Download To achieve level 3 you need to

Number
To achieve level 3 you need to
understand about –
Rounding to the Nearest 10 or 100
Level 4
Place Value
Millions
Hundreds of
Thousands
Tens of
Thousands
Thousands
Hundreds
Tens
Unit (ones)
9
6
7
4
1
0
8
Example
30
36 is between 30 and 40
40
The number 9674108 is shown above written on a place value chart
36 is closer to 40 than 30
Example
36 rounded to the nearest ten is 40
What is the place value of the 7 in these
If the number is between, we round up
a.
35 is halfway between 30 and 40, 35 to the nearest ten is 40
Answer
Checking Calculations
a.
In 675 the 7 is in the tens place. We say that the place value of the 7 is ten
38 + 57 = 85, Round to check 40 + 60 = 100
b.
In 870031 the 7 is in the tens of thousands place. We say that the place value of the 7 is tens of thousands.
Always check your answer even when using a calculator
Large numbers are read in groups of three
675
MILLIONS
THOUSANDS
Hundred Tens Units Hundreds Tens Units Hundreds Tens
Negative Numbers
We often use negative numbers as well as positive numbers
9
The number –2 is a negative number
-2 means 2 less than zero
-5
-4
-3
-2
-1
0
b. 870031
6
7
4
1
0
Units
8
The number 9674108 is read as
1
2
3
4
5
6
7
8
nine million, six hundred and seventy four thousand, one hundred and eight.
We can show negative and positive numbers on a number line
The number system we use is a decimal system.
Place Value
The place value of each number is ten times as large as the place value of the number immediately to the right.
We read 432 as four hundred and thirty two.
Pencil and Paper Methods
The number 52 has 5 tens and 2 ones.
Adding and Subtracting 3 Digit Numbers
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, are called digits.
Always check your answers are reasonable
The place value of a digit tells us its value.
For instance if we add 237 and 122 we must get an answer greater than 300
Putting whole Numbers in Order
Multiplying and Dividing 2 digit Numbers by 1 digit Numbers
Example
Reading Calculator Displays
Rearrange this list of numbers into order of size starting with the largest number
There is a maximum numbers of digits that can be displayed on any calculator screen.
86, 104, 79, 88, 114, 200,
On a calculator with an 8 digit screen display the answer to the division 7 ÷ 3 is displayed as 2.3333333. Other calculators
may display more than or fewer than 8 digits.
Answer
200, 114, 104, 88, 86, 79,
We often need to give the answer to a calculation to the nearest whole number
Greater Than Less Than
384 is greater than 296
427 is less than 449
Instead of writing greater than we can use >
Examples
1.
A calculator displayed the answer to 7 ÷ 3 as 2.3333333
To the nearest whole number the answer to 7 ÷ 3 is 2
2.
A calculator displayed the answer to 53 ÷ 7 as 7.5714286
To the nearest whole number the answer to 53 ÷ 7 is 8
Instead of writing less than we can use <
384 > 296
427 < 449
Addition - ADD, PLUS, TOGETHER, SUM
Subtraction - MINUS, SUBTRACT, TAKE AWAY, LESS
Multiplication
Multiplication is the same as adding a number again and again
For instance 4 x 3 means 4 lots of 3 or 3 + 3 + 3 + 3
Division
Sometimes we need to give the answer to a calculation to the closest but smaller whole number.
Example
Oranges are 52p each. How many can be bought with £5.00
Answer
On the calculator the answer to 500 ÷ 52 is 9.615384615
Only 9 oranges can be bought
Dividing a number is the same as sharing
In the previous example the answer was given as the closest but smaller whole number.
Example - Divide 35 by 7 is the same as sharing 35 by 7
Using the Calculator for Calculations
We say one number is divisible by another if there is no remainder
The calculator is often used for calculations. It is very useful when the numbers are large or have many digits
A number is divisible by 2 if it is an even number
Always have a rough idea of the size of the answer you expect to get. It is very easy to press a wrong key or to forget to
press the = key at the end of a calculation
A number is divisible by 5 if it ends in a 0 or 5
Level 5
Using Negative Numbers
Inverse Operations - Use these to check operations. Inverse operations “undo” each other.
Example
Adding and subtracting are inverse operations as are multiplying and dividing.
Indices
The temperature at 6a.m. was -5°C. By 9a.m. the temperature has risen by 7°. What was the temperature at
9a.m.
Square Roots and Cube Roots
Answer
The temperature must rise 5° to 0°. By the time the temperature has risen a further 2°, it would then be 2°C.
Negative Numbers on the Calculator

The answer to “what number squared gives 4” is 2, 2 is called the “square root of 4”
The sign for a square root is
4.
To get a negative number displayed on the calculator we use the +/- key
For instance 4 means “the square root of 4”
Squaring and finding the square root are inverse operations. One “undoes” the other
Example
To get -4 displayed. Key 4 +/-
2
For instance 2 = 4 and 4 = 2
Square roots are found using the calculator

Example
Use the calculator to find the answers to these.
The answer to “what number cubed gives 64” is 4, 4 is called the “cube root of 64”
The sign for a cube root is
3
a. 2 + (-6)
64 .
b. 2 – (-6)
c. –2 – (-6)
For instance 3 64 means “the cube root of 64”
Cubing and finding the cube root are inverse operations. One “undoes” the other
Answer
a. Key
c. Key
For instance 43 = 64 and 3 64 = 4
Cube roots are found using the calculator
Adding and Subtracting without the Calculator
2 + 6 +/- = to get answer of –4
2 +/- - 6 +/- = to get answer of 4
b. Key
d. Key
d. –2 + (-6)
2 - 6 +/- = to get answer of 8
2 +/- - 6 +/- = to get answer of –8
We can use a number line to add or subtract
Negative Numbers - Scales and Number Lines
To add a positive number, move to the right. To add a negative number, move to the left.
Scales often have positive and negative numbers on them.
For example, on a temperature scale 10° below 0° is shown as -10°.
Addition and subtraction are inverse operations. To subtract, we must move in the opposite direction to that in
which we move to add.
Positive numbers such as +4 are often written without the + sign. For instance, +4 is often written as
4.
Negative numbers, such as –4, are always written with the – sign.
To subtract a positive number, move to the left.
To subtract a negative number, move to the right.
The positive and negative numbers are called integers. The integers include both the positive whole
numbers and the negatives and also zero. Zero is neither positive nor negative.
Ordering Negative Numbers
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
We can often use multiplication by 10 or 100 or 1000 etc. to help us when we are multiplying by 80 or 800 or
8000 etc.
Example
We can often use division by 10 or 100 or 1000 etc. to help us when we are dividing by 40 or 400 or 4000 etc.
Insert < or > to make these statements true.
a.
6
-6
Answer
a. 6 > -6 (Since 6 is greater than –6)
Interpreting Negative Numbers
b.
–5
-6
b. –5 > -6 (Since -5 is greater than –6)
Example
Heights above sea level can be described with positive numbers and heights below sea level with
negative numbers.
a. +260m
b. –6m
a. 260m above sea level
b. 6m below sea level
Calculator Errors
When using the calculator, it is a good idea to have a rough idea of the answer.
It is easy to key a calculation into the calculator incorrectly.
If the answer given by the calculator is very different from the rough estimate, the calculation should be keyed in
again
Estimating Answers using Approximation
We can check that the answer to a calculation such as 423 x 76 is about the right size by approximating.
We can approximate 423 as 400. We can approximate 76 as 80
So we estimate 423 x 76 to be about 32000
Writing Negative Numbers
Example
Heights above sea level are to be described with positive numbers and heights below sea level are to
be described with negative numbers. Use positive or negative numbers to describe the height of a
place which is
a. 5m below sea level
b. at sea level
c. 5m above sea level
Answer
a. –5m
4–1=3
4–1=3
4–1=3
4–1=3
Using Place value 23 x 10 = 230, 23 x 100 = 2300
The further to the right a number is, the larger it is.
The further to the left a number is, the smaller it is.
Answer
Examples
4–1
Begin at 4, move 1 to the left
1–3
Begin at 1, move 3 to the left
1 – (-3) Begin at 1, move 3 to the right
-1 – (-3) Begin at -1, move 3 to the right
Using Place Value of Multiply and Divide
b. 0m
c. +5m
Always estimate answers when using the calculator.
Long Multiplication without the calculator
There are many methods for finding the answer when two large numbers are multiplied together.
Long Division without the calculator
Most methods for long division are similar but the setting out is different.
Approximation can be used in the working.