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
 2 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 righthand column, in order, are,, 3,... What is the connection between the answers to parts (i) and (ii)? The numbers in the righthand 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 informationThe angle sum property of triangles can help determine the sum of the measures of interior angles of other polygons.
Interior Angles of Polygons The angle sum property of triangles can help determine the sum of the measures of interior angles of other polygons. The sum of the measures of the interior angles of a triangle
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 informationMathematics 31 Precalculus and Limits
Mathematics 31 Precalculus 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 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 informationTHE GOLDEN RATIO AND THE FIBONACCI SEQUENCE
/ 24 THE GOLDEN RATIO AND THE FIBONACCI SEQUENCE Todd Cochrane Everything is Golden 2 / 24 Golden Ratio Golden Proportion Golden Relation Golden Rectangle Golden Spiral Golden Angle Geometric Growth, (Exponential
More information3 rd 5 th Grade Math Core Curriculum Anna McDonald School
3 rd 5 th Grade Math Core Curriculum Anna McDonald School Our core math curriculum is only as strong and reliable as its implementation. Ensuring the goals of our curriculum are met in each classroom,
More information1 of 69 Boardworks Ltd 2004
1 of 69 2 of 69 Intersecting lines 3 of 69 Vertically opposite angles When two lines intersect, two pairs of vertically opposite angles are formed. a d b c a = c and b = d Vertically opposite angles are
More informationGeometry Progress Ladder
Geometry Progress Ladder Maths Makes Sense Foundation Endofyear objectives page 2 Maths Makes Sense 1 2 Endofblock objectives page 3 Maths Makes Sense 3 4 Endofblock objectives page 4 Maths Makes
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 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 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 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 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 informationDecide how many topics you wish to revise at a time (let s say 10).
1 Minute Maths for the Higher Exam (grades B, C and D topics*) Too fast for a firsttime use but... brilliant for topics you have already understood and want to quickly revise. for the Foundation Exam
More informationPolygons are figures created from segments that do not intersect at any points other than their endpoints.
Unit #5 Lesson #1: Polygons and Their Angles. Polygons are figures created from segments that do not intersect at any points other than their endpoints. A polygon is convex if all of the interior angles
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 informationEVERY DAY COUNTS CALENDAR MATH 2005 correlated to
EVERY DAY COUNTS CALENDAR MATH 2005 correlated to Illinois Mathematics Assessment Framework Grades 35 E D U C A T I O N G R O U P A Houghton Mifflin Company YOUR ILLINOIS GREAT SOURCE REPRESENTATIVES:
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 informationAssociative Property The property that states that the way addends are grouped or factors are grouped does not change the sum or the product.
addend A number that is added to another in an addition problem. 2 + 3 = 5 The addends are 2 and 3. area The number of square units needed to cover a surface. area = 9 square units array An arrangement
More information0018 DATA ANALYSIS, PROBABILITY and STATISTICS
008 DATA ANALYSIS, PROBABILITY and STATISTICS A permutation tells us the number of ways we can combine a set where {a, b, c} is different from {c, b, a} and without repetition. r is the size of of the
More informationYear 10 Homework Task for Maths
Year 10 Homework Task for Maths How to be Successful in Maths If you want to do yourself justice you need to: Pay attention at all times Do the classwork set by your teacher Follow your teacher s instructions
More informationGrade 1 Math Expressions Vocabulary Words 2011
Italicized words Indicates OSPI Standards Vocabulary as of 10/1/09 Link to Math Expression Online Glossary for some definitions: http://wwwk6.thinkcentral.com/content/hsp/math/hspmathmx/na/gr1/se_9780547153179
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 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 information1 foot (ft) = 12 inches (in) 1 yard (yd) = 3 feet (ft) 1 mile (mi) = 5280 feet (ft) Replace 1 with 1 ft/12 in. 1ft
2 MODULE 6. GEOMETRY AND UNIT CONVERSION 6a Applications The most common units of length in the American system are inch, foot, yard, and mile. Converting from one unit of length to another is a requisite
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 informationA convex polygon is a polygon such that no line containing a side of the polygon will contain a point in the interior of the polygon.
hapter 7 Polygons A polygon can be described by two conditions: 1. No two segments with a common endpoint are collinear. 2. Each segment intersects exactly two other segments, but only on the endpoints.
More informationMath Dictionary Terms for Grades K1:
Math Dictionary Terms for Grades K1: A Addend  one of the numbers being added in an addition problem Addition  combining quantities And  1) combine, 2) shared attributes, 3) represents decimal point
More informationIntroduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test
Introduction The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are
More informationMaths Level Targets. This booklet outlines the maths targets for each sublevel in maths from Level 1 to Level 5.
Maths Level Targets This booklet outlines the maths targets for each sublevel in maths from Level 1 to Level 5. Expected National Curriculum levels for the end of each year group are: Year 1 Year 2 Year
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 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 informationTYPES OF NUMBERS. Example 2. Example 1. Problems. Answers
TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime
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 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 informationSome sequences have a fixed length and have a last term, while others go on forever.
Sequences and series Sequences A sequence is a list of numbers (actually, they don t have to be numbers). Here is a sequence: 1, 4, 9, 16 The order makes a difference, so 16, 9, 4, 1 is a different sequence.
More informationI know when I have written a number backwards and can correct it when it is pointed out to me I can arrange numbers in order from 1 to 10
Mathematics Targets Moving from Level W and working towards level 1c I can count from 1 to 10 I know and write all my numbers to 10 I know when I have written a number backwards and can correct it when
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 information7.3 & 7.4 Polygon Formulas completed.notebook January 10, 2014
Chapter 7 Polygons Polygon 1. Closed Figure # of Sides Polygon Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 2. Straight sides/edges 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon 15 Pentadecagon
More information2000 Solutions Fermat Contest (Grade 11)
anadian Mathematics ompetition n activity of The entre for Education in Mathematics and omputing, University of Waterloo, Waterloo, Ontario 000 s Fermat ontest (Grade ) for The ENTRE for EDUTION in MTHEMTIS
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 informationPreAlgebra Interactive Chalkboard Copyright by The McGrawHill Companies, Inc. Send all inquiries to:
PreAlgebra Interactive Chalkboard Copyright by The McGrawHill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGrawHill 8787 Orion Place Columbus, Ohio 43240 Click the mouse button
More information15 Polygons. 15.1 Angle Facts. Example 1. Solution. Example 2. Solution
15 Polygons MEP Y8 Practice Book B 15.1 Angle Facts In this section we revise some asic work with angles, and egin y using the three rules listed elow: The angles at a point add up to 360, e.g. a c a +
More informationVocabulary Cards and Word Walls Revised: June 2, 2011
Vocabulary Cards and Word Walls Revised: June 2, 2011 Important Notes for Teachers: The vocabulary cards in this file match the Common Core, the math curriculum adopted by the Utah State Board of Education,
More informationof surface, 569571, 576577, 578581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433
Absolute Value and arithmetic, 730733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property
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 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 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 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 information61 Angles of Polygons
Find the sum of the measures of the interior angles of each convex polygon. 1. decagon A decagon has ten sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle measures.
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 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 informationIngleby Manor Scheme of Work : Year 7. Objectives Support L3/4 Core L4/5/6 Extension L6/7
Autumn 1 Mathematics Objectives Support L3/4 Core L4/5/6 Extension L6/7 Week 1: Algebra 1 (6 hours) Sequences and functions  Recognize and extend number sequences  Identify the term to term rule  Know
More informationGeometry. Unit 6. Quadrilaterals. Unit 6
Geometry Quadrilaterals Properties of Polygons Formed by three or more consecutive segments. The segments form the sides of the polygon. Each side intersects two other sides at its endpoints. The intersections
More informationISAT Mathematics Performance Definitions Grade 4
ISAT Mathematics Performance Definitions Grade 4 EXCEEDS STANDARDS Fourthgrade students whose measured performance exceeds standards are able to identify, read, write, represent, and model whole numbers
More informationSQUARESQUARE ROOT AND CUBECUBE 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 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 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 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 informationThird Grade Illustrated Math Dictionary Updated 91310 As presented by the Math Committee of the Northwest Montana Educational Cooperative
Acute An angle less than 90 degrees An acute angle is between 1 and 89 degrees Analog Clock Clock with a face and hands This clock shows ten after ten Angle A figure formed by two line segments that end
More informationTarget To know the properties of a rectangle
Target To know the properties of a rectangle (1) A rectangle is a 3D 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 informationGeometry. Geometry is the study of shapes and sizes. The next few pages will review some basic geometry facts. Enjoy the short lesson on geometry.
Geometry Introduction: We live in a world of shapes and figures. Objects around us have length, width and height. They also occupy space. On the job, many times people make decision about what they know
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 informationOral and Mental calculation
Oral and Mental calculation Read and write any integer and know what each digit represents. Read and write decimal notation for tenths and hundredths and know what each digit represents. Order and compare
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 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 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 informationEdexcel Maths Linear Topic list HIGHER Edexcel GCSE Maths Linear Exam Topic List  HIGHER
Edexcel GCSE Maths Linear Exam Topic List  HIGHER NUMBER Add, subtract, multiply, divide Order numbers Factors, multiples and primes Write numbers in words Write numbers from words Add, subtract, multiply,
More information2015 Junior Certificate Higher Level Official Sample Paper 1
2015 Junior Certificate Higher Level Official Sample Paper 1 Question 1 (Suggested maximum time: 5 minutes) The sets U, P, Q, and R are shown in the Venn diagram below. (a) Use the Venn diagram to list
More informationPERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various twodimensional figures.
PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various twodimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the
More informationGrade 9 Mathematics Unit 3: Shape and Space Sub Unit #1: Surface Area. Determine the area of various shapes Circumference
1 P a g e Grade 9 Mathematics Unit 3: Shape and Space Sub Unit #1: Surface Area Lesson Topic I Can 1 Area, Perimeter, and Determine the area of various shapes Circumference Determine the perimeter of various
More informationor (ⴚ), to enter negative numbers
. 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
More informationUnit 8. Ch. 8. "More than three Sides"
Unit 8. Ch. 8. "More than three Sides" 1. Use a straightedge to draw CONVEX polygons with 4, 5, 6 and 7 sides. 2. In each draw all of the diagonals from ONLY ONE VERTEX. A diagonal is a segment that joins
More informationStanford Math Circle: Sunday, May 9, 2010 SquareTriangular Numbers, Pell s Equation, and Continued Fractions
Stanford Math Circle: Sunday, May 9, 00 SquareTriangular 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 information4. An isosceles triangle has two sides of length 10 and one of length 12. What is its area?
1 1 2 + 1 3 + 1 5 = 2 The sum of three numbers is 17 The first is 2 times the second The third is 5 more than the second What is the value of the largest of the three numbers? 3 A chemist has 100 cc of
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 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 informationA of a polygon is a segment that joins two nonconsecutive vertices. 1. How many degrees are in a triangle?
8.1 Find Angle Measures in Polygons SWBAT: find interior and exterior angle measures in polygons. Common Core: G.CO.11, G.CO.13, G.SRT.5 Do Now Fill in the blank. A of a polygon is a segment that joins
More informationSecond Grade Math Standards and I Can Statements
Second Grade Math Standards and I Can Statements Standard CC.2.OA.1 Use addition and subtraction within 100 to solve one and twostep word problems involving situations of adding to, taking from, putting
More informationSOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The OddRoot 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 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 informationSCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self Study Course
SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self Study Course MODULE 17 MATRICES II Module Topics 1. Inverse of matrix using cofactors 2. Sets of linear equations 3. Solution of sets of linear
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 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 informationSMMG December 2 nd, 2006 Dr. Edward Burger (Williams College) Discovering Beautiful Patterns in Nature and Number. Fun Fibonacci Facts
SMMG December nd, 006 Dr. Edward Burger (Williams College) Discovering Beautiful Patterns in Nature and Number. The Fibonacci Numbers Fun Fibonacci Facts Examine the following sequence of natural numbers.
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 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 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 informationMathematics Second Practice Test 1 Levels 46 Calculator not allowed
Mathematics Second Practice Test 1 Levels 46 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 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 information81 Geometric Mean. or Find the geometric mean between each pair of numbers and 20. similar triangles in the figure.
81 Geometric Mean or 24.5 Find the geometric mean between each pair of numbers. 1. 5 and 20 4. Write a similarity statement identifying the three similar triangles in the figure. numbers a and b is given
More information5.4 The Quadratic Formula
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
More informationStudy Guide. 6.g.1 Find the area of triangles, quadrilaterals, and other polygons. Note: Figure is not drawn to scale.
Study Guide Name Test date 6.g.1 Find the area of triangles, quadrilaterals, and other polygons. 1. Note: Figure is not drawn to scale. If x = 14 units and h = 6 units, then what is the area of the triangle
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 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 informationFind the sum of the measures of the interior angles of a polygon. Find the sum of the measures of the exterior angles of a polygon.
ngles of Polygons Find the sum of the measures of the interior angles of a polygon. Find the sum of the measures of the exterior angles of a polygon. Vocabulary diagonal does a scallop shell illustrate
More informationFactors and Products
CHAPTER 3 Factors and Products What You ll Learn use different strategies to find factors and multiples of whole numbers identify prime factors and write the prime factorization of a number find square
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 information