# Lesson C - Algebra 1/2 (AT)

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```					                                                                                Lesson C - Texas Version
Lesson C - Algebra 1/2 (AT)
Fractional Sequences (Graphical Calculators)

Objectives
Generate terms of a linear sequence using term-to-term and position-to-term definitions of the
sequence, on paper and using a spreadsheet or graphical calculator (C); find the next term and the nth
term of quadratic sequences and functions and explore their properties (E).
Oral/Mental Starter
Hand out sets of cards (1 between 2) with the following expressions written on them: 1/x, 1/2x, 2/x2,
1/(x + 1).
‘Order the value of these expressions from smallest to largest if:
(i) x =1 (ii) x = 2 (iii) x = 3 (iv) x = 4 (v) x = 1/2 .’
‘Which expression has moved the least? Why? Which expression has moved the most? Why?’
Main Activity
 ‘We are going to learn how to program a Graphical Calculator to generate, first of all some
simple, and then some more complex sequences. In order to do this we need to make sure we
can derive the DIRECT RULE for a given sequence.’
Recap how to do this for the sequences: 2, 4, 6, 8, 10, … and 15, 10, 5, 0, -5, …

   Demonstrate how to generate these sequences on the Calculator. For the first sequence
above:

Press 0       STO            X, T, Θ, n        ENTER

X, T, Θ, n    + 1          STO         X, T, Θ, n      2nd    : (this is the decimal
point
key) 2        X, T, Θ, n

Press     ENTER       10 times and watch the pattern.

   Explain how the sequence is generated.
0     X         sets X to zero (ready for a new sequence.)
X+1        X means X will increase by 1 each time ENTER is pressed (i.e to get the term
number.)
:               separates 2 commands, to be carried out at the same time.
2X              the DIRECT RULE.

   Repeat the above process to enter the DIRECT RULE for the second sequence i.e. - 5X + 20

Replace 2        X, T, Θ, n        with   - 5      X, T, Θ, n   + 20

   Pupils work through the Activity Sheet.
Lesson C - Texas Version
Plenary
 ‘What have you discovered about how to work out the DIRECT RULE for a sequence of
fractions?’ (To look at the numerator and denominator separately, and work out the direct rule
for each; don’t think of it as a decimal.)

   ‘Sometimes in Maths it is helpful to think about the problem in a less obvious way, to make the
working out a lot easier!’

   Write the following on the board for the pupils to enter in their calculators:
‘You will have to use the ALPHA key lots of times!’

1        B:1          C      ENTER

B        A:C          B:A+B            C

‘Press     ENTER       lots of times and write down the sequence of numbers.’

   ‘Why does this sequence appear?’ ‘What is the term-to-term rule?’ ‘This is known as the
Fibonacci Sequence.’
   ‘Now have a go at programming your calculator to generate this Fibonaci Sequence:
9, 10, 19, 29, …

K Wallis 8/03
Lesson C - Texas Version

Sequences and Graphical Calculators

A. Whole Number Sequences
1.

Press 0      STO             X, T, Θ, n      ENTER

X, T, Θ, n    + 1         STO       X, T, Θ, n     2nd    : (this is the

decimal point key) 2      X, T, Θ, n

Press     ENTER        10 times and watch the pattern.

2.      Repeat the above process to enter the DIRECT RULE for the sequence -5X + 20.
Make sure you generate the sequence 15, 10, 5, 0, -5, -10, …

3.      Use a similar method to produce the following sequences.
In each case write down the DIRECT RULE you used on the answer sheet provided.

SEQUENCE                                                     DIRECT RULE
1    The multiples of 5                                        5, 10, 15, 20, …            5X
2    The square numbers                                        1, 4, 9, 16, …
3    The negative whole numbers                                -1, -2, -3, -4, …
4    Powers of 2                                               2, 4, 8, 16, …
5    Numbers counting down from 100                            100, 99, 98, 97, …
6    Numbers counting down from 100 in steps of 10.            100, 90, 80, 70, …

B. Fraction Sequences
To generate fraction sequences we simply need to tell the Graphical Calculator to display all the term
values as fractions. We do this by pressing  MATH        1 at the end of our instructions. Turn over
to see how.
Lesson C - Texas Version
1.

Press 0      STO          X, T, Θ, n        ENTER

X, T, Θ, n   + 1        STO           X, T, Θ, n   2nd   : (this is the

decimal point key) 1 ÷      X, T, Θ, n     MATH       1

Press    ENTER       10 times and write down the sequence of numbers displayed on your

2.      Enter the sequence shown here.
Press      ENTER
8 times and write down

the sequence of fractions displayed on your

C. Harder Fractions Sequences
1.      Enter the sequence shown here.
Press      ENTER    8 times and write down

the sequence as top heavy fractions on

2.      Using a similar method, try to produce this sequence:

1/2, 2/3, 3/4, 4/5, …

Write the direct rule you used on the sheet.

3.      Using a similar method, try to produce this sequence:

1/2, 3/4, 5/6, 7/8, …

Write the direct rule you used on the sheet.
Lesson C - Texas Version

Sequences and Graphical Calculators

A.       Whole Number Sequences

SEQUENCE                                           DIRECT RULE
1     The multiples of 5                               5, 10, 15, 20, …         5X
2     The square numbers                               1, 4, 9, 16, …
3     The negative whole numbers                       -1, -2, -3, -4, …
4     Powers of 2                                      2, 4, 8, 16, …
5     Numbers counting down from 100                   100, 99, 98, 97, …
6     Numbers counting down from 100 in steps of 10.   100, 90, 80, 70, …

B.       Fraction Sequences

(1)      1/X      ________________________________

(2)      (½)X ________________________________

C.       Harder Fraction Sequences

(1)      (X + 1)/X      _________________________________

(2)      1/2, 2/3, 3/4, 4/5, … has DIRECT RULE:_________________

(3)      1/2, 3/4, 5/6, 7/8, … has DIRECT RULE: _________________

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