Secondary Mathematics Content Page
Monday, 20 January 2020
S1T1 - Numbers, Power, Place Value
NUMBERS
Integers
Whole numbers including negative whole number
Example: …,-4, -3, -2, -1, 0, 1, 2, 3, 4,…
A positive number is any number greater than zero.
Example: 1, 2, ½, 6.4
Negative Numbers
We read -1 as negative one.
A negative number is any number less than zero.
Example: -1, -2.5, -4/7
Large Numbers
One Million : 1 000 000 [6 zeros, 1000 times of thousands]
One Billion: 1000 000 000 [9 zeros, 1000 times of millions]
Place Values
Digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are used to form numbers.
The value the digit is the position of the digit in the number
487
7 è Value of the digit 7 is 7
80 è Value of 8 is 80
400 è Value of 8 is 400
Index/Power
The small, raised number next to a normal letter or number, to be multiplied by itself
Example
b2= b x b
43 = 4 x 4 x 4
Square power 2 : 2
- multiplying by itself
Example
(1) Square of 4 = 42 = 4 x 4 = 16
(2) 32 = 3 x 3 = 9
(3) Square of (-5)2 = (-5) x (-5) = 25
Perfect Square are the squares of whole numbers:
(2 x 2) 4, (3 x 3) 9, (4 x 4) 16, …
Square Root - the Opposite of Square
A number that when multiplied by itself, gives the number
Symbol: √
5 --> square 52 --> 25
5 <-- Square root √25 <-- 25
Example
(1) 16 = 4 x 4
√16 = 4
(2) a2 = 25
a = + √5 x 5 = = +5
Why +5 when a number is square-root?
(-5) x (-5) = 25 ( -ve x -ve = + ve)
5 x 5 = 25
=> 25 = (-5)2 = 52 = 25
a2 = + √a x a = -a or a
Cube - Power of 3 : 3
- Multiplying the number by 3 times
Example:
(1) Cube of 3 = 33 = 3 x 3 x 3 = 27
(2) 43 = 4 x 4 x 4 = 64
(3) cube of (-5) 3 = (-5) x (-5) x (-5) = = -125
Cube Root
A value that when ‘cubed’ gives the original number
Symbol: 3√
4 --> cube 43 --> 64
4 <-- cube root 3√64 <-- 64
Example:
3√8 = 3√2 x 2 x 2 = 2
3√216 = 3√6 x 6 x 6 = 6
Prime Number
Greater than 1 that cannot be formed by multiplying 2 smaller natural number.
Example: 2, 3, 5, 7, …
2 = 1 x 2, 13 = 1 x 13
* 1 is not a prime number because prime numbers are greater than 1
* All even numbers are not prime number except 2.
Example
List all the prime number that are greater than 10 and less than 20
Step 1: List all the numbers
11, 12 , 13 , 14, 15, 16, 17, 18, 19
Step 2: Strike out the non-prime number
11, 12 , 13 , 14, 15, 16, 17, 18, 19
Step 3: List out the prime number
11, 13, 17, 19
1. Write the following in numerals
(a) 7.18 billion. (b) 19.05 million
Refresh and Revise
Conversion of Unit of Measurement
Length
(mm = millimeter cm = centimeter m = metre)
1 cm = 10 mm
1 m = 100 cm
1km = 1000 m
1km = 1000m = 100000cm (1000 x 100)
Area
1 cm2 = 1cm x 1cm = 10mm x 10mm = 100mm2
1 m2 = 1m x 1m = 100 cm x 100 cm = 10000 cm2
1km2 = 1000m x 1000m = 1 000 000 m2
Mass
1 g = 1000mg
1kg = 1000g
1 ton = 1000kg
TIME
1 min = 60 seconds (s)
1 hour (h) = 60 min (m)
1h = 60 x 60 = 3600s
1 s = 1/60 x 1/60 = 1/3600 h
1 s = 1/3600 h
1 h = 3600 s
0.5 or ½ hr = 30 mins
Volume
1 litre = 1000ml = 1000cm3
1cm3 = 1ml
Practice:
1. Convert 125 km to m
2. Convert 299 m to km
3. Convert 700 g to kg
4. Convert 2 ton to g
5. Convert 2 hrs to seconds
6. Convert 1260 seconds to minutes
7. Convert 100 kilometres per hour into metres per second. (2/13/1/2/T)
S1T1 Number Line, Ordering , Simple Inequalities
The number line
Whole numbers, fractions and decimals can be represented on the number line.
The numbers are placed at their correct positions, equal distance apart.
Negative Numbers
We read -1 as negative one.
A negative number is any number less than zero.
Example: -1, -2.5, -4/7
Any pair of numbers eg: 3 and -3 are same distance from the origin.
Using Number Line
Example
What is the value of 3 - 5?
3 - 5 = -2
Ascending and Descending Orders
Ascending : Increasing in value; Moving higher/becoming bigger
[Tip to remember : A - side of A is going up, from small to big ]
Descending : Moving down in value/becoming smaller
[D => Down , from big to small]
Example
Arrange the following in ascending order
5.3, 5.25, 5.205
Step1: Arrange by the decimal point
5.3
5.25
5.205
Step2: Ascending => small to big.
Compare the number values from left to right
5.205 has the smallest hundredth value,
5.25 has the smaller tenth value
Step3: Arrange the numbers
5.205, 5.25, 5.3
Comparing and Ordering Numbers
Symbols
> greater than
< less than
≥ greater than or equal to
≤ less than or equal to
= equal
≠ not equal to
Tip to remember
4 > 3
greater = "open mouth" > “point” = less than
Similarly,
3 < 4
less than = “point” < greater = "open mouth"
Using number line and Inequalities
Using the dot : o excluding the number => < or > symbol
• including the number => ≤ or ≥ symbol
Using the symbol : > => right arrow ------>
< => left arrow <-------
Example ( > symbol)
Draw x > 2 on the number line.
Step1 : "stand" at 2, draw o (> symbol)
Step2 : > (greater than) symbol => Right arrow
[To check if arrow direction is correct, use a bigger number (eg 3), and move the arrow in that direction]
Example ( < symbol )
Draw x < 1 on the number line
Steps : "stand" at 1, draw o (< symbol); then left arrow (<)
Example (≥ symbol )
Draw x ≥ 2 on the number line.
(Draw a solid circle at 2 and then right arrow direction)
Example ( ≤ )
Draw x ≤1 on the number line.
(Draw a solid circle at 1 and left arrow direction)
Practice
1. Arrange the following numbers in ascending order.
(a) 1.14, 1.106, 1.13, 1.1
(b). 0.52, 0.543, 0.5, 0.506
2. Arrange the following numbers in descending order.
(a) 6.07, 6.32, 6.3, 6.301
(b). 9.123, 9.09, 9.3, 9.15
3. Round 965.27 km to
a. 1 decimal place
b. the nearest 10 km
c. 3 significant figures
4. The following are temperatures, in C, over 6 days in Iwate.
-1.4, -2, 0, -3, -0.5
(a) List all the temperatures that are >= -1 C
(b) List all the temperatures that are < -0.5 C
5. Use <, > or = to complete each of these statements. (1/14/2/4/T)
a. 1/3 ___ 0.3
b. 12 ½% ______ 1/8
c. 7/12 _______ 5/9
6. Complete the number line.
<——|——|——|——|——|——|——|——|——|
-6 ___ ___ 3. 6. ___ 12 15 ____
S1T1 Approximation and Significant Digits
Rounding is about approximating a number to a given value or number of decimal places.
Round Off Numbers
If the digit is LESS than 5, Round DOWN [drop all digits to the right]
If the digit is GREATER/EQUAL to 5, Round UP [ADD 1 to left, drops all digits to right]
Method
Step1: Underline the required Rounding place value. <round-off to 10, 8456>
Step2: ( ) right number and Compare :
If equal or greater than 5 (5, 6,7,8,9), ROUND-UP, ADD
If less than 5 (4,3,2,1,0), ROUND-DOWN, NO CHANGE
Step 3: Change the numbers to the right to 0
Example:
(1) Round 36.586 to nearest 1 decimal place.
36.546 (Step 1 : Underline round off)
36.5(4)6 (Step 2 : 2nd decimal 4 is less than 5, drop all to right] )
= 36.5 (1 d.p) (Step 3: Change the numbers to the right to 0)
Example
(2) Round 36.875 to nearest 2 decimal places.
36.875 = 36.88 (2 d.p) [third decimal 5, add one to 7, and drop all to right]
Significant Digits
Number is Significant for
1. Every non-zero : 42549 (5 sig digits)
2. Zeros in between digits : 1001, 1.003 (4 sig digits)
3. Zeros at end of numbers with decimals : 300.100 (6 sig digits)
Number is not significant for
1. Zeros to left of numbers : 00003467 (4 sig digits)
2. Zeros at end of non-decimal numbers : 823000 (3 sig digits)
3. Zeros to right of decimal number < 1 : 0.0005 (1 sig digits)
[ For Decimal, remember to put a trailing zero for specified significant figure.
Eg: 0.200 is 3 significant figures]
Round Off Numbers to Significant Figures
1. Start counting from first significant figure (number other than 0),
2. Check the digit to the right of the rounding digit
3. Apply the rounding rule, and replace all digits to right with 0.
Example:
Round off 81267 to 3,2,1 significant figure/s.
Step1: Look at significant figure => 3, count 3 sig fig numbers from left and underline.
<< 8 1 2 6 7 are the 3 significant figures >>
Step2: Round off number to right
<< 8 1 2 6 7 (round off 4th , number 6 => +1 to 2, 8 1 3). >>
Step3: Replace number to right with 0
8 1 3 0 0 (to 3 sig fig)
To 2 significant figure: 81000 (2 sig fig)
To 1 significant figure: 80000 (1 sig fig)
Example:
Round 52.8975 to 4 significant figures
52.8975. (Step 1 : 4 s.f at 9)
=> change 52.8 to 52.9 (Step 2 : Round off number to right , 5th - 7)
52.90 (Step 3 - Put trailing 0)
52.8297 ~ 52.90 (4 sig fig)
More Examples
(1) 00384.12 has 5 significant digits.
(2) 56.045 has 5 significant digits
(3) 3.0120 has 5 significant digits (the 0s are to be counted)
(4) 00036540 has 5 significant digits (36540) [Zeros to left are not counted]
(5) 0.0006 has 1 significant digits [Decimal number less than 1]
(6) 4 significant digits: 12.34, 01.234, 01234, 1.002, 4.020, 0.001234
Practice
1. Round off each of the following numbers to 1 decimal place, 2 decimal places and 3 decimal places.
(a) 36.7247 (b) 2.8734 (c) 100.283
(d) 1.0049 (e) 0.9999 (f) 9.959
(f) 0.2139 (g) 5.3027
2. Round off each of the following numbers to 2 significant figures, 3 significant figures and 4 significant figures.
(a) 483.86 (b) 248999 (c) 15.795
(d) 0.59 974 (e) 18932 (f) 37.649
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