Number Systems

4.5 NUMBER SYSTEMS

NUMBER SYSTEMS LESSON IDEA 1

TITLE: COUNTDOWN ­ ADAPTED FROM THE TV PROGRAMME OF THE SAME NAME!

TOPIC: BASIC ARITHMETIC WITH NATURAL NUMBERS

AIM: Practice in +, ­, ×,÷, and brackets. (This provides good numeracy practice without a calculator but could be used also to reinforce the importance of order of operations with a calculator.)

RESOURCES: Make out fourteen cards as shown (or simply write the numbers onto the board). The top row contains four large numbers and the other three rows contain the numbers from 1 to 10.

METHOD:

1. If one is using cards, mix them up and turn them face down (keeping the four large numbers in the top row).
2. Pick six numbers at random from the cards (one from the top and five from anywhere else).
3. Pick a three-digit target number at random.
4. By adding, subtracting, multiplying, dividing some or all of the six chosen numbers, try to reach the target number, or to get as close as possible. Brackets can be used.

EXAMPLE : Numbers chosen: 100, 5, 7, 9, 4, 1

Target number: 654

Solution: 100×7-5×(9+1)+4=654

If students are weak at arithmetic, the "random numbers" could be pre-arranged!

EXAMPLE: Numbers chosen: 100, 5, 2, 3, 7, 1

Target number: 512

Solution: (100×5)+2+3+7=512

CLASSROOM MANAGEMENT IMPLICATIONS: The students can work in teams or individually, and a time limit can be imposed (e.g. 60 seconds), as appropriate to the class.

NOTE : This activity can be incorporated into the teaching of natural numbers, or it can be used as light relief after completing a difficult topic.

NUMBER SYSTEMS LESSON IDEA 2

TITLE: NUMBER PUZZLE

TOPIC: BASIC ARITHMETIC WITH NATURAL NUMBERS

AIM: Practice in adding natural numbers in an entertaining way.

RESOURCES: The puzzle opposite can be photocopied or reproduced on overhead or blackboard.

METHOD:

Try to fill in the missing numbers. The missing numbers are integers between 0 and 9. Numbers in each row add up to the totals on the right of the row. Numbers in each column add up to the totals at the bottom of the columns. The diagonal lines (to top right and bottom right) also add up to the totals given.

Solution:

CLASSROOM MANAGEMENT IMPLICATIONS:

This activity can be done in pairs or in small teams of three or four in a quiz-like fashion. An interesting followup exercise could be to ask students to produce such a puzzle themselves. Some discussion may lead to the conclusion that the best way of producing such a puzzle is to prepare a full solution first and then to leave out certain numbers.

NUMBER SYSTEMS LESSON IDEA 3

TITLE: BUZZ GAME

TOPIC: NUMBERS

AIM: To revise and reinforce number patterns.

RESOURCES: None required.

METHOD: (Many variations)

1. The class is divided into small groups (say, less than 7 to a group). In each group, students in sequence rattle off the natural numbers. When a multiple of, say, 7 is met, the word "buzz" is used instead of the number and the order of students is reversed.
   For example: 1, 2, 3, 4, 5, 6, buzz (reverse), 8, 9, 10, 11, 12, 13, buzz (reverse), ...
2. The person who errs is "knocked out" and the next person starts a new sequence.

CLASSROOM MANAGEMENT IMPLICATIONS

None

NOTE

If it is not convenient to form small groups, the "reverse" move may be omitted. The game can also be used with multiples, divisors of numbers such as 48, primes, sequences, etc. and even combinations of these (with other "buzz" words). This is a useful mainly with First Years.

NUMBER SYSTEMS LESSON IDEA 4

TITLE: THE PLUS AND MINUS GAME

TOPIC: INTEGERS

AIM: To help students in the addition and subtraction of integers.

RESOURCES: None.

METHOD:

1. This activity helps to present the addition and subtraction of integers without having another set of rules to learn.
2. Consider the following example:
   Evaluate: -3 + 4 ­ 1 + 5 + 2
   The student creates a scoreboard and fills in the scores for the "+" team and the " - " team as follows:
3. The scores are then added for the two teams and the answer to the question arrived at by asking who wins and by how much.
   In this case the "+" team wins by 7 so the answer to the problem is +7.
   If the game is a draw, then no one wins and the answer is zero.

CLASSROOM MANAGEMENT IMPLICATIONS:

None

NOT E :

1. This method can be carried on to simplifying algebra, where a separate "plus and minus game" is played for each group of like terms.
2. One of the most frustrating things in dealing with integers is the ease with which the rules for adding and subtracting integers can be confused with those for multiplying and dividing integers. Many students look at -3 ­ 2 and, incorrectly, get +5 thinking "like signs give plus". It is important that students learn a full rule rather than a partial one: "when multiplying or dividing two integers, like signs give plus". Studying patterns like the following, may serve to situate the rule on firm ground:
   2 ×3 = 6
   2 ×2 = 4
   2 × 1 = 2
   2 ×0 = 0
   2 ×-1 = -2
   2 ×-2 = -4
3. Throughout the lesson ideas in these Guidelines, and in the proposed new-style examination papers, a distinction is drawn between the negation sign (-) and the symbol for the operation of subtraction (­); the latter is longer in appearance.

NUMBER SYSTEMS LESSON IDEA 5

TITLE: THE GOLDEN RATIO: AN INTRODUCTION TO RATIO AND PROPORTION FOR FIRST YEAR STUDENTS

TOPIC: RATIO AND PROPORTION

AIM: To help the students to develop an understanding of ratio and proportion.

RESOURCES:

The main resource is something very close to the students' hearts ­ their own bodies. Also required: metre sticks, calculators, basic geometry sets and space for the students to measure themselves!

INTRODUCTION:

Students will find from this practical work that nearly all of us are built to mathematical formulae! The underlying ratio is the Golden Ratio (or Golden Mean), approximately 1.6:1. The Greeks utilised this geometrical ratio in designing their buildings; it was also known to them as the Divine Ratio.

Students should be directed to research the Golden Ratio in other subject areas ­ they will find the Art department a great fund of information.

METHOD:

1. 1. Students should measure and note the distance afrom their navel to their toes and the distance b from their navel to the top of their head. They should find that, no matter what their height, the ratio of the two numbers will usually give a result very close to 1.6.
   
   For some young students the ratio will not be too near to 1.6 (it is not unusual to have a range 1.49 to 1.72 but in general the majority are within 1.55 to 1.67). This is easily explained as they are still growing and the ratio will be achieved in later years.
2. Proportion as a concept arises when the ratios are compared.
3. During this work there are excellent opportunities to debate matters like:
   (i) How to measure accurately using metres and centimetres.
   (ii) How to write the measurements in decimal form (one student recorded a measurement of length as "one metre fifteen inches"!).
   (iii) How to round up or down the calculator's over-accurate answer.
   (iv) What would be the result if one did the reverse division ?
4. 4 A very important concept will arise when students realise the nuance of using units, metres, centimetres and parts. The art class might come to the rescue where often, in figure drawing, the "head" is used as the unit of measurement, i.e. the onepart, and the body is then divided into different numbers of parts no matter what the length of the head is.
   
   The mystery of this ratio grows when students compare the ratio of length to width of their hand and head; it also conforms to the Golden Mean.

This exercise can be turned into a more abstract geometrical form by drawing a rectangular shape of sides in the ratio 1.6:1, say 8 cm : 5cm, enlarging this to 16 cm : 10 cm, and then examining the result of the ratio of any of the comparable lengths in between these limits. This drawing can be used to help students begin to understand the trigonometric ratio "tan" as the ratio of the lengths of two line segments.

Finally, a unique one-to-one ratio for a human being is given by the Vitruvian Man. This is Leonardo da Vinci's famous drawing that will be familiar to nearly all of the students from television's World in Action programme. If they measure their height and then the width of their outstretched arms, they will find the ratio of the two lengths is 1. It gives a whole new meaning to being called "a square"! For the more abstract thinkers the geometrical figure in the square can be related to

CLASSROOM MANAGEMENT IMPLICATIONS:

Students can work in pairs for the practical work.

NUMBER SYSTEMS LESSON IDEA 6

TITLE: I HAVE ... YOU HAVE

TOPIC: NUMBER SYSTEMS: NUMBER OPERATIONS

AIM: To give practice in mental arithmetic

RESOURCES: A set of cards of the form:

• I have [a number].
• Who has [some function of it]?

Examples might be:

1. I have 10; who has half that number?
2. I have 5; who has 40 more?
3. I have 45; who has the result of dividing this by 5?
4. I have 9; who has the square root of my number?

The cards may form a "cycle"; for the above set, the last card might read

• I have 50; who has one-fifth of this?

Alternatively, there can be a first card and a final card, the latter reading (in this case):

• I have 50; that is all.

There should be enough cards for each student in the class to have two or three.

The teacher may have a printed list of all the cards,in sequence.

METHOD:

1. The cards are shuffled and given out to the students. The teacher retains two successive cards in the "cycle" (orthe first and final cards). The students hold their cards in such a way that they can see the writing on all of them but so that their neighbours cannot.
2. The teacher starts by reading the second card of the two successive ones s/he holds (or the first card of the set): saying slowly and clearly (for example) "I have 10 [pause]; who has half that number?"
3. All students have to do the calculation to see if theyhold the next card. The student who has that card should then read it: "I have 5 [pause]; who has 40 more?" The student then places that card face downwards on his/her desk.
4. The game proceeds. If a student speaks out of turn as a result of doing a calculation incorrectly, hopefully the teacher and various students will notice and (for example) call out "No!" The student who holds the correct card should also be speaking up, reading the card. The teacher may feel more at ease if s/he has thecomplete list of cards in sequence, but the game isperhaps more fun for all concerned if the teacher is alsorelying on his/her ability to calculate mentally.
5. If all goes well, the game ends when the teacher hears his/her cue: the calculation that gives the first number (10, for the example described here) or the final one. At this point, all the cards should have been used, and so should be face downwards on the desks. If any student still holds a card, a mistake has been made.
6. The cards can be collected, shuffled, and dealt again. For the second round of the game, the cards might be read more quickly.

CLASSROOM MANAGEMENT IMPLICATIONS:

If each student has (say) three cards initially, most students remain "in the game" for a considerable period of time. Those who have played all their cards can stay involved by watching out for mistakes by others, but students are more likely to stay on task if they are in danger of "missing their cue".

NOTE :

1. The first time the game is played, a "trial run" may be needed to ensure that students know what to do (and to encourage them to read their cards slowly and clearly).
2. The exercise may not occupy a whole class. Especially when the game is familiar, it can be a "warm-up activity" or an end-of-lesson reward for good work (on Friday afternoon?) ­ not necessarily with the same set of cards each time.
3. For some weaker or more nervous students, arrangingthe cards in their correct sequence could provide practice without the fear of going wrong in front of their peers.
4. Sets of cards may be available commercially, but teachers can make their own, pitching them at an appropriate level for a particular class (and perhaps concentrating on particular number operations).
5. When the game is familiar, students might be asked tomake a set of cards (perhaps working in pairs, perhaps for homework).
6. With very weak students, the game might be played with calculators available ­ the emphasis being on "what do I do to get 'half'?", "what does '40 more' mean?", and so forth.

NUMBER SYSTEMS LESSON IDEA 7

TITLE: MENTAL MATHEMATICS

TOPIC: VARIOUS

AIM: Revision of material covered

RESOURCES: It is advisable to have questions previously prepared, but it is not essential. A sample sheet of fifteen questions is included below.

METHOD:

1. The class is split into groups (these could be based on rows, depending on how the classroom is organised).
2. These groups (teams) compete against each other, with 3 points awarded for a correct answer; a wrong answer gets passed onto the next team for a bonus of 2 points or to the next team after that for a bonus of 1 point.
3. The winning team has the carrot of no homework that night! The teacher can also throw in "open" questions answerable by the first person to raise a hand: correct answer 5 points, say; incorrect answer -5 points.
4. The idea should be to use one class period to deliver the questions, and to mark and score the results. It is unlikely that the teacher will get through more than twenty questions. Only essential information and items relevant to each question should be written on the board. Questions may have to be read out twice and the importance of listening emphasised.

SAMPLE WORKSHEET: MENTAL MATHEMATICS

1. What is two thirds of 60?
2. 16 ×0.2 = ?
3. Calculate 5.4 ­ 2.8
4. 5.7 divided by 0.3 = ?
5. Find I of 60
6. If I travel 10 km in 10 minutes, what is the average speed in km/h?
7. 80% of a number is equal to 64. What is the number?
8. Find the mean of 1, 2, 4 and 5.
9. What is one third of I of 8?
10. Calculate 29 + 14.8.
11. What is the last prime number before 50?
12. 20% of a number is 15. What is the number?
13. Two thirds of a number is 16. What is one quarter of this same number?
14. A car uses 5 litres of petrol to travel 80 km. How many litres are required to travel 320 km?
15. 8 people can build a wall in 15 days. How long would it take 5 people to build the same wall?

CLASSROOM MANAGEMENT IMPLICATIONS:

None (limit the excitement!)

NOTE:

1. Questions need to be simple enough to be done in the head yet broad enough to cover the material dealt with either on a topic by topic basis or covering a series of topics. This exercise is good with all groups in the junior cycle.
2. A variation of a quiz can occur when students make up questions that have a given answer. For example, if the answer is 7, what is the question? This simple exercise/game aims to make the student and teacher think in a different way and requires few or no resources. If the questions are numerical, a scoring system can be drawn up assigning for example: One point for using + or ­, two points for using ×or ÷, and three points for using .