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 (D.Watkins@mgfl.net)
Contents Times Tables... 4 Addition... 5 Subtraction (Taking Away)... 6 Multiplication... 7 by a single digit by a two digit number by 10, 100 or 1000 Order of Operations (BODMAS and Brackets)... 9 Division... 10 by a single digit by 10, 100 or 1000 Measurement: Key Facts... 11 Fractions... 12 Meaning of fractions Working out a fraction Fractions of shapes Fractions/Decimal equivalents Equivalent Fractions Simplifying Fractions Fraction Number Line Adding and Subtracting Fractions Decimals... 20 Number Lines Number Patterns Percentages... 22 Basic Facts With a Calculator Without a Calculator Negative Numbers... 25 Number Line Multiplication and Division Addition and Subtraction
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 1 1 1 1 1 1 4 + 9 = 13 7 + 8 + 1 = 16 0 + 6 + 1 = 7 3 + 0 = 3 some people write their carries underneath the sum: this is fine! 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 7 1 7 1 we can't do 4 7 take one off the 8 and put it in front of the 4 to make 14 14 7 = 7 7 5 = 2 and 6 0 = 6 Final Answer: 627 Example 2 Take away: 813 269 0 1 0 1 7 1 0 1 7 1 0 1 we can't do 3 9, so we borrow 13 9 = 4 we can't do 0 6, so we borrow 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' Split up 23 to be '20' and '3' 20 3 10 4 Step two - multiply to get a number in each of the four squares 20 10 20 4 10 4 20 200 80 3 30 12 3 10 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 4 = 20 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 1 1 1 1 2 = 0 remainder 1 15 2 = 7 remainder 1 Change 15 to be 15 0 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 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"
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 You can find equivalent fractions by multiplying the top and bottom by the same number. 2 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 3 2. 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 2 5 = 10 3 1 = 3 SMILE 3 5 = 15 10 3 = 7 Final Answer: the number on the bottom stays the same
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 (or 7) b) Write (or 3.6) Write (or 2) 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". 100% means "the whole amount" Pupils should know the following Examples
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 first find 10% then half it 10% of 6000 = 600 5% of 6000 = 300 2 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 12 5 100 3% as a decimal is 0 03 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 take away a positive number e.g. 2 + (-5) = -3 start at 2 count down 5 downwards downwards 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) To add a negative number count downwards Start at -5, count down 2 Add: (-13) + 5 To add a positive number count upwards 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.