### Shape and Space

Symmetry:

- The quality of being made up of exactly similar parts facing each other or around an axis.
- Exact correspondence of form and constituent configuration on opposite sides of a dividing line or plane or about a center or an axis.

### Length

Various instruments are used to measure length. For example:

To convert length from a larger unit into a smaller unit, multiply by the relevant power of 10.

To convert length from a smaller unit into a larger unit, divide by the relevant power of 10.

Convert the following measurements to the units indicated:

a. 7 cm to mm

b. 8 m to cm

c. 9 km to m

- Rulers and tape measures are marked in millimetres or centimetres to measure shorter lengths accurately.
- A trundle wheel is used to measure length to the nearest metre. Note that a counter may be attached to the trundle wheel to count metres.
- A car's odometer often measures distance in tenths of a kilometre.

To convert length from a larger unit into a smaller unit, multiply by the relevant power of 10.

To convert length from a smaller unit into a larger unit, divide by the relevant power of 10.

__Example.__Convert the following measurements to the units indicated:

a. 7 cm to mm

b. 8 m to cm

c. 9 km to m

**Solution:**### Whole Number: Division

- In mathematics, especially in elementary arithmetic, division (÷) is an arithmetic operation.
- Another basic view of division is sharing equally.

**Example:**Danesse has 3 friends. She has 12 cookies to share with them. How many cookies does each of the four people get?

- Danesse probably shared her cookies by saying one for you, one for you, so on. Thus the 12 cookies are shared equally by four people, and each gets three cookies.
- Or, you might say that the 12 cookies have been partitioned into four sets of the same size.

**Exercises:**- 21 ÷ 3 = ________
- 24 ÷ 4 = ________
- 42 ÷ 7 = ________
- 32 ÷ 8 = ________
- 24 ÷ 8 = ________
- 54 ÷ 6 = ________
- 72 ÷ 6 = ________
- 27 ÷ 9 = ________
- 72 ÷ 9 = ________
- 56 ÷ 7 = ________

### Fractions

- A fraction is a number that is written in the form:

- The a is the numerator, and the b is the denominator.
- The line separating the numerator and denominator is a fraction bar.
- Fractions are used when representing numbers that describe the parts of a whole. The fraction a/b also can be read as "
**a out of b**" ,"**a over b**" or "**a divided by b.**"

- Both a and b must be integers, meaning positive and negative whole numbers.
- The denominator, or b, cannot be zero. This is because one cannot divide by zero.

**Example:**If there are 18 students in a classroom, and 6 of the students wear glasses, what fraction of the students wear glasses?

- A fraction can be thought of as "a out of b."
- Total number of students is 18.
- Number of students with glasses is 6.
- Number of students with glasses out of the whole class is 6/18.

### Whole Numbers: Multiplication

- Multiplication (often denoted by the cross symbol "×") is the mathematical operation of scaling one number by another.
- Whole-number products greater than 1 can be computed by repeated addition;

**for example:**

- 7 multiplied by 3 (often said as "7 times 3") can be calculated by adding 7 copies of 3 together: 7 x 3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 = 21
- Here 7 and 3 are the "factors" and 21 is the "product".
- 7 multiplied by 3 can also be calculated by adding 3 copies of 7 together: 7 x 3 = 7 + 7 + 7 = 21

__Exercises:__- 7 x 1 =
- 6 x 5 =
- 7 x 8 =
- 3 x 9 =
- 6 x 6 =
- 9 x 8 =
- 5 x 7 =
- 6 x 9 =
- 2 x 8 =
- 9 x 9 =

### Whole Number : Subtraction

**How to subract number:**Example: 99876 -57894 = ?

- Line up the numbers:

9|9|8|7|6|

__- 5|7|8|9|4|__

________

- Subtract the numbers starting from place value ones to ten thousands:

8 17 17

9|

__- 5|7|8|9|4|__

4 1 9 8 2

**Exercises:**- 203 - 35 =
- 72 - 16 =
- 2180 - 726 =
- 62731 - 12735 =
- 7261 - 3815 =
- 92 - 25 =
- 18 - 10 =
- 8271 - 941 =
- 51426 - 903 =
- 6251 - 170 =

### Whole Numbers: Addition

**First, we need to work on place values:**

45 971

| | | | |___ Ones 1's

| | | |____ Tens 10's

| | |_____ Hundreds 100's

| |_______Thousands 1 000's

|_________Ten Thousands 10 000's

**If we expand this out, we get:**

45971 = 40 000 + 5 000 + 900 + 70 + 1

**Write this from up to down:**

4|0|0|0|0|

|5|0|0|0|

|9|0|0|

|7|0|

__+ |1|__

__4 5 9 7 1__

**How to add up numbers:**Example: 53 681 + 7927 = ?

- Line up the numbers:

__+ |7|9|2|7|__

________

- Add up the numbers starting from place value ones to ten thousands:

5|3|6|8|1|

__+ |7|9|2|7|__

__6 1 6 0 8__

### Whole Numbers

**Whole Numbers**

- Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, … (and so on)

No fraction!

**Counting Numbers**

- Counting Numbers are Whole Numbers, but without the zero. Because you can't "count" zero. So they are 1, 2, 3, 4, 5, … (and so on).

**Natural Numbers**

- Mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject.

**Place Value till Ten Thousands**

- The position, or place, of a digit in a number written in standard form determines the actual value the digit represents.
- Below shows the place value for various positions:

__Place (underlined)____Name of Position__1 00

__0__Ones (units) position

1 0

__0__0 Tens

1

__0__00 Hundreds

__1__000 Thousands

__1__0 000 Ten Thousands

1

__0__00 000 Hundred Thousands

__1__000 000 Millions

Place Value

**Example**

The number 21,040:

- Has a 2 in the ten thousands place,
- a one in the thousands place,
- a 4 in the tens place,
- and a 0 in both the hundreds and ones place.
- Can be written as twenty-one thousand and forty.