Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
History of mathematical notation wikipedia , lookup
Collatz conjecture wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Elementary mathematics wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Elementary algebra wikipedia , lookup
System of polynomial equations wikipedia , lookup
Level 5 mathematics 0011 0010 1010 1101 0001 0100 1011 1 2 4 Expectations and activities Aims: 0011 0010 1010 1101 0001 0100 1011 • To become familiar with Level 5 expectations in mathematics • To develop understanding of Level 5 mathematical knowledge 1 2 4 Are children at Key Stage 2 getting the learning experiences they need to meet the Level 5 expectations? 0011 0010 1010 1101 0001 0100 1011 1 How can activities be developed to create these opportunities for Level 5 children? 2 4 Task: • Using the Level 4 – 5 APP assessment guidelines, identify the additional skills, knowledge and concepts needed to achieve a Level 5 from a Level 4 0011 0010 1010 1101 0001 0100 1011 1 2 4 • Highlight which of the identified aspects the children find most tricky to learn Here is an equilateral triangle inside a rectangle. 0011 0010 1010 1101 0001 0100 1011 Not to scale 1 x 12° Calculate the value of angle x. Do not use a protractor (angle measurer). Show your method. You may get a mark. º 2 4 2 marks It is not enough for children to know the facts, they need to be able to apply their knowledge to the problem, in what may be an unfamiliar context for them, in order to succeed at Level 5 (argue and reason) 0011 0010 1010 1101 0001 0100 1011 1 2 4 Algebra 0011 0010 1010 1101 0001 0100 1011 1 2 4 Level 5 algebra expectations 0011 0010 1010 1101 0001 0100 1011 • Use letter symbols to represent unknown numbers or variables • Know and use the order of operations and understand that algebraic operations follow the same conventions and order as arithmetic operations • Simplify or transform linear expressions by collecting like terms; multiplying a single term over a bracket • Substitute integers into simple formulae • Use and interpret coordinates in all four quadrants • Plot the graphs of simple linear functions 1 2 4 Algebraic Conventions 0011 0010 1010 1101 0001 1011 the • Recognise and0100 explain 3x + 5 = 11 p + q = 20 2l + 2b = p y = x/2 – 7 use of symbols Represent an unknown value in equations with a unique solution Represent unknown values in equations with a set of solutions Represent variables in formulae Represent variables in functions • Identify equivalent terms and expressions 2x + x + 5 ax + 5 7(x + 2) (x + 2)(x + 5) x³ × x simple chains of operations some with unknown coefficients brackets (linear) brackets (quadratic) positive indices • Identify types and forms of formulae a/b = l, a = l × b a = l × b, 2l + 2b = p equivalence of formula dimensions of a formula 1 2 4 Mints A teacher has 5 full packets of mints and 6 single mints. 0011 0010 1010 0001 The 1101 number of 0100 mints1011 inside each packet is the same. 1 2 The teacher tells the class: “Write an expression to show how many mints there are altogether. Call the number of mints inside each packet y” 4 Here are some of the expressions that the pupils write: 5+6+y 5y6 5y + 6 5 + 6y (5 + 6) x y 6 + 5y Write down two expressions that are correct. 0011 0010 1010 1101 0001 0100 1011 Jill took 2 marbles out of one of the Jill has 3 bags of marbles. bags, and none out of the other bags. Each bag has p marbles inside. Jill takes some marbles out. Now the total number of marbles in Jill’s 3 bags is 3p – 6 Some of the statements on the right could be true. Put a tick by each statement which could be true. Jill took 2 marbles out of each of the bags. 1 Jill took 3 marbles out of one of the bags, and none out of the other bags. 2 4 Jill took 3 marbles out of each of two of the bags, and none out of the other bag. Jill took 6 marbles out of one of the bags, and none out of the other bags. Jill took 6 marbles out of each of two of the bags, and none out of the other bag. Solving Linear equations • One-step linear equations One-step linear equations the unknown 0011 0010 1010 1101 0001with 0100 1011 in a ‘standard’ position: x+4=7 positive integer solutions x/4 = 6 x – 9 = 34 8x = 56 3x = 5 non-integer solutions x + 14 = 9 negative integer solutions One-step linear equations with the unknown perceived to be in a ‘harder’ position: 13 = 8 + x positive integer solutions 20/x = 10 20/x = 3 non-integer solutions 13 = 8 – x negative integer solutions • Equations involving brackets 3(x + 4) = 27 (x – 5)/3 = 7 positive integer solutions • Inequalities 5x < 10 –4 < 2x < 10 5x + 3 < 10 one boundary to solution set, one-step solution two boundaries to solution set two-step solution 1 2 4 Algebra Pairs Join pairs of algebraic expressions that have the same value when a = 3, b = 2 and c = 6 0011 One 0010 pair 1010 0001 0100 1011 is 1101 joined for you. ab 3c – 2b 3c 1 2c + b Which expressions have the same value when a = b = c a2 2 4 a+c 2a Sort these equations into “always true”, “sometimes true” or “never true”: 0011 0010 1010 1101 0001 0100 1011 a + 5 = 12 b + 12 = b + 16 2c + 3 = 3 + 2c 2d – 5 = 5 – 2d f + 12 = g + 12 k + 5 < 20 4 + 2e = 6e p2 = 10p 3(m + 3) = 3m + 3 4(3 + n) = 12 + 3n 1 2 4 Sequences • Linear sequences: find and describe in words and symbols 0011 0010 1101 1011the nth term of a sequence the 1010 rule for the0001 next 0100 term and Use the context of a sequence to generate the related numerical terms Notice and describe how the sequence is growing term by term and relate this to the context Notice and describe a general term in the sequence and relate this to the context Appreciate different forms for the general term and relate each to the context Generate different forms for the general term and relate each to the context • Quadratic sequences: 1 2 find and describe in symbols the rule for the next term and the nth term of a sequence Repeat the progression described above 4 • Recognise and describe types of sequences: for example, arithmetical sequences and multiples, triangular numbers, square numbers… Use knowledge of related geometrical patterns Use differences to test for types of sequence Sequences 0011 0010 1010 1101 0001 0100 1011 1 2 4 1. Use the context of a sequence to generate the related numerical terms 2. Notice and describe how the sequence is growing term by term and relate this to the context 3. Notice and describe a general term in the sequence and relate this to the context 4. Appreciate different forms for the general term and relate each to the context 5. Generate different forms for the general term and relate each to the context Number Chains 0011 0010 1010 1101 0001 0100 1011 Each number chain has a similar property to the Fibonacci sequence – that is, each term is the sum of the previous two. 1 Find the missing terms: 3 4 … … … … 18 … … 2 4 … 36 Functions and graphs • Interpret graphs of functions • Generate graphs of functions • Interpret graphs arising from real-life problems • Generate graphs arising from real-life problems 0011 0010 1010 1101 0001 0100 1011 1 2 4 Level 5 expectations 0011 0010 1010 1101 0001 0100 1011 • Generate and plot pairs of coordinates for: y = x + 1, y = 2x • Plot graphs such as: y = x, y = 2x • Plot and interpret graphs such as: y = x, y = 2x and y = x + 1, y = x – 1 1 2 4 Are children at Key Stage 2 getting the learning experiences they need to meet the Level 5 expectations? 0011 0010 1010 1101 0001 0100 1011 1 How can activities be developed to create these opportunities for Level 5 children? 2 4