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Year 8 Autumn 2
Negative numbers and inequality statements
4 hours
Key concepts
The Big Picture: Number and Place Value progression map
 order positive and negative integers, decimals and fractions
 use the symbols =, ≠, <, >, ≤, ≥
Possible learning intentions
Possible success criteria
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Compare numbers
Order numbers
Place a set of negative numbers in order
Place a set of mixed positive and negative numbers in order
Identify a common denominator that can be used to order a set of fractions
Order fractions where the denominators are not multiples of each other
Order a set of numbers including a mixture of fractions, decimals and negative numbers
Use inequality symbols to compare numbers
Make correct use of the symbols = and ≠
Prerequisites
Mathematical language
Pedagogical notes
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Positive number
Negative number
Integer
Numerator
Denominator
Zero is neither positive nor negative. The set of integers includes the natural numbers {1,
2, 3, …}, zero (0) and the ‘opposite’ of the natural numbers {-1, -2, -3, …}.
Pupil must use language correctly to avoid reinforcing misconceptions: for example, 0.45
should never be read as ‘zero point forty-five’; 5 > 3 should be read as ‘five is greater than
3’, not ‘5 is bigger than 3’.
Ensure that pupils read information carefully and check whether the required order is
smallest first or greatest first.
The equals sign was designed by Robert Recorde in 1557 who also introduced the plus (+)
and minus (-) symbols.
NCETM: Glossary
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Understand that negative numbers are numbers less than zero
Order a set of decimals with a mixed number of decimal places (up
to a maximum of three)
Order fractions where the denominators are multiples of each
other
Order fractions where the numerator is greater than 1
Know how to simplify a fraction by cancelling common factors
Notation
The ‘equals’ sign: =
The ‘not equal’ sign: ≠
The inequality symbols: < (less than), > (greater than), ≤ (less than or
equal to), ≥ (more than or equal to)
Common approaches
Teachers use the language ‘negative number’ to avoid future confusion with calculation
that can result by using ‘minus number’
Every classroom has a negative number washing line on the wall
Reasoning opportunities and probing questions
Suggested activities
Possible misconceptions
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KM: Farey Sequences
KM: Decimal ordering cards 2
KM: Maths to Infinity: Fractions, decimals and percentages
KM: Maths to Infinity: Directed numbers
NRICH: Greater than or less than?
YouTube: The Story of Zero
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Jenny writes down 0.400 > 0.58. Kenny writes down 0.400 < 0.58.
Who do you agree with? Explain your answer.
Find a fraction which is greater than 3/5 and less than 7/8. And
another. And another …
Convince me that -15 < -3
Learning review
www.diagnosticquestions.com
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Some pupils may believe that 0.400 is greater than 0.58
Pupils may believe, incorrectly, that:
- A fraction with a larger denominator is a larger fraction
- A fraction with a larger numerator is a larger fraction
- A fraction involving larger numbers is a larger fraction
Some pupils may believe that -6 is greater than -3. For this reason ensure pupils avoid
saying ‘bigger than’
Formulate and evaluate expressions
10 hours
Key concepts
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The Big Picture: Algebra progression map
use and interpret algebraic notation, including: a²b in place of a × a × b, coefficients written as fractions rather than as decimals
understand and use the concepts and vocabulary of factors
simplify and manipulate algebraic expressions by taking out common factors and simplifying expressions involving sums, products and powers, including the laws of indices
substitute numerical values into scientific formulae
rearrange formulae to change the subject
Possible learning intentions
Possible success criteria
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Understand the notation of algebra
Manipulate algebraic expressions
Evaluate algebraic statements
Know how to write products algebraically
Use fractions when working in algebraic situations
Identify common factors (numerical and algebraic) of terms in an expression
Factorise an expression by taking out common factors
Simplify an expression involving terms with combinations of variables (e.g. 3a²b + 4ab2 + 2a2 – a2b)
Know the multiplication (division, power, zero) law of indices
Understand that negative powers can arise
Substitute positive and negative numbers into formulae
Be aware of common scientific formulae
Know the meaning of the ‘subject’ of a formula
Change the subject of a formula when one step is required
Change the subject of a formula when a two steps are required
Prerequisites
Mathematical language
Pedagogical notes
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Product
Variable
Term
Coefficient
Common factor
Factorise
Power
Indices
Formula, Formulae
Subject
Change the subject
During this unit pupils should experience factorising a quadratic expression such as 6x2
+ 2x.
Collaborate with the science department to establish a list of formulae that will be
used, and ensure consistency of approach and experience.
NCETM: Algebra
NCETM: Departmental workshop: Index Numbers
NCETM: Departmental workshops: Deriving and Rearranging Formulae
NCETM: Glossary
Know basic algebraic notation (the rules of algebra)
Simplify an expression by collecting like terms
Know how to multiply a single term over a bracket
Substitute positive numbers into expressions and formulae
Calculate with negative numbers
Notation
See key concepts above
Common approaches
Once the laws of indices have been established, all teachers refer to ‘like numbers
multiplied, add the indices’ and ‘like numbers divided, subtract the indices. They also
generalise to am × an = am+n, etc.
When changing the subject of a formula the principle of balancing (doing the same to
both sides) must be used rather than a ‘change side, change sign’ approach.
Reasoning opportunities and probing questions
Suggested activities
Possible misconceptions
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KM: Missing powers
KM: Laws of indices. Some useful questions.
KM: Maths to Infinity: Indices
KM: Scientific substitution (Note that page 2 is hard)
NRICH: Temperature
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Convince me a0 = 1.
What is wrong with this statement and how can it be corrected:
52 × 54 = 58 ?
Jenny thinks that if y = 2x + 1 then x = (y – 1)/2. Kenny thinks
that if y = 2x + 1 then x = y/2 – 1. Who do you agree with?
Explain your thinking.
Learning review
KM: 8M3 BAM Task, 8M7 BAM Task, 8M8 BAM Task
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Some pupils may misapply the order of operation when changing the subject of a
formula
Many pupils may think that a0 = 0
Some pupils may not consider 4ab and 3ba as ‘like terms’ and therefore will not
‘collect’ them when simplifying expressions
Linear equations
4 hours
Key concepts
The Big Picture: Algebra progression map
 solve linear equations with the unknown on both sides of the equation
 find approximate solutions to linear equations using a graph
Possible learning intentions
Possible success criteria
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Solve linear equations with the unknown on one side
Solve linear equations with the unknown on both sides
Explore connections between graphs and equations
Prerequisites
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Identify the correct order of undoing the operations in an equation
Solve linear equations with the unknown on one side when the solution is a negative number
Solve linear equations with the unknown on both sides when the solution is a whole number
Solve linear equations with the unknown on both sides when the solution is a fraction
Solve linear equations with the unknown on both sides when the solution is a negative number
Solve linear equations with the unknown on both sides when the equation involves brackets
Recognise that the point of intersection of two graphs corresponds to the solution of a connected equation
Check the solution to an equation by substitution
Mathematical language
Choose the required inverse operation when solving an equation Algebra, algebraic, algebraically
Solve linear equations by balancing when the solution is a whole Unknown
Equation
number or a fraction
Operation
Solve
Solution
Brackets
Symbol
Substitute
Graph
Point of intersection
Notation
The lower case and upper case of a letter should not be used
interchangeably when worked with algebra
Juxtaposition is used in place of ‘×’. 2a is used rather than a2.
Division is written as a fraction
Pedagogical notes
This unit builds on the wok solving linear equations with unknowns on one side in
Stage 7. It is essential that pupils are secure with solving these equations before
moving onto unknowns on both sides.
Encourage pupils to ‘re-present’ the problem using the Bar Model.
NCETM: The Bar Model
x
x
x
x
8
NCETM: Algebra
x
14
NCETM: Glossary
x
Common approaches
All pupils should solve equations by
balancing:
4x + 8 = 14 + x
-x
-x
3x + 8 = 14
-8
-8
3x = 6
÷3
÷3
x = 2
Reasoning opportunities and probing questions
Suggested activities
Possible misconceptions
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KM: Solving equations
KM: Stick on the Maths: Constructing and solving equations
NRICH: Think of Two Numbers
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Show me an (one-step, two-step) equation with a solution of -8
(negative, fractional solution). And another. And another …
Show me a two-step equation that is ‘easy’ to solve. And
another. And another …
What’s the same, what’s different: 2x + 7 = 25, 3x + 7 = x + 25, x
+ 7 = 7 – x, 4x + 14 = 50 ?
Convince me how you could use graphs to find solutions, or
estimates, for equations.
Learning review
KM: 8M10 BAM Task
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x
8
14
x
x
6
x
x
2
Some pupils may think that you always have to manipulate the equation to have
the unknowns on the LHS of the equal sign, for example 2x – 3 = 6x + 6
Some pupils think if 4x = 2 then x = 2.
When solving equations of the form 2x – 8 = 4 – x, some pupils may subtract ‘x’
from both sides.
Expressions and Equations from real-life contexts
Video
https://www.teachingchannel.org/videos/teaching-algebra
Worksheets
https://www.tes.com/teaching-resource/forming-and-solving-equations-6334898
DCSF National Strategies Handbook - contains guidance and worksheets
Linear sequences: nth term
4 hours
Key concepts
The Big Picture: Algebra progression map
 generate terms of a sequence from either a term-to-term or a position-to-term rule
 deduce expressions to calculate the nth term of linear sequences
Possible learning intentions
Possible success criteria
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Explore sequences
Generate a sequence from a term-to-term rule
Understand the meaning of a position-to-term rule
Use a position-to-term rule to generate a sequence
Find the position-to-term rule for a given sequence
Use algebra to describe the position-to-term rule of a linear sequence (the nth term)
Use the nth term of a sequence to deduce if a given number is in a sequence
Generate a sequence using a spreadsheet
Prerequisites
Mathematical language
Pedagogical notes
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Sequence
Linear
Term
Difference
Term-to-term rule
Position-to-term rule
Ascending
Descending
Using the nth term for times tables is a powerful way of finding the nth term for any
linear sequence. For example, if the pupils understand the 3 times table can ne
described as ‘3n’ then the linear sequence 4, 7, 10, 13, … can be described as the 3
times table ‘shifted up’ one place, hence 3n + 1.
Exploring statements such as ‘is 171 is in the sequence 3, 9, 15, 21, 27, ..?’ is a very
powerful way for pupils to realise that ‘term-to-term’ rules can be inefficient and
therefore ‘position-to-term’ rules (nth term) are needed.
NCETM: Algebra
NCETM: Glossary
Use a term-to-term rule to generate a sequence
Find the term-to-term rule for a sequence
Describe a sequence using the term-to-term rule
Notation
T(n) is often used when finding the nth term of sequence
Common approaches
Teachers refer to a sequence such as 2, 5, 8, 11, … as ‘the three times table minus one’,
to help pupils construct their understanding of the nth term of a sequence.
All students have the opportunity to use spreadsheets to generate sequences
Reasoning opportunities and probing questions
Suggested activities
Possible misconceptions
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KM: Spreadsheet sequences
KM: Generating sequences
KM: Maths to Infinity: Sequences
KM: Stick on the Maths: Linear sequences
NRICH: Charlie’s delightful machine
NRICH: A little light thinking
NRICH: Go forth and generalise
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Show me a sequence that could be generated using the nth term
4n ± c. And another. And another …
What’s the same, what’s different: 4, 7, 10, 13, 16, …. , 2, 5, 8,
11, 14, … , 4, 9, 14, 19, 24, …. and 4, 10, 16, 22, 28, …?
The 4th term of a linear sequence is 15. Show me the nth term of
a sequence with this property. And another. And another …
Convince me that the nth term of the sequence 2, 5, 8, 11, … is
3n -1 .
Kenny says the 171 is in the sequence 3, 9, 15, 21, 27, … Do you
agree with Kenny? Explain your reasoning.
Learning review
KM: 8M9 BAM Task
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Some pupils will think that the nth term of the sequence 2, 5, 8, 11, … is n + 3.
Some pupils may think that the (2n)th term is double the nth term of a linear
sequence.
Some pupils may think that sequences with nth term of the form ‘ax ± b’ must start
with ‘a’.
Further extension material includes:
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T-totals
Explore non-linear sequences