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MATHEMATICS
SECONDARY I - IV
ALBUM
Created by Darin M. Bicknell 2013
Montessori Observation
Big Picture Montessori
This work is licensed under a Creative Commons Attribution Non Commercial - Share Alike 3.0 Unported License.
Math Secondary I - IV
Contents
1.
Overview
1.1.
1.2.
1.3.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
Aims
The Elementary Mathematics Curriculum*
Presentation: The Great Story of Mathematics Continues
Number, Set Notation and Language
Squares and Cubes
Directed Numbers
Vulgar and Decimal Fractions and Percentages
Ordering
Standard Form
The Four Rules
Estimation
Limits of Accuracy
Ratio, Proportion, Rate
Percentages
Use of an Electronic Calculator
Measures
Time
Money
Personal and Household Finance
Graphs in Practical Situations
Graphs of Functions
Straight Line Graphs
Algebraic Representation & Formulae
Algebraic Manipulation
Functions
Indices
Solutions of Equations & Inequalities
Linear Programming
Geometrical Terms & Relationships
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
Geometrical Constructions
Geometrical Constructions
Symmetry
Angle Properties
Locus
Mensuration
Trigonometry
Statistics
Probability
Vectors in Two Dimensions
Matrices
Transformations
1. Overview
1.1.
Aims
The aims of the curriculum are the same for all candidates. The aims are set out
below and describe the educational purposes of a course in Mathematics. They
are not listed in order of priority. The aims are to enable candidates to:
o Develop their mathematical knowledge and oral, written and practical
skills in a way which encourages confidence and provides satisfaction and
enjoyment;
o Read mathematics, and write and talk about the subject in a variety of
ways;
o Develop a feel for number, carry out calculations and understand the
significance of the results obtained;
o Apply mathematics in everyday situations and develop an understanding
of the part which mathematics plays in the world around them;
o Solve problems, present the solutions clearly, check and interpret the
results;
o Develop an understanding of mathematical principles;
o Recognise when and how a situation may be represented mathematically,
identify and interpret relevant factors and, where necessary, select an
appropriate mathematical method to solve the problem;
o Use mathematics as a means of communication with emphasis on the use
of clear expression;
o Develop an ability to apply mathematics in other subjects, particularly
science and technology;
o Develop the abilities to reason logically, to classify, to generalise and to
prove;
o Appreciate patterns and relationships in mathematics;
o Produce and appreciate imaginative and creative work arising from
mathematical ideas;
o Develop their mathematical abilities by considering problems and
conducting individual and co-operative enquiry and experiment, including
extended pieces of work of a practical and investigative kind;
o Appreciate the interdependence of different branches of mathematics;
o Acquire a foundation appropriate to their further study of mathematics
and of other disciplines.
1.2.
The Elementary Mathematics Curriculum*
1.2.1.
1.2.2.
1.2.3.
1.2.4.
1.2.5.
1.2.6.
1.2.7.
1.2.8.
1.2.9.
1.2.10.
1.2.11.
1.2.12.
1.2.13.
1.2.14.
The Story of Numbers
Numeration
Multiplication
Division
Fractions
Decimal Fractions
Squaring and Cubing
Square Root and Cube Root
Powers of Numbers
Negative Numbers
Non-decimal Bases
Word Problems
Ratio and Proportion
Algebra
* The Elementary Math Album licensed under a Creative Commons Attribution Non Commercial - Share Alike 3.0 Unported License by Jonathan Feagle can be
found at http:/www.freemontessori.org
1.3
Presentation: The Great Story of Mathematics Continues
2.
Number, Set Notation and Language
2.1.
Natural numbers
Definition:
Numbers used for counting i.e., numbers beginning from 1 to infinity.
1, 2, 3, 4
Proficiency & Practice with Natural Numbers
Student uses BODMAS (BEDMAS)
Student practice putting numbers in order
Students practice rounding to the nearest ten, hundred, and thousand.
Student practice solving simple problems, using the four operations of
arithmetic: addition, subtraction, multiplication and division
Presentation 1: ORDER OF OPERATIONS
We have used natural numbers in many ways in the classroom. We often
use the numbers with operators like addition, subtraction, multiplication,
and division and at times we have also used two operators in one
problem. Like when we used them for binomial work (2+3) x 4.
Suppose you have to work out the answer to 4 + 5 × 2. What do you
think the correct answer is?
 The student may say the answer is 18, but the correct answer is 14.
There is an order of operations which you must follow when working out
calculations like this.
Multiplication × is always done before addition +.
In 6 + 5 × 3 this gives 6 + 15 = 21.
What would be the answer if you have to work out (4 + 3) × (7 –2).
Give the students the correct answer of 35.
What did you realise? Did you see additions and subtraction within the
brackets have to be done first, giving 7 × 5 = 35?
Have the students work on their own from questions the make up for
each one another.
Presentation 2: BODMAS (BEDMAS)
So, how do you work out a problem such as 64 ÷ 8 + 12 × 3?
To answer questions like this, there is the simple BODMAS (BEDMAS)
rule. This tells you the order in which you do the operations.
B
O
D
M
A
S
Brackets
Order
Division
Multiplication
Addition
Subtraction
B
E
D
M
A
S
Brackets
Exponents (powers)
Division
Multiplication
Addition
Subtraction
For example, to work out 64 ÷ 8 + 12 × 3:
First divide: 64 ÷ 8 = 8; giving 8 + 12 × 3
Then multiply: 12 × 3 = 36 giving 8 + 36
Then add: 8 + 36 = 44.
Give another problem 34 – 5 + 4 × 3 ÷ 2. Have students identify the
correct order.
First 12 ÷ 2 = 6; giving 34 – 5 + 4 x 6
Second 4 x 6 = 24; giving 34 – 5 + 24
Third 5 + 24 = 29; giving 34 – 29
Finally 34 – 9; giving 5
Hand the lesson off to the student(s) once you see they understand the
order and let them create their own equations.
Extension
From state, provincial, local or national exams provide questions with
answers in a timed fashion to allow the students to challenge their skills.
The teacher or students can mock up speed test cards with 10 questions
on one side and answers. The students should be provided with a stop
watch or egg timer to count down trying to improve their accuracy and
time taken to complete the test.
Presentation 3: Hierarchy of Numbers Revision
The teacher writes out the following numbers using the green, red, blue
and green for each hierarchy. Have the students arrange the numbers in
order from the smallest first to the largest.
8045
4085
48058504
8540
5840
Look at the thousands first and then each of the other hierarchies in turn.
The correct order is
8504, 8504, 8045, 5840, 4805, and 4085.
Presentation 4: Hierarchy of Numbers Revision
Write each of the following numbers in natural numbers using the colour
code for the hierarchies, green for units, blue for tens, and red for
hundreds.
Eight million, two hundred thousand, fifty-eight
Nine million, four hundred and six thousand, one hundred, seven
One million, five hundred, two
Two million, seventy-six thousand, forty
Presentation 5: Approximation, Rounding with Natural Numbers
Prepare newspaper articles with estimates of sizes, also include
statements about your approximate time to travel to the school written
down as well as getting directions for cooking, baking times from a box of
frozen food or pre-packaged food.
“It takes me about 15 minutes to get to school without traffic and if
it is raining it could take me as much as 30 minutes or more.”
“Cook in the microwave for 5 minutes on high”
“Boil hot water and pour it into in the container of noodles for 5
minutes or until soft.”
“The demonstration in Jakarta had over 5000 people.”
We use rounded information all the time. Look at these examples. All of
these statements use rounded information. Each actual figure is either
above or below the approximation shown here.
When rounding is done correctly, you can find out what the maximum
and the minimum. For example, if you know that the amount of time
people demonstration in Jakarta is rounded to the nearest 1000.
The smallest figure to be rounded up to 5000 is 4500, and the largest
figure to be rounded down to 5000 is 5400 (because 5500 would be
rounded up to 6000).
So there could actually be anywhere from 4500 to 5400 people at the
demonstration.
Examples:
Round each of these numbers to the nearest 10.
57
34
12
21
58
107
20
60
110
Answers rounded to the nearest 10.
60
30
10
Give these cooking times to the nearest 5 minutes. Note that smallest
figure to be rounded up to 5 minutes is 2 minutes and 30 seconds, and
the largest figure to be rounded down to 5 minutes is 7 minutes and 29
seconds (because 7 minutes and 30 seconds would be rounded up to 10
minutes).
34 minutes, 57 minutes,
13 minutes,
7 minutes and 50 seconds
Answers rounded to the nearest 5 minutes.
35 minutes,
55 minutes,
15 minutes,
10 minutes
Extension
From state, provincial, local or national exams provide questions with
answers in a timed fashion to allow the students to challenge their skills.
The teacher or students can mock up speed test cards with 10 questions
on one side and answers. The students should be provided with a stop
watch or egg timer to count down trying to improve their accuracy and
time taken to complete the test.
2.2.
Integers
Definition:
Integers are whole numbers i.e., they can be positive negative or zero.
-3, -2, -1, 0, 1, 2, 3
Positive integers: 1, 2, 3, 4
Negative Integers: -1, -2, -3, -4
Proficiency & Practice with Integers
Students will use Integers in real life applications
Students will explore the meaning of the Integer
Students will practice using Integers within inequalities
Students will practice how to do arithmetic with Integers
Presentation 1: The Integer Number Line
Prepare an Integer number line either as a single bead on a line that can be
slid left or right with the integers clearly marked behind the fishing line. Or
you can create from thick cardboard a vertical / horizontal number line with
beads placed in holes.
We have presented the Integer number line material before so this is a
review for some and maybe new to others.
Look at the number line.
Notice that the negative numbers are to the left of 0, and the positive
numbers are to the right of 0.
Numbers to the right of any number on the number line are always bigger
than that number.
Numbers to the left of any number on the number line are always smaller
than that number.
–3
Negative
–2
–1
0
1
2
3
Positive
The Integers can also be represented vertically on a number line.
Look at the number line.
Notice that the negative numbers are below of 0, and the positive numbers
are above 0.
Numbers above any number on the number line are always bigger than that
number.
Numbers below any number on the number line are always smaller than that
number.
5
Positive
4
3
2
1
0
-1
-2
-3
-4
-5
Negative
2.3.
Prime numbers
A number which has two unique factors i.e., 1 and the number itself.
2, 3, 5, 7, 11, 13, 17
e.g., 3 = 3 × 1, 7= 1 × 7.
Note that: 1 is not a prime number as it has two factors but both are same.
2.4.
Square numbers
12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36, 72 = 49
These number obtained by squaring integers are called square numbers.
Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81
2.5
Factors
Factor is a whole number which divides exactly into a whole number,
leaving no remainder.
e.g., 1)
2)
Factors of 4
1×4=4
2 × 2 = 4,
so 1, 2, 4 are factors of 4.
Factors of 12
1 × 12 = 12
3 × 6 = 12
3 × 4 = 12, so 1, 2, 3, 4, 6, 12 are factors of 12.
2.6
Prime Factors
Factors that are Prime.
e.g., Prime Factors of 252: 22 × 32 × 7
2.7
Common Factors
Common factors of numbers.
e.g., 1)
2.8
Common factor of 15 and 12
Factors of 12 = 1, 2, 3, 4, 6, 12
Factors of 18 = 1, 2, 3, 6, 9, 18
Common Factor of 12 & 18 are 1, 2, 3, and 6
Highest common Factor: 6
Multiples
A multiple of a number is what you get when you multiply that number
by some other whole number.
e.g., 1) Multiplying 3 by 5 gives 15 so 15 is a multiple of 3.
2) Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, and 81
2.9
Common Multiple
Common multiples of numbers.
e.g., 1)
2.10
Common multiples of 2 and 4
Multiples of 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24
Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40
Common multiples of 2 & 4 are 4, 8, 12, 16, 18, 20, and 24
Lowest Common Multiple: 4
Rational numbers
A rational number is a number which can be expressed in the form of p/q
where p and q are whole numbers.
Recurring decimals are rational. e.g., 0.666666666 i.e., 0.6'
Terminating decimals (that end) are rational. e.g., 0.8896 is a rational
number.
More Examples: -3/5, -2, 6/9, 12, 0.9', Square root of 4/9 = 2/3
2.11
Irrational numbers
An irrational number is a number which cannot be represented in the
form of p/q where p and q are integers.
2.12
Real numbers
All the numbers that can be represented on number line are called Real
numbers. i.e., all the integers, rational and irrational numbers.
2.13
Sequences
A set of numbers which follows a certain pattern are said to be in
sequence.
e.g., 2, 4, 6, 8, 10, and 12. This is a sequence of numbers which have a
common difference of 2 between every consecutive number.
Each number of a sequence is said to be the term i.e., term 1(t1) = 2
t1 = 2(1) = 2
t2 = 2(2) = 4
t3 = 2(3) = 6
So, tn = 2(n) is the formula for nth term.
Different types of sequence:
Sequence with same difference between terms:
The nth term (tn) of sequences with same common difference is:
tn = an + b; Where a = common difference, b = preceding number
of the sequence
e.g., find the formula of nth term for the sequence 9, 16, 23, 30.
Here common difference (t2 – t1 = t3 – t2 = t4 – t3) a =7
Preceding number of the first term of sequence i.e., (9 – 7), b = 2,
Hence formula for nth term is tn = 7n + 2.
Sequence with consecutive terms having same ratio between them
e.g., 1, 4, 16, 64, 256.
In this sequence each term is four times the previous term.
As, t1 = 1,
t2 = 4,
So, t2 = 4 (1) gives t2 = 4 t1
Which can now be used to define formula for nth term:
tn = 4 t(n-1) , since the nth term is four times the (n-1)th term.
Known Sequences:
Prime Numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23
(In this sequence you won't find any specific difference between
the consecutive terms)
Square Numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81
(Here it can be seen that there is a difference of 2 increasing with
every consecutive term)
Triangular Numbers: 1, 3, 6, 10, 15
(Here the difference between the terms increases by 1 each time)
Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 is the Fibonacci sequence.
(Here the first two terms are both 1 and then every consecutive
term is obtained by adding the two previous terms)
2.14
Sets
A set is a collection of well-defined objects.
Defining a set:
E.g.
i) Number of Montessori schools in Indonesia
ii) Number of the best schools in Indonesia
Statement 1 is a set: Number of Montessori schools can be
listed easily so it is well-defined.
Statement 2 is not a set: This statement is not well defined
because it can be argued, what is meant by "best schools", it
open to interpretation.
These well-defined objects are often called as "elements" of the set.
Also a set is represented by listing all these elements within ‘curly’
brackets or braces { }.
E.g., A set of vowels can be written as {a, e, i, o, u}.
Other form of set representation:
{x : x is a vowel of English alphabets},
Read as a set consisting of all elements of x, such that x is a vowel of the
English alphabet.
In this representation, "x" stands for all the elements that can be written
according to the definition of the set which here is the vowel from the
English alphabet.
Usually a set is identified by naming it with an alphabet written in capital.
Some Important Definitions:
A = {x : x is a natural number} = {1, 2, 3, 4, 5, 6.........}
B = {x : x is an even number } = {2, 4, 6, 8, 10 ..........}
C = {x : x is a prime number } = {2, 3, 5, 7, 11 ..........}
D = {x : 2 < x < 8} = {3,4,5,6,7}
E = {(x,y): y=2x + 1, x and y are natural number} = {(1,3), (2,5), (3,7)...}
Method:
As x and y are natural numbers
Taking x=1, substituting it in the formula y = 2x + 1 gives
Y = 2(1) + 1 = 3 and hence the first element is (1,3),
Similar can be done to find out taking other natural numbers as
value of x or y.
2.15
Venn Diagram
A Venn Diagram is another easy way to represent sets.
A Venn Diagram is a large rectangle with circles or ovals inside to
represent different sets and their properties.
In a Venn Diagram, the circles may or may not overlap depending on the
similarity of elements in those sets. i.e.,
If the sets have some of the elements same then the circles overlap and
the same elements are written within that overlapped part. As shown in
the figure the three circles are the sets and the overlapping parts show
the common elements in them.
A Universal set is the set
consisting of all the other sub-sets. As shown in figure,
the set
consisting of all elements of the sets A, B and C.
If an element is not in the given sets represented as a circle then it is lies
inside the rectangle but outside the circles representing the sets.
Set Notation:
E.g; Let set A be set of Prime numbers less than 11
i.e., A = {2, 3, 5, 7}
Venn Diagram for A':