Year 5 Calculator Program

Year 5 Calculator Program

Year 5 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 that students need to develop proficiency with;

many students will have already developed some proficiency with the written algorithms and student generated algorithms

the algorithm study within this program will certainly help to deepen understandings already possessed;

pattern searching is one of the key mathematical processes which this program will aim to develop;

some proficiency with pattern searching relies on the ability to analyse, infer and validate ideas, and helps students gain a better ‘feel’ for mathematics;

where the focus of a lesson is not the practice of some algorithm or formal setting out of some application (e.g. perimeter), then the calculator should be used for any calculation which cannot be carried out mentally;

mental calculation is a crucial prerequisite for successful calculator use;

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

students may be exposed to a wider variety of mathematical experiences through investigations with calculators.

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

clear keys (C/CE)

operations keys ( + – x )

decimal point

memory keys (M+ M– CM RM)

equals key (=)

Students should be led through a series of investigations which help students to learn the particular characteristics of their calculator. They should also be allowed the freedom to share their own discoveries with their classmates, and perhaps a class list of helpful keying sequences could be made and displayed. Once useful discoveries are made they should be practiced so that they become part of the students’ habits with the calculator.

The activities which follow are simply an outline of the types of investigations that could be pursued with calculators. They are designed around a variety of purposes, including those of enriching students’ experiences with number, showing how and when the calculator’s facilities are useful.

ACTIVITY 1Number Study

This activity will describe a number of processes students can be experiencing to aid their understanding of number and numeration concepts, particularly related to place value. The students should be using their calculators in conjunction with other teaching aids such as base 10 blocks.

Students will need to develop a firm understanding of numbers up to four digits so that the patterns for developing further place value ideas are well established. Once those patterns are understood, students can move much more easily on to five digit numbers.

• The teacher should read out numbers so that the students can enter them on their calculators. If pairs are entered and added, the teacher can pinpoint students who have incorrect totals and watch them more closely as similar examples are given. It is important that the students experience success with these early activities, so numbers which involve zeros and the ‘teen’ numbers should not be used in the initial stages.
  

e.g.Key in4293 + 1856 =
Display shows6149

• Once the students have mastered the above, they should be asked to operate within numbers, i.e. given 6745, they could be asked to:

• make the seven into five using the subtraction key;
• make the digit in the tens place into an eight;
• think of two ways to make the hundreds digit into a zero.

• The teacher can read out parts of numbers and the students can enter those parts and read out the new number.

NOTE: Use the ‘+’ key each time and the ‘=’ key at the end.

i.e.Key2 tens + five thousands + nine ones + seven hundreds =
Display shows5729

• This method can be a way of moving on to the more difficult numbers involving ‘zeros and teens’.

i.e.Keyfour ones + six hundreds + one ten (press =)
Display shows614
Read out what your display shows (‘Six hundred and fourteen’)
Key + 3000 =
Now read out the new number (‘Three thousand six hundred and fourteen’).

This method can be repeated as many times as necessary and then the teacher should then involve the ‘teen’ numbers in the addition types described earlier.

Numbers involving zeros can cause some difficulties for students, but the calculator display always inserts zeros if the correct keying sequence is used.

i.e.Keytwo hundred + nine thousand + five ones
Display shows9205

Examples can be selected which feature the zero in the hundreds place and the ones place also. The students will benefit from having to listen to numbers and enter them in the calculator. They could also be asked to enter a number of their own choice which has a zero in a particular place. Several could be chosen to write their choices on the blackboard. These could be read aloud and perhaps added by the students on their calculators.

• The calculator can be very useful when studying numbers which are larger or smaller than given numbers by 1’s, 10’s, 100’s or 1000’s.

i.e. given 7800
Which number is one less than this?

Students can key in the number they choose then self-correct by adding one. This gives them the opportunity to analyse their own mistakes, and to compare their answer with the correct answer. Obviously, the teacher must be able to provide students with strategies for arriving at answers in the first place. Calculators also provide students with opportunities to speak about their strategies (wrong or right) for arriving at their choices.

Other types of examples might include:

Which numbers are:

• one hundred less than 2000
• ten more than 1590
• ten less than 4200
• one more than 3149
• one thousand more than 3649

ACTIVITY 2Investigating the algorithms (with Whole Numbers)

• Many students are able to effectively deal with some or all of the operations (+ – x ÷)despite the fact that they do not understand very much about how or why they work. More modern approaches which are based on the use of base 10 blocks (MAB) do attempt to develop a sound basis, and the activities which follow, back-up and reinforce those efforts. They make students think and talk about each of the algorithms, and if students have to explain what they are doing, perhaps they will come to understand more about the processes.

Addition

• Students need to be helped to realise the commutative and associative properties within addition. Investigations may be along the following lines:

36 / 48 / 52 / 38 / 32 / 56
+ 48 / + 36 / + 36 / + 42 / + 46 / + 48
52 / 52 / 48 / 56 / 58 / 32

All of the above arrangements result in the same answer, and the students may like to infer why so many different configurations are possible. Involving 3 or 4 digit numbers makes many more combinations possible, so students should check to see if the pattern continues. If they are interested, the students may decide to check on the other operations for any similar properties.

• Students learn quite a deal about the algorithmic process through completing partially given algorithms.

e.g. / Supply the missing numbers (there may be more than one correct answer).
34*
+ **2 / 92*
+ *75 / 737
+ 5*8 / 4*
*9
+ *7
540 / 13*1 / 1*2* / *26

Subtraction

• These examples and those in the previous exercises on addition should be used to reinforce the relationships between the two operations. Subtraction should be seen as the inverse of addition, and that very often one can be used to provide missing information in a situation which involves the other.

e.g. / **
– 16 / This situation involves addition to provide the missing number in a subtraction example.
39

Students should have ample opportunity to investigate these situations with their calculators, verifying that in fact they are able to add the two parts to arrive back at the original number. It may be helpful to some students if they are able to identify with the idea that addition and subtraction involve a ‘total and two ‘parts’.

Addition

/ Part
+ Part /

Subtraction

/ Total
– Part
Total / Part

If one is given the parts in either situation, then they can always be added to give the total. If the whole and a part are given, then subtraction will give the other part.

• A study of partially completed subtraction algorithms can be an interesting investigation for students.

e.g. / * * 3
– 1 9 * / 5 1 1
– * * 2
2 2 9 / 1 2 *
* 0 0
– 3 * * / 4 1 * 0
– 1 * 9 *
3 4 6 / * 6 2 7

Multiplication

• During this year students will have to come to fully understand multiplication by a single digit and then by two digits. The activities can help students realise the inverse relationships between multiplication and division and the idea of ‘wholes’ and ‘parts’ can be discussed to decide whether to multiply or divide.

e.g. / * *
x 7 / Since we have been given the whole (182) and one of the parts (7), division must be used to discover the other part. The students should then multiply the parts (26 x 7) to verify that they have found the required number.
182

The position of the missing line of numbers should be varied.

• To help students gain a better feel for multiplication they should be given a number and asked to write different ‘multiplication sentences’. In this way, they may understand factors and multiples a little better also.

e.g. given 240, write a number of multiplication sentences

:2 x 10 x 12
:6 x 4 x 10
:24 x 2 x 5
:15 x 4 x 4
:30 x 4 x 2and so on

The students should be encouraged to look for relationships between the sentences offering new answers after mental calculation, perhaps using the calculator for verification.

• Leading to a study of multiplication by two digits, students should have considerable experience with multiplying by 10, noticing a pattern and then multiplying by multiples of ten.
  After multiplying five or six numbers by 10, students should be able to supply answers mentally, after noticing how the pattern develops. They should then be asked why the pattern works, otherwise multiplying by 10 becomes linked with ‘adding a zero’ to the end of the number.
  While this reasoning may ‘work’ with whole numbers, it falls down when decimals are involved. So, because it cannot be relied on consistently, students should look further for a sounder reason. Joint investigations with the calculator and the place value chart may help to make the pattern more evident. Once this pattern is well understood, dividing by powers of 10 becomes the reverse of that pattern. If students have relied on the ‘adding a zero’ pattern, they do not have a sound basis to begin reasoning from.
  In the long run, it is worth spending the time to have students notice that when numbers are multiplied by 10, each digit moves one place to the left. The only role that ‘zeros’ play is to fill up any places which then have no other digits. But these place value aspects need to be brought out and discussed. The pattern can be continued then to multiplying by 100.

• As part of the algorithm study, students should be asked to complete given ones presented in a variety of forms and stages of completedness. Some examples are:

(a) / **
x * / (b) / 86
x 6 / (c) / * 2 *
x 9
5 2 5 / * *
* * * / 3 8 * 3
5 1 6

Students should be free to conduct trials on an individual basis to try and arrive at answers, but there are obviously signs they should look for during their deliberations. In example (c) above, the missing ’ones’ digit can only be a ‘7’ because it is the only digit which gives a number ending in ‘3’ when multiplied by nine.
The discussions which take place during and after these investigations can be very useful for students who are looking for strategies. The students must be encouraged to share their methods with the class, because misconceptions can also be rectified.

Part of the study of multiplication should feature the distributive relationships within the operation. The students should be led to ‘discover’ and then further investigate such relationships.

e.g. / Given: / 57
x 46
Notice that: / 57
x 46 / = / 50
x 46 / + / 7
x 46
and / = / 57
x 40 / + / 57
x 6
and / = / 30
x 46 / + / 27
x 46
and / = / 57
x 24 / + / 57
x 22
and so on

Students will be able to suggest many alternatives to the above, but will need some help from teachers in drawing all of the ideas together. The distributive relationship above is relatively straight forward because only one of the factors is being changed. Some students may try to change both at the same time and will need some advice regarding the limitations of the investigation. (i.e. keep one factor the same each time).

Division

• While we may lower our immediate expectations regarding students’ performances with the written division algorithm, the calculator is able to cope with the difficult examples as easily as it handles the basic facts. Consequently this study will include a mixture of small and large numbers because students do not have to actually perform the written computations. They do have to begin to understand the processes involved, particularly the place value aspects and the relationships between division and multiplication.
  The students will need to understand these relationships with smaller and more familiar numbers (such as the basic facts) so that they can build up patterns which can be followed later.

e.g. / 6
x 4 / Supply the missing numbers in the following:
24
(a) / *
x 4 / (b) / 6
x * / (c) / 6
4 * * / (d) / 6
* 2 4
24 / 24

Which operation is used each time to supply the missing number?

By using numbers which the students can deal with mentally, operating within the above formats can become almost automatic for most students. When larger numbers are involved they can think back to these situations and plan their methods of solution.

e.g. Work out the missing numbers:

(a) / 2 6
x * *
312 / (b) / * *
x 2 3
782 / (c) / 5 1
* 1 4 7 9
(d) / * *
16 7 5 2 / (e) / 42
39 **** / (f) / 2 6
* * 9 1 0

• Students may be involved with relatively simple examples which have remainders. This brings a different dimension into calculator use because they do not automatically handle remainders. Students have to accept the whole number part of answers and work from there. But the thinking that goes on is quite valuable.

ACTIVITY 3Estimating and Approximating

• These two processes must become an integral part of the students’ thinking with number. The calculator can be most useful in the development of them. Once the teachers and students have discussed the strategies associated with the applications of these processes, the students can be given a number of exercises to test their ability. When the real focus of a mathematics lesson is estimating/approximating, then students should not be expected to perform time consuming computations as well. The student should estimate then verify with the calculator.
  Using the calculator means that the students can do a great deal more actual estimating in a wider variety of situations because of the saving in time. When larger numbers or numbers involving decimal fractions are used, students can actually learn quite a deal by having to estimate then verify. It may teach, by discovery, relationships between numbers particularly when the operations of multiplication are involved.
  Additionally, the use of calculators relies on the students having the ability to estimate with reasonable accuracy. This ability, in turn, relies on speedy and accurate recall of basic facts. Consequently examples must be chosen in relation to the students’ abilities with recall of basic facts. It is foolish to expect that students estimate 643 x 68 if they are having difficulties with their multiplication facts (particularly the 6 and 7 times tables). On the other hand approximation and then estimation exercises can be organised as a useful way to practise sets of facts which have recently been learned or revised.

• Many of the activities which practise these processes can be applied to games situations such as Calculator Bingo. With this game, a grid of numbers (which are the answers to calculations involving a specific operation such as multiplication) are supplied. The students are then given a list of numbers which can be used to make up the given numbers on the grid.

e.g.
253 / 221 / 391 / 437 /

Key Numbers

247 / 429 / 621 / 143 / 11 / 27 / 29
1053 / 741 / 663 / 299 / 17 / 19 / 23
323 / 209 / 459 / 513 / 13

How to play:
Using the Key Numbers, multiply any pair together. If the product is on the grid, mark it off.

• Look at other numbers on the same line (horizontal or vertical) and see if you can estimate which key numbers could be used to make up those numbers.
• Try to make ‘4 in a line’ as in a game of ‘Bingo’.
• Once you have achieved that, pick another line and solve it.

e.g. Supply the missing parts:

(a) / * * rem *
583 / (b) / * * rem *
7164
(c) / 23 rem 4

6* * * / (d) / 16 rem 6

*134

Hints:

1. Remember your basic facts for multiplication.
   If 7 x 9 = 63, then 27 x 19 will also end in 3;
   If 3 x 7 = 21, then 13 x 27 will also end in 1; and so on.
2. Use your estimation skills to try to work out the key numbers. First look at the ‘ones’ digit’, then select a pair of numbers which could work.
   i.e. 17 x 19 would end in ‘3’; but could not equal 1053

• Depending on the students’ abilities, the above activity may be too difficult, in which case, the numbers could be more carefully selected, or grids organised around addition and subtraction.
  As their skills increase, examples with division can be included. If the students become very good, they could be asked to nominate the pairs of key numbers for a line or lines, then check their estimations with the calculator.

ACTIVITY 4Searching for Patterns

• As a crucial part of their work with number, students will be working with quite a few number patterns including squares, triangular numbers, and multiples. Additionally, a study of factors will help to highlight the sets of primes, composites (especially square numbers).
  As soon as children can count, they have begun a study of patterns. In counting the small child is imitating patterns with sounds in an order which matches what they have previously heard. The teacher has to add meaning to any patterns the student is expected to use. While it is great if the child is able to discover number patterns for themselves, most will have to be led towards them. During this year level, the calculator can be used to perform most of the calculations which are part of the study.
  Multiples can be generated very easily using the addition logic of the calculator. Most machines will ‘remember’ the second number entered during addition. So, to generate the multiples of 8, students could enter the following:
  

8 + 8 = = = = = and so on

The display for the above would read: 8 16 24 32 ......
Of course multiplication could also be used for the multiples. The students would have to key the following: