Calculators in Year 4
Calculators in Year 4
The teachers task of helping students to use calculators effectively and efficiently will be made much easier if all students have the same machine.
The activities which follow are simply an outline of the types of investigations students could be involved with. Where appropriate, advice regarding particular points to note are highlighted and some keying sequences are included. The teacher will have to use other sources (including their imagination) to support the suggestions in this program. Some of the activities may have to be adapted to suit the particular needs of students. At present there are very few suitable resource books available for teachers to use. None of those resources will, by itself, provide a method of integrating calculators into the mathematics program. Teachers will, through programs such as this one and their own experiences, decide just how much they wish to use the power of calculators in their classrooms.
ACTIVITY 1Number study
This activity will outline a number of experiences students could have to help them to further understand number and numeration concepts. The calculator should be used in conjunction with other learning aids such as base 10 blocks, counters, place value charts, etc. It is imperative that students gain a sound understanding of the numbers in the study so that the patterns for developing further place value ideas are well established.
Once the teacher is convinced that the students have had sufficient opportunity to familiarise themselves with the calculator, particularly the clear keys, operations keys and the equals key, the activities below can be commenced. It should be remembered that students need to experience numbers in a variety of ways, including making them with various materials, saying the numbers they ‘see’ in concrete form, written form and numerical form, and writing the numbers in numerical and word forms.
The students can enter numbers that the teacher says aloud on to the calculator. Each one could be entered, read back by the students, written on the blackboard by the teacher and checked by the students for a match. Students as well as the teacher could read out numbers to the class, particularly those who might need special practice or encouragement. Instead of clearing the calculator each time, the students could press the addition key after the numbers and combine this activity with practise in addition. After three numbers have been entered, the totals could be checked to test accuracy.
e.g. / Enter 24 / Display reads / 24Press the addition key
Enter 36
Press the addition key / 60
Enter 18
Press the equals key / 78
Larger numbers can be used once proficiency has been reached with two digit numbers. Special notice should be taken of students’ abilities to cope with the ‘teen’ numbers and numbers involving zeros. The teacher or a child should often write the number/s on the blackboard so that the other students can have a visual check on the correctness of their answers.
After the students are confident and accurate with the entering of numbers into the calculator, other activities can be begun which allow students to practise and reinforce their knowledge of place value with whole numbers.
Enter 247Which digit is in the tens place?
Using the addition or subtraction keys only,
make that digit into a 6 (e.g. +20 or –80)
Read out the new number (either 267 or 167)
e.g.
This type of work should be revisited as often as necessary to maintain proficiency and clear thinking as to the relative values of each place in the number sequence. Most probably, students of Year 4 will use the addition key to change to a larger digit (change a 4 to a 6) and the subtraction key if the digit is to become smaller (change a 9 to a 2). If the class all uses these methods, discuss the possibility of using the alternative key, allowing the students to experiment to find the answer.
When efforts are being spent to have students recognise and read numbers, they could be given relatively simple exercises involving the operations.
Enter 68 x 7Read the number in the display [476]
e.g.
Once all students are able to read three digit numbers confidently, the examples could involve four digit numbers. These exercises also train students to listen more carefully, and efforts should be made to train them to listen to a number as it is read aloud, visualise the digits and enter it on the calculator. It is most important that the students become very accurate with entries into the calculator.
As another method of reinforcing knowledge of place value, the students could be asked to use the calculator to add up the parts of numbers which the teacher or another child supplies.
Make up a number from these parts:three tens
four hundreds
six ones
e.g.
[NOTE: Directions should instruct students to press the addition key if another number is to be entered, and the equals key if the last part has been entered]
Individual students could be given a ‘compact’ number (e.g. 926) and be instructed to read out each part (in various orders) for the remainder of the class to enter on the calculator. This gives practise in performing both types of place value activities.
The calculator can be used to investigate how numbers appear when they are referred to in non-standard forms, e.g. 38 tens. These examples can be entered on to the calculator as multiplication types (38 x 10). The students can be given practise in keying in other examples of the same type until they are able to see a pattern and key in the given number straight away. They can experiment with hundreds and even thousands when the teacher feels that they are ready.
Each of the types of exercises described in this activity may need to be revised at different stages of the year and should be combined with activities using other concrete materials such as base 10 blocks, bundling sticks and place value charts.
It must be remembered that the calculator is only viewed as a learning tool, and at different times, students will choose not to use it, finding that mental calculations and inferences from known data are quicker or more appropriate. Those who do choose to rely on the calculator for longer periods must be made to feel that their decision is fully supported by the teacher. One of the advantages of using calculators is the growth of more positive attitudes towards mathematics and every effort should be made to maintain this.
ACTIVITY 2Learning about the Algorithms
The calculator can be used to help students think about and understand more about the written algorithms they are learning. These investigations aim to help students understand the place value concepts which are involved and also the inverse and other relationships between the operations. When the students are freed from the actual paper and pencil computations, they have the opportunity to think about the rules they are following. They should be asked to explain the steps of the algorithms so that the teacher can clearly see where any misconceptions lie.
Activities can begin with the students just keying in numbers which are presented orally or on written sheets. The object here is to promote quick and accurate entry skills, while improving visual memory or listening skills depending on how the numbers are presented. Students should be progressively trained to memorise entire numbers (up to 4 digits) or to remember them in convenient parts (e.g. all of the thousands) as the numbers get larger. This makes them concentrate on the place values of the digits, particularly if the ‘teen’ numbers or ‘zeros’ are involved.
As an alternative, the students can be given a number of computations to check, circling any wrong answers and writing down what the answer should have been.
Addition on the calculator goes hand in hand with estimation exercises. The students could be given several addition examples and be asked, through estimation, to match them up with answers which are given as a mixed group.
e.g. / 348+ 194 / 517
+ 308 / 527
+ 384 / 416
+ 189 / 255
+ 607 / 265
+ 316
ANSWERS: / 911 605 542 581 862 825
Students can also be given the opportunity to play ‘Addition Bingo’ or bingo organised around any of the operations. In these situations they are provided with a grid on which numbers have been written. These numbers will be the answers to addition sums involving two or three addends from a given set. The students have to estimate which addends have been used to make up the numbers on a particular line of the grid, then verify that estimation with the calculator. Conversely, they could be given a time limit to go for a ‘full house’, but that should be limited to grids where numbers are made up of only two addends.
A grid of numbers using two addends only would be:
Addition Bingo / Grid for Answers61 / 137 / 64 / 72 / 111 / 46
+15
97 / 112 / 91 / 99 / 106
130 / 55 / 51 / 45 / 118
67 / 133 / 103 / 152 / 84
90 / 36 / 121 / 101 / 96
Addends required for solution are:3621468263
751555997
A grid which uses a mixture of two and three addends to make up the numbers, and is consequently more difficult, would be:
Addition Bingo / Grid for Answers184 / 95 / 108 / 114 / 109
153 / 55 / 126 / 165 / 123
104 / 88 / 140 / 84 / 129
77 / 171 / 60 / 26 / 43
82 / 35 / 93 / 194 / 162
Addends required for solution are:2719416885
7293101636
The students should also be given partially completed algorithms and be required to fill in the missing numbers. In some instances it may be possible to have more than one correct answer and students should have experience with these types also.
2 * 6+ 5 1 *
764 / 3 7 9
+ * * *
5 1 4 / * 8 8
+ 3 9 *
9 * 2
2 *
* 9
3 *
1 0 8 / * * 9
4 1 *
* 2 6
2 3 *
1 6 2 0
The strategies students use should be discussed because there are many which apply in the cases above. One strategy would involve the students estimating which digits would provide the required results, using the calculator to test those, and adjusting where necessary. Another strategy for some examples would be to add up all of the addends which are visible, taking careful note of the relative place values, subtracting that number from the total and distributing the difference to those blank parts in the original example.
These activities also allow students to experience the relationship between addition and subtraction, with many addition examples being solved by subtraction.
Other addition games can be played on the calculator including that of ‘Target Numbers’. A target number, perhaps 31 is set up and the students, in pairs, can add on, beginning at zero, any of the single digit counting numbers (1 to 9). Once a target number is set, the students may grow to realise that there are strategies which help a player to win. When the target is known, there becomes a crucial number to reach so that a player should not lose. If 31 is the target and only single digit numbers are being added on, 21 becomes the crucial number. The first player to reach 21 (and who realises the strategy) will win because whatever single digit the opponent adds on, the next player can always reach 31 in the next turn.
For variety, change the digits which are to be added on (say 1 to 5 only). The strategy and the crucial number will change accordingly.
Partially completed subtraction algorithms will also provide students with opportunities to think about the process of subtraction. With some of the examples below, the students will have to use addition to provide the missing digits, exposing them to the inverse relationships of the two operations.
e.g. / 4 2 6– 1 3 7
* * * / 6 3 9
– 1 * *
4 7 6 / 4 * *
+ 1 5 4
2 9 2
5 * 7
2 9 *
2 3 4 / * * *
– 2 8 4
2 6 / 6 5 9
– * * *
1 7 3
Larger numbers can be introduced as the students progress, but care should be taken not to exceed their capabilities. They are far better off spending a little more time with numbers that they can be successful with, than progressing too fast. Many students should be challenged to mentally solve activities such as the ones above, using the calculator only to verify their answers. If mistakes are found, the original answers should be adjusted and then checked again using the calculator.
As an alternative to using ‘Bingo’ with subtraction, the students could become involved with an estimation exercise called ‘Target Subtraction’.
Once again, a grid of numbers is supplied and the students are put into teams (or pairs). The object of the game is to pick two numbers from the grid whose difference is close to the target number (say 50). Once each side has chosen a pair of numbers (using mental estimation), the calculator is used to determine whose choice is closest to the target.
The grid could be:
82 / 583 / 105 / 270 / 943 / 625149 / 196 / 409 / 562 / 311 / 732
629 / 1000 / 908 / 441 / 871 / 146
350 / 87 / 825 / 144 / 793 / 399
28 / 386 / 394 / 281 / 492 / 117
Targets such as 50; 100; 150 would be reasonable numbers for students of this age who are still building up place value ideas. For groups of students finding even those differences difficult, numbers which are smaller should be used.
To further reinforce the inverse relationship between addition and subtraction, addition grids which require both operations to be applied could be set up.
e.g. / Complete all of the missing parts on this table.Use the calculator.
+ / 83 / 119 / 382
29 / 112
474
148 / 223
437
314
As another activity, the students could be asked to pick out the examples requiring subtraction for solution, from a given page of story problems (say 10 in all). This helps the teacher highlight those students who are having difficulty understanding the concept of subtraction and its applications. Once the subtraction examples have been successfully isolated, the calculator should be used to solve them. Similar investigations can be carried out with the other operations at appropriate times.
Multiplication facts are easily generated, practised and reinforced using the calculator. These types of activities will be discussed within a later section. The first sets of activities here aim at helping students to understand the concept of multiplication, to improve their ability to estimate with multiplication and to better understand the written algorithms for multiplication of whole numbers.
The students can be given various numbers and be asked to write as many sets of factors as their explorations reveal.
e.g. / Given 168Possible sets include / 84 x 2
8 x 3 x 7
6 x 14 x 2
4 x 6 x 7 / and so on
The students could also be given a grid of factors and be asked to circle sets which multiply together to make a given number.
i.e.Circle sets which would make up 96.
[One such set would be 4 x 6 x 4].
16 / X / 2 / X / 3 / X / 8 / X / 4X / 4 / X / 6 / X / 4 / X / 24 / x
6 / X / 2 / X / 8 / X / 5 / X / 3
X / 24 / X / 4 / X / 4 / X / 2 / x
2 / X / 48 / X / 4 / X / 6 / X / 32
X / 6 / X / 12 / X / 8 / X / 3 / X
12 / X / 2 / X / 3 / X / 16 / X / 2
X / 16 / X / 5 / X / 3 / X / 32 / X
8 / x / 3 / X / 4 / X / 4 / X / 6
The calculator should be used as a validating tool after students have made decisions about a set of factors.
The multiplication algorithm can be studied using the calculator. These activities enable students to observe the relationships between multiplication and the operations of addition and division. Only single digit multipliers are used in this study and investigations using larger multipliers occur in later programs.
(a) / 4 2x *
2 9 4
This type of example can be solved using either addition or division, and the students should experience both methods.
i.e. / Key / Display reads
4 2 += / 42
= / 84
= / 126
= / 168
= / 210
= / 252
= / 294
The students can observe that seven lots of ‘42’ were needed to make the required number.
Using their basic tables as a guide, the students can discover that the example could also be solved using division.
(b) / * 3
x 5
3 1 * / 2
x *
4 3 2 / * 8
x *
1 7 4
With examples such as this, the teacher should draw the students attention back to the basic multiplication facts to observe patterns. Through observation and investigation, the students should grow to realise that particular inferences can be made about the ‘ones’ digits in multiplication examples. For example, in number 3 above, only two digits will multiply ‘8’ to give a result of ‘4’ in the ones place. They are 3 and 8 (which results in 24 and 64). So, the students can try each and inspect the results. They will have to make a decision about which digit to try in the ‘tens’ place. Once they choose a number and test it, their next decision should be based on the amount of error in the first.
The students must be encouraged to try numbers and not be threatened if they make mistakes. The point is that we must try to help them know how to learn from those mistakes. Once they learn to make adjustments which move them closer to the correct solution, we know that their ‘feel’ for number and relationships is improving. Therefore, these types of exercises and investigations are very useful for the teacher to observe how well the students are understanding and progressing in number study.
In Year four, students will be beginning to memorise division facts, but their work with the written algorithm may not yet have begun. Consequently, any division examples which are beyond their knowledge of basic facts will be solved using the calculator or some other means devised by the teacher or students.
The exercises which follow aim at helping students to gain proficiency with the use of the calculator in division operations, to better understand the inverse relationships between multiplication and division and to give experiences which involve estimation with division.
The students can be given a 5 x 5 grid and be asked to write in the numbers 1 to 25 in any order. The teacher then calls out division examples which give an answer in the range from 1 to 25. As the students hear each one, they work out the answer on the calculator and then cross out the appropriate square.
i.e.One way to write the numbers would be:
16 / 18 / 3 / 13 / 71 / 21 / 12 / 9 / 24
11 / 4 / 14 / 19 / 15
2 / 5 / 20 / 25 / 23
10 / 22 / 6 / 17 / 8
Examples to be called out could be:
378 ÷18 / 340 ÷20 / 161 ÷23
208 ÷16 / 44 ÷44 / 104 ÷13
144 ÷16 / 253 ÷23 / 322 ÷14
345 ÷23 / 570 ÷57 / 525 ÷25
81 ÷27 / 456 ÷24 / 105 ÷21
The first student to fill a line (horizontal, vertical, diagonal) wins.
More may need to be called out to achieve a winner.
The students can be asked to work out division examples themselves to match the numbers on the grid. They can then take turns in calling out the examples for the remainder of the class. The numbers on the grid can also be changed, resulting in the need for a new set of examples for solution. For example, the grid could use the first 25 prime numbers or the set of numbers from 51 to 75. In the quest for creating appropriate division examples, the teacher and students should be discussing ways of making up those examples, i.e.