Tag Archives: learning

Long multiplication

You can use short multiplication if you’re multiplying one number by another that’s in your times tables (up to 12). However, if you want to multiply by a higher number, you need to use long multiplication.

  • Write down the numbers one on top of the other with the smaller number on the bottom and a times sign on the left (just as you would normally), then draw three lines underneath to hold three rows of numbers.
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  • Multiply the top number by the last digit of the bottom number as you would normally.
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  • Write a zero at the end of the next answer line (to show that you’re multiplying by tens now rather than units).
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  • Multiply the top number by the next digit of the bottom number, starting to the left of the zero you’ve just added.
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  • Add the two answer lines together to get the final answer.
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Notes:

  • Some people write the tens they’ve carried right at the top of the sum, but that can get very confusing with three lines of answers!
  • Don’t forget to add the zero to the second line of your answer. If it helps, you can try writing it down as soon as you set out the sum (and before you’ve even worked anything out).
  • At 11+ level, long multiplication will generally be a three-digit number multiplied by a two-digit number, but the method will work for any two numbers, so don’t worry. If you have to multiply two three-digit numbers, say, you’ll just have to add another line to your answer.

Sample questions:

Have a go at these questions. Make sure you show your working – just as you’d have to do in an exam.

  1. 216 x 43
  2. 17 x 423
  3. 23 x 648
  4. 782 x 28
  5. 127 x 92

Homophones

Homophones are words that sound the same even though they’re spelt differently and mean different things. Getting them right can be tricky, but it’s worth it in the end.

The reason why homophones are important is not just to do with the general need to spell correctly. Many people think getting them wrong is a ‘worse’ mistake than simply mis-spelling a word because it means that you don’t really know what you’re doing. Anyone can make a spelling mistake, but using completely the wrong word somehow seems a lot worse. That may not sound fair, but that’s just how a lot of people think, so it’s worth learning the common homophones so you don’t get caught out.

Here’s a list of the main ones:

SpellingsMeanings
a lot/allotmuch/distribute
ad/addnotice/put together
ads/adds/adzenotices/puts together/type of axe
aid/aidehelp/assistant
ail/aletrouble (verb)/beer
air/heiratmospheric gas/inheritor
aisle/isle/I’llcorridor/island/I will
all ready/alreadyeverything set/by a certain time
all together/altogethereveryone in the same place/absolutely
all/awltotally/piercing tool
allowed/aloudpermitted/out loud
alter/altarchange/church table
ant/auntinsect/parent’s sister or brother’s wife
arc/arkpart of circle/Biblical boat
assent/ascentagreement/rise
assistance/assistantshelp/helper
ate/eightconsumed/8
aural/oralto do with hearing/to do with mouths
away/aweighoff/up (eg anchors aweigh)
ay/aye/eye/Iyes/yes/organ of sight/1st person
bail/balescoop water out/jump out
bait/batefood on hook/hold (eg bated breath)
ball/bawlsphere/
band/bannedgroup/illegal
bard/barredpoet/banned
bare/bearnaked/tolerate or grizzly
baron/barrenlord/arid
base/bassfoundation/low note
be/beeexist/flying insect
beach/beechsandy area/type of tree
beat/beethit/beetroot
beau/bowmale admirer/bend down or front
bell/bellemusical instrument/pretty girl
berry/buryfruit/inter
berth/birthsleeping place/arrival of baby
billed/buildinvoiced/construct
bite/bytenibble/unit of data
blew/blueforced air/colour
bloc/blockgroup of countries/cuboid
boar/borewild pig/boring person
board/boredflat object/weary
boarder/borderlodger/edge
bode/bowedimply/curved
bolder/boulderbraver/rock
born/bornecreated/tolerated
bough/bowbranch/bend down or front of ship
boy/buoymale child/floating marker
brake/breakslow down/shatter
breach/breechgap/part of gun
bread/bredfood made with yeast/brought up
brewed/broodfermented/family
brews/bruiseferments/scar
bridle/bridalleather strap/to do with brides
broach/broochopen (a question)/piece of jewellery
browse/browslook through/hair above the eyes
but/buttalthough/water tank
buy/by/byepurchase/by means of/goodbye
cache/cashhidden hoard/notes and coins
callous/callusheartless/blister
cannon/canongun/collection of artistic works
canvas/canvasssail fabric/ask questions of
capital/capitolupper case or city/government building
carat/carrot/caret/karatweight/vegetable/arrow/share of gold
carol/carrelsong/cubicle
cast/castethrown/social class
cede/seedgive up/reproductive unit
ceiling/sealingroof/making watertight
cell/sellpart of body/exchange for money
cellar/sellerbasement/person selling
censor/sensorban (film etc)/measurement device
cent/scent/sentUS penny/perfume/dispatched
cents/scentsUS pennies/perfumes
cereal/serialbreakfast dish/TV show
cession/sessiongiving up/period of course
chance/chantsluck/songs
chased/chastepursued/like a virgin
cheap/cheepinexpensive/bird sound
chews/choosenibbles/select
chilly/chillicold/hot food
choir/quiregroup of singers/unit of paper
chord/cordgroup of notes/string
chute/shootdisposal passage/take shot at goal
cite/sight/sitequote/seeing/location
clause/clawsparagraph/talons
coarse/courserough/track or route
colonel/kernelarmy rank/stone in fruit
complement/complimentgo well with/say something nice
coo/coupsound of dove/revolution
core/corpscentre/army unit
correspondence/correspondentsletters/letter-writers
council/counselgoverning body/advice
councillor/counsellorgovernor/advisor
creak/creekwhine/stream
crews/cruiseteams/voyage
cue/queuesnooker tool/line of people waiting
currant/currentdried fruit/electric flow
cymbal/symbolmusical instrument/icon
dam/damnriver barrier/damnation
days/daze24-hour periods/confuse
dear/deerexpensive/type of mammal
defused/diffusedmade safe/circulated
desert/dessertsandy zone/pudding
dew/do/duewater on grass/finish/owed
die/dyeexpire/colour (verb)
discreet/discretenot talkative/separate
doe/dough/dohfemale dear/unbaked bread/oh, no
done/dunfinished/grey-brown
draft/draughtpractise writing/on tap (eg beer)
dual/duelin two parts/fight with swords etc
earn/urn/ern or ernemake money/vase/type of bird
ewe/you/yewfemale sheep/2nd person/type of tree
faint/feintlose consciousness/fake attack
fair/farejust/food
fated/feteddestined/celebrated
faun/fawnrural god/beige or young deer
faze/phase disturb/stage
feat/feetachievement/plural of foot
find/finedseek/told to pay money
fir/furtype of tree/animal hide
flair/flaretalent/bullet making bright light
flea/fleetype of insect/run away
flew/flu/fluepast tense of fly/influenza/chimney
flocks/phloxherds/type of plant
flour/floweringredient for bread/plant
for/four/foreto the benefit of/4/in front
fort/fortecastle/speciality
forth/fourthforwards/4th
foul/fowldisgusting/birds
friar/fryer monk/pan
gait/gateway of walking/door outside
gene/jeanDNA unit/trousers
gild/guildcover in gold/organisation
gilt/guiltcovered in gold/having done wrong
gored/gourdholed/fruit or water container
gorilla/guerrillatype of ape/freedom fighter
grate/greatfireplace/grand
grease/Greecelubrication/a country
groan/grownmoan/past tense of grow
guessed/guestpast tense of guess/invitee
hail/halecelebrate/healthy
hair/harestrands growing on head/rabbit
hall/haulroom/pull
hangar/hangerstorage for aircraft/hook in wardrobe
hay/heydried grass/oy
heal/heel/he’llmake well/back of foot/he will
hear/herelisten/in this place
heard/herdpast tense of hear/group of animals
heed/he’dpay attention to/he would or he had
hertz/hurtsfrequency unit/causes pain
hew/hue/Hughcut/colour/a name
hi/highhello/raised
higher/hiremore raised/rent
him/hymna pronoun/religious song
hoard/hordecollection/mass of people
hoarse/horserough (of voices)/an animal
hoes/hosegarden tools/tube
hold/holedkeep or carry/past tense of hole
hole/wholespace/entire
holey/holy/whollywith holes/sacred/completely
hour/our60 minutes/a pronoun
humorous/humerusfunny/arm bone
idle/idollazy/religious statue
illicit/elicitillegal/draw out
in/inninside/hotel
instance/instantsexample/moments
intense/intentsfierce/purposes
it’s/itsit is/belonging to it
jam/jambfruit spread/door frame
kernel/colonelcore/army rank
knap/napcrest/doze
knead/kneed/needmix dough/hit with knee/require
knight/nightwarrior/dark time
knit/nitfit together/egg of louse
knot/nottied rope/negative
know/no/Nohbe aware of/negative/type of drama
knows/noseis aware of/facial feature
laid/ladepast tense of lay/load ship
lain/lanepast participle of lay/alley
lay/leiplace/flower necklace
leach/leechleak/blood-sucking worm
lead/ledheavy metal/past tense of lead
leak/leekdrop out/vegetable
leased/leastpast tense of lease/superlative of less
lee/leashadow of wind/meadow
lessen/lessonmake less/teaching session
levee/levyembankment/tax
liar/lyreperson who lies/musical instrument
license/licencepermit (verb)/permission
lichen/likenmould/compare
lie/lyefalsehood/alkali solution
links/lynxconnections/wild cat
load/lodeput into/vein of metal in ground
loan/lonelending/single
locks/loxsecures/smoked salmon (American)
loot/lutemoney/musical instrument
made/maidcreated/young woman
mail/malepost/masculine
main/mane/Mainechief/hair/state in USA
maize/mazecorn/labyrinth
manner/manorway/lord’s house
mantel/mantlementalpiece/coat
marshal/martialarmy rank/to do with war
massed/mastbrought together/upright post on ship
maybe/may beperhaps/might be
meat/meet/metetype of food/get together/distribute
medal/meddleaward/interfere
metal/mettleshiny material/spirit
might/mitemay/tiny spider
mince/mintsground beef/plural of mint
mind/minedbrain/dug up
miner/minor/mynahdigger/junior/type of bird
missed/mistpast tense of miss/fog
moan/mowngroan/past participle of mow
mode/mowedway/past tense of mow
moose/mousseelk/foam
morn/mournmorning/regret
muscle/musselpart of body/sea creature
mustard/musteredspicy dressing/broughted together
naval/navelto do with the navy/belly button
nay/neighno (dated)/sound of horse
none/nunnot one/female monk
oar/or/oreblade/alternatively/metal source
ode/owedpoem/due
oh/owe/oah/have a debt of/oh (poetic)
overseas/overseesforeign/manages
pail/palebucket/faint
pain/paneache/window panel
pair/pare/pearcouple/shave/type of fruit
palate/palette/palletpart of mouth/artist’s tray/platform
passed/pastpast tense of pass/in the past
patience/patientstolerance/people in hospital
pause/pawsbreak/animal hands and feet
pea/peevegetable/urinate
peace/pieceharmony/bit
peak/peek/piquesummit/look quickly/annoyance
peal/peelsound of bells/take skin off
pearl/purlprecious stone/knitting stitch
pedal/peddlefoot lever/sell
pedalled/peddledcycled/sold
peer/pierlook carefully/jetty
per/purrfor each/sound of a cat
pi/pie3.14/dish topped with pastry
plain/planeunexciting/2D object
pleas/pleaserequests (noun)/if it pleases you
plum/plumbtype of fruit/measure water depth
pole/pollrod/election or survey
pore/pourconcentrate on/flow
practice/practiserehearsal/rehearse
pray/preytalk to God/victim
presence/presentsbeing somewhere/gifts
prince/printsson of monarch/printed photographs
principal/principlemain/rule of conduct
profit/prophetmoney made/religious seer
rack/wrackwire tray/shipwreck
rain/reign/reinwater from clouds/rule/control strap
raise/rays/razelift/plural of ray/destroy
rap/wraphit/pack up (eg a present)
rapped/rapt/wrappedpast tense of rap/spellbound/past tense of wrap
read/redpast tense of read/scarlet
read/reedstudy/type of plant
real/reelgenuine/cylinder for fishing line etc
reek/wreaksmell bad/cause
rest/wrestrelax/wrench away
retch/wretchvomit/poor soul
review/revuelook over/stage performance
right/rite/writecorrect/ritual/form Leopardtters
ring/wringsound of bell/squeeze out water
road/rode/rowedstreet/past tense of ride/past tense of row
roe/rowfemale deer/use oars
role/rollpart in play/type of bread
root/routepart of plant/roads to take
rose/rowstype of flower/tiers
rote/wroterepetition/past tense of write
rough/ruffcoarse/Elizabethan collar
rung/wrungpast tense of ring/squeezed water out
rye/wrycereal plant/mocking
sail/salecanvas propulsion/selling
scene/seensituation/past tense of see
scull/skullrow alone/head of skeleton
sea/seeocean/be aware of
seam/seemsewn connection/appear
seas/sees/seizeoceans/is aware of/grab
serf/surfagricultural worker/waves
sew/so/sowconnect with thread/thus/plant
shear/sheercut/complete
shoe/shoofootwear/chase away
side/sighededge/past tense of sigh
sighs/sizebreathes out/dimensions
slay/sleighkill/sled
sleight/slightdeceptive skill/faint
soar/sorerise/painful
soared/swordpast tense of soar/bladed weapon
sole/soulonly/spirit
some/suma few/total
son/sunmale child/star in the sky
staid/stayedunadventurous/past tense of stay
stair/starestep/look hard
stake/steakwooden post/joint of meat
stationary/stationerymotionless/writing materials
steal/steelrun off with/metal compound
step/steppestair/European plains
stile/stylefence steps/manner
straight/straitnot bending/narrow strip of water
suite/sweethotel rooms/sugary
summary/summerybrief account/to do with summer
surge/sergerush/type of cloth
tacks/taxnails/levy
tail/taleback end/story
taught/tautpast tense of teach/tight
tea/teemeal/golf ball holder
team/teemgroup of players/swarm
tear/tierteardrop/row
tern/turntype of bird/go round a corner
their/there/they’rebelonging to them/in that direction/they are
theirs/there’sthe one belonging to them/there is
threw/throughpast tense of throw/in and out of
thrown/thronepast participle of throw/royal chair
thyme/timetype of herb/progress of days or years
tic/tickhabit/mark (if correct)
tide/tiedflow of water/past tense of tie
to/too/twotowards/as well/2
toad/towedfrog/past tense of tow
toe/towpart of foot/pull
told/tolledpast tense of tell/rang
trussed/trustbound (with rope)/belief
vain/vane/veinproud/fin/artery
vale/veilvalley/lace face covering (for brides etc)
vial/viletest tube/evil
wade/weighedwalk in water/past tense of weigh
wail/whalehowl/type of ocean mammal
waist/wastemiddle of body/use carelessly
wait/weightdelay or stay/mass
waive/wavegive up/breaker
ware/wear/wherepottery/put on clothes/which place
way/weigh/wheymanner/measure weight/part of milk
ways/weighsmanners/measures weight
we/weepronoun/urination or little
weak/weekfeeble/seven days
weather/whetherclimatic conditions/if
we’d/weedwe would or we had/unwanted plant
we’ll/wheelwe will or we shall/round component
wet/whetliquid/sharpen
we’ve/weavewe have/make cloth
which/witchpronoun/wizard
while/wileas/ruse or cunning plan
whine/winewhimper/alcoholic grape drink
who’s/whosewho is or who has/of whom the
wood/wouldtree material/conditional marker
yoke/yolkpart of plough/yellow part of egg
yore/your/you’reformer times/belonging to you/you are
you’ll/Yuleyou will or you shall/Christmas

Children’s reading list

Books, books, books…

I’m often asked by parents what books they should try to get their children to read, but I don’t think I’ve been much help so far, so this is my attempt to do better! If you’re still not convinced, there are a number of reading lists on my Useful Links page.

Tastes differ, obviously, so perhaps the best thing I can do is to list all the books that I loved when I was a boy. I can’t remember exactly how old I was when I read them, so you’ll have to use your common sense, but they did at least provide me with happy memories.

Ronald Welch

My favourite series of books when I was a child was the one written by Ronald Welch about the Carey family. He wrote about the men in the family over the course of around 500 years, from 1500 up to the First World War. Each novel focused on one character in one particular period – rather like Blackadder, and there was a clear formula: whatever the period, he would have to fight a duel, he would do something heroic and he would win the fair lady! The duels started with a dagger and a sword and then moved on to rapiers and then finally pistols as the years rolled on. I loved the military aspect to the books – as most boys would – and I read just about every single one I could get my hands on. Unfortunately, they’re almost impossible to find in print nowadays, but it’s always worth a look…

CS Forester

CS Forester wrote the ‘Hornblower’ novels. I was interested in both sailing and military history when I was young, and this sequence of novels about a naval officer called Horatio Hornblower in the Revolutionary and Napoleonic Wars from 1792-1815 was a perfect blend of the two.

Alexander Kent (Douglas Reeman)

Alexander Kent was the pen name of Douglas Reeman, who wrote a series of novels about Richard Bolitho. I first came across him after finishing all the CS Forester novels, and he provided a similar mix of nautical and military history during the same period. They weren’t quite as good as the Hornblower novels, but I still enjoyed them.

Enid Blyton

I didn’t read absolutely all the Enid Blyton books when I was a boy, but the one that I do remember is The Boy Next Door. Among other things, I loved the name of the character (‘Kit’), I loved the bits about climbing trees and I also loved the word ‘grin’, which I never understood but thought was somehow magical!

Roald Dahl

Again, I don’t remember reading all the Roald Dahl novels, but James and the Giant Peach left a big impression. The characters were so interesting, and the idea of escaping from home on an enormous rolling piece of fruit was very exciting to me in those days…!

Sir Arthur Conan Doyle

I read The Complete Adventures of Sherlock Holmes when I was a boy, and it’s probably still the longest book I’ve ever read. I remember vividly that the edition I read was 1,227 pages long! I listened to the whole thing again recently in a very good audiobook edition read by Stephen Fry, and it was just as good second time around. I loved the mystery of the stories, and I still read a lot of crime fiction even now. I’ve always had a very analytical mind, so Holmes’s brilliant deductions were always enjoyable to read about.

Charlie Higson

The Young Bond novels weren’t around when I was young, but I read the first few as an adult, and I enjoyed them. James Bond is a classic fictional creation that appeals to boys in particular, and I think I would’ve lapped it up as a teenager. The first one is called Silverfin. Once you’ve read it, you’ll be hooked!

Jane Austen

Jane Austen introduced me to irony with the immortal opening line from Pride and Prejudice, but the first of her novels that I read was actually Emma. I had to read it at school as part of my preparation for the Oxford entrance exam, and I didn’t like it at first. However, that was just because I didn’t understand what was going on. Once my English teacher Mr Finn had explained that the character of Emma is always wrong about everything, I found it very funny and enjoyable. They say that ‘analysing’ a book can sometimes ruin it, but in this case it was quite the opposite.

Ernest Hemingway

“If Henry James is the poodle of American literature, Ernest Hemingway is the bulldog. What do you think?” I was once asked that question in an interview at the University of East Anglia, and I had no idea how to reply! As it happens, Hemingway was one of my favourite authors. My interviewer called his style ‘macho’, but that wasn’t the appeal for me. I simply liked the stories and the settings. I particularly loved the bull-fighting scenes in The Sun Also Rises, and there was just a glamour to the characters and the period that I really enjoyed. If you don’t know where to start, try The Old Man and the Sea. It’s very simple and very short, but very, very moving.

Algebra

Nothing makes the heart of a reluctant mathematician sink like an algebra question.algebra

Algebra is supposed to make life easier. By learning a formula or an equation, you can solve any similar type of problem whatever the numbers involved. However, an awful lot of students find it difficult, because letters just don’t seem to ‘mean’ as much as numbers. Here, we’ll try to make life a bit easier…

Gathering terms

X’s and y’s look a bit meaningless, but that’s the point. They can stand for anything. The simplest form of question you’ll have to answer is one that involves gathering your terms. That just means counting how many variables or unknowns you have (like x and y). I like to think of them as pieces of fruit, so an expression like…

2x + 3y – x + y

…just means ‘take away one apple from two apples and add one banana to three more bananas’. That leaves you with one apple and four bananas, or…

x + 4y

Here are a few practice questions:

  1. 3x + 4y – 2x + y
  2. 2m + 3n – m + 3n
  3. p + 2q + 3p – 3q
  4. 2a – 4b + a + 4b
  5. x + y – 2x + 2y

Multiplying out brackets

This is one of the commonest types of question. All you need to do is write down the same expression without the brackets. To take a simple example:

2(x + 3)

In this case, all you need to do is multiply everything inside the brackets by the number outside, which is 2, but what do we do about the ‘+’ sign? We could just multiply 2 by x, write down ‘+’ and then multiply 2 by 3:

2x + 6

However, that gets us into trouble if we have to subtract one expression in brackets from another (see below for explanation) – so it’s better to think of the ‘+’ sign as belonging to the 3. In other words, you multiply 2 by x and then 2 by +3. Once you’ve done that, you just convert the ‘+’ sign back to an operator. It gives exactly the same result, but it will work ALL the time rather than just with simple sums!

Here are a few practice questions:

  1. 2(a + 5)
  2. 3(y + 2)
  3. 6(3 + b)
  4. 3(a – 3)
  5. 4(3 – p)

Solving for x

Another common type of question involves finding out what x stands for (or y or z or any other letter). The easiest way to look at this kind of equation is using fruit again. In the old days, scales in a grocery shop sometimes had a bowl on one side and a place to put weights on the other. To weigh fruit, you just needed to make sure that the weights and the fruit balanced and then add up all the weights. The point is that every equation always has to balance – the very word ‘equation’ comes from ‘equal’ – so you have to make sure that anything you do to one side you also have to do to the other.

There are three main types of operation you need to do in the following order:

  1. Multiplying out any brackets
  2. Adding or subtracting
  3. Multiplying or dividing

Once you’ve multiplied out any brackets (see above), what you want to do is to simplify the equation by removing one expression at a time until you end up with something that says x = The Answer. It’s easier to start with adding and subtracting and then multiply or divide afterwards (followed by any square roots). To take the same example as before:

2(x + 3) = 8

Multiplying out the brackets gives us:

2x + 6 = 8

Subtracting 6 from both sides gives us:

2x = 2

Dividing both sides by 2 gives us the final answer:

x = 1

Simple!

Here are a few practice questions:

  1. b + 5 = 9
  2. 3y = 9
  3. 6(4 + c) = 36
  4. 3(a – 2) = 24
  5. 4(3 – p) = -8

Multiplying two expressions in brackets (‘FOIL’ method)

When you have to multiply something in brackets by something else in brackets, you should use what’s called the ‘FOIL’ method. FOIL is an acronym that stands for:

First
Outside
Inside
Last

This is simply a good way to remember the order in which to multiply the terms, so we start with the first terms in each bracket, then move on to the outside terms in the whole expression, then the terms in the middle and finally the last terms in each bracket. Just make sure that you use the same trick we saw earlier, combining the operators with the numbers and letters before multiplying them together. For example:

(a + 1)(a + 2)

First we multiply the first terms in each bracket:

a x a

…then the outside terms:

a x +2

…then the inside terms:

+1 x a

…and finally the last terms in each bracket:

+1 x +2

Put it all together and simplify:

(a + 1)(a + 2)

= a² + 2a + a + 2

=a² + 3a + 2

Here are a few practice questions:

  1. (a + 1)(b + 2)
  2. (a – 1)(a + 2)
  3. (b + 1)(a – 2)
  4. (p – 1)(q + 2)
  5. (y + 1)(y – 3)

Factorising quadratics (‘product and sum’ method)

This is just the opposite of multiplying two expressions in brackets. Normally, factorisation involves finding the Highest Common Factor (or HCF) and putting that outside a set of brackets containing the rest of the terms, but some expressions can’t be solved that way, eg a² + 3a + 2 (from the previous example). There is no combination of numbers and/or letters that goes evenly into a², 3a and 2, so we have to factorise using two sets of brackets. To do this, we use the ‘product and sum’ method. This simply means that we need to find a pair of numbers whose product equals the last number and whose sum equals the multiple of a. In this case, it’s 1 and 2 as +1 x +2 = +2 and +1 + +2 = +3. The first term in each bracket is just going to be a as a x a = a². Hence, factorising a² + 3a + 2 gives (a + 1)(a + 2). You can check it by using the FOIL method (see above) to multiply out the brackets:

(a + 1)(a + 2)

= a² + 2a + a + 2

=a² + 3a + 2

Subtracting one expression from another*

Here’s the reason why we don’t just write down operators as we come across them. Here’s a simple expression we need to simplify:

20 – 4(x – 3) = 16

If we use the ‘wrong’ method, then we get the following answer:

20 – 4(x – 3) = 16

20 – 4x – 12 = 16

8 – 4x = 16

4x = -8

x = -2

Now, if we plug our answer for x back into the original equation, it doesn’t balance:

20 – 4(-2 – 3) = 16

20 – 4 x -5 = 16

20 – -20 = 16

40 = 16!!

That’s why we have to use the other method, treating the operator as a negative or positive sign to be added to the number before we multiply it by whatever’s outside the brackets:

20 – 4(x – 3) = 16

20 – 4x + 12 = 16

32 – 4x = 16

4x = 16

x = 4

That makes much more sense, as we can see:

20 – 4(4 – 3) = 16

20 – 4 x 1 = 16

20 – 4 = 16

16 = 16

Thank Goodness for that!

Here are a few practice questions:

  1. 30 – 3(p – 1) = 0
  2. 20 – 3(a – 3) = 5
  3. 12 – 4(x – 2) = 4
  4. 24 – 6(x – 3) = 6
  5. 0 – 6(x – 2) = -12

Other tips to remember

  • If you have just one variable, leave out the number 1, eg 1x is just written as x.
  • When you have to multiply a number by a letter, leave out the times sign, eg 2 x p is written as 2p.
  • The squared symbol only relates to the number or letter immediately before it, eg 3m² means 3 x m x m, NOT (3 x m) x (3 x m).