equations.pdf

Students can sometimes discover the question themselves and attempt to answer before knowing the specific question.

Comfort with answer at the end being an uncommon format.

Common Misconception

Linear Equations

SLIDE NUMBER 25

April 2019

It is not uncommon for equations to be given in a format that could leave the variable on the right instead of the left.

Consider the previous question:

34 = 4x + 10

24 = 4x

6 = x

-10

÷4

Having the variable on the right does not alter the outcome of the answer.

Consider:

2 + 3 = 5 and 5 = 2 + 3

https://www.mathsgenie.co.uk/resources/64_forming-and-solving-equations.pdf

Students can sometimes discover the question themselves and attempt to answer before knowing the specific question.

Comfort with answer at the end being an uncommon format.

Review of main ideas from above:

Consider this angle problem from the ‘Increasingly Difficult Questions’ website.

SLIDE NUMBER 26

April 2019

CONSIDER…

http://taylorda01.weebly.com/uploads/4/2/3/8/42387051/angles_in_triangles_and_quadrilaterals_01_v2.pdf

Consider Slide - Answer

Two base angles must be equal (Isosceles Triangle)

Angles in a triangle sum to 180 degrees.

An equation could be formed to show this problem…

Linear Equations

SLIDE NUMBER 27

April 2019

3n – 10 + 3n – 10 + ? = 180

6n -20 + ? = 180

6n + ? = 200

? = 200 - 6n

Pattern

Misconception

Comfort zone

Oracy

Session Objectives

The purpose of the session is to:

Differentiate between Expressions, Equations and Formula

Solve simple Linear equations

Solve linear equations involving multi-step skills

Solve linear inequalities

SLIDE NUMBER 28

April 2019

Solve Linear Inequalities

Notation

Examples and approaches

Exam Style questions

Linear Equations

SLIDE NUMBER 29

April 2019

Notation

Confidence in understanding notation used to show inequalities is key!

Linear Equations

SLIDE NUMBER 30

April 2019

Essential information!

Less than

Greater than

Less than or equal to

Greater than or equal to

Notation continued

Linear Equations

SLIDE NUMBER 31

April 2019

Notation

Number line

Inequaltiy

Representation

Examples and Approaches

Linear Equations

SLIDE NUMBER 32

April 2019

The direction of the inequality sign is essential! So do we ever change it?... When manipulating inequalities most rules are similar to standard linear equations

Okay to do:

Simplify a side

Add/subtract from both sides

Multiply/divide both sides by a positive number.

Actions that flip the sign:

Multiply/divide both sides by a negative number.

Swapping left and right sides.

Tip!

More ink in the symbol means more on the diagram

Examples and Approaches – Solving Example 1

Linear Equations

SLIDE NUMBER 33

April 2019

+10

÷2

2t < 20

t < 10

Number line

Negative

Solve

Inequality

Multi Step

Examples and Approaches – Solving Example 2

Linear Equations

SLIDE NUMBER 34

April 2019

-5

÷-5

- 5x ≤ - 15

x ≥ 3

Don’t forget to flip the sign here!

Number line

Negative

Solve

Inequality

Multi Step

Examples and Approaches – Example 3 - Number line representation

Linear Equations

SLIDE NUMBER 35

April 2019

Can you show the following inequality on a number line?

Number line

Negative

Solve

Inequality

Multi Step

Task: Complete the following 5 questions from www.CorbettMaths.com

SLIDE NUMBER 36

April 2019

CONSIDER…

5. Show the following inequality on a number line.

4. Write the inequality shown on the number line

https://corbettmaths.com/wp-content/uploads/2018/12/Inequalities-pdf.pdf

Session Objectives

Consider task - Answers

SLIDE NUMBER 37

April 2019

5. Show the following inequality on a number line.

4. Write the inequality shown on the number line

X ≤ -10

X is less than or equal to -4

X is less than -3

https://donsteward.blogspot.com/2012/09/linear-inequalities.html

Promotion of discussion: Is this about inequalities?

What question could you ask relating to inequalities?

What key words in the question help you model the problem?

Session Objectives

Exam Style Questions

From Fluency and Reasoning to Problem Solving.

SLIDE NUMBER 38

April 2019

For what values of x is the perimeter of the square greater than the perimeter of the rectangle?

P of S = 12x - 8

P of R = 6x + 22

https://donsteward.blogspot.com/2012/09/linear-inequalities.html

Promotion of discussion: Is this about inequalities?

What question could you ask relating to inequalities?

What key words in the question help you model the problem?

Session Objectives

Exam Style Questions

From Fluency and Reasoning to Problem Solving.

SLIDE NUMBER 39

April 2019

For what values of x is the perimeter of the square greater than the perimeter of the rectangle?

P of S =

P of R =

12x - 8

6x + 22

12x - 8

6x + 22

Session Objectives

Exam Style Questions

SLIDE NUMBER 40

April 2019

12x - 8

6x + 22

-6x

6x - 8

÷6

Session Objectives

SLIDE NUMBER 41

April 2019

The purpose of the session was to:

Differentiate between Expressions, Equations and Formula

Solve simple Linear Equations

Solve Linear Equations involving multi-step skills

Solve Linear Inequalities

SLIDE NUMBER 42

April 2019

CONSIDER…

End of Presentation

At this point it would be advisable to go back over the presentation. Ensure that you are fully able to deal accurately and effectively with each session objective.

You should supplement the content of this session with suitable reading, research and discussion with others.

End of presentation

Elliott Wade

SLIDE NUMBER 43

April 2019

GCSE Mathematics – Algebra – Linear Equations