or (ⴚ), to enter negative numbers


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1 . Using negative numbers Add, subtract, multiply and divide positive and negative integers Use the sign change key to input negative numbers into a calculator Why learn this? Manipulating negativ e numbers is a crucial skill for anyone working in ﬁ nance. You can understand adding or subtracting numbers by imagining them on a number line. ⴙ ⴚ5 ⴝ ⴚ 4 ⴚ ⴚ ⴝ Did you know? 5 If you add a negative number, the result is smaller. So, adding a negative number is the same as subtracting a positive number. If you subtract a negative number, the result is bigger. So, subtracting a negative number is the same as adding a positive number. You can use the sign change key, into your calculator. ⴙ ⴚ or (ⴚ), to enter negative numbers When you multiply or divide a positive number by a negative number, the answer is negative. The earliest known written use of negative numbers is in an Indian manuscript from the seventh century CE but, confusingly, it uses ⴙ as a symbol to mean negative! When you multiply or divide a negative number by a positive number, the answer is negative. When you multiply or divide a negative number by a negative number, the answer is positive. I can add and subtract positive integers to/from negative integers a ⴚ ⴚ 5 b ⴚ8 ⴙ 4 c ⴚ8 ⴚ 4 d ⴚ5 ⴙ 7 e ⴚ5 ⴚ 7 f ⴚ ⴚ g ⴚ ⴙ h ⴚ ⴙ 0 ⴚ I can use the sign change key to enter negative numbers into a calculator Use the sign change key on your calculator to help with these calculations. a ⴚ0 ⴙ ⴚ6 b ⴚ48 ⴚ ⴚ c ⴚ ⴛ ⴚ5 d ⴚ48. ⴜ 8. e Hannah s bank statement shows her balance as ⴚ 585 at the end of January. In February, she makes a deposit of 00 and withdraws 75. What is her bank balance at the end of February? I can multiply and divide a negative number by a positive number ⴚ ⴛ 5 4 = 0 a ⴛ ⴚ5 b ⴛ ⴚ4 c ⴚ ⴜ d ⴜ ⴚ e 5 ⴜ ⴚ5 f ⴛ ⴚ6 g ⴚ4 ⴜ h ⴚ ⴛ ⴜ 4 Getting things in order add divide integer multiply negative
2 a b 4 5 c 5 d 9 e 8 f 5 6 g h 5 On Monday night the temperature was C. By 4.0 a.m. Tuesday, the temperature had dropped by 4 degrees. At 8 a.m. Tuesday, the temperature was C. a What was the temperature at 4.0 a.m.? b What was the temperature change between 4.0 a.m. and 8 a.m.? I can add or subtract any integers Tip Use a number line to help you. 8 = 4 a 5 b 4 8 c 0 5 d 9 e 7 f g 9 9 h 4 Copy and complete. a 4 6 b 8 48 c 7 8 d 7 e f 60 Find two different pairs of numbers that multiply to make 8 and have a difference of. Learn this I can multiply and divide any integers When multiplying or dividing with two integers: if the signs are the same, the answer is positive if the signs are different, the answer is negative. these ii by evaluating the brackets first, and ii by expanding the brackets first. Do you get the same solution each time? ( 5) ii ( + 5) = 8 = 6 ii ( + 5) = + 5 = = 6 a (4 7) b (0 ) c 5 ( 4) d ( 8) 7 I can evaluate expressions with negative numbers and bracket A Pattern spotting Copy and continue this pattern to find the answer to 4. 0 Write out another pattern to help you work out 5. B Power play ( ) a ( ) b ( ) 4 c ( ) 7 d ( ) 0 e ( ) 7 Look for a rule for the value of ( ) n, where n is any positive integer. Write down your rule. positive sign sign change key subtract. Using negative numbers 5
3 . Indices and powers Find square numbers, square roots, cube numbers and cube roots Write numbers using index notation Use the square, square root, cube and cube root keys on a calculator Understand and use the index laws for multiplication and division of numbers in index form Use the index laws for positive powers of letters Why learn this? Indices are used in formulae to measure the amount of space in shapes. Square numbers are used to calculate areas, and cube numbers are used to calculate volumes. When you multiply a number by itself, you are squaring it. For example is a square number. Finding the square root of a number is the inverse, or opposite, of squaring. 6 4 because is a square root of 6. 5 is five cubed which means is a cube number. The inverse of cubing is finding the cube root. _ 5 is 5 because is the cube root of 5. You can write repeated multiplication of numbers using index notation and There are special rules (or laws ) for working with numbers written using index notation. When multiplying, you add the powers: 5. When dividing, you subtract the powers: & Level 7 Why are the numbers floating? Because they re indaseas! Joke! Watch out! A positive integer has two square roots, one positive and one negative, but by convention the square root sign positive root only. refers to the Did you know? A 6thcentury writer suggested that the 4th power should be called zenzizenzic, and the 8th power should be called zenzizenzizenzic! Without using a calculator, write these squares and square roots. a 64 b 5 c d e 00 f 9 _ g h 6 i 7 j 5 Use the squares you know to mentally calculate these. 5 5 = 5, so 5 = 5 = 9 5 = 5 a 4 b 6 c 0 d Estimate these square roots. Use the square root key on your calculator to check the exact answer. 8 = 4 and = 9, so 8 lies between and and is closer to. Estimate: 8 =.8. a b 7 c d 74 _ I can recall the first twelve square numbers and their square roots I can use the squares I know to calculate others mentally I can estimate the square roots of nonsquare numbers 6 Getting things in order cube cubed cube number (e.g. ) cube root (e.g. 8 ) index (indices)
4 Write these numbers as squares, cubes or powers of = 0 because 0 0 = 00 a 8 b 64 c 000 d e I can give the positive and negative square roots of a number a b the square roots of 8 c the square roots of 4 d 49 I can use index notation to write squares, cubes and powers of 0 _ Rewrite these using index notation. a ⴛⴛⴛⴛⴛⴛⴛ b ⴛⴛⴛ c 7ⴛ7ⴛ7ⴛ7ⴛ8ⴛ8ⴛ8 d 5ⴛ5ⴛ5ⴛ a 4 c 7 b I can rewrite numbers using index notation _ e d 0 Use the cube numbers you know to mentally calculate these. a 6 c ⴚ9 b 8 Estimate these cube roots. a 9 d (0.) _ b Tip 0. ⴝ ⴜ 0 I can estimate the cube roots of noncube numbers _ c 50 d 90 I can use the index laws for multiplying and dividing numbers in index form I can use a calculator to ﬁnd squares, square roots, cubes and cube roots Simplify, leaving your answers in index form. a ⴛ b 7 ⴛ 75 c 64 ⴛ 6 d 9 ⴛ 9 e 55 ⴜ 5 f 79 ⴜ 74 g 6 ⴜ 6 h 4 ⴜ 4 Use a calculator to write these in order, smallest ﬁrst (ⴚ5) 8 ⴜ 6 Level 7 Simplify these, leaving your answers in index form. A a c6 ⴛ c5 b d8 ⴜ d c z ⴛ z4 d t5 ⴛ t ⴛ t6 e (r ⴛ r 5) ⴜ r f (u9 ⴜ u4) ⴛ u B Squared away Keith writes the numbers to 6 on cards and begins to lay them out. Two cards next to each other always add up to make a square number. 8ⴙⴝ9 ⴙ 5 ⴝ 6 5 ⴙ 0 ⴝ 5 etc. Lay out the rest of the cards so that this rule continues. index law index notation inverse power I can recall the cubes of to 5 and 0, and their roots I can use the cube numbers I know to calculate others mentally I can use the index laws for multiplying and dividing letters in index form Binary Computers often use binary strings to store and process information. A binary string uses only the digits 0 and, for example How many different binary strings are there with a one digit b two digits c three digits? List them in each case. d How many different binary strings are there with n digits? square number square root. Indices and powers 7
5 . Prime factor decomposition Find the lowest common multiple and the highest common factor Find and use the prime factor decomposition of a number Understand and use the index laws for multiplication and division of numbers in index form Use the index laws for numbers Why learn this? Just like the element s in chemistry, prime nu mbers are the building blocks that combine to make ever y other number. The lowest common multiple (LCM) of two numbers is the lowest number that is a multiple of them both. & The highest common factor (HCF) of two numbers is the highest number that is a factor of them both. & You can write any number as the product of its prime factors. For example 90 ⴝ ⴛ ⴛ ⴛ 5 or ⴛ ⴛ 5. You can use the prime factor decomposition to ﬁnd the HCF and LCM of two numbers quickly. Did you know? To multiply powers of the same number, add the indices. ⴛ ⴝ ⴝ 4 4ⴚ To divide powers of the same number, subtract the indices. ⴜ ⴝ ⴝ & Level 7 ⴙ4 4 6 Any number to the power zero is. For example 0 ⴝ, 50 ⴝ, 50 ⴝ. Level 7 Negative powers can be written as unit fractions or decimals. For example 0ⴚ ⴝ ⴝ 0., 0ⴚ ⴝ ⴝ 0.0, 0ⴚ ⴝ _ ⴝ Level The 0thcentury composer Messiaen wrote a piece of music that used prime numbers to create unpredictable rhythms. Find all the factor pairs for these numbers. a 56 b 7 c 48 I can ﬁnd all the factor pairs for any whole number d 0 I can ﬁnd the HCF of two numbers Find the highest common factor (HCF) of these numbers. a 0 and 4 b 7 and 45 c 8 and 66 d 96 and 44 Find the lowest common multiple (LCM) of these numbers. 7 and 9 Multiples of 7: 7, 4,, 8, 5, 4, 49, 56, Multiples of 9: LCM = 6 a and 0 I can ﬁnd the LCM of two numbers 6, 9, 8, 7, 6, 45, 54, 6, b 5 and 5 c 0 and 4 d and Find the prime factor decomposition of these numbers Look for a pair of factors, neither Getting things in order 5 of which is. Circle the factor if it is a prime number. Continue until no further factor pairs are possible. 80 = 5 or 5 a 0 b 4 c 7 d 99 factor highest common factor (HCF) I can ﬁnd the prime factor decomposition of a whole number Tip Check a prime tion by multiplying the orisa fact factors back together. index (indices) index law index form
6 Simplify these, leaving your answers in index form. a b c d (4 4 5 ) 4 e 4 5 f 7 7 g h 5 Use prime factor decomposition to find the HCF of these numbers. 4 and 54 Complete prime factor decomposition: 4 = 7 54 = 7 Identify common factors, and multiply them: HCF = 7 = 4 a 0 and 48 b 0 and 00 c 80 and 0 d 76 and 50 Use prime factor decomposition to find the LCM of these numbers. 84 and 08 Complete prime factor decomposition: 84 = 7 08 = 7 Multiply together all the factors but only include overlaps once: 7 = 94 a 5 and 4 b 0 and 00 c 80 and 80 d 76 and 50 a Write down the value of. What is the value of? b Use an index law to simplify. c What is the value of 0? I can use the index laws for multiplying and dividing numbers in index form I can find the HCF of two numbers using their prime factor decompositions I can find the LCM of two numbers using their prime factor decompositions I can prove that any number to the power zero is Work these out, writing your answers as decimals. a b c 0 0 d 0 0 a Use an index law to simplify. b What is the value of? c A whole number raised to a negative power is smaller than. True or false? Use prime factor decomposition to simplify these. Give your answers in index form = 5 and 48 = 4 so = 5 4 = 4 5 a 4 b 60 c d 6 0 Level 7 I can understand and use negative indices I can use the index laws to simplify multiplication and division calculations A Factor line Calculate the HCF of and 0. Draw a pair of axes and join the points (0, 0) and (, 0) with a straight line. How many points with wholenumber coordinates (not including (0, 0)) does the line pass through? What do you notice? Make a prediction: how many points with wholenumber coordinates (not including (0, 0)) will the line connecting (0, 0) to (, 5) pass through? Test your prediction to see if you are correct. B Highest common formula Using your answers from Q6 and Q7, or otherwise, copy and complete this table. Number a Number b HCF a b LCM Write down a formula for the LCM. Can you explain why it works? lowest common multiple (LCM) prime factor prime factor decomposition. Prime factor decomposition 9
7 .4 Sequences Generate and describe integer sequences Generate and predict terms from practical contexts Why learn this? A mathematical sequence is a list of numbers which follow a rule or pattern. The numbers in a sequence are called the terms of the sequence. A termtoterm rule tells you what to do to each term to obtain the next term in a sequence. An arithmetic sequence starts with a number, a, and adds on or subtracts a constant difference, d, each time. The numbers change in equalsized steps To find the rule for a sequence, look at the differences between consecutive terms the difference pattern. Not all sequences have equalsized steps. For example Sequences can describe in numbers how things grow or develop from the size of an insect population to the spread of a forest fire. Super fact! The Fibonacci sequence is a set of numbers which appears all over nature. It can be used to express the arrangement of a pine cone or how fast some species reproduce. For each sequence, identify the termtoterm rule and write the next two terms. Termtoterm rule: add The next two terms are + = and 6 + = 9 a, 5, 9,, b 9, 6,, 0, c 5, 6.5, 8, 9.5, d.5,.6,.7,.8, e, 4, 8, 6, f,,,, g 00, 00, 50, 5, Look at these growing rectangles. h 9,,, _, Watch out! A termtoterm rule could contain add, subtract, multiply or divide. I can continue or generate a sequence and use a termtoterm rule a Draw the next rectangle in the sequence. b Write down the number of squares in each rectangle. c Does this sequence increase in equal steps? Describe what is happening each time. d How many squares will be in the 5th and 6th rectangles? The first term of a sequence is, and the termtoterm rule is square the number and add. What are the next two terms in the sequence? 0 Getting things in order arithmetic sequence difference pattern flow chart
8 Which of these sequences are arithmetic sequences? Copy them and identify the values of a and d for each sequence. a c b, 5, 7, 9,, 4, 9, 6, d 0, 7, 4,, 7, 7, 7, 7, Write the sequence generated by this ﬂow chart. Is it an arithmetic sequence? Start with 4 Is your answer bigger than 50? Add Yes Learn this In an arithmetic sequence, a is the ﬁrst term and d is the constant difference that is added on each time. If the sequence is decreasing, d is a negative number. STOP I can recognise and describe an arithmetic sequence I can generate a sequence from a ﬂow chart I can recognise and describe an arithmetic sequence I can generate a sequence from a practical context No Steph is making some terraced houses out of rods. a Draw the next picture. b Copy and complete the table. Number of houses Number of rods c Describe this sequence with a termtoterm rule. d Is this sequence arithmetic? If so, write down the values of a and d. George wants to model the spread of a forest ﬁre. He starts by colouring in one square in his book. The ﬁre will spread to another square if it shares an edge with a square that is already on ﬁre. I can predict and test the next term in a practical sequence a Copy and complete the table. Term number Squares on ﬁre b c d e f Predict the number of squares on ﬁre for the 4th and 5th terms. Draw the fourth term to test your prediction. What is happening each time? Explain your answer. Describe this sequence with a termtoterm rule. Is this sequence arithmetic? If so, write down the values of a and d. Copy these sequences and write the next two terms. A a, 4, 6, 9, b 00, 99, 97, 94, c 5, 7,, 7, d, 8, 8, 6, e 49, 6, 5, 6, f, 8, 0, 7, B Look and say A sequence begins like this.,,,,,... What is the next number in the sequence? Hint: There is a clue in the title. generate predict sequence I can continue a nonarithmetic sequence Fired up George used a squarebased system to model a forest ﬁre in Q7. Repeat his experiment using equilateral triangles. (You may like to use isometric paper.) Does the ﬁre spread faster or slower? What realworld situations might this model? term termtoterm rule.4 Sequences
9 .5 Generating sequences using rules Generate a sequence using a termtoterm rule Generate a sequence using a positiontoterm rule Why learn this? Positiontoterm rules can help you predict future instances of events that follow sequences such as solar eclipses. The term number tells you the position of that term in the sequence. A positiontoterm rule tells you what to do to the term number to obtain that term in the sequence. & A positiontoterm rule can be written in words or in algebra. For example, n 5: n is the term number, so to find a term multiply its term number by and add 5. & Each of the arithmetic sequences below has one mistake. Rewrite the sequence correctly and identify a (the first term) and d (the constant difference). a, 7,, 6, b 5,, 0, 6, c, 0, 6,, I can recognise and describe an arithmetic sequence In an arithmetic sequence, the rd term is 5 and the 5th term is. What are the values of a and d for this sequence? In another arithmetic sequence, the rd term is 8 and the 7th term is 0. What are the values of a and d for this sequence? Use the positiontoterm rules to find the st, nd, rd and 0th terms of these sequences. Multiply the term number by. When term number =, term = =. When term number =, term = = 4. When term number =, term = = 6. When term number = 0, term = 0 = 0. a Multiply the term number by 5. b Multiply the term number by and add. c Multiply the term number by 7 and subtract 4. d Divide the term number by and add 5. Watch out! I can find a term given its position and a positiontoterm rule Don t confuse termtoterm and positiontoterm rules! Getting things in order arithmetic sequence decrease generate
10 Use the positiontoterm rules to find the st, nd, rd and 00th terms of these sequences. a Multiply the term number by. b Multiply the term number by and subtract 7. c Subtract from the term number and then multiply by 8. d Subtract the term number from 0. e Divide the term number by 4. Use the positiontoterm rules to find the st, nd, rd and 7th terms of these sequences. 5n st term = 5 + = 7 nd term = 5 + = rd term = 5 + = 7 7th term = = 7 a 4n b 00 n c 7n 99 d 6n 8 e n f n Tip If the number before n is positive, the sequence is increasing. If it is negative, the sequence is decreasing. I can find a term given its position and a positiontoterm rule using positive and negative numbers I can find a term given its position and an algebraic positiontoterm rule a For each of these positiontoterm rules, write the first five terms. 5n 7 4n 6n n b What do you notice about the termtoterm rule and the positiontoterm rule in each case? Match each termtoterm definition to a positiontoterm definition. Termtoterm Positiontoterm a Start at, add each time. b Start at 4, subtract each time. ii Multiply the term number by 4 and subtract. Multiply the term number by and add 7. Tip Find the first few terms of each sequence. c Start at, add 4 each time. Double the term number and add. True or false? a The sequence 7n produces the multiples of 7. b Every term in the sequence n is odd. c The sequence 5 n is a decreasing sequence. d The sequences (n 4) and n 8 are not the same. e Every term in the sequence (00 n) is positive. I can understand and use algebraic positiontoterm rules A Can you digit? The digits to 8 can be arranged as an arithmetic sequence of twodigit numbers like this., 4, 56, 78 (start at, add each time) Find another way to arrange the digits to 8 as an arithmetic sequence of twodigit numbers. B Rows by any other name Calculate the first five terms of the sequence with the positiontoterm rule _ n(n ). By what name are these numbers better known? increase positiontoterm rule sequence term.5 Generating sequences using rules
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