Numbers and the Number System 4

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Numbers and the Number System - 4
Algebra and Equations
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Qualifying to Teach 2.1b
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Objectives

• To continue to develop knowledge and
  understanding of numbers and the number system.

• Develop knowledge and understanding of
• Directed Numbers
• Substitution
• Simplifying and Expanding
• Algebra/Equations

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Directed Numbers
A directed number has a or sign in front of
it.
To add or subtract directed numbers, find the
starting position, then move right () or left
(-) on the number line.
-6 -5 -4 -3 -2 -1 0 1 2 3
4 5 6
Examples 2 3 5 (or just 5) Start at 2
and move right 3 places to 5. -3 6 3 (3)
Start at -3 and move right 6 places to 3. -1
4 -5 Start at -1 and move left 4 places to
-5.
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Multiplying/Dividing Directed Numbers
To multiply or divide directed numbers, multiply
or divide the numbers and then attach the sign
according to these rules.

• If the signs are the same, the answer will be
  positive.
• If the signs are opposite, the answer will be
  negative.

Examples 3 x -4 -12 -6 2 -3 -8
-4 2
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Have a go at these

• Q1.
• 2 8 b) 9 3 c) 2 5 d) 7 - 2
• e) 3 - -4 f) 11 -2 g) 10 - -4
• Q2.
• 3 x 2 b) 7 x 3 c) 6 x 5 d) 12 3
• e) 16 -2 f) 10 4 g) 4 x 3 x -1

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Substitution
Substitution means replacing the letters in a
formula or expression by the given numbers.
Example Find the value of 3a 4b 5c where a
3, b 7 and c -4. We get 3a 4b 5c
3(3) 4(7) 5(-4) 9 28 - -20 9
28 20 57
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Have a go at these

• If a 3, c 2 and d 5. Evaluate the
  following
• 3a 2 2. 4c d 3. 2c 3a
• 5d a 5. d 2c 6. d - 2a
• 7. 4c 2d 8. 7a 5d 9. c - d

• If a 6, c -3 and d -4. Evaluate the
  following
• 3a 2 2. 4c d 3. 2c 3a
• 5d a 5. d 2c 6. d - 2a
• 7. 4c 2d 8. 7a 5d 9. c - d

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Simplifying Expressions

• An algebraic expression is a collection of
  algebraic terms along with their and signs.
• Like Terms are numerical multiples of the same
  algebraic quantity.
• You can add or subtract like terms.
• The process of adding and subtracting like terms
  in an expression or equation is called
  simplifying.
• You can also simplify terms and expressions by
  multiplying or dividing.
• When you are dividing terms, you can simplify
  them by cancelling.

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Have a go at these

• Simplify the following
• 4a 7b 2a b
• 5 3y 6y 2
• 2x 7y 2x 3y
• 5a 6b 6a c 12 8c
• 4x 2a 9y 9a 16x y 7b

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Expanding
Brackets are used to group algebraic terms. The
process of removing brackets from an expression
(or an equation) is called expanding. When
expanding brackets in an expression, you must
multiply all the terms inside the brackets by the
term just before the bracket.
Example Expand the following expression. 3(5a
2b) 3(5a 2b) 3 x 5a 3 x 2b 15a
6b
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Have a go at these
Expand the following 1) 3(a b) 9) 7b(a
2) a(3b 3) 2) 4(2f 3k) 10) 3(x - 2)
(x - 2) 3) -5(3x 6) 4) 3x(5 4) 5)
-2y 6(3x 6y) 6) -3(a 3b 6c) 7) 2(-x
3y) -3(6y 3x) 8) -2x(3 4) 6(2x y)
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What is Algebra?
In algebra we are trying to find an unknown or
unknowns.
These unknowns are usually represented by letters
such as a or x.
e.g. 5 x 9 x 4
In an algebraic context 3a 4b means 3 times
one number plus 4 times another number.
REMEMBER a and b stand for unknown numbers!!
If a was 5 and b was 2, in the equation 3a
4b, then 3a 4b would be 3x5 4x2 158 23
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In the Primary School
In the primary school it may be better to
introduce other signs for the unknown(s) at first.
Usually a box is used to represent the unknown.
Children are then expected to put the appropriate
number in the box.
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In the National Numeracy Strategy
Algebra is a compact language which follows
precise conventions and rules. Formal algebra
does not begin until Key Stage 3 but you need to
lay the foundations in Key Stages 1 and 2 by
providing early algebraic activities from which
later work in algebra can develop.
Look in the NNS Introducing the Framework -
Laying the Foundations for Algebra Pages 9-10.
This section highlights the early algebraic
activities which lay the foundations for algebra.
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Keeping the Balance
REMEMBER What ever you do to one side of an
equation (or sum) you need to do the same to the
other side to keep the equation balanced.
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Have a go at these -

• 2a 5 11
• 2x 6 20
• 8 7 3y
• -7 2d 10
• 12 15 2m
• -4 -7 3b

• 3x 7 20
• 5a 10 60
• 12 2y 8
• 3k 7 -10
• 5 6t 7
• 2b 4 b 3