Download Methods Taught in Numeracy - Newbattle Community High School

This booklet outlines the methods we teach pupils
for place value, times tables, addition, subtraction,
multiplication, division, fractions, decimals,
percentages, negative numbers and basic algebra
Any queries relating to the learning and teaching of maths at Newbattle High School
may be directed to the Principal Teacher (Mathematics and Numeracy),
Mr D Watkins ([email protected])
Contents
Times Tables .......................................................
4
Addition ............................................................. 5
Subtraction (Taking Away) ........................................
6
Multiplication .......................................................
by a two digit number
by a single digit
7
by 10, 100 or 1000
Order of Operations (BODMAS and Brackets) ..................
9
Division .............................................................
by 10, 100 or 1000
by a single digit
10
Measurement: Key Facts .........................................
11
Fractions ...........................................................
12
Meaning of fractions
Equivalent Fractions
Working out a fraction
Simplifying Fractions
Fractions of shapes
Fraction Number Line
Fractions/Decimal equivalents
Adding and Subtracting Fractions
Decimals ...........................................................
Number Patterns
Number Lines
20
Percentages .......................................................
With a Calculator
Basic Facts
22
Without a Calculator
Negative Numbers ................................................
Number Line
Addition and Subtraction
Multiplication and Division
25
Example 1
Add 356 and 78:
This is laid out
correctly.
This is not correct. The 8
must go underneath the 6
as they are both units
Example 2
Add 3074 and 689:
1
4 + 9 = 13
1
1
7 + 8 + 1 = 16
some people write their carries
underneath the sum: this is fine!
1
1
0+6+1=7
1
1
3+0=3
Final Answer: 3763
Pupils should know how to take away numbers using a written
method.
We do not teach the method of "borrow and pay back" that some
parents/grandparents may have learnt when they were at school
Line the numbers up by place value (like Example 1 on the
addition page) and begin taking away from the right (the units).
When the number on the top is smaller than the one on the
bottom, "borrow" 1 from the column to the left.
Example 1
Take away: 684 – 57
7 1
we can't
do 4 – 7
take one off the 8
and put it in front of
the 4 to make 14
7 1
14 – 7 = 7
7 1
7–5=2
and
6–0=6
Final Answer: 627
Example 2
Take away: 813 – 269
0
1
we can't
do 3 – 9, so
we borrow
0
1
13 – 9 = 4
7
1
0
1
we can't
do 0 – 6, so
we borrow
7
1
0
1
10 – 6 = 4
and
7–2=5
Final Answer: 544
Pupils should know how to multiply by a single digit
Example 1
Multiply 36 × 4
2
Step one - work out 4 × 6 (= 24)
Write down the units (4)
carry the tens (2)
Step two - work out 4 × 3 (= 12)
add on the 2 that was carried (to get 14)
Final Answer: 144
Pupils should know how to multiply by 10, 100 or 1000
To multiply by
To multiply by
To multiply by
Example 2
Multiply 50•6 × 100
move every digit two
places to the left.
The decimal point does
not move
we can't leave the units
column empty so we have
to write a zero here
Final answer: 5060
(or 5060·0)
Multiplication: two two-digit numbers
Pupils should know how to multiply two two-digit numbers.
There are a number of ways to do this, and pupils are encouraged to
use a way that works for them. On this page we will explain the
method that we teach most often.
Example 1
Multiply 23 × 14
Step one - make a tables square with two
numbers along the top and two along the side
Split up 14 to be
'10' and '4'
10
Split up 23 to
be '20' and '3'
4
20
3
Step two - multiply to get a number in each of
the four squares
20 × 10
20 × 4
10
4
20 200
80
3
12
3 × 10
30
3 × 4
Step three - add the four answers to get the
final answer
200 + 80 + 30 + 12
Final Answer: 322
When doing a calculation with more than one operation, we have to
complete the calculation in the following order:
some teachers use the word Order instead
ultiply
This can be remembered using the word BODMAS.
If a calculation contains brackets, all sums in brackets must be
done first.
Example 1
Calculate ( 2 + 3 ) × ( 6 - 2 )
=
5
=
×
20
4
Final Answer: 20
Where a calculation does not contain brackets, we must follow
the order given by BODMAS.
Example 2
Calculate 14 - 2 × 5
multiply becomes before subtract in BODMAS,
so we must do the multiply sum first
=
=
Final Answer: 4
In written calculations the
decimal points must stay in line
5÷3=
1 remainder 2
26 ÷ 3 =
8 remainder 2
24 ÷ 3 = 8
If you have a remainder at the end of the calculation, add a
zero (or •0) at the end of the number and continue as normal.
In written calculations the
decimal points must stay in line
1
1
1 ÷ 2 = 0 remainder 1
15 ÷ 2 = 7 remainder 1
1
1
Change 15
to be 15•0
1
1
10 ÷ 2 = 5
Pupils should know how to divide by 10, 100 or 1000
To divide by
To divide by
To divide by
right.
right.
Example 1
Divide 123 ÷ 100
move every digit two
places to the right.
The decimal point does
not move
Final answer: 1•23
VOLUME
There are
The picture below shows four quarters of a pizza which is the same
as one WHOLE pizza
Any fraction with the same number on the top and the bottom is
equal to 1 whole
of a number means "split
the number into 5 equal groups
and then say how much is in 1 group"
of a number means "split
the number into 7 equal groups
and then say how much is in 3 group"
of a number means "split
the number into 5 equal groups
and then say how much is in 1 group"
Example 1
Shade in
of this shape:
There are 15 squares, so we work out
of 15:
Final Answer:
Every decimal number can be re-written as a fraction.
The number on the bottom of the fraction will usually be
10, 100 or 1000 before simplifying.
Decimals with one digits after the point are equivalent to fractions with '10' on the bottom
Decimals with two digits after the point are equivalent to fractions with '100' on the bottom
Decimals with three digits after the point are equivalent to fractions with '1000' on the bottom
6)
0•051 =
Example 1
×2
You can find equivalent fractions
by multiplying the top and bottom
by the same number.
×2
Example 2
Find fractions that are equivalent to
:
We can choose any number we like to multiply by. If we
randomly choose 2, 7 and 10, we get the following fractions
×2
×7
× 10
×2
×7
× 10
Final Answers:
Pupils should be able to simplify a fraction.
Fractions can be simplified by dividing the top and bottom by the
same number.
The top and bottom of any fraction must always be a whole
number. So the number you divide by must be a factor of both
numbers. This means that it must divide into both numbers
exactly with no remainders.
There are two ways to do this:
1. Keep dividing both the top and bottom of the fraction (trying
2, 3, 5, 7 etc.) until you can't go any further
Example 1
Simplify the fraction
÷2
÷2
÷3
Final Answer:
÷2
2.
÷2
÷3
Divide the top and the bottom by the Highest Common Factor
Example 2
Simplify the fraction
÷ 12
Final Answer:
÷ 12
Example 1
some fractions
(in this case we simplified
Example 2
8–1=7
the number on the
bottom stays the same
3+2=5
because
the number on the
bottom stays the same
8–5=3
the number on the
bottom stays the same
Fractions: Harder Adding and Subtracting
If the denominators are not the same, we have to first find
an equivalent fraction before we can add.
(these are harder examples. Not all pupils will do them in S1-S3)
Example 1
Add:
2+1=3
rewrite
as
because they are
equivalent
the number on the
bottom stays the same
Final Answer:
One way of finding equivalent fractions with a common
denominator is to use a method called "kiss and smile".
This is because of the shape made if we draw
lines joining pairs of numbers that are multiplied.
Example 2
Take away:
KISS
SMILE
2 × 5 = 10
3×1=3
3 × 5 = 15
10 – 3 = 7
the number on
the bottom
stays the same
Final Answer:
Pupils should know how to indicate a given number on a number line.
Example 1
Draw an arrow to show where the number
Answer:
The number 5•6 is six tenths along from 5.
5
Example 2
Draw an arrow to show where the number
Answer:
The number 1•23 is slightly higher than 1•2
Example 3
Draw an arrow to show where the number
4·25
·
·
·
·
Answer:
The number 4•265 is exactly mid-way between 4•26 and 4•27
4·25
(4·250)
·
(4·260)
·
(4·270)
·
(4·280)
·
(4·290)
Pupils should be able to continue a sequence of numbers.
Example 1
a)
Write
•
b)
•
•
(or 7)
•
Write
•
Write
•
•
•
•
•
•
•
•
(or 2)
•
(or 3.6)
•
•
•
Pupils should be able to write lists of numbers in order.
Example 2
To help answer this question, it is easiest to think of 4 as 4.0 so that all
of the numbers have one decimal place
To help answer this question, it is easiest to think of 3.01 as 3.010 so
that all of the numbers have three decimal places
means out of a hundred
0% means "nothing".
Pupils should know the following
Examples
100% means "the whole amount"
Percentages: without a Calculator
1% =
, so to find 1% we divide by 100.
Example 1
Find 1% of 300
×1
Final Answer: 3
We can use 1% to help us find other percentages.
Example 2
Find 4% of £2000
first find 1%
then multiply by 4
1% of 2000 = 20
4% of 2000 = 80
Final Answer: £80
We can also use 10% to help us find other percentages.
Example 3
Find 30% of £700
first find 10%
then multiply by 3
10% of 700 = 70
30% of 700 = 210
Final Answer: £210
Example 4
Find 5% of £6000
10% of 6000 = 600
first find 10%
then half it
÷2
5% of 6000 = 300
Final Answer: £300
An alternative method with a calculator is to change the percentage
into a decimal (by dividing by 100) and to multiply.
3 ÷ 100
3% as a decimal is 0•03
12•5 ÷ 100
12•5% as a decimal is 0•125
The temperature in Aberdeen was -4°C during the
night. The next day it rose 6°C. What was the
temperature the next day?
Start at -4 on the number line and count up 6 numbers.
To add and take away sums, we must count up or down the
number line.
add a positive number
e.g.
2+5=7
start at 2
count up 5
upwards
start at -3
count up 7
start at -8
count up 1
add a negative number
downwards
take away a positive number
downwards
e.g.
2 + (-5) = -3
start at 2
count down 5
start at -3
count down 7
start at -8
count down 1
Adding a negative number is the same as
taking away a positive number.
Add: (-5) + (-2)
Add: (-13) + 5
To add a negative number
count downwards
To add a positive number
count upwards
Start at -5, count down 2
Start at -13, count up 5
Taking away a negative number is the same as adding.
Examples 1a and 1b
Take away:
(a) 5 - (-2)
(b) (-3) - (-1)
taking away a negative
is the same as adding
Final Answer: 7
Final Answer: -2
If one number in a multiplication or division sum is
negative, the answer is negative.
If two numbers in a multiplication or division sum are
negative, the answer is positive.
Examples 2 and 3
one number is negative
so the answer is
negative
two numbers are
negative so the answer
is positive
We are often asked this question by parents.
Some possible suggestions are below:
1.
Help your child with questions they are stuck on for their
maths homework. Maths homework will usually be set weekly.
2.
Read through this booklet and refer to it when helping your
child with a maths question.
3.
Practice times tables with your child: see the advice on page 4.
4.
Help your child to learn the common measurement facts on
page 11 and talk about them when measuring in the home (e.g.
during cooking or making something)
5.
Help your child to learn the common fraction/percentage/
decimal equivalences on page 22.
6.
Ask your child to identify a topic from this booklet that they
find hard and work on it with them at home. Worksheets can
be downloaded from the website www.dynamicmaths.co.uk.
Your child's maths teacher could help them to identify areas
they need more practice on.
7.
There are a number of good Apps and online games that are
designed to develop maths skills.