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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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 KM: Solving equations KM: Stick on the Maths: Constructing and solving equations NRICH: Think of Two Numbers 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 x 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 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 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 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 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 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: T-totals Explore non-linear sequences