Sharp EL-531RH User manual

SCIENTIFIC CALCULATOR
TEACHER’S GUIDE
JULY 1999EL-531RH

1
Contents
Generating Sequences………………………………………………………………………p.39
Train Journeys………………………………………………………………………………p.41
Simulated Dice……………………………………………………………………………p.43
Mean Dice Scores…………………………………………………………………………p.45
Again and Again……………………………………………………………………………p.47
Fibonacci……………………………………………………………………………………p.49
Factorizing Quadratics……………………………………………………………………p.51
Triples………………………………………………………………………………………p.53
Teacher’s Guide Part 1has already been completed.This guide presents Part 2 beginning on page
39 (marked with ). We encourage you to put this guide to good use.
Introduction………………………………………………………………………………………………p.2
How to Operate…………………………………………………………………………………p.3
Number of Bowling…………………………………………………………………………………p.4
Down to One…………………………………………………………………………………p.6
Reverse the Order…………………………………………………………………………………p.8
Different Products…………………………………………………………………………………p.10
Sums and Products…………………………………………………………………………………p.12
Target 100……………………………………………………………………………………p.14
Ordering Fractions…………………………………………………………………………………p.16
Addting Fractions…………………………………………………………………………………p.18
Halfway Between…………………………………………………………………………………p.20
Near Integers…………………………………………………………………………………p.22
Reshaping Cuboids…………………………………………………………………………………p.24
Function Tables…………………………………………………………………………………p.26
Palindromes…………………………………………………………………………………p.28
Trial and Improvement………………………………………………………………………p.30
Last Digits…………………………………………………………………………………p.32
A Question and Interest……………………………………………………………………p.34
Getting Even…………………………………………………………………………………p.37

2
The use of calculators as a classroom teaching tool is becoming more and more
popular.Contrary to the belief that their use encourages dependency and inhibits
the development of mental skills,research has proven that calculators are highly
unlikely to harm achievement in mathematics and using them can actually improve
the students’ performance and attitude.* Calculators allow students to quickly gen-
erate large amounts of data from which patterns can be spotted,and predictions can
be made and tested.This is an important aspect of the development of mental meth-
ods of calculation.Therefore,priority must be given to create new ways to exploit
the potential of the calculator as an effective learning tool in the classroom.
ThisTeacher’s Guide presents several classroom activities that make use of Sharp
scientific calculators.The purpose of these activities is not to introduce the calcula-
tor as a device to relieve the burden of performing difficult calculations,but rather
to develop the students understanding of mathematical concepts and explore areas
of mathematics that would otherwise be inaccessible.Mental methods should al-
ways be considered as a first resort when tackling calculations introduced in these
activities.The development of trial and improvement methods are supported by the
activities as well.We hope you will find them interesting and useful for reinforcing
your students’ understanding of mathematical concepts.
* MikeAskew & DylanWilliams (1995) Recent Research in Mathematics Education HMSO
Introduction

3
How to Operate
2nd function key
Pressing this key will enable the functions written
in yellow above the calculator buttons.
ON/C,OFF key
Direct function
Mode key
This calculator can operate in three different
modes as follows.
<Example>
Written in yellow
above the ON/C key
<Power on> <Power off>
1.KEY LAYOUT
•Mode = 0;normal mode for
performing normal arithmetic
and function calculations.
•Mode = 1;STAT-1 mode for
performing 1-variable
statistical calculations.
•Mode = 2;STAT-2 mode for
performing 2-variable
statistical calculations.
If the calculator fails to operate normally,press the reset
switch on the back to reinitialise the unit.The display format
and calculation mode will return to their initial settings.
RESET
2. RESET SWITCH Reset switch
RESET
2nd function
[Normal mode]
[STAT-1 mode]
[STAT-2 mode]
NOTE:
Pressing the reset switch will erase any data stored in memory.
≈Read Before Using≈

4
Number Bowling Junior high school
•••••••••••••••••Using the calculator • • • ••••••••••••••
Calculator functions used:Subtraction,addition,last answer memory
•••••••••••••••••••••Objective •••••••••••••••••••••
Read whole numbers and understand that the position of a digit signifies its value.
Understand and use the concept of place value in whole numbers.
•••••••••••••••Explanation of the activity ••••••••••••••
Think of a 3-digit number and enter it into your calculator.
Pretend each digit is a“bowling pin.”
Knock down each pin one at a time,so that your calculator display shows 0.
A: Using subtraction
B:Using addition
(1) Enter a 3-digit number.
(2) Knock down one digit,or“pin”;i.e.change the last
digit to a 0.
(3) Knock down the next pin;i.e.change the tens column
digit to 0.
(4) Knock down the pin of the hundreds column.
A: Using subtraction
638
8
30
600
Press the following buttons and then start operation.
638= DEG
ANS-8= DEG
ANS-600=DEG
ANS-30= DEG

5
Junior high school
(1) Enter a 3-digit number.
(2) Knock down one digit,or pin;i.e.change the last digit
to a 0,except this time,do so by adding a number to
the last digit to make it 0.
(3) Knock down the next pin;i.e.change the tens column
digit to 0.
(4) Knock down the pin of the hundreds column.
B:Using addition
Number Bowling
• • • • • • • • • • • Using the activity in the classroom • • • • • • • • • • •
This activity is a good game for students to play in pairs.
One student enters a number in the calculator,and the other student has to knock each digit,or
“pin,” down.
Example:
638 - 8 = 630
630 - 30 = 600
600 - 600 = 0
•••••••••••••••Points for students to discuss • • • •••••••••••
It is important for students to talk about what they are doing and use the appropriate language,for
example:“six hundred and thirty,minus thirty,equals six hundred.” Students should be challenged
to vary the starting point;i.e.sometimes starting with the hundreds digit and sometimes with the
tens digit.
Further Ideas
• Play the game using 2-,4-,or 5-digit numbers according to the ability of the students.
2
60
300
Press the following buttons and then start operation.
638 638= DEG
ANS+2= DEG
ANS+60= DEG
ANS+300=DEG

6
Down to One Junior high school
•••••••••••••••••Using the calculator • • • ••••••••••••••
Calculator functions used:Subtraction,division,last answer memory
•••••••••••••••••••••Objective •••••••••••••••••••••
Develop a variety of mental methods of computation.
Develop the use of the four operations to solve problems.
Use sequence methods of computation when appropriate to a problem.
Estimate and approximate solutions to problems.
•••••••••••••••Explanation of the activity ••••••••••••••
Use the calculator to generate a 3-digit random number.
The aim is to get the calculator to display the number 1 .
Players can use any of the numbers 1 – 9 together with any of the keys below:
, , , , , ,
You cannot put numbers together to make 2- or 3-digit numbers.
You can use each number only once.
The first player to get his/her calculator display to show 1 scores five points.
If after an agreed time limit no player has reached 1 ,the player who is closest scores two points.
While working on this activity,students should develop their skills of mental mathematics and their
fluency with numerical calculations.
Suppose the random number you generate is 567.
Example A:
The answer is 1 and the game is finished.
567 9
7
8
Press the following buttons and then start operation.
567÷9= DEG
ANS÷7= DEG
ANS-8= DEG

7
Junior high school
Down to One
• • • • • • • • • • • Using the activity in the classroom • • • • • • • • • • •
Students should be encouraged to estimate the results of calculations and think about the appro-
priate operations and numbers to use during the game.Let’s start with 864,for example.This
number is divisible by 9,6,3 and 2.The equation 846 ÷9 could therefore be a possible first step.
This will prompt students to test the divisibility of numbers.Students should also be encouraged to
think about the various strategies they use.
The game could be played between small group of students.
•••••••••••••••Points for students to discuss • • • •••••••••••
For some students,it may be more appropriate to start with a 2-digit number.In this case,the calcula-
tor should be set to fixed decimal place mode by pressing the [2ndF] key once and then pressing the
[ .] key,which has FSE written in yellow above it,until FIX is displayed at the top of the calculator
screen.And press [2ndF] [TAB] and [0] keys.Doing this will round answers to 0 decimal places.The
starting number can then be generated by multiplying a random number by 1 00.
Further Ideas
• Play the game using decimal starting numbers.
• Give the students a shuffled set of cards numbered from 1 to 9 and a shuffled set of
cards numbered 1 0,20,30,40,50.Students choose five cards from the first set,and
two cards from the second set.The calculator is then used to generate a random three
digit integer,and the students have to make this total by using the numbers on the cards.
Example B:
You want to subtract 8 from 9,but you cannot since you have already used 8 once.
So...
The calculator displays 1 and the game is finished.
9
8
3
2

8
Reverse the Order
Junior high school
•••••••••••••••••••••Objective •••••••••••••••••••••
Develop a variety of mental methods of computation.
Estimate and approximate solutions to problems.
•••••••••••••••Explanation of the activity ••••••••••••••
Enter any 2-digit number into the calculator.
Reverse the order of the digits through simple calculator operations.
While working on this activity, students should develop their skills of mental mathematics.
They should also be interpreting and generalizing their answers.
Example A:
To reverse the order of 58:
Solution:Add 27 to 58 to get 85.
Now try using a 3-digit number.
Example B:
Enter 432 into the calculator
Solution:Add 1 98 to 234 to get 432.
58 27
432 234
234
1 98
Press the following buttons and then start operation.
85 58
•••••••••••••••••Usingthecalculator • • • ••••••••••••••
Calculator functions used: Addition, subtraction
DEG
DEG
DEG
DEG

9
Junior high school
Reverse the Order
• • • • • • • • • • • Using the activity in the classroom • • • • • • • • • • •
This activity is probably best introduced orally to a group of students.Ask the students to enter
any two digit number into their calculators.Then,ask them to find a simple way to reverse the
order of the digits of these numbers.Students may do this by using inverse operations.
•••••••••••••••Points for students to discuss • • • •••••••••••
After trying an example,the students can talk about the operations and numbers that they used.
This discussion should lead to the generalization that one way to reverse the order of the digits is
to add or subtract a multiple of 9.More able students could be asked to try and prove this gener-
alization:
( 1 0a+ b) + N = ( 1 0b+ a)
N = ( 1 0b+ a) - ( 1 0a+ b)
N = 9b- 9a= 9(b- a)
Further Ideas
Try using the activity with 3-digit numbers,4-digit numbers,etc.
Choose any 2-digit number,reverse it,and then add the reversed number to the original.
What happens?Try this with 3-digit numbers or 4-digit numbers,etc.

10
Different Products Junior high school
•••••••••••••••••••••Objective •••••••••••••••••••••
Estimate and approximate solutions to problems.
•••••••••••••••Explanation of the activity ••••••••••••••
Have the class make up multiplication problems using the digits 1 ,2,3 and 4.Each digit can only be
used once.Find out what the largest product among the possible answers will be.
While working on this activity,students should practice their skills of mental estimation.They
should also be interpreting and generalizing their answers.
What is the largest number you can make by pressing the keys and
once and only once?
Example:
Can you make a larger number?
Using algebra,for any four digits a, b, c, d,where
a< b< c< d,the largest product is given by:
( 1 0d+ a) x ( 1 0c+ b).
Ans:The largest product is given by
1 2 34
2 34 1
4 1 3232

11
Junior high school
Different Products
• • • • • • • • • • • Using the activity in the classroom • • • • • • • • • • •
This activity could be introduced to the whole class by asking students to individually make up any
multiplication using only the digits 1 ,2,3 and 4.The different multiplication problems and their
answers can then be compared and students can be set the task of finding the largest product.
Students should be encouraged to estimate the answers to the various multiplication problems.
•••••••••••••••Points for students to discuss • • • •••••••••••
Students can explore other sets of four numbers,generalizing the rule to find the largest product
using words or symbols.After generalizing,explain the rule that for any four digits a, b, c, d,where
a< b< c< d,the largest product is given by:
( 1 0d+ a) x ( 1 0c+ b).
If the investigation is extended to the five digits 1 ,2,3,4,5,then the largest product is given by:
43 1 x 52 = 224 1 2.
For some students it may be appropriate to begin with only three digits.
Further Ideas
• Find the largest product for any number of digits.
• Find the smallest product for any number of digits.
• Find the different sums that can be made by adding the digits 1 ,2 and 3 once and only
once.For example 1 2 + 3 = 1 5.What happens for other sets of 3-digit numbers?

12
Sums and Products Junior high school
•••••••••••••••••••••Objective •••••••••••••••••••••
Calculate with decimals and understand the results.
Select suitable sequences of operations and methods of computation,including trial-and-improve-
ment methods,to solve problems involving integers and decimals.
•••••••••••••••Explanation of the activity ••••••••••••••
Choose two numbers whose sum is 1 0.
Find out what the product of those two numbers would be.
Find the products of other pairs of numbers whose sum is 1 0.
Find out which number pair gives the largest possible product.
This activity helps to reinforce students’ understanding of the mathematical terms‘sum’ and‘prod-
uct’ and develops trial-and-improvement methods.
•••••••••••••••••Using the calculator • • • ••••••••••••••
Calculator functions used: Addition,multiplication,subtraction,parentheses
Try to find the largest product of any two numbers whose sum is 1 0.
Example:
2 8
2 8
You can also calculate this as 2 x ( 1 0 - 2) = 1 6.
2 1 0 2
What two numbers give the largest product?
Try multiplying various combinations of numbers whose sum is 1 0.
Ans: 5 x 5 = 25
Press the following buttons and then start operation.
2+8= DEG
2X8= DEG
2X(10-2)=DEG

13
Junior high school
Sums and Products
• • • • • • • • • • • Using the activity in the classroom • • • • • • • • • • •
This activity could be introduced orally.
The largest product is 25,given by 5 x 5.Some students may need to be encouraged to consider
decimal numbers to verify that the largest product is 25.More able students should be encouraged
to try and prove that this is the largest product.
One method of using the calculator is to enter the product as two numbers that can be edited.
Some students may prefer to enter the product as an expression such as 2 x ( 1 0 - 2),which can be
edited.
•••••••••••••••Points for students to discuss • • • •••••••••••
Students could be encouraged to devise similar problems to give to each other involving numbers
with different sums.
Further Ideas
• Investigate products of 3,4,5...numbers which have the same sum.This could be explored
graphically.
(Generally,for two numbers whose sum is n,the largest product is given by (n/2)2,for three
numbers whose sum is n,the largest product is given by (n/3)3...The nearest integer to (n/e)
where e= 2.7 1 8 is the number of numbers which will give the maximum product.)
• The problem of finding two numbers whose product is a given total can be turned into a
game where students score points according to the number of trials they perform to
identify the solution.For example:The sum of two numbers is 1 0 and their product is
1 9.7 1 . What are the two numbers?
Ans:The two numbers whose product is 1 9.7 1 are 7.3 and 2.7.

14
Target 100 Junior high school
•••••••••••••••••••••Objective •••••••••••••••••••••
Understand and use the concept of place value in whole numbers and decimals,relating this to
computation.
Calculate with decimals and understand the results;e.g.multiplying by numbers between 0 and 1 .
Mentally estimate and approximate solutions to numerical calculations.
•••••••••••••••Explanation of the activity ••••••••••••••
A game for two players.
• Player 1 enters any 2-digit number into the calculator.
• Player 2 then multiplies this by another number so that the answer is as close as possible to 1 00.
• Players score points according to how close they are to 1 00:
within 1 0 = 1 point
within 5 = 2 points
within 1 = 5 points
exactly 1 00 = 1 0 points
• Player 2 then enters a number and the game continues.
•The first player to score 20 points wins.
While working on this activity,students will be extending their understanding of decimals and
improving their estimation skills.
•••••••••••••••••Using the calculator • • • ••••••••••••••
Calculator functions used: Multiplication
Example:
Player 1 enters 28.
28
Player 2 multiplies this by 3.5.
3.5
Player 2 scores two points.
The game continues until one player reaches 20 points.
Press the following buttons and then start operation.
DEG
28X3.5= DEG

15
Junior high school
Target 100
• • • • • • • • • • • Using the activity in the classroom • • • • • • • • • • •
This activity could be given to students with little introduction from the teacher.Alternatively,the
game could initially be played between the teacher and a large group of students.It is important
that students are encouraged to think carefully about the numbers they choose and that the
teacher focuses on the students’ mental skills.Most benefit is obtained from the activity when
students are playing together in small teams,discussing their choices of a number to multiply by.
•••••••••••••••Points for students to discuss • • • •••••••••••
At the end of the activity,students’ strategies should be discussed and compared.
Further Ideas
• Play the game with different target numbers.For example,students could multiply or divide
a random number to reach a target of 1 .
• The first player multiplies a random number to aim for a target of 1 00.The second player
then multiplies this answer to try and get even closer to 1 00.The player who gets the
calculator to display a number between 99 and 1 0 1 wins.

16
Ordering Fractions Junior high school/
Elementary school
(upper grades)
•••••••••••••••••••••Objective •••••••••••••••••••••
Understand and use fractions.
Understand the interrelationship between fractions and decimals.
•••••••••••••••Explanation of the activity • • • • • •••••••••
Estimate where a given fraction would be located on a numerical line.
Check the answer using the calculator.
While working on this activity,students will be developing their understanding of the relative sizes of
common fractions.The activity suggests an approach to teaching equivalence of common fractions.
•••••••••••••••••Using the calculator • • • ••••••••••••••
Calculator functions used: Addition,division,fractional calculation
You will need a 0 – 2 number line.
Estimate where the following common fractions should be placed on the number line
and then record estimates.
, , , , , , , , ,
Use a calculator and a ruler to check your estimates.
ExampleA:
Find the value of the fraction .
Using division:
1 2
Using fractional calculation:
1 2 on the calculator display means .
1 2
Convert to decimal notation.

17
Ordering Fractions
• • • • • • • • • • • Using the activity in the classroom • • • • • • • • • • •
This activity may be introduced orally.The number line could be copied onto an overhead projec-
tor transparency or written on the board.Divide students into small groups and give each group a
fraction card.Have the groups discuss where to place their given fraction on the line.Groups then
take turns marking their fractions on the number line.Solutions can be discussed,together with
methods of checking the solutions.This should lead into converting common fractions to
decimal notation, and students can be shown how to do this on the calculator. It is important
that students are aware of the general method of converting common fractions into decimal
notation (dividing the numerator by the denominator), as well as the use of the fraction key
on the calculator.
•••••••••••••••Points for students to discuss • • • •••••••••••
It will be discovered that some of the fractions are equivalent to each other and this leads into the
second part of the activity.When the fraction 4/6 is entered into the calculator,pressing‘=’ simpli-
fies the fraction to 2/3.Students should explore the results of entering different fractions,thus
generating sets of equivalent fractions.It is important that students are encouraged to understand
the concept of equivalence.
Further Ideas
Small groups of students are given a pack of cards with a different fraction on each card.The

18
Adding Fractions
•••••••••••••••••••••Objective •••••••••••••••••••••
Understand and use fractions.
Calculate with fractions and understand the results.
•••••••••••••••Explanation of the activity ••••••••••••••
Using the calculator,find the sum of two given fractions each having 1 in the numerator.
Look for patterns to help understand how to add the fractions without using the calculator.
This activity suggests an approach to teaching addition of common fractions.
•••••••••••••••••Using the calculator • • • ••••••••••••••
Calculator functions used: Addition,fractional calculation
Example:
Using fractional calculation, find the sum of and .
1 2 1 3
+ = on the calculator display means .
Convert to decimal notation.
0.83333··· on the calculator display means .
Find the sums of other common fractions.
1 5 1 7
Junior high school/
Elementary school
(upper grades)
Press the following buttons and then start operation.
15+1 7= DEG
15+1 7= DEG
12+1 3= DEG
12+1 3= DEG

19
Adding Fractions
• • • • • • • • • • • Using the activity in the classroom • • • • • • • • • • •
This activity should be presented after studying equivalence of common fractions.
The activity is best introduced orally.Ensure that the students know how to add two common
fractions on the calculator.Ask them to add 1 /2 and 1 /3 and record the answer (5/6).Ask the
students if they can see any connection between the answer and the original two fractions.Stu-
dents may note that 2 + 3 = 5 and 2 x 3 = 6.Allow students to explore other unit fractions and
encourage them to generalize.Students should be asked to try and explain what is happening.It
should be noted that the pattern may appear to break down when fractions with a common
denominator are added.
•••••••••••••••Points for students to discuss • • • •••••••••••
Students can then explore what happens when other common fractions are added.For some
students,it may be appropriate to begin by considering a pair of fractions that includes one unit
fraction.
It is important that students are encouraged to understand what is happening,and that reference
is made to equivalent fractions.
Further Ideas
•Investigate subtracting,multiplying or dividing common fractions.
•The Babylonians mostly used fractions which had 1 as the numerator.For example,5/6
could be written as 1 /2 + 1 /3.Investigate Babylonian fractions.
Junior high school/
Elementary school
(upper grades)