MEP Pupil Text 12. A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued.
|
|
- Holly Ramsey
- 8 years ago
- Views:
Transcription
1 MEP Pupil Text Number Patterns. Simple Number Patterns A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued. Worked Example Write down the next three numbers in each sequence., 4, 6, 8, 0,... 3, 6, 9,, 5,... This sequence is a list of even numbers, so the next three numbers will be, 4, 6. This sequence is made up of the multiples of 3, so the next three numbers will be 8,, 4. Worked Example Find the next two numbers in each sequence. 6, 0, 4, 8,,... 3, 8, 3, 8, 3,... For this sequence the difference between each term and the next term is 4. Sequence 6, 0, 4, 8,,... Difference So 4 must be added to obtain the next term in the sequence. The next two terms are + 4 = 6 and = 30, giving 6, 0, 4, 8,, 6, 30,... For this sequence, the difference between each term and the next is 5. Sequence 3, 8, 3, 8, 3,... Difference Adding 5 gives the next two terms as = 8 and = 33, giving 3, 8, 3, 8, 3, 8, 33,... 6
2 MEP Pupil Text Exercises. Write down the next four numbers in each list., 3, 5, 7, 9,... 4, 8,, 6, 0,... (c) 5, 0, 5, 0, 5,... (d) 7, 4,, 8, 35,... (e) 9, 8, 7, 36, 45,... (f) 6,, 8, 4, 30,... (g) 0, 0, 30, 40, 50,... (h),, 33, 44, 55,... (i) 8, 6, 4, 3, 40,... (j) 0, 40, 60, 80, 00,... (k) 5, 30, 45, 60, 75,... (l) 50, 00, 50, 00, 50,.... Find the difference between terms for each sequence and hence write down the next two terms of the sequence. 5, 8,, 4, 7,..., 0, 8, 6, 34,... (c) 7,, 7,, 7,... (d) 6, 7, 8, 39, 50,... (e) 8, 5,, 9, 36,... (f), 3, 3, 4, 4,... (g) 4, 3,, 3, 40,... (h) 6, 3, 0, 7, 4,... (i) 0, 6,, 8, 4,... (j) 8, 4, 0, 6,,... (k), 8, 5,,,... (l) 5, 8,, 4, 7, In each part, find the answers to (i) to (iv) with a calculator and the answer to (v) without a calculator. (i) =? (i) 99 =? (ii) =? (ii) 999 =? (iii) =? (iii) 9999 =? (iv) =? (iv) =? (v) =? (v) =? (c) (i) 88 =? (d) (i) 7 9 =? (ii) 888 =? (ii) 7 99 =? (iii) 8888 =? (iii) =? (iv) =? (iv) =? (v) =? (v) =? 4. (i) Complete the following number pattern: = = =?? =? (ii) Look at the numbers in the right hand column. Write down what you notice about these numbers. 63
3 . MEP Pupil Text Use your calculator to work out the next line of the pattern. What do you notice now? (NEAB) 5. 9 = = = = = = = = 8 (i) Complete the statement, The units digits in the right hand column, in order, are 8, 7, 6,... (ii) (iii) Complete the statement, The tens digits in the right-hand column, in order, are,, 3,... What is the connection between the answers to parts (i) and (ii)? The numbers in the right-hand column go up by 9 each time. What else do you notice about these numbers? (MEG). Recognising Number Patterns This section looks at how the terms of a sequence are related. For example the Fibonacci sequence:,,, 3, 5, 8, 3,... is obtained by adding together two consecutive terms to obtain the next term. Worked Example + = + = = = = 3 The sequence of square numbers is, 4, 9, 6, 5, 36,... Explain how to obtain the next number in the sequence. The next number can be obtained in one of two ways. 64
4 MEP Pupil Text () The sequence could be written,,, 3, 4, 5, 6,... and so the next term would be 7 = 49. () Calculating the differences between the terms gives Sequence, 4, 9, 6, 5, 36,... Difference The difference increases by each time so the next term would be = 49. Worked Example Describe how to obtain the next term of each sequence below. 3, 0, 7, 4, 3,... 3, 6,, 8, 7,... (c), 5, 6,, 7, 8,... Finding the differences between the terms gives Sequence 3, 0, 7, 4, 3,... Difference All the differences are the same so each term can be obtained by adding 7 to the previous term. The next term would be = 38. Here again the differences show a pattern. Sequence 3, 6,, 8, 7,... Difference Here the differences increase by each time, so to find further terms add more than the previous difference. The next term would be 7 + = 38. (c) The differences again help to see the pattern for this sequence. Sequence, 5, 6,, 7, 8,... Difference If the first difference, 4, is ignored, the remaining pattern of the differences is the same as the sequence itself. This shows that the difference between any two terms is equal to the previous term. So a new term is obtained by adding together the two previous terms. The next term of the sequence will be =
5 . MEP Pupil Text Worked Example 3 A sequence of shapes is shown below. Write down a sequence for the number of line segments and explain how to find the next number in the sequence. The sequence for the number of line segments is 4, 8,, 6,... The difference between each pair of terms is 4, so to the previous term add 4. Then the next term is = 0. This corresponds to the shape opposite, which has 0 line segments. Exercises. Find the next two terms of each sequence below, showing the calculations which have to be done to obtain them. 5,, 7, 3, 9,... 6, 0, 5,, 8,... (c), 9, 6, 3, 0,... (d) 30,, 5, 9, 4,... (e) 50, 56, 63, 7, 80,... (f),, 4, 8, 4,,... (g) 6,, 6,, 6,... (h) 0, 3, 8, 5, 4,... (i) 3, 6,, 4, 48,... (j), 4, 0,, 46,... (k) 4,,, 6, 46,.... A sequence of numbers is, 8, 7, 64, 5,... By considering the differences between the terms, find the next two terms. 3. Each sequence of shapes below is made up of lines which join two points. For each sequence: (i) (ii) (iii) write down the number of lines, as a sequence; explain how to obtain the next term of the sequence; draw the next shape and check your answer. 66
6 MEP Pupil Text (c) (d) (e) 4. Write down a sequence for the number of dots in each pattern. Then explain how to get the next number. (c) (d) 67
7 . MEP Pupil Text (e) (f) 5. Describe how the sequence is formed., 4, 9, 6, 5, What is the relationship between the sequence in and the sequences below? For each sequence explain how to calculate the terms. (i) 3, 6,, 8, 7, 38,... (ii),, 7, 4, 3, 34,... (iii), 8, 8, 3, 50, 7,... (iv) 4, 6, 36, 64, 00, 44, Describe how the sequences below are related. (i),,, 3, 5, 8, 3,,... (ii) 4, 4, 5, 6, 8,, 6, 4,... (iii),, 0,, 3, 6,, 9,... Describe how to find the next term of each sequence. 7. A computer program prints out the following numbers When one of these numbers is changed, the numbers will form a pattern. Circle the number which has to be changed and correct it. Give a reason why your numbers now form a pattern. (SEG) 8. Here are the first four numbers of a number pattern. 7, 4,, 8,... Write down the next two numbers in the pattern. Describe, in words, the rule for finding the next number in the pattern. (LON) 68
8 MEP Pupil Text 9. To generate a sequence of numbers, Paul multiplies the previous number in the sequence by 3, then subtracts. Here are the first four numbers of his sequence.,, 5, 4,... Find the next two numbers in the sequence. Here are the first four numbers of the sequence of cube numbers., 8, 7, 64,... Find the next two numbers in the sequence. (MEG) 0. Row q Row p The numbers in Row of the above pattern are found by using pairs of numbers from Row. For example, 7 = 9 9 = 8 9 By considering the sequence in Row, write down the value of p. Find the value of q. (MEG). A number pattern begins 4, 8,, 6, 0, 4,... Describe the number pattern. Another number pattern begins, 4, 9, 6, 5, 36,... (i) Describe this number pattern. (ii) What is the next number in this pattern? Each number in this pattern is changed to make a new number pattern. The new number pattern begins (iii),, 7, 4, 3, 34,... What is the next number in the new pattern? Explain how you found your answer. (SEG). (i) Write down the multiples of 5, from 5 to 40. (ii) Describe the pattern of the units digits. Sequence P is 3, 6, 9,, 5, 8,,... Explain how sequence P is produced. 69
9 . MEP Pupil Text (c) Copy and complete the table below. Sequence P Add and then multiply by Sequence Q = 4, 4 = = 7, 7 = 4 4 9?????? 5??? 8??? (d) (i) Find the next two terms in the sequence, 4, 0, 9, 3, 46, 64,... (ii) Explain how you obtained your answer to part (d) (i). (MEG).3 Extending Number Patterns A formula or rule for extending a sequence can be used to work out any term of a sequence without working out all the terms. For example, the 00th term of the sequence,, 4, 7, 0, 3,... can be calculated as 98 without working out any other terms. Worked Example Find the 0th term of the sequence 8, 6, 4, 3,... The terms of the sequence can be obtained as shown below. This pattern can be extended to give Worked Example st term = 8 = 8 nd term = 8 = 6 3rd term = 3 8 = 4 4th term = 4 8 = 3 0th term = 0 8 = 60 Find the 0th and 00th terms of the sequence 3, 5, 7, 9,,... 70
10 MEP Pupil Text The terms above are given by st term = 3 nd term = 3 + = 5 3rd term = 3 + = 7 4th term = = 9 This can be extended to give 5th term = = 0th term = = 00th term = = 0. Worked Example 3 Find the 0th term of the sequence, 5, 0, 7, 6, 37,... The terms of this sequence can be expressed as st term = + nd term = + 3rd term = 3 + 4th term = 4 + Extending the pattern gives 5th term = 5 + 0th term = 0 + = 40. Exercises. Find the 0th and 0th terms of each sequence below. 4, 8,, 6, 0,... 5, 0, 5, 0, 5,... (c),, 3, 4, 5,... (d) 7, 9,, 3, 5,... (e) 5, 9, 3, 7,,... (f) 0, 9, 8, 7, 6,... (g) 50, 44, 38, 3, 6,... (h), 5, 8, 3,... (i) 8, 7, 6, 5, 4,... (j) 4, 0, 4, 8,,... (k) 7,, 7,, 7,... (l) 3,, 7,,... 7
11 .3 MEP Pupil Text. Find the 0th term for each of the two sequences below. (i) 3, 6,, 8, 7,... (ii) 5, 6, 7, 8, 9,... Hence find the 0th term of the sequences, (i) 8,, 8, 6, 36,... (ii) 5, 36, 77, 44, 43,... (iii) 0, 4, 0, 8, Find the 0th term of the sequences below. (i) 4, 9, 4, 9, 4,... (ii) 3, 5, 7, 9,,... Use your answers to to find the 0th term of the sequence, 45, 98, 7, 64, By considering the two sequences, 4, 9, 6, 5,...,, 3, 4, 5,... find the 0th term of the sequence 0,, 6,, 0,... Find the 0th term of the sequence 0, 6, 4, 60, 0, For each sequence of shapes below find the number of dots in the 0th shape. (c) 6. Patterns of triangles are made using sticks. The first three patterns are drawn below. Pattern Pattern Pattern 3 7
12 MEP Pupil Text Pattern number 3 Number of sticks How many sticks has Pattern 4? A pattern needs 33 sticks. What is the number of this pattern? (c) (i) How many sticks are needed to make Pattern 00? (ii) Explain how you found your answer. (SEG).4 Formulae and Number Patterns This section considers how the terms of a sequence can be found using a formula and how a formula can be found for some simple sequences. The terms of a sequence can be described as u, u, u 3, u 4, u 5,... where u is the first term, u is the second term and so on. Consider the sequence, 4, 9, 6, 5,... u = = u = 4 = u 3 = 9 = 3 3 u 4 = 6 = 4 4 u 5 = 5 = 5 5 This sequence can be described by the general formula Worked Example un = n. Find the first 5 terms of the sequence defined by the formula un = 3n + 6. u = = 9 u = = u 3 = = 5 u 4 = = 8 u 5 = = So the sequence is 9,, 5, 8,,... Note that the terms of the sequence increase by 3 each time and that the formula contains a ' 3n '. 73
13 .4 MEP Pupil Text Worked Example Find the first 5 terms of the sequence defined by the formula un = 5n 4. u = 5 4 = u = 5 4 = 6 u 3 = = u 4 = = 6 u 5 = = So the sequence is, 6,, 6,,... Here the terms increase by 5 each time and the formula contains a ' 5n '. In general, if the terms of a sequence increase by a constant amount, d, each time, then the sequence will be defined by the formula where c is a constant number. Worked Example 3 un = dn + c Find a formula to describe each of the sequences below. 3, 0, 7, 34, 4, 48,...,, 3, 34, 45, 56,... First find the differences between the terms. Sequence 3, 0, 7, 34, 4, Difference As the difference between each term is always 7, the formula will contain a ' 7n' and be of the form un = 7 n + c. To find the value of c consider any term. The first one is usually easiest to use. Here, using n = and u = 3 gives so the formula is 3 = 7 + c 3 = 7 + c c = 6, un = 7 n
14 MEP Pupil Text Note Always check that the formula is correct for other terms, e.g. n = and n = 3. In this case, u = = 0 so the formula holds. and u 3 = = 7 Again start by finding the differences between the terms of the sequence. Sequence,, 3, 34, 45, 56,... Difference The difference is always so the formula will contain 'n' and will be un = n + c. Using the first term, i.e. n = and u =, gives so the formula is = + c = + c c = 0, un = n 0. [Check: u = 0 = and u 3 = 3 0 = 3, which are correct.] Worked Example 4 The 7th, 8th and 9th terms of a sequence are 6, 69 and 77 respectively. Find a formula to describe this sequence. Looking at the differences, Sequence 6, 69, 77, Difference 8 8 you can conclude that the sequence must be of the form un = 8 n + c. For n = 7, u7 = 56 + c, giving c = 5. Thus the formula to describe the sequence is un = 8n
15 .4 MEP Pupil Text Note A more general way of tackling this type of problem is to fit the linear sequence to the given information. un = dn + c In this case, u7 = 7d + c = 6 and u8 = 8d + c = 69. Subtracting u 7 from u 8 gives d = 8 and substituting for d in either of the two equations gives c = 5. Thus, as before, un = 8n + 5. Exercises. Use the formulae below to find the first 6 terms of each sequence. un = 4n + un = 5n 7 (c) un = 0n + (d) un = n (e) un = n + n (f) u n =. The sequences described by the formulae, un = 8n, un = 3n+ 5, un = n +, un = 3 n are given below. Select the formula that describes each sequence. 0, 7, 6, 63, 4,... 8,, 4, 7, 0,... (c) 6, 4,, 30, 38,... (d), 5, 0, 7, 6, Find the 0th term for each sequence below. un = 4 n u n = 3 n 50 (c) un = 4 n + 7 (d) u n 40 (e) u n n = n = n 4 (f) u n = Consider the formula for the sequence below. 8, 5,, 9, 36, Explain why the formula contains 7n. Find the formula for the sequence. 76
16 MEP Pupil Text 5. Find the formula which describes each sequence below. 4, 9, 4, 9, 4,..., 4, 7, 0, 3,... (c), 4, 0, 6,,... (d) 00, 98, 96, 94, 9,... (e),,,, 3,... (f) 5,, 9, 6, 3,... (g) 0, 9, 9, 8, 8, Which of the sequences below are described by a formula of the form un = dn + c? Where possible, give the formula.,,, 3, 5, 8,..., 0, 8, 8, 46, 74,... (c) 0, 3, 8, 5, 4, 35, 48,... (d),, 4, 7,, 6,... (e),.,.,.3,.4,.5,... (f),.,.,.33,.464,.605,... (g) 3, 7,, 5, 9, 3, Write down the first 6 terms of the sequence un n. Then use your answer to write down formulae for the following sequences. 3, 6,, 8, 7, 38,... 4,, 4,, 0, 3,... (c), 8, 8, 3, 50, 7,... (d) 0, 6, 6, 30, 48, 70,... (e) 99, 96, 9, 84, 75, 64, By considering the sequence described by un n, find formulae to describe the following sequences. 0, 7, 6, 63, 4,... 0, 7, 36, 73, 34,... (c) 99, 9, 73, 36, 75, The 0th, th and th terms of a sequence are 50, 54 and 58. Find a formula to describe this sequence and write down the first 5 terms. 0. The 00th, 0st and 0nd terms of a sequence are 608, 64 and 60. Find a formula to describe this sequence and find the 0th term.. The 0th, th, 4th and 6th terms of a sequence are 5, 6, 7 and 8. Find a formula to describe this sequence and find its first term. 77
17 .4 MEP Pupil Text. Consider the sequence,, 5, 9, 3, 7,, 5,... Find the next term in the sequence and explain how you obtained your answer. The n th term in the sequence is 4n 3. Solve the equation 4 n 3 = 397 and explain what the answer tells you. (MEG) 3. Here are the first four terms of a number sequence. 7,, 5, 9, Write down the n th term of the sequence. (LON) 4. Sheep enclosures are built using fences and posts. The enclosures are always built in a row. Post Fence One enclosure Two enclosures Three enclosures Sketch (i) four enclosures in a row (ii) five enclosures in a row. Copy and complete the table below. Number of enclosures Number of posts 6 9 (c) (d) Work out the number of posts needed for 0 enclosures in a row. Write down an expression to find the number of posts needed for n enclosures in a row. (LON) 5. Patterns of squares are formed using sticks. The first three patterns are drawn below. Pattern Pattern Pattern 3 The table shows the number of sticks needed for each pattern. 78
18 MEP Pupil Text Pattern 3 Number of sticks (i) Draw Pattern 4. (ii) How many sticks are needed for Pattern 4? How many more sticks are needed to make Pattern 5 from Pattern 4? (c) There is a rule for finding the number of sticks needed to make any of these patterns of squares. If the number of squares in a pattern is s, write down the rule. (SEG) 6. Sticks are arranged in shapes. Shape Shape Shape 3 7 sticks sticks 7 sticks The number of sticks form a sequence. (i) Write down a rule for finding the next number in the sequence. (ii) Find a formula in terms of n for the number of sticks in the n th shape. Find a formula, in terms of n, for the area of the n th rectangle in this sequence (SEG) Just for Fun John and Julie had a date one Saturday. They agreed to meet outside the cinema at 8 pm. Julie thought that her watch was 5 minutes fast but in actual fact it was 5 minutes slow. John thought that his watch was 5 minutes slow but in actual fact it was 5 minutes fast. Julie deliberately turned up 0 minutes late while John decided to turn up 0 minutes early. Who turned up first and how long had he/she to wait for the other to arrive? 79
19 MEP Pupil Text.5 General Laws This section considers sequences which are formed in various ways and uses iterative formulae that describe how one term is obtained from the previous term. The behaviour of the sequences with huge numbers of terms is also considered to see whether they increase indefinitely or approach a fixed value. Worked Example Find a formula to generate the terms of these sequences: 3, 3 5, 4 7, 5 9, 6,... 4, 6, 9, 3.5, 0.5,... What happens to these sequences for large values of n? This can be approached by looking at the numerators and denominators of the fractions. The numbers, 3, 4, 5, 6,... are from the sequence un= n +. The numbers 3, 5, 7, 9,,... are from the sequence un = n +. Combining these gives the formula for the sequence as 3, 3 5, 4 7, 5 9, 6,... u n = n + n + As the value of n becomes larger and larger, this sequence produces terms that get closer and closer to. Consider the terms below: 0 u 00 = = u 000 = = u 0000 = = u = = We say that the sequence converges to. 80
20 MEP Pupil Text Note A more rigorous approach is to divide both the numerator and denominator of the expression for u n by n, giving n un = + = n + Now as n becomes larger, the term 0 n, giving u n = + n. + n We write u n as n (infinity) and say that ' u n tends to as n tends to infinity.' The terms of this sequence are multiplied by a factor of.5 to obtain the next term Considering each term helps to see the general formula. 0 u = 4 = 4 5. u = 4. 5 = 4. 5 u 3 = = u 4 = = u 5 = = 4 5. So the general term is u n n 4 5 =.. The terms of this sequence become larger and larger, never approaching a fixed value as in the last example. We say the sequence diverges. In fact, any sequence which does not converge is said to diverge. Worked Example Find iterative formulae for each of the following sequences. 7,, 5, 9, 3, 7,... 6,, 4, 48, 96,... (c),,, 3, 5, 8, 3,,... 8
21 .5 MEP Pupil Text This group of sequences show how one term is related to the previous term, so the formulae give u n+ in terms of the previous term, u n. u = 7 u = 6 u u u = = u + 4 u = 6 = u = + 4 = u + 4 u = = u = = u + 4 u = 4 = u So u (c) u = u = n = un + 4 with u = 7. So un+ = un with u = 6. u = + = u + u 3 u = + = u + u 4 3 u = + 3 = u + u u = = u + u So un+ = un + un, with u = and u =. Worked Example 3 Find the first 4 terms of the sequence defined iteratively by un+ = un + 4 u n starting with u =. Show that the sequence converges to. u = u 4 = + 5 =. u = = 05. u = = These terms appear to be getting closer and closer to. If the sequence does converge to a particular value then as n becomes larger and larger, u n+ becomes approximately the same as u n. So if un+ = un = d, say, then u n + = un + 4 u n 8
22 MEP Pupil Text becomes d = 4 d + d d d = d + = 4 d d = 4 4 d d = or. So the sequence does converge to. If the sequence had started with u =, then it would have converged to, the other possible value of d. Exercises. Find formulae to generate the terms of each sequence below. 0, 4, 9, 6 3, 5 4,... 5, 7 3, 9 4, 5, 3 6,... (c), 6, 8, 54, 6,... (d), 0.9, 0.8, 0.79, 0.656,... (e),.,.44,.78,.0736,... (f), 3, 9, 7, 8,... (g), 4, 8, 6, 3,... (h) 3, 5, 9, 7, 33,... (i) 3 4, 6 5, 9 6, 7, 5 8,... Which of the above sequences converge and which diverge? For those which converge, find the value to which they converge.. Find iterative formulae for each of the following sequences. 8, 5,,, 4,... 5, 0, 80, 30, 80,... (c), 3, 5, 9, 7,... (d) 4000, 000, 000, 500, 50,... (e) 3, 3, 6, 9, 5, 4, 39,... (f),,, 3, 5, 9, 7, 3, The iterative formula un+ = un + 6 u n can be used with u = to define the sequence. Find the first 5 terms of the sequence. Show that the sequence converges to 6. 83
23 .5 MEP Pupil Text 4. Does the sequence defined by un+ = un u with u = converge? If it does, find the value to which it converges. What happens if u is a different value? n 5. Find the first 5 terms of the sequence u + = u 7u ( ) n n n starting with u = 0.. To what value does this sequence converge? (Give your answer as a fraction.) 6. To what value does the sequence ( ) u + = u 3u n n n with u =, converge? Calculate the first four terms of the sequence to check your answer. 7. Javid and Anita try to find different ways of exploring the sequence 4, 0, 8, 8, 40,... Javid writes ( ) ( ) ( ) ( ) st number 4 = 4 = + 3 nd number 0 = 5 = + 3 3rd number 8 = 3 6 = th number 8 = 4 7 = How would Javid write down (i) the 5th number (ii) the nth number? Anita writes st number 4 = 3 nd number 0 = 3 4 3rd number 8 = 4 5 4th number 8 = 5 6 How would Anita write down (i) the 5th number (ii) the nth number? (c) Show how you would prove that Javid's expression and Anita's expression for the nth number are the same. (MEG) 8. Write down the next number in this sequence.,, 4, 8, 6, 3,... Describe how the sequence is formed. (c) One number in the sequence is 04. Describe how you can use the number 04 to find the number in the sequence which comes just before it. (SEG) 84
24 MEP Pupil Text 9. Row Sum = Row 3 5 Sum = 8 = 3 Row Sum = 7 = 3 3 Write down the numbers and sum which continue the pattern in Row 4. Which row will have a sum equal to 000? (c) What is the sum of Row 0? (d) The first number in a row is x. What is the second number in this row? Give your answer in terms of x. (MEG).6 Quadratic Formulae Consider the sequence generated by the formula un = n + n., 6,, 0, 30, 4,... The differences between terms can be considered as below., 6,, 0, 30, 4,... First differences Second differences The first differences increase, but the second differences are all the same. Whenever the second differences are constant a sequence can be described by a quadratic formula of the form un = an + bn + c where a, b and c are constants. Worked Example Find the formula which describes the sequence, 8,, 40, 65, 96. First examine the differences., 8,, 40, 65, 96,... First differences Second differences As the second differences are constant, the sequence can be described by a quadratic formula of the form un = an + bn + c. To find the values of a, b and c consider the first 3 terms. 85
25 .6 MEP Pupil Text Using u = gives = a + b + c () Using u = 8 gives 8 = 4a + b + c () Using u 3 = gives = 9a + 3b + c (3) These are three simultaneous equations. To solve them, subtract equation () from equation () to give equation (4) and from equation (3) to give equation (5) as below. 8 = 4a + b + c () = 9a + 3b + c (3) subtract = a + b + c () = a + b + c () 7 = 3a + b (4) 0 = 8a + b (5) Then subtracting equation (4) from equation (5) gives 0 = 8a + b (5) 4 = 6a + b (4) 6 = a so a = 3. Substituting for a in equation (4) gives 7 = b so b =. Finally, substituting for a and b in equation () gives = 3 + c so c = 0. The sequence is then generated by the formula Note un = 3n n. You should, of course, check the first few terms: n = u = 3 = n = u = 3 = 8 n = 3 u = =. 3 Investigation Find the next term in the sequence,, 9 4, 7 5, 7 86
26 MEP Pupil Text Alternative Approach Solving sets of simultaneous equations like these can be quite hard work. Examining carefully the differences leads to an easier method. The first four terms of the sequence are un = an + bn + c u = a + b + c u = 4a + b + c u3 = 9a + 3b + c u4 = 6a + 4b + c Consider the differences for these terms. a + b + c, 4a + b + c, 9a + 3b + c, 6a + 4b + c First difference 3a + b 5a + b 7a + b Second difference a a Note that the second difference is equal to a, the first of the first differences is 3a + b and the first term is a + b + c. This can be used to create a much easier approach to finding a, b and c, as shown in the next example. Worked Example Find the formula which generates the sequence 6,, 8, 7, 38, 5,... First find the differences. 6,, 8, 7, 38, 5,... First differences Second differences As the second differences are constant, the sequence is generated by the quadratic formula un = an + bn + c. Using the results from the differences considered in the alternative approach in Worked Example gives a = 3a + b = 5 a + b + c = 6. This gives a =, b =, and c = 3, so the formula is un = n + n
27 .6 MEP Pupil Text Exercises. Show that each sequence below has a constant second difference and use this to find the next terms. 5, 0, 7, 6, 37,... 0, 0, 8, 54, 88,... (c), 4, 34, 6, 98,... (d), 4, 6, 0, 4,... (e),,,, 7,.... The third, fourth and fifth terms of a quadratic sequence are 6, 0 and 6. Find the first, second and sixth terms of the sequence. 3. A sequence begins, 4, 4, 66,... Find the 0th term of this sequence. 4. The first and third terms of a sequence are 6 and 48. If the second difference is constant and equal to 0, find the second term of the sequence. 5. For each of the following sequences, state whether or not they are generated by a quadratic formula and if they are, give the formula. 3,, 3, 6,, 8,... 0, 4, 6, 66, 36, 600,... (c), 3, 7, 3,, 3,... (d), 3, 4, 7,, 8,... (e) 5, 0, 7, 6, 7, 40,... (f), 3, 7,, 5, 9,... (g),, 7, 9, 37, 6, The nd, 3rd and 4th terms of a quadratic sequence are 0, 3 and 8. Find the st and 5th terms of the sequence. 7. In a sequence, the nd term is 0 more than the st term, the 3rd term is 5 more than the nd term, the 4th term is 0 more than the 3rd term. Show that the sequence is quadratic and find a formula for the sequence if the first term is. 8. The terms of a particular cubic sequence are given by u n. Find the first 6 terms of this sequence and then the first, second and third differences. What do you notice? n = 3 Check that the result you noted in Part is true for a cubic sequence of your own choice. 3 (c) By considering un = an + bn + cn + d, show that the third difference of any cubic sequence is a constant and give its value in terms of a. 88
28 MEP Pupil Text (d) Then find the formula which describes the cubic sequence 4, 5, 40, 85, 56, 59, The table shows how the first 5 triangle, square and pentagon numbers are formed Triangle Square Pentagon 5 35 Show that all the sequences formed are quadratic and find expressions for them. The hexagon numbers give the sequence, 6, 5, 8, 45,... Show that the terms of this sequence are given by un = n n. (c) By looking at the expressions you have obtained so far, predict formulae for the heptagonal and octagonal numbers. Use the facts that the 8th heptagonal number is 48 and the 8th octagonal number is 76 to check your formulae. (d) Show that the 8th decagonal number is Look at the three sequences below. Sequence p 4, 6, 8, 0,,... Sequence q 3, 8, 5, 4, 35,... Sequence r 5, 0, 7,... The sequence r is obtained from sequences p and q as follows. and so on = 5, = 0, = 7 89
29 .6 MEP Pupil Text (i) Use the numbers 0 and 4 to calculate the fourth term of sequence r. (ii) Calculate the fifth term of sequence r. (i) Find the tenth term of sequence p. (ii) Find the sixth term of sequence q. (c) (i) Write down the nth term of sequence p. (ii) The nth term of sequence q is n + kn where k represents a number. Find the value of k. (NEAB). The first four diagrams of a sequence are shown below. Diagram Diagram Diagram 3 Diagram 4 The table below shows the number of black and white triangles for the first three diagrams. Diagram number Number of white triangles 3 6 Number of black triangles 0 3 Total number of triangles 4 9 Complete the table, including the column for a fifth diagram. What will be the total number of triangles in diagram 0? (c) (i) On a grid, plot the number of white triangles against the diagram numbers. (ii) On the same grid, plot the number of black triangles for each diagram number in your table. (iii) What do you notice about the two sets of points? (d) Two pupils are trying to find a general rule to work out the number of white triangles. One rule they suggest is Number of white triangles = d( d + ) where d is the diagram number. Is this rule correct? Show any calculations that you make. (NEAB) 90
ALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite
ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,
More informationUNIT H1 Angles and Symmetry Activities
UNIT H1 Angles and Symmetry Activities Activities H1.1 Lines of Symmetry H1.2 Rotational and Line Symmetry H1.3 Symmetry of Regular Polygons H1.4 Interior Angles in Polygons Notes and Solutions (1 page)
More informationGeometry Progress Ladder
Geometry Progress Ladder Maths Makes Sense Foundation End-of-year objectives page 2 Maths Makes Sense 1 2 End-of-block objectives page 3 Maths Makes Sense 3 4 End-of-block objectives page 4 Maths Makes
More informationCBA Fractions Student Sheet 1
Student Sheet 1 1. If 3 people share 12 cookies equally, how many cookies does each person get? 2. Four people want to share 5 cakes equally. Show how much each person gets. Student Sheet 2 1. The candy
More informationNumber Patterns, Cautionary Tales and Finite Differences
Learning and Teaching Mathematics, No. Page Number Patterns, Cautionary Tales and Finite Differences Duncan Samson St Andrew s College Number Patterns I recently included the following question in a scholarship
More informationDecimals and Percentages
Decimals and Percentages Specimen Worksheets for Selected Aspects Paul Harling b recognise the number relationship between coordinates in the first quadrant of related points Key Stage 2 (AT2) on a line
More informationMathematics 31 Pre-calculus and Limits
Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationSequences. A sequence is a list of numbers, or a pattern, which obeys a rule.
Sequences A sequence is a list of numbers, or a pattern, which obeys a rule. Each number in a sequence is called a term. ie the fourth term of the sequence 2, 4, 6, 8, 10, 12... is 8, because it is the
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationYear 9 set 1 Mathematics notes, to accompany the 9H book.
Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More information1.2. Successive Differences
1. An Application of Inductive Reasoning: Number Patterns In the previous section we introduced inductive reasoning, and we showed how it can be applied in predicting what comes next in a list of numbers
More informationEVERY DAY COUNTS CALENDAR MATH 2005 correlated to
EVERY DAY COUNTS CALENDAR MATH 2005 correlated to Illinois Mathematics Assessment Framework Grades 3-5 E D U C A T I O N G R O U P A Houghton Mifflin Company YOUR ILLINOIS GREAT SOURCE REPRESENTATIVES:
More informationFractions of an Area
Fractions of an Area Describe and compare fractions as part of an area using words, objects, pictures, and symbols.. Circle the letter of each cake top that shows fourths. A. D. Each part of this rectangle
More informationTo discuss this topic fully, let us define some terms used in this and the following sets of supplemental notes.
INFINITE SERIES SERIES AND PARTIAL SUMS What if we wanted to sum up the terms of this sequence, how many terms would I have to use? 1, 2, 3,... 10,...? Well, we could start creating sums of a finite number
More informationSect 6.7 - Solving Equations Using the Zero Product Rule
Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred
More informationIllinois State Standards Alignments Grades Three through Eleven
Illinois State Standards Alignments Grades Three through Eleven Trademark of Renaissance Learning, Inc., and its subsidiaries, registered, common law, or pending registration in the United States and other
More informationISAT Mathematics Performance Definitions Grade 4
ISAT Mathematics Performance Definitions Grade 4 EXCEEDS STANDARDS Fourth-grade students whose measured performance exceeds standards are able to identify, read, write, represent, and model whole numbers
More informationPaper 1. Calculator not allowed. Mathematics test. First name. Last name. School. Remember KEY STAGE 3 TIER 5 7
Ma KEY STAGE 3 Mathematics test TIER 5 7 Paper 1 Calculator not allowed First name Last name School 2009 Remember The test is 1 hour long. You must not use a calculator for any question in this test. You
More informationAcquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A3d,e Time allotted for this Lesson: 4 Hours
Acquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A3d,e Time allotted for this Lesson: 4 Hours Essential Question: LESSON 4 FINITE ARITHMETIC SERIES AND RELATIONSHIP TO QUADRATIC
More informationTarget To know the properties of a rectangle
Target To know the properties of a rectangle (1) A rectangle is a 3-D shape. (2) A rectangle is the same as an oblong. (3) A rectangle is a quadrilateral. (4) Rectangles have four equal sides. (5) Rectangles
More informationIB Maths SL Sequence and Series Practice Problems Mr. W Name
IB Maths SL Sequence and Series Practice Problems Mr. W Name Remember to show all necessary reasoning! Separate paper is probably best. 3b 3d is optional! 1. In an arithmetic sequence, u 1 = and u 3 =
More informationGrade 5 Math Content 1
Grade 5 Math Content 1 Number and Operations: Whole Numbers Multiplication and Division In Grade 5, students consolidate their understanding of the computational strategies they use for multiplication.
More informationKey Topics What will ALL students learn? What will the most able students learn?
2013 2014 Scheme of Work Subject MATHS Year 9 Course/ Year Term 1 Key Topics What will ALL students learn? What will the most able students learn? Number Written methods of calculations Decimals Rounding
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationFree Inductive/Logical Test Questions
Free Inductive/Logical Test Questions (With questions and answers) JobTestPrep invites you to a free practice session that represents only some of the materials offered in our online practice packs. Have
More informationMATHEMATICS TEST. Paper 1 calculator not allowed LEVEL 6 TESTS ANSWER BOOKLET. First name. Middle name. Last name. Date of birth Day Month Year
LEVEL 6 TESTS ANSWER BOOKLET Ma MATHEMATICS TEST LEVEL 6 TESTS Paper 1 calculator not allowed First name Middle name Last name Date of birth Day Month Year Please circle one Boy Girl Year group School
More informationMathematics Second Practice Test 1 Levels 4-6 Calculator not allowed
Mathematics Second Practice Test 1 Levels 4-6 Calculator not allowed Please read this page, but do not open your booklet until your teacher tells you to start. Write your name and the name of your school
More informationWhich shapes make floor tilings?
Which shapes make floor tilings? Suppose you are trying to tile your bathroom floor. You are allowed to pick only one shape and size of tile. The tile has to be a regular polygon (meaning all the same
More informationWhich two rectangles fit together, without overlapping, to make a square?
SHAPE level 4 questions 1. Here are six rectangles on a grid. A B C D E F Which two rectangles fit together, without overlapping, to make a square?... and... International School of Madrid 1 2. Emily has
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationSTRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
More informationIn mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.
MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target
More informationof surface, 569-571, 576-577, 578-581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433
Absolute Value and arithmetic, 730-733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property
More informationStanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions
Stanford Math Circle: Sunday, May 9, 00 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationGAP CLOSING. 2D Measurement. Intermediate / Senior Student Book
GAP CLOSING 2D Measurement Intermediate / Senior Student Book 2-D Measurement Diagnostic...3 Areas of Parallelograms, Triangles, and Trapezoids...6 Areas of Composite Shapes...14 Circumferences and Areas
More informationPaper 1. Calculator not allowed. Mathematics test. First name. Last name. School. Remember KEY STAGE 3 TIER 4 6
Ma KEY STAGE 3 Mathematics test TIER 4 6 Paper 1 Calculator not allowed First name Last name School 2009 Remember The test is 1 hour long. You must not use a calculator for any question in this test. You
More informationSQUARE-SQUARE ROOT AND CUBE-CUBE ROOT
UNIT 3 SQUAREQUARE AND CUBEUBE (A) Main Concepts and Results A natural number is called a perfect square if it is the square of some natural number. i.e., if m = n 2, then m is a perfect square where m
More informationVISUAL ALGEBRA FOR COLLEGE STUDENTS. Laurie J. Burton Western Oregon University
VISUAL ALGEBRA FOR COLLEGE STUDENTS Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS TABLE OF CONTENTS Welcome and Introduction 1 Chapter 1: INTEGERS AND INTEGER OPERATIONS
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationBlack Problems - Prime Factorization, Greatest Common Factor and Simplifying Fractions
Black Problems Prime Factorization, Greatest Common Factor and Simplifying Fractions A natural number n, such that n >, can t be written as the sum of two more consecutive odd numbers if and only if n
More information3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
More informationNumeracy Targets. I can count at least 20 objects
Targets 1c I can read numbers up to 10 I can count up to 10 objects I can say the number names in order up to 20 I can write at least 4 numbers up to 10. When someone gives me a small number of objects
More informationMATHS LEVEL DESCRIPTORS
MATHS LEVEL DESCRIPTORS Number Level 3 Understand the place value of numbers up to thousands. Order numbers up to 9999. Round numbers to the nearest 10 or 100. Understand the number line below zero, and
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationSOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The Odd-Root Property
498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1
More informationMATHEMATICS Y3 Using and applying mathematics 3810 Solve mathematical puzzles and investigate. Equipment MathSphere www.mathsphere.co.
MATHEMATICS Y3 Using and applying mathematics 3810 Solve mathematical puzzles and investigate. Equipment Paper, pencil, ruler Dice, number cards, buttons/counters, boxes etc MathSphere 3810 Solve mathematical
More informationfor the Common Core State Standards 2012
A Correlation of for the Common Core State s 2012 to the Common Core Georgia Performance s Grade 2 FORMAT FOR CORRELATION TO THE COMMON CORE GEORGIA PERFORMANCE STANDARDS (CCGPS) Subject Area: K-12 Mathematics
More information2.1. Inductive Reasoning EXAMPLE A
CONDENSED LESSON 2.1 Inductive Reasoning In this lesson you will Learn how inductive reasoning is used in science and mathematics Use inductive reasoning to make conjectures about sequences of numbers
More informationGCSE MATHEMATICS. 43602H Unit 2: Number and Algebra (Higher) Report on the Examination. Specification 4360 November 2014. Version: 1.
GCSE MATHEMATICS 43602H Unit 2: Number and Algebra (Higher) Report on the Examination Specification 4360 November 2014 Version: 1.0 Further copies of this Report are available from aqa.org.uk Copyright
More information11.3 Curves, Polygons and Symmetry
11.3 Curves, Polygons and Symmetry Polygons Simple Definition A shape is simple if it doesn t cross itself, except maybe at the endpoints. Closed Definition A shape is closed if the endpoints meet. Polygon
More informationArithmetic 1 Progress Ladder
Arithmetic 1 Progress Ladder Maths Makes Sense Foundation End-of-year objectives page 2 Maths Makes Sense 1 2 End-of-block objectives page 3 Maths Makes Sense 3 4 End-of-block objectives page 4 Maths Makes
More informationGrade 3 Core Standard III Assessment
Grade 3 Core Standard III Assessment Geometry and Measurement Name: Date: 3.3.1 Identify right angles in two-dimensional shapes and determine if angles are greater than or less than a right angle (obtuse
More informationPERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.
PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationPaper 1. Calculator not allowed. Mathematics test. First name. Last name. School. Remember KEY STAGE 3 TIER 4 6
Ma KEY STAGE 3 Mathematics test TIER 4 6 Paper 1 Calculator not allowed First name Last name School 2007 Remember The test is 1 hour long. You must not use a calculator for any question in this test. You
More informationAnswer: Quantity A is greater. Quantity A: 0.717 0.717717... Quantity B: 0.71 0.717171...
Test : First QR Section Question 1 Test, First QR Section In a decimal number, a bar over one or more consecutive digits... QA: 0.717 QB: 0.71 Arithmetic: Decimals 1. Consider the two quantities: Answer:
More informationDiscrete Mathematics: Homework 7 solution. Due: 2011.6.03
EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that
More informationSuch As Statements, Kindergarten Grade 8
Such As Statements, Kindergarten Grade 8 This document contains the such as statements that were included in the review committees final recommendations for revisions to the mathematics Texas Essential
More informationMathematics K 6 continuum of key ideas
Mathematics K 6 continuum of key ideas Number and Algebra Count forwards to 30 from a given number Count backwards from a given number in the range 0 to 20 Compare, order, read and represent to at least
More information7.7 Solving Rational Equations
Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate
More informationEveryday Mathematics CCSS EDITION CCSS EDITION. Content Strand: Number and Numeration
CCSS EDITION Overview of -6 Grade-Level Goals CCSS EDITION Content Strand: Number and Numeration Program Goal: Understand the Meanings, Uses, and Representations of Numbers Content Thread: Rote Counting
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationCharlesworth School Year Group Maths Targets
Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation
More informationIf A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question
More informationCAHSEE on Target UC Davis, School and University Partnerships
UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationMajor Work of the Grade
Counting and Cardinality Know number names and the count sequence. Count to tell the number of objects. Compare numbers. Kindergarten Describe and compare measurable attributes. Classify objects and count
More informationPigeonhole Principle Solutions
Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such
More informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More informationMD5-26 Stacking Blocks Pages 115 116
MD5-26 Stacking Blocks Pages 115 116 STANDARDS 5.MD.C.4 Goals Students will find the number of cubes in a rectangular stack and develop the formula length width height for the number of cubes in a stack.
More informationFactoring Polynomials and Solving Quadratic Equations
Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3
More informationPaper 1. Calculator not allowed. Mathematics test. First name. Last name. School. Remember KEY STAGE 3 TIER 6 8
Ma KEY STAGE 3 Mathematics test TIER 6 8 Paper 1 Calculator not allowed First name Last name School 2009 Remember The test is 1 hour long. You must not use a calculator for any question in this test. You
More informationEquations, Inequalities & Partial Fractions
Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities
More informationFOREWORD. Executive Secretary
FOREWORD The Botswana Examinations Council is pleased to authorise the publication of the revised assessment procedures for the Junior Certificate Examination programme. According to the Revised National
More informationSection 7.1 Solving Right Triangles
Section 7.1 Solving Right Triangles Note that a calculator will be needed for most of the problems we will do in class. Test problems will involve angles for which no calculator is needed (e.g., 30, 45,
More informationFourth Grade Math Standards and "I Can Statements"
Fourth Grade Math Standards and "I Can Statements" Standard - CC.4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationBridging Documents for Mathematics
Bridging Documents for Mathematics 5 th /6 th Class, Primary Junior Cycle, Post-Primary Primary Post-Primary Card # Strand(s): Number, Measure Number (Strand 3) 2-5 Strand: Shape and Space Geometry and
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More informationNF5-12 Flexibility with Equivalent Fractions and Pages 110 112
NF5- Flexibility with Equivalent Fractions and Pages 0 Lowest Terms STANDARDS preparation for 5.NF.A., 5.NF.A. Goals Students will equivalent fractions using division and reduce fractions to lowest terms.
More informationPURSUITS IN MATHEMATICS often produce elementary functions as solutions that need to be
Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions 2007 Ron Doerfler http://www.myreckonings.com June 27, 2007 Abstract There are some of us who enjoy using our
More informationCurriculum Overview YR 9 MATHS. SUPPORT CORE HIGHER Topics Topics Topics Powers of 10 Powers of 10 Significant figures
Curriculum Overview YR 9 MATHS AUTUMN Thursday 28th August- Friday 19th December SUPPORT CORE HIGHER Topics Topics Topics Powers of 10 Powers of 10 Significant figures Rounding Rounding Upper and lower
More informationConvert between units of area and determine the scale factor of two similar figures.
CHAPTER 5 Units of Area c GOAL Convert between units of area and determine the scale factor of two. You will need a ruler centimetre grid paper a protractor a calculator Learn about the Math The area of
More informationMTN Learn. Mathematics. Grade 10. radio support notes
MTN Learn Mathematics Grade 10 radio support notes Contents INTRODUCTION... GETTING THE MOST FROM MINDSET LEARN XTRA RADIO REVISION... 3 BROADAST SCHEDULE... 4 ALGEBRAIC EXPRESSIONS... 5 EXPONENTS... 9
More informationMathematics Florida Standards (MAFS) Grade 2
Mathematics Florida Standards (MAFS) Grade 2 Domain: OPERATIONS AND ALGEBRAIC THINKING Cluster 1: Represent and solve problems involving addition and subtraction. MAFS.2.OA.1.1 Use addition and subtraction
More informationGAP CLOSING. 2D Measurement GAP CLOSING. Intermeditate / Senior Facilitator s Guide. 2D Measurement
GAP CLOSING 2D Measurement GAP CLOSING 2D Measurement Intermeditate / Senior Facilitator s Guide 2-D Measurement Diagnostic...4 Administer the diagnostic...4 Using diagnostic results to personalize interventions...4
More informationStandards and progression point examples
Mathematics Progressing towards Foundation Progression Point 0.5 At 0.5, a student progressing towards the standard at Foundation may, for example: connect number names and numerals with sets of up to
More informationApplications for Triangles
Not drawn to scale Applications for Triangles 1. 36 in. 40 in. 33 in. 1188 in. 2 69 in. 2 138 in. 2 1440 in. 2 2. 188 in. 2 278 in. 2 322 in. 2 none of these Find the area of a parallelogram with the given
More informationYear 9 mathematics test
Ma KEY STAGE 3 Year 9 mathematics test Tier 6 8 Paper 1 Calculator not allowed First name Last name Class Date Please read this page, but do not open your booklet until your teacher tells you to start.
More informationThe Australian Curriculum Mathematics
The Australian Curriculum Mathematics Mathematics ACARA The Australian Curriculum Number Algebra Number place value Fractions decimals Real numbers Foundation Year Year 1 Year 2 Year 3 Year 4 Year 5 Year
More informationMathematics 2540 Paper 5540H/3H
Edexcel GCSE Mathematics 540 Paper 5540H/3H November 008 Mark Scheme 1 (a) 3bc 1 B1 for 3bc (accept 3cb or bc3 or cb3 or 3 b c oe, but 7bc 4bc gets no marks) (b) x + 5y B for x+5y (accept x+y5 or x + 5
More informationScope and Sequence KA KB 1A 1B 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B
Scope and Sequence Earlybird Kindergarten, Standards Edition Primary Mathematics, Standards Edition Copyright 2008 [SingaporeMath.com Inc.] The check mark indicates where the topic is first introduced
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationIntermediate Math Circles October 10, 2012 Geometry I: Angles
Intermediate Math Circles October 10, 2012 Geometry I: Angles Over the next four weeks, we will look at several geometry topics. Some of the topics may be familiar to you while others, for most of you,
More information10 th POLISH SUDOKU CHAMPIONSHIP INSTRUCTION BOOKLET. February 22, 2015 IMPORTANT INFORMATION:
10 th POLISH SUDOKU CHAMPIONSHIP February 22, 2015 INSTRUCTION BOOKLET IMPORTANT INFORMATION: 1. Answer form can be sent many times, but only the last version will be considered. 2. In case of a tie, the
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationSample Problems. Practice Problems
Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these
More information