Year 7 Calculator Program
Year 7 Calculator Program
Calculators are used in the program of activities outlined on the following pages based on a number of assumptions:
Ø calculating is just one process amongst many which students should master during the primary years;
Ø too much time is normally spent in practising computational skills which the great majority of students have already mastered;
Ø students sho have not mastered the computational algorithms would benefit greatly from the algorithm study incorporated in this program;
Ø calculators enable students to think more carefully about the task at hand because they remove the burden of computing;
Ø using calculators enables teachers to present students with a wider variety of situations which involve numbers;
Ø all students will need to be taught how to use the calculator to maximum efficiency through a series of introductory activities.
The year seven student show use this program will need to gain facility with the following keys:
On / off keys
Clear keys (C / CE)
Operation keys (+ - X ¸)
Memory keys (M+ M- CM RM)
Special keys (% √ )
In conjunction with the calculator manual, students should be placed in a variety of ‘what happens if you . . .?’ situations to explore the capabilities of their calculators
e.g. / · If you press two operations keys in succession . . . ?· If your calculation overflows the display space . . . ?
· Does the memory clear if you switch the machine off?
· When you key in an operation, what happens if you keep pressing the equals key?
· What does ‘E” mean on the display? How do you get rid of it?
The following investigations cover an entire years activities and need to be subdivided and sequenced in relation to the students’ needs and abilities. A wide variety of sources need to be ‘tapped’ to supplement these suggested activities, and some of these are suggested in the term plan at the rear.
ACTIVITY 1 Understanding the algorithms
Despite the use of calculators to remove the burden and the time factor relating to pencil and paper computations, students still do need to display competent and accurate use of them. However, when the purpose of a lesson is not the practice of one of those algorithms, the calculator should be used. For example, if the students are practising a particular formula (such as the perimeter of a rectangle), use of the calculator allows students to focus their full attention to the application of that formula, allows many more to be practised in a much shorter time and allows the slower computers to compete with the better ones. The ability to think quickly becomes the main factor involved.
Another useful activity for the students is actually thinking about the algorithms themselves. By supplying only part of an algorithm and requiring the students to complete it, we place students in the position of really having to consider how each algorithm works.
i.e. Supply the missing numbers. (Check for more than one solution)ADDITION / 1. / 3 4 8 / 2. / 3 * 4 / 3. / 1 8 *
+ * * * / + 1 3 * / 2 0 9 8
6 2 4 / * * * / + * 5 *
9 1 5 / 3 6 4
3 * 0 9
MULTIPLICATION / 1. / 6 7 3 / 2. / * * *
X 4 5 / X * * *
* * * * / 3 8 8 8
2 6 9 2 0 / 4 3 2 0
* * * * * / * * * *
Examples can be prepared in various stages of completion for the other algorithms as well. The degree of complexity will depend on the capabilities of the students.
ACTIVITY 2 Estimate then validate
Estimating helps students gain a better ‘feel’ for number particularly if the quality of the estimate can be checked immediately and with very little effort. The calculator is most helpful in this latter aspect relating to computations with number.
Students should be given experiences using the four operations where they estimate answers then check using the calculators. These activities should also give students the opportunity to try various ideas on the calculator and inspect and perhaps discuss the results.
Teachers should discuss tolerances either side of the actual answer, building up a respect for the degree of accuracy demanded by each individual situation.
For example, shopping experiences demand an estimate which should always be on the higher side of the actual answer. On many other occasions the positive or negative aspect of the estimate does not matter.
Examples should involve numbers, money and metric measures taken (where practicable) from newspapers (advertising etc) and from situations the students can easily identify with. Some of the types of activities which might be attempted are outlined below.
1. You are allowed to use the following keys: 5; +; = .
You are able to press the keys as often as you like and in any order.
Make the calculator display show 100.
See if you can do it in 15 presses or less.
CHALLENGE
What numbers can you make if you only press the keys 7 times?
2. John and Joy managed to obtain jobs during the school holidays.
John worked for 15 days. On the first day, he was paid one cent. Each day after that, he was paid twice as much as the day before. / DAY / JOHN / JOY / Joy worked for 15 days. On the first day, she was paid $5 and on each day after that, she was paid $2 more than the day before.1
2
3
4
5
6
7
8
9
10
11
12
13
14
15 / 1c
2c
4c
8c / $5
$7
$9
$11
Who do you think earned the most money?
Copy the table and complete the information.
3. For all of the questions below, use only the digits 1, 2, 3 and 4.
[ ] [ ] / + / [ ] [ ] / The answer is 37.
[ ] [ ] / + / [ ] [ ] / The answer is the smallest possible.
[ ] [ ] / - / [ ] [ ] / The answer is 7.
[ ] [ ] / X / [ ] [ ] / The answer is 943.
[ ] [ ] / X / [ ] [ ] / The answer is the largest possible.
[ ] [ ] [ ] / X / [ ] / The answer is 642.
[ ] [ ] / - / [ ] [ ] / The answer is the smallest possible.
[ ] [ ] [ ] / ¸ / [ ] / The answer is the largest possible.
4. Find a number so that when it is multiplied by itself, the result is 8.
i.e. [ ] x [ ] = 8
Record each trial and the result in a table:
2 x 2 / 4 – too small
A square has an area of 30 square centimetres.
What is the length of each side?
5. Estimate the number required to complete the statements below. Use the calculator to verify your try. If it does not fall within the prescribed limits, then adjust your answer and check again.
STATEMENT / LIMITS FOR THE ANSWER. . . x 34 / Between 60 and 70
73 + . . . / Between 150 and 160
976 - . . . / Between 650 and 700
. . . x 56 / Between 200 and 250
100 - . . . / Between 30 and 40
. . . + 751 / Between 1000 and 1100
. . . - 72 / Between 100 and 110
6. This exercise examines estimates which are both less than and greater than the actual answer. The students are required to round up and also to round down and to calculate these estimates mentally. The calculator is then used to compute the exact result.
3 X 7.6 / 21 / 24
4 X 1.7
5 X 9.3
6.2 X 12
4.1 X 5.2
6.3 X 8.8
9.9 X 5.6
7 X 3.25
5 X 4.75
0.8 X 1.6
0.2 X 8.8
0.3 X 10.5
7. This example asks the students to rapidly round numbers and operate on them. Up to 20 examples are given in a relatively short period of time, compelling the students to calculate mentally an approximation of the answer.
This skill then becomes one they will use to determine the reasonableness of answers which are worked on the calculator in other situations.
Do not spend too much time on each example.
When finished, use the calculator to check how well you estimated.
(a) 528 + 394 / (b) 29 x 11 / (c) 486 ¸ 7
(d) 4438 - 2293 / (e) 431 x 29 / (f) 6245 + 1923 + 238
(g) 3718 - 2494 / (h) 2249 ¸ 19 / (i) 108 x 48
(j) 8293 - 5814
ACTIVITY 3 Plane Shapes (Perimeters)
The study of perimeters is enhanced through the students actually being involved in the estimation and measuring of the distances around shapes in the environment. The role of the calculator is that of storing information and calculation. As the students determine a distance, the number is entered into the machine and added to the other distances as they are found. This reinforces the idea that perimeter is a linear measure of the distance around a polygon.
(a) Perimeters of Rectangles (including squares)
· Practise adding up lengths and breadths to find perimeters, (e.g. L + L + B + B). Check other methods such as (L + B) X 2 and 2L + 2B. Use of memory may be needed for the latter.
· Investigate the relationship between lengths, breadths and perimeter. What happens if the length increases by 1 as the breadth remains constant? What happens if the length increases by 1 and the breadth decreases by 1? How does the shape of the figure change? Increase both length and breadth and what happens to the perimeter and the shape? (Geogoards might be used to demonstrate).
· Investigate reversals of formulae. Using perimeters of squares, divide by 4 to obtain the side. Using perimeters of rectangles and one of the sides, reverse the formula to obtain the other side. The students’ methods should be discussed, though most will use either of the following:
(i) Enter perimeter; divide by 2; subtract length; or
(ii) Enter perimeter; subtract length twice; divide by 2.
(b) Circumferences of Circles
· Investigate the relationship between the diameter and the circumference to reach a generalised notion and value of pi. During estimation exercises and even during the introductory written and calculator exercises, pi should assume an approximate value of 3. As the students begin to understand better the concepts associated with pi, a more accurate approximation of 3.14 can be assigned. It may be interesting to some students to ascertain a value of pi to 5 or 6 decimal places for some of the calculator examples.
· Using the calculator, estimates of circle circumference can be checked rapidly. Estimations should result in real thought being given to the idea that circumference is approximately three times the diameter. The students should be given ample opportunity to check this by measuring. Too often, the students are moved too rapidly to the abstract pencil and paper notions involving pi, diameters, circumferences, etc.
· Obtaining the diameter from the circumference should follow, not by the manipulation of formula, but from thinking about the relationships involved. All of the above work with circles can take place without a written formula if the students have had opportunity to think about the relationships between the measures. In this study, the actual formulae come right at the end of the investigations, well after the students have been manipulating the data through estimations and on their calculators. In the past, formulae have been introduced first, in the hope that understanding will eventually come. For many students, that approach is too abstract and they have no sound base upon which to construct their thinking.
· Additionally, the formal setting out of these examples does not go hand in hand with the investigations outlined above. If some value is seen in this formal work, then some other time should be set aside to concentrate on that aspect. The activities above aim at letting students have rapid practice in the estimation and validation of their thoughts relating to circumferences of circles.
(c) The students should use the calculator to compute rapidly the perimeters (or circumferences) of a wide range of plane shapes, even irregular shapes where lengths of sides are given.
(d) Once the above are handled proficiently by students, perimeters of combined figures can be attempted. Strategies for finding these perimeters should be discussed by the students themselves in groups and then as a class group.
Above all, the exercises should help to reinforce the idea that perimeter refers to the length of the boundary.
ACTIVITY 4 Plane Figures (Area)
(a) Areas of Rectangles (including squares)
§ After the formula has been generalised and understood, students can investigate the relationship between area and shape. Given a constant area, what different shaped rectangles can be drawn?
§ Investigate reversals of formula. Given the area and one side (or just the area in the case of squares) find the missing dimension.
§ Combine the study of areas of rectangles with a study of multiplying and dividing by decimals. For example predict what happens when you multiply by 0.8
§ Using rectangles and the diagonal, the area of triangles should be investigated. Develop a rule from this study.
(b) Areas of Circles
§ Establish the general idea of formula for area of circles through an established method such as the sum of the areas of many small triangles.
§ Practise application of the formula with the calculator particularly using circular shapes the students can estimate with firstly (using pi = 3).
§ Develop a pattern of areas using whole number radii. See if students can detect that the area is directly related to the square of the radius. This pattern may be more apparent if the more general approximation of pi (equals 3) is used.