Year 6 Calculator Program

Year 6 Calculator Program

This yearly program is based on a number of assumptions regarding the use of calculators:

Calculating is just one of many mathematical processes students need to develop proficiency with;

Most students will have already developed proficiency with the written algorithms;

Algorithmic-based thinking does not prepare students for high school mathematics which necessitates far more divergence in mathematical thought;

The pattern searching study will help to provide students with a better ‘feel’ for number and with helpful background knowledge for problem solving;

Where the focus of lessons involving number work is not the practising of computational algorithms, then the calculator should be used for any calculations which cannot be carried out mentally;

Students inevitably show more interest in mathematical thinking and greater motivation and perseverance with their work;

A wider variety of types of mathematical experiences is possible through investigations with calculators;

Where particular rules need to be practiced (e.g. perimeters, areas) then they can be handled many times in a short space of time;

Using the calculator places much more emphasis on the ability to calculate and estimate mentally so that some check on the reasonableness of answers is always carried out.

For this program, students would need to become proficient with the following keys:

clear keys (C /CE / AC)
operation keys (+ – x )
decimal point
memory keys (M+ M– CM RM)

Students will need to be led through a series of investigations which help to reveal the particular characteristics of their calculator. Once discoveries are made these will need to be practised so that they become part of the students’ habits with the calculator when similar situations arise at a later time.

The activities which follow are simply an outline of the types of investigations which could be given. Where appropriate, advice regarding particular points to note and some keying sequences are included. Many sources in addition to these activities will need to be ‘tapped’ to provide a full program with integrated calculator use.

ACTIVITY 1Investigating the algorithms

It is imperative that students be able to accurately use the operations keys. Many examples can be given as part of games situations, e.g. solving secret codes; number crosswords.

Completing partially given algorithms

In an endeavour to help students fully comprehend the algorithms, this study makes students come to terms with the relationships within each rule. Very often, more than one answer will be possible and students should be encouraged to notice this fact.

The students should have ample opportunity to work at relatively simple levels before progressing to the more complicated examples.

e.g. /

ADDITION

1 / * *
+ 2 7 / 2 / 4 3 6
+ 2 * * / 3 / 5 *
* 9
+ 3 1
64 / 7 2 5
1 4 8
4 / 3 * 9
* * 5 *
6 3 * 4 / 5 / * * *
4 * *
+ * 5 *
8 3 0 6 / 1 1 3 2

SUBTRACTION

1 / 3 2 0
– 1 * * / 2 / * * *
– 1 6 4 / 3 / 3 8 3
– * * *
1 4 3 / 4 3 2 / 1 0 7
4 / 6 2 4
– 3 9 7 / 5 / 3 * * 1
– * 4 4 2 / 6 / * 0 0 0
– 3 2 * 4
* * * / 1 9 3 * / 1 * 3 *

MULTIPLICATION

1 / 4 9
x 2 3 / 2 / 2 4 6
x * *
* * *
* * 0 / 9 8 4
* * * *
* * * * / 8 3 6 4

DIVISION

1. / 48 rem *
7 339 / 2. / 3 5 8 rem 13
15* * * *
3. / 1 4 rem 13

* *2 7 9 / 4. / * * * rem *

193 6 9 2

Discerning relationships within the algorithms

Given data, students should investigate relationships such as how to arrange five given digits into a 3 digit number and a two digit number to obtain the largest product.

i.e. arrange 5, 2, 9, 6, 1; into the following multiplication set up.
* * * x * * = LARGEST POSSIBLE PRODUCT
Should it be
or
or / 965 x 21
521 x 96
some other arrangement

Similar investigations could be carried out with the other operations.

e.g. / Arrange 8, 6, 5, 0, 1 such that it forms a number divisible by 746.
Arrange 6, 1, 2, 4, 7, 9 into a subtraction sum so that the difference between the numbers is 229.
Arrange 1, 3, 4, 5, 6, 7 into an addition sum of 3 two digit numbers such that the answer is 152.

ACTIVITY 2Pattern Searching

Incorporated in this activity will be the process of investigating what constant functions are included in the calculator’s logic.

Multiplication

When a sequence involving multiplication is keyed into the calculator, the machine will often ‘remember’ one of the numbers. For example, if we key in 100 x 8 – then either the 100 or the 8 may be stored in memory. Usually it is the first number and this is useful for patterns. Therefore if we wish to multiply a series of numbers by 100 to study the results, then we should key the 100 in first.

e.g. / To multiply 100 by 6; 10; 7; 3; 5; use the following keying sequence:
KEY100 X 6 = 10 = 7 = 3 = 5 =
As each equals key is pressed, the display shows each new answer.

This sequence is useful for revising and reinforcing multiplication facts, and as an aid for teaching place value work in whole numbers and decimal fractions.

e.g. / 9 x 1 = 2 = 3 = and so on, will generate the nine times table.

The multiplication key is also useful for a study of square numbers, which must become part of the students’ instant recall (for at least the first 12). Students must be able to recognise numbers as being square because very often this knowledge is useful in solving non-routine problems.

The keying sequence for generating square is:

KEY / 3 x = / NOTE:Their is no need to clear the display each time.
4 x =
and so on

Students should generate at least the first 30 squares and study any patterns which appear to evolve.

Such patterns will include:

a) / the units digits are in a repeating pattern
1 4 9 6 5 6 9 4 1
b) / the pattern is palindromic about the ‘5’
c) / the difference between consecutive square numbers is the set of odd numbers.

Following the study of squares, other exponents can be discovered using a similar keying sequence.

e.g. / for cubes, / 3 x = = / displays 27
4 x = = / displays 64
to raise numbers to the fourth power,
3 x = = = / displays 81
4 x = = = / displays 256

These activities help students to get a feel for the power of exponential notation and extends their knowledge of the relationships between numbers.

Some of the most interesting patterns with multiplication result from squaring numbers which are themselves in a pattern.

e.g. / 12 / = / 92 / = / 81
112 / = / 121 / 992 / = / 9801
1112 / = / 12321 / 9992 / = / 998001
11112 / = / 99992 / =
111112 / = / 999992 / =
1012 / = / 10201 / 62 / = / 36
10012 / = / 1002001 / 662 / = / 4356
100012 / = / 6662 / =
1000012 / = / 666662 / =

The variations of the above which can be investigated are limitless. All students need is the motivation to try and then they may like to infer results which are beyond the limits of their calculator display.

DIVISION

One of the most interesting studies with the calculator concerns division with whole numbers (i.e. decimal representation of common fractions 1/7 = 1  7).

Students should be given opportunity to investigate division of whole number by whole numbers.

Guided discovery should suggest that they look at division by such numbers as 2, 5, 10, 4, 8 and other powers of 2 and 5 together or individually. These divisions will always be finite decimals.

In comparison, division by 3, 6, 7, 9, 11, 12 or any other than the powers of 2 and 5, will result in a recurring decimal answer. These divisions will result in patterns which the students should notice and respond to. Very often, a number of explorations can help the student infer further answers which can be verified through the medium of the calculator.

e.g. / Division by 3
1  3 / = / 0.3
2  3 / = / 0.6
Division by 6
1  6 / = / 0.16
2  6 / = / 0.3 (1  3)
3  6 / = / 0.5 (1  2)
4  6 / = / 0.6 (2  3)
4  6 / = / 9.83
Division by 7
1  7 / = / 0.142 857 / NOTE
:a pattern of six repeating digits
:each digit begins the pattern at one time
:the first digit increases in magnitude each time
2  7 / = / 0.285 714
3  7 / = / 0.428 571
4  7 / = / 0.571 428
5  7 / = / 0.714 285
6  7 / = / 0.857 142
Division by 9
1  9 / = / 0.1
2  9 / = / 0.2
3  9 / = / 0.3 (1  3; 2  6)
4  9 / = / 0.4
and so on
Division by 11
1  11 / = / 0.09 / This pattern becomes obvious as the multiples of 9. Students should be able to complete the pattern without the calculator.
2  11 / = / 0.18
3  11 / = / 0.27
4  11 / = / 0.36
5  11 / = / 0.45
and so on

Just as multiplication patterns can be found with numbers such as 9; 99; 999; and so on, patterns exist when these numbers are used with division activities.

i.e. / 1  9 / = / 0.1 / 8  9 / = / 0.8
1  99 / = / 0.01 / 8  99 / = / 0.08
1  999 / = / 0.001 / 8  999 / = / 0.008

Students should investigate and explain the differences when dividing by the following numbers;

a) / 9; 99; 999; . . .
b) / 9; 90; 900; . . .
i.e. / 5  9 / = / 0.5 / 5  9 / = / 0.5
5  99 / = / 0.05 / 5  90 / = / 0.05
5  999 / = / 0.005 / 5  900 / = / 0.005

They should notice that with the pattern on the left, both zeros and fives recur, while with the pattern on the right only the fives repeat. This reasonable because 5  90 is a tenth of 5  9 and so the pattern of repeating fives should start in one place to the right.

This study can be looked at further by dividing by such numbers as 990; 909; 9909; etc.

It may be worthwhile looking at division by all of the multiples of 3, and within that pattern, look particularly at division by the multiples of 9 (SHOWN *).

i.e. / 1  3 / = / 0.3
1  6 / = / 0.16
1  9 / = / 0.1 / *
1  12 / = / 0.083
1  15 / = / 0.06
1  18 / = / 0.05 / *
1  21 / = / 0.047619
1  24 / = / 0.0416
1  27 / = / 0.037 / *
and so on

NOTE:

While it may appear that the ‘0.037’ for 1  27 is unrelated to any of the others, students might notice that it is in fact one third of 0.1.

i.e. If we write 0.1 as 0.111 111 111 etc and then divide by 3, we will obtain 0.037 037 037 . . .

Looking at the divisors and the decimal fractions can reveal some interesting points. As mentioned earlier:

(a)If the divisor is a power of 2 or 5 (such as 20 = 2x2x5), then the decimal will have no repeating part.

(b)If the divisor has factors of numbers other than 2 or 5 then the decimal fraction will be in a repeating pattern.

(c)NOW, if the divisor has factors which are a combination of (a) and (b) such as 6 (2x3) or 12 (2x2x3), then the decimal fraction will have both a non-repeating part AND a repeating part.

i.e.1  24 = 0.416
only the ‘6’ repeats

ACTIVITY 3Place Value Concepts

Students can be read numbers which they must enter on the calculator.

e.g. Key in seventy-seven thousand, four hundred and fifty.

The children then read the number back to the teacher who may wish to write it on the blackboard to reinforce the setting out (spacing etc).

The teacher (or a child) can read out parts of numbers which the students enter on the calculator, in conjunction with the addition key. (They will need to press the ‘equals’ key or the addition key after the final part has been entered in order to obtain the completed number).

i.e. What number is made up of:
nine hundreds (enter 900+); 8 ones; 2 tens; and 1 thousand?

Students should have experience with changing parts of numbers which have been entered on the calculator.

i.e. Key 2 364
Make the digit in the hundreds place into a seven (using the addition key).

NOTE:The idea is that children recognise that the ‘3’ is in the hundreds place and that they must add ‘400’ to make it into a seven. Some students may only add ‘4’ before they realise their error.

To reinforce the concept of a base 10 numeration system, students should have the opportunity of multiplying and dividing numbers by 10, discussing the results and generalising their ideas through inferences.

NOTE:If the patterns below with whole numbers and decimals are studied in conjunction with place value charts, students begin to realise that zeros are NOT simply added to the end of existing numbers when they are multiplied by 10 or 100 and so on. They should see that each of the digits moves a place to the left when multiplied by 10. The zero simply is used as a space filler if required.
Adding zeros to the need of numbers does not help place value understanding, nor does it ‘work’ with decimals. It is important that students reach a degree of understanding with these place value activities, and after that, they may be able to infer short cuts which speed up their thinking.
After examining the patterns which are formed below, students should be given frequent practice in mental calculations involving multiplication and division with powers of 10.

(a)Most calculators will store the first number keyed in during a multiplication example. So if students are to study a pattern of numbers multiplied by 10:

KEY / 10 / X 6 / = / Other numbers can be multiplied by 10 by entering the number and pressing =.
7 / =
1.1 / =
11 / =
7.2 / =
and so on

(b)The above pattern looks at whole numbers and decimals multiplied by 10. It can be extended to study the pattern of multiplying by 100, 1000 and further powers of 10.

(c)Division by 10 using a calculator can be studied using the fact that most calculators will store the divisor. So the following keying sequence can be used to investigate the patterns which are formed:

DISPLAY
KEY / 6 /  10 / = / 0.6
7 / = / 0.7
15 / = / 1.5
4 / = / 0.4
20 / = / 2
22 / = / 2.2
and so on

To bring out the place value notions, students would need to notice that each digit moves a place to the right.
The short cut of ‘moving the decimal point’ should not be used as a teaching technique, because it is the digits tghat move in relation to the decimal point. More understanding of the place value system will result if students use the suggested strategy rather than a mixture of both. Usually, it is only when decimals are introduced that the ‘moving the decimal point’ technique is used. Students should stick with the notions they have already developed with whole numbers.

(d)Once a degree of proficiency has been reached with inferring answers mentally regarding division by 10, the investigations can be extended to include division by 100; 1000 and further.

ACTIVITY 4Problem Solving (Routine problems only)

When students are faced with routine problems which involve calculations, it is an appropriate opportunity for them to use calculators. The vital element of these situations is the thinking involved, particularly the logic which determines the relationships and order in which calculations need to be performed.

One step problems

Situations which involve one step only are very much suite to oral presentation and calculator use. Students are required to listen carefully for the mathematical relationship involved (i.e. which operation).

Examples given in this activity should range from ones which involve more complex numbers such as decimals (e.g. money).

i.e. / (a)I began the day with 46 marbles in my bag. I lost 18 at lunch hour. How many did I have left?
(b)If I bought 3 shirts at a sale for $4.55 each, how much would I be charged altogether?
(c)Four people shared a total pool prize of $500 in the Casket. How much would each receive?
(d)During sports practice, the students of a school formed two circles with 120 children in the first and 59 in the second. How many children attended the practice?

This is good opportunity for students to make up problems around the four operations. They could be given two numbers (say 117 and 13) and be asked to make up a mathematical story about those numbers. The student who creates the story should have the opportunity to correct and explain about the solution to the remainder of the class. The numbers given should be chosen carefully so that if division is used, allowances are made for whole numbers quotients, at least in the beginning.

Two stop problems

Once students have reached a high degree of proficiency with the single step problems, they should have experience with situations which have a second part to them. These can still be presented orally, but this will depend on the complexity of the numbers which are used. Students should be permitted to jot down information or partly worked answers during this stage of calculator use. Most will not have full facility with memory usage as yet (see next section) and would be better off jotting down appropriate figures if they need to be remembered. Very often, calculations are performed in an order which does not always suit.

e.g.I go to a store and buy three kilograms of margarine for $1.25 each. What change would I receive from $10.00.

Most students would perform $1.25 x 3 to ascertain the cost of the margarine as the first calculation. Unless they have facility and understanding of either memory or change sign keys, they cannot immediately use that answer. They must enter the $10.00 (which clears the $3.75 from the screen) and then subtract.

Teachers must decide whether the mathematical and contextual details within a problem are too confusing or ‘wordy’ for students to deal with orally. If so, then they should be presented in written form for students to deal with. Students must be given the opportunity to explain their solution methods to other students, and use of the calculator means that many problems can be dealt with in short spaces of time. Sessions of this type should rarely be for any longer than 10 minutes of actual working time (usually less) but more time may be spent in the discussion of solutions and methods.