Flipped Learning: Will It Totally Change The Way Maths Is Taught?

What Is Flipped Learning?

Flipped learn­ing turns con­ven­tional teach­ing on its head. Instead of deliv­er­ing a ‘one size fits all’ lec­ture to the whole class and then set­ting ques­tions for home­work, stu­dents are asked to learn new top­ics for home­work (typ­i­cally via online video tuto­ri­als) and then tackle ques­tions in the classroom.

Flipped Learning

Flipped Learn­ing

This allows the teacher to spend less time in front of the whole class and more time assist­ing stu­dents one to one. Class­room time may also be used to pro­mote inter­ac­tive, cre­ative and engag­ing shared experiences.

Stu­dents learn a new topic at their own pace and only move on when they’ve mas­tered it. The tech­nol­ogy enables teach­ers to pre­cisely mon­i­tor the progress of each stu­dent. Teach­ers are able to focus their efforts on help­ing stu­dents when they’re stuck.

Ori­gins of Flipped Learning

I’m not sure it’s pos­si­ble to exactly pin down who “invented” the con­cept of ‘Flipped Learn­ing’ but Salman Khan cer­tainly pop­u­larised it through his Khan Acad­emy and this Ted Talk: Let’s Use Video To Rein­vent Education -

At the time of writ­ing this video has over 3.3 mil­lion views. The Khan Acad­emy has inspired teach­ers in the USA and across the world– see these class­room case stud­ies. The major­ity (all?) of those case stud­ies are videos and they make inspir­ing viewing.

One of the first adopters was the Los Altos School Dis­trict (in Cal­i­for­nia, but the case stud­ies include exam­ples from Spain, Ire­land and Peru), here’s its video:-

Advan­tages of Flipped Learning

I’m sure that Flipped Learn­ing is not a panacea and much still depends on the qual­ity of teach­ing but I think the case for flipped learn­ing is com­pelling. Based on some of the infor­ma­tion and video sources listed above, I can see the fol­low­ing advantages:-

Stu­dents learn at their own pace - Stu­dents can be asked to study tuto­ri­als that are appro­pri­ate to their cur­rent level.  They can then work on that topic at their own pace. Able stu­dents can race ahead, less able stu­dents can make steady progress. In his video (see above) Salman Khan high­lighted the fact that most (all?) stu­dents strug­gle at some point. Some stu­dents may get stuck at the early stages but, once they mas­ter a topic, they can catch up and even over­take their class­mates. Nobody is dis­counted due to a poor start.

Stu­dents mas­ter top­ics - Stu­dents do not move on from a topic until they have mas­tered it. This avoids gaps in under­stand­ing and ensures that all stu­dents mas­ter the basics.

Indi­vid­ual Per­for­mance can be mon­i­tored - The tech­nol­ogy enables teach­ers to track each stu­dents per­for­mance pre­cisely. If a stu­dent is stuck, the teacher can inter­vene promptly or even ask another stu­dent (who has mas­tered the topic in ques­tion) to help.

Inde­pen­dent Learn­ing - Stu­dents learn how to learn independently.

Self-reliant - I guess this is related to inde­pen­dent learn­ing. The tra­di­tional method of teach­ing can make it dif­fi­cult when a stu­dent is stuck. Many peo­ple find it chal­leng­ing to admit they don’t under­stand some­thing, espe­cially in front of their peers and if they feel that they should really know it already. Flipped Learn­ing gives stu­dents the oppor­tu­nity to go back and revise trou­ble­some top­ics with­out pres­sure from their teacher and peers.

Engage­ment - Stu­dents enjoy the chal­lenge of work­ing through the video and ques­tions and mov­ing onto the next level.

Improved Teach­ing - Less time deliv­er­ing ‘one size fits all’ lec­tures, less time mark­ing and more time one to one teach­ing and more time on chal­leng­ing and engag­ing activ­i­ties and devel­op­ing math­e­mat­i­cal think­ing and problem-solving skills.

Equal Oppor­tu­nity - All stu­dents, no mat­ter what their back­ground, past expe­ri­ence or con­fi­dence lev­els have access to the tools to enable them to reach their poten­tial. Flipped Learn­ing empow­ers students.

Mentoring/Parents - Tech­nol­ogy makes it pos­si­ble for stu­dents to be men­tored by any­body. A stu­dent in Mum­bai can be assisted by a teacher or vol­un­teer in Lon­don. This clearly raises ques­tions of safety and qual­ity, but these are not insur­mount­able. Par­ents can get involved and learn along­side their children.

Con­tin­u­ous Improve­ment - Because Flipped Learn­ing is dri­ven by tech­nol­ogy and data it offers the poten­tial to bench­mark and imple­ment best prac­tice across schools and countries.

Dis­ad­van­tages of Flipped Learming

As you’ve prob­a­bly guessed, I’m more than a lit­tle biased, so I find it dif­fi­cult to think of many dis­ad­van­tages. I’d be inter­ested to hear about any neg­a­tive expe­ri­ences asso­ci­ated with Flipped Learning.

Tech­nol­ogy avail­abil­ity — May require stu­dents to have access to inter­net at home.

Reliance on home­work - If the Flipped Learn­ing model used requires stu­dents to study at home, some stu­dents (for a mul­ti­tude of rea­sons) may not do this work.

Mis­ap­pli­ca­tion - It’s pos­si­ble (as is the case with all teach­ing tech­niques) for Flipped Learn­ing to be poorly exe­cuted. For exam­ple, all stu­dents could be asked to study the same tuto­ri­als rather than tai­lor­ing to suit indi­vid­ual needs.  It would also be pos­si­ble to focus on just push­ing stu­dents through the syl­labus and ignore the oppor­tu­ni­ties for rich learn­ing and shared experiences.

The Future of Flipped Learning

I’m sure that Flipped Learn­ing will become increas­ingly pop­u­lar. As its effec­tive­ness becomes clear, stu­dents will expect it and par­ents demand it. At the moment the Khan Acad­emy dom­i­nates the flipped learn­ing land­scape but teach­ers will develop local solu­tions. For exam­ple, two young Maths teach­ers in Eng­land have devel­oped an impres­sive web­site with videos, ques­tions and check­lists cov­er­ing Key Stage 3, GCSE and A level maths (see my review of Hegarty Maths here).

Most of the flipped learn­ing appli­ca­tions I’ve seen seem to start at Year 5 or Year 6. Why not start them young? I’m sure that many 5 year-olds would be com­fort­able learn­ing maths on a tablet.

What would cement learn­ing bet­ter than mak­ing a video? Why not let chil­dren teach other via videos? Educreations.com hosts edu­ca­tional videos and half of them are made by chil­dren– here’s an excel­lent exam­ple. Videos made by stu­dents are more pow­er­ful than pol­ished videos made by teach­ers– they can high­light both mis­takes and break­through moments (as per the example).

Flipped Learn­ing — Summary

In my opin­ion Flipped Learn­ing has huge poten­tial. If it’s imple­mented cor­rectly, with the appro­pri­ate tech­nol­ogy and use of data, I’m sure it could allow the vast major­ity of stu­dents to reach their poten­tial. How­ever, I’m not a teacher, just a par­ent who has become fas­ci­nated by this approach. What do you think about Flipped Learning?

 

 

 

 

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Maths Curriculum Changes September 2014 Key Stage 1 & Key Stage 2

Overview of Ages, School Years, Key Stages and Testing

This table maps ages, years, key stages and assess­ment points. For more details refer to gov­ern­ment web­site:- https://www.gov.uk/national-curriculum/overview

Maths Overview

Ages, School Years, Key Stages and Test­ing
Age
Year
Key Stage
Test­ing
Aver­age Level of Attainment
3 to 4Early Years
4 to 5Recep­tionEarly Years
5 to 6Year 1KS 1
6 to 7Year 2KS 1Teacher Assess­ment2
7 to 8Year 3KS 2
8 to 9Year 4KS 2
9 to 10Year 5KS 2
10 to 11Year 6KS 2
National tests and teacher assessments4
11 to 12Year 7KS 3
12 to 13Year 8KS 3
13 to 14Year 9KS 3Teacher Assess­ments5/6
14 to 15Year 10KS 4Some chil­dren take GCSE’s
15 to 16Year 11KS 4Most chil­dren take GCSE’s

Changes to the Maths Curriculum

The national cur­ricu­lum is chang­ing from Sep­tem­ber 2014. For­tu­nately the changes do not impact the frame­work out­lined in the above table but they do alter the cur­ricu­lum within each year. As with all changes, this cre­ates the need for tran­si­tional arrange­ments. In the school year 2014/2015, Year 2 and Year 6 pupils will be taught the cur­rent pro­gramme of study and sit the cur­rent key stage 1 and key stage 2 tests respec­tively. See here for more details:- https://www.gov.uk/government/collections/national-curriculum

The Times Edu­ca­tion Sup­ple­ment is a superb resource of online resources pri­mar­ily aimed at teach­ers — tes.co.uk - but any­one may reg­is­ter (teach­ers, par­ents and stu­dents). I found this page– NC2014 Maths Com­pared to 2006 Pri­mary Frame­work - which has links to detailed analy­ses of the changes for each of the Pri­mary School years (1 through 6). You do have to reg­is­ter to gain access to them (in pdf for­mat) but if you want a thor­ough, com­plete and easy to under­stand write-up, this is the best I’ve seen.

Why Change?

This arti­cle in the Tele­graph sets out Michael Gove’s rea­son­ing:- “I want my chil­dren, who are in pri­mary school at the moment, to have the sort of cur­ricu­lum that chil­dren in other coun­tries have, which are doing bet­ter than our own,” he said.

Michael Gove

Michael Gove Attri­bu­tion new3dom3000 — Flickr Commons

Because, when my son and daugh­ter grad­u­ate from school and then either go on to uni­ver­sity or into the work­place, they’re com­pet­ing for col­lege places and jobs with folk from across the globe, and I want my chil­dren to receive an edu­ca­tion as rig­or­ous as any country’s.”

Some of the head­line changes to maths:-

  • In the first year of school, chil­dren will be expected to read and write num­bers up to 100, count in mul­ti­ples of twos, fives and tens and know sim­ple sums off by heart.
  • Alge­bra to be taught at age 10.
  • Learn up to the twelve times table by age 9 — as opposed to learn­ing up to the ten times table before they leave pri­mary school.

The changes have not been uni­ver­sally wel­comed. In this arti­cle Rebecca Han­son (a pri­mary maths adviser) argues that  “The new cur­ricu­lum has no ratio­nale and no prece­dent. It is unique in demand­ing that sub­stan­tial quan­ti­ties of abstract math­e­mat­ics be taught to six-year-olds — some­thing which all other cur­ric­ula pro­tect against as the dam­age doing so causes to the men­tal devel­op­ment of some chil­dren is well understood.

With­out proper time spent on mas­ter­ing the foun­da­tion skills needed to apply the Sin­ga­pore curriculum’s approach, chil­dren will strug­gle and fail under the new math­e­mat­ics cur­ricu­lum. Its imple­men­ta­tion needs to be imme­di­ately suspended.”

Despite this resis­tance, the changes are going ahead and they are more exten­sive than the head­line changes high­lighted above. The fol­low­ing tables show a sum­mary of the changes only (what’s been removed and what’s been added) to each year:-

Sum­mary of Changes to Each Year (1 to 6 inclusive)

Please note the tables below only list the CHANGES (what’s been removed and what’s been added).

Year 1

Removed
Added
Data handling/Statistics is removed from Y1
Count­ing & writ­ing numer­als to 100
No spe­cific require­ment to describe patternsWrite num­bers in words up to 20
No spe­cific require­ments to describe ways of solv­ing
prob­lems or explain choices
Num­ber bonds secured to 20
Use of vocab­u­lary such as equal, more than, less than, fewer, etc.

Year 2

Removed
Added
Round­ing two-digit num­bers to the near­est 10Solv­ing prob­lems with sub­trac­tion
Halving/doubling no longer explic­itly required Finding/writing frac­tions of quan­ti­ties (and lengths)
Using lists/tables/diagrams to sort objects
Adding two 2-digit num­bers
Adding three 1-digit num­bers
Demon­strat­ing com­mu­ta­tiv­ity of addi­tion & mul­ti­pli­ca­tion
Describ­ing prop­er­ties of shape (e.g. edges, ver­tices)
Mea­sur­ing tem­per­a­ture in °C
Tell time to near­est 5 min­utes
Make com­par­isons using < > = symbols
Recog­nise £ p sym­bols and solve sim­ple money problems

Year 3

Removed
Added
Spe­cific detail of problem-solving strate­gies (although the require­ment to solve prob­lems remains) Adding tens or hun­dreds to 3-digit numbers
Round­ing to near­est 10/100 moves to Year 4
For­mal writ­ten meth­ods for addition/subtraction
Reflec­tive sym­me­try moves to Year 4 8 times tables replaces 6 times tables
Con­vert­ing between met­ric units moves to Year 4Count­ing in tenths
No require­ment to use Carroll/Venn dia­grams
Com­par­ing, order­ing, adding & sub­tract­ing frac­tions
with com­mon denominators
Inden­ti­fy­ing angles larger than/smaller than right angles
Inden­tify hor­i­zon­tal, ver­ti­cal, par­al­lel and per­pen­dic­u­lar lines
Tell time to the near­est minute, includ­ing 24-hour clock and using Roman numer­als
Know the num­ber of sec­onds in a minute and the num­ber of days in each month, year and leap year

Year 4

Removed
Added
Spe­cific detail on lines of enquiry, rep­re­sent­ing prob­lems and find strate­gies to solve prob­lems and explain­ing meth­ods (i.e. largely from old Ma1)
Solv­ing prob­lems with frac­tions and dec­i­mals to two dec­i­mal places
Using mixed num­bers (moved to Y5)
Round­ing dec­i­mals to whole num­bers
Most ratio work moved to Y6
Roman numer­als to 100
Writ­ten divi­sion meth­ods (moved to Y5)Recog­nis­ing equiv­a­lent fractions
All cal­cu­la­tor skills removed from KS2 PoS Know­ing equiv­a­lent dec­i­mals to com­mon frac­tions
Mea­sur­ing angles in degrees (moved to Y5) Divid­ing by 10 and 100 (incl. with dec­i­mal answers)
Using fac­tor pairs
Trans­la­tion of shapes
Find­ing perimeter/area of com­pound shapes
Solve time con­ver­sion prob­lems

Year 5

Removed
Added
Detail of problem-solving process and data han­dling
cycle no longer required
Under­stand & use dec­i­mals to 3dp
Cal­cu­la­tor skills moved to KS3 Solve prob­lems using up to 3dp, and fractions
Prob­a­bil­ity moves to KS3Write %ages as frac­tions; frac­tions as dec­i­mals
Use vocab­u­lary of primes, prime fac­tors, com­pos­ite num­bers, etc.
Know prime num­bers to 20
Under­stand square and cube num­bers
Use stan­dard mul­ti­pli­ca­tion & divi­sion meth­ods for up
to 4 dig­its
Add and sub­tract frac­tions with the same
denominator
Mul­ti­ply proper frac­tions and mixed num­bers by whole numbers
Deduce facts based on shape knowl­edge
Dis­tin­guish reg­u­lar and irreg­u­lar poly­gons
Cal­cu­late the mean aver­age

Year 6

Removed
Added
Detail of problem-solving processes no longer explicit
Com­pare and order­ing frac­tions greater than 1
Divis­i­bil­ity tests
Long divi­sion
Cal­cu­la­tor skills move to KS3 PoS
4 oper­a­tions with frac­tions
Rota­tion moves to KS3
Cal­cu­late dec­i­mal equiv­a­lent of frac­tions
Prob­a­bil­ity moves to KS3
Under­stand & use order of operations
Median/Mode/Range no longer required
Plot points in all 4 quad­rants
Con­vert between miles and kilo­me­tres
Name radius/diameter and know rela­tion­ship
Use for­mu­lae for area/volume of shapes
Cal­cu­late area of tri­an­gles & parallelograms
Cal­cu­late vol­ume of 3-d shapes
Use let­ters to rep­re­sent unknowns (algebra)
Gen­er­ate and describe lin­ear sequences
Find solu­tions to unknowns in prob­lems

What Do You Think About the Changes?

I’ve tried to sum­ma­rize how the changes will be phased in, rea­sons for the changes, crit­i­cisms of the changes and a sum­mary of what’s been added and what’s been removed.

What do you think about the changes? Are they nec­es­sary? Are they easy to under­stand? Will they help schools, par­ents and chil­dren to improve maths standards?

 

 

 

 

 

 

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How Numeracy is Taught in Primary Schools

Number Square

Num­ber Square

This should be more accu­rately titled “How My Chil­dren Were Taught Numer­acy By Their Pri­mary School”. I know that dif­fer­ent schools have dif­fer­ent strate­gies and every­thing is under con­stant review. How­ever I think it is true to say that all pri­mary schools use tech­niques that were not taught to the par­ents of today when they were at school. I’ve used my children’s expe­ri­ence to

  • Under­stand some of the new techniques
  • Under­stand why numer­acy teach­ing has changed.

Why Has Numer­acy Teach­ing Changed?

When I were a lad we were taught how to count, add up and take away. We learned our times tables and then we tack­led long mul­ti­pli­ca­tion and divi­sion. That’s how I seem to remem­ber  it — but it was a few years ago. Now there appears to be a bewil­der­ing range of addi­tional tech­niques; num­ber lines, count­ing on, num­ber squares, num­ber bonds, chunk­ing, the grid method and the lat­tice method to name a few.  When my chil­dren were taught using these meth­ods, I couldn’t under­stand the logic. Why are all these addi­tional tech­niques necessary?

For­tu­nately my local pri­mary school sent par­ents “A guide for par­ents on how addi­tion, sub­trac­tion, mul­ti­pli­ca­tion and divi­sion are taught”.  After read­ing this guide, I began to under­stand not only the tech­niques but also the logic of teach­ing a range of dif­fer­ent meth­ods to achieve the same end. This is how the four rules (addi­tion, sub­trac­tion, mul­ti­pli­ca­tion and divi­sion) are taught:

Addi­tion

Count­ing On — Hold one num­ber in your head and count on.

Using a Known Fact — Rapid response to a fact known by heart, for example–

Num­ber Bonds — Learn pairs of num­bers that make 10 or 20 and recall them immediately.

Using a Derived Fact — Using a known fact to work out a new one, for exam­ple if 15 + 5 = 20 (num­ber bond) then 15 + 6 = 21.

Hun­dred Squares - Use knowl­edge of mul­ti­ples of ten to add, for exam­ple 30 + 50 or 32 + 50. Num­ber squares maybe used.

Adding Sev­eral Num­bers - A num­ber of tech­niques; for exam­ple — start­ing with the largest num­ber or look­ing for pairs of num­bers that make 10.

Par­ti­tion­ing and Recom­bin­ing - Split­ting num­bers up. Add the ‘tens’ first and then add the ‘units’. For example:-

46 + 63 = (40 + 60) + (6 + 3) = 100 + 9 = 109

Count­ing on in Mul­ti­ples of 100, 10 or 1 - Start with the largest num­ber and count on by par­ti­tion­ing the sec­ond num­ber and adding the tens and then the units. Even the units can be split to make it eas­ier. For exam­ple:- 76 + 47 = 76 + 40 + 7

 

Adding Sig­nif­i­cant Dig­its First - This is an exten­sion of par­ti­tion­ing; add hun­dreds, then tens and then units. For example:-

Adding Significant Digits First

Adding Sig­nif­i­cant Dig­its First

Although this is set out for­mally in columns, chil­dren are expected to develop men­tal strate­gies to do these sums in their heads.

Adding Near Mul­ti­ples of 10 - Using the fact that to add 9 is the same as to add 10 and then sub­tract 1. It’s a men­tal strat­egy, for example:-

53 + 9 = (53 + 10) — 1 = 63 — 1 = 62

76 + 39 = (76 + 40) — 1 = 116 — 1 = 115

328 + 199 = (328 + 200) — 1 = 528 — 1 = 527

Com­pen­sa­tion - This builds on ‘adding near mul­ti­ples of 10′. Add too much and then sub­tract the dif­fer­ence. For example:-

754 + 86 = (754 + 100) — 14 = 854 — 14 = 840

367 + 92 = (367 + 100) — 8 = 467 — 8 = 459

Stan­dard Writ­ten Meth­ods with Exchang­ing - Finally the stan­dard method I remem­ber from school– care­fully writ­ing in columns so that units, tens, hun­dreds etc line up and car­ry­ing below the line:-

Addition:- Standard Written Method with Exchanging

Addi­tion:- Stan­dard Writ­ten Method with Exchanging

Sub­trac­tion

Count­ing Out - To find the answer to 9 — 3, hold up 9 fin­gers and fold down three.

Count­ing Back From — To find the answer to 9 — 3 count back three num­bers from 9.

Count­ing Back To — Count back from the largest num­ber to the lowest.

Num­ber Bonds - If the num­ber bonds to 1o and 20 are known (see addi­tion, above), these facts can be used for sub­trac­tion. For exam­ple. if 6 + 4 = 10 then 10 — 4 = 6.

Using Derived Facts - If it’s known (from num­ber bonds) that 20 — 7 = 13 then 21 — 7 must be 14.

Par­ti­tion­ing into Tens and Units - Split­ting num­bers up into tens and units

For exam­ple:-

47 — 24 = (40 and 7) — (20 and 4)

40 — 20 = 20

7 — 4 = 3

20 + 3 =23

A vari­a­tion is to only par­ti­tion the num­ber to be subtracted:-

47 — 24 = 47 — 20 — 4 = 27 –4 = 23

Par­ti­tion­ing can be extended to thou­sands and decimals.

Count­ing Up — Sub­trac­tion by addi­tion! Start with low­est of the two num­bers and count up to the high­est. Best explained with an example:

132 — 56

Counting Up - Subtraction by Addition!

Count­ing Up — Sub­trac­tion by Addition!

Then just add 70 + 4 + 2 = 76.

This method can be used with larger num­bers and decimals.

Sub­tract­ing Near Mul­ti­ples of 10 — Sub­tract­ing by 9 is the same as sub­tract­ing by 10 and adding 1.

54 — 9 = (54 — 10) + 1 = 44 + 1 = 45

76 — 39 = (76 — 40) + 1 = 36 + 1 = 37

438 — 199 = (438 — 200) + 1 = 238 + 1 = 239

Same Dif­fer­ence - Add or sub­tract an amount to make one of the num­bers more man­age­able as long as you do the same to both num­bers. Best explained with some examples:-

83 — 47 - it’s eas­ier to sub­tract 50 so add three to both num­bers– the dif­fer­ence is the same:-

83 — 47 = 86 — 50 =36

Neg­a­tive Num­bers — A method to avoid “bor­row­ing” (see below) by par­ti­tion­ing the num­bers in the sum:-

Subtraction by using Negative Numbers

Sub­trac­tion by using Neg­a­tive Numbers

Bor­row­ing / Tra­di­tional Method — This is the method I remem­ber from school.

This is best explained by see­ing all the work­ings, step by step. This video from the Khan Acedemy is great. It not only shows how to use the bor­row­ing method — it also starts by break­ing it down and show­ing the logic that lies behind it.

Mul­ti­pli­ca­tion

Repeated Addi­tion — This method shows that 4 x 3 = 3 + 3 + 3 + 3

Dou­bling — First dou­ble the tens, then the units and add the two answers together.

For exam­ple: 47 x 2 = (40 x 2) + (7 x 2) = 80 + 14 = 94

Mul­ti­ply­ing by Mul­ti­ples of 10 — Dig­its move one place to the left when mul­ti­ply­ing by 10, two places to the left when mul­ti­ply­ing by 100 and three places to the left when mul­ti­ply­ing by 1000. For example:

9 x 10 = 90

9 x 100 = 900

0.3 x 1000 = 300

6 x 30 = 6 x 3 x 10 = 18 x 10 = 180

Mul­ti­ply by 4, 5, 8, 20 and 25

To mul­ti­ply by 4, dou­ble and dou­ble again.

To mul­ti­ply by 5, mul­ti­ply by 10 and then halve.

To mul­ti­ply by 20, mul­ti­ply by 10 and then dou­ble (or vice versa)

To mul­ti­ply by 8, mul­ti­ply by 4 and double.

To mul­ti­ply by 25, mul­ti­ply by 100, halve and then halve again.

Mul­ti­ply­ing by Near Mul­ti­ples of 10 — To mul­ti­ply by 9 is the same as mul­ti­ply­ing by 10 and then sub­tract­ing the num­ber you mul­ti­plied. For example:

15 x 9 = (15 x 10) — 15 = 150 –15 = 135

13 x 29 = (13 x 30) — 13 = 390 — 13 = 377

Mul­ti­ply­ing using Fac­tors — Num­bers that appear com­pli­cated can often be bro­ken down into more man­age­able num­bers by find­ing their fac­tors. For example:-

35 x 22

35 = 7 x 5

22 = 2 x 11

so 35 x 22 = 7 x 5 x 2 x 11

Then look for easy mul­ti­pli­ca­tions within the new sum:

7 x 5 x 2 x 11 = 7 x 11 x 5 x 2 = 77 x 10 = 770

so 35 x 22 = 770

Mul­ti­ply­ing by Dou­bling and Halv­ing — In a mul­ti­pli­ca­tion prob­lem, it’s pos­si­ble to achieve the same answer by dou­bling on of the num­bers and halv­ing the other. For example:

24 x 25 = 12 x 50 = 6 x 100 = 600

Writ­ten Methods

The Grid Method

I explained this in detail here — The Grid Method

There’s also:- The Lat­tice Method

Long Mul­ti­pli­ca­tion

The tra­di­tional method that I remember:

43 x 36

Long Multiplication

Long Mul­ti­pli­ca­tion

Divi­sion

Shar­ing

If you have 6 sweets and you are shar­ing them a friend, how many do you get each? Often done with the actual objects– in this case– sweets!

Group­ing

To solve 8 ÷ 2 put 8 objects into groups of 2 and see how many groups there are.

Alter­na­tively use a num­ber line. For exam­ple 18 ÷ 3 could be read as “How many groups of 3 are needed to reach 18?”

Use Number Line to Divide by Grouping

Use Num­ber Line to Divide by Grouping

Divid­ing by 10, 100, 1000 — To divide by 10, move the dig­its one place to the right, the dec­i­mal point remains fixed. To divide by 100, move the dig­its two places to the right and to divide by 1,000 move the dig­its three places to the right.

90 ÷ 10 = 9

90 ÷ 100 = 0.9

90 ÷ 1000 = 0.09

Find­ing Quarters

To divide by 4, half and half again.

Find­ing Frac­tions of Amounts

To find a frac­tion of an amount is to per­form a divi­sion. For example:

¼ of 32 = 32 ÷ 4 = 8

To find vul­gar frac­tions, divide by the denom­i­na­tor (the bot­tom num­ber) and mul­ti­ply by the numer­a­tor (the top num­ber). For example:

¾ of 72 = (72 ÷ 4) x 3 = 18 x 3 = 54

Using Fac­tors With Division

A divi­sion can be made more man­age­able by find­ing the fac­tors of the num­ber you are divid­ing by. For example:

90 ÷ 6   6 = 3 x 2 so 2 and 3 are fac­tors of 6

90 ÷ 3 = 30 and 30 ÷ 2 = 15 there­fore 90 ÷ 6 = 15

Writ­ten Methods

Chunk­ing

Another case where a video is worth a mil­lion words:-

The Bus Stop Method/Long Division

This is the tra­di­tional, for­mal method I remem­ber from school. This is cer­tainly best explained  with a video, here’s a link to the Khan Academy’s intro­duc­tion to long divi­sion.

So..Why Has Numer­acy Teach­ing Changed in Pri­mary Schools

The above details how my children’s pri­mary school approaches numer­acy but it doesn’t explain why meth­ods have changed. I must admit, before I went through this infor­ma­tion, I didn’t under­stand why it was nec­es­sary to teach any­thing other than the for­mal meth­ods I remem­bered such as car­ry­ing over, bor­row­ing, long mul­ti­pli­ca­tion and long divi­sion. I thought that these tra­di­tional meth­ods were the most robust and teach­ing other meth­ods was con­fus­ing and counter productive.

I now see it dif­fer­ently. I can see that once a child has mas­tered count­ing and is start­ing to learn times tables, it’s a huge jump to these for­mal meth­ods. The inter­me­di­ate meth­ods out­lined above act as step­ping stones to help chil­dren fully under­stand num­bers. They give all chil­dren the best chance of mak­ing progress, not just the minor­ity that just find numer­acy easy. It’s dif­fi­cult, when you know how to add up, to fully appre­ci­ate the ben­e­fit of this step by step approach. Your nat­ural incli­na­tion is to be impa­tient and say why not just teach the final method. I now under­stand how these steps should allow chil­dren of all abil­i­ties to build their numer­acy skills in a rel­a­tively con­sis­tent and reli­able manner.

I can see some chal­lenges with these meth­ods. Firstly it must be painful for some chil­dren to go through all these steps– espe­cially if they have been taught the fun­da­men­tals at home. I sup­pose this makes the case for some “flipped learn­ing” even at pri­mary school– so all chil­dren can learn at their own pace. Sec­ondly, when teach­ing meth­ods change from gen­er­a­tion to gen­er­a­tion, it’s very help­ful for schools to pro­vide as much infor­ma­tion as pos­si­ble to par­ents. In addi­tion, par­ents may have to “go back to school” if they want to best sup­port their children.

These are just my views based on a sam­ple size of just two — my chil­dren! What do you think? Do you think that these new meth­ods are bet­ter than the tra­di­tional, for­mal methods?

Posted in Primary School Maths | Leave a comment

Review Hegarty Maths

Hegarty Maths — The Future of Maths Learning?

I selected HegartyMaths.com as one of my ten favourite maths web­sites– see here. I think it pro­vides such an excel­lent ser­vice and range of resources that I decided to write a full review.

Hegarty Maths– Home­page Screenshot

Hegarty Maths homepage

Hegarty Maths homepage

Overview

The web­site is free to use and owned by two Lon­don based Maths teach­ers; Mr Hegarty and Mr. Arnold and includes Key Stage 3, GCSE and A Level Maths. There are nearly 1,000 videos cov­er­ing all top­ics for each syl­labus and fully explained — these teach­ers prac­tice what they preach; they show all their work­ings — exam solu­tions of past papers.

Design

Mr. Hegarty and Mr. Arnold have very sen­si­bly used a cou­ple of young web design and web devel­op­ment experts. As a result the web­site is well pre­sented and extremely easy to navigate.

Videos

The videos use screen cap­ture and voice-over– so you see a screen as Mr. Hegarty or Mr. Arnold work on it and they talk you through the tutorial/solution. They are both con­fi­dent, flu­ent speak­ers and bring their own per­son­al­i­ties to the videos. Top­ics are explained thor­oughly at a rea­son­able pace and there are usu­ally a few exam­ples to rein­force learning.

Areas Cov­ered

All areas likely to be taught in sec­ondary school are cov­ered– Key Stage 3, GCSE and A Level. In my opin­ion this full range helps to make the web­site inclu­sive and acces­si­ble– only a very lim­ited prior knowl­edge is assumed. If you’re strug­gling with GCSE, you can go back for a KS3 refresher. There is as much empha­sis on foun­da­tion GCSE as there is on higher GCSE.

Flipped Learn­ing

Both teach­ers are strong advo­cates of the “Flipped Learn­ing” method. I’m not sure who invented this method but the Khan Acad­emy has cer­tainly cham­pi­oned it:-

The clue’s in the name– the tra­di­tional method of learn­ing is flipped– instead of a tuto­r­ial fol­lowed by ques­tions for home­work — flipped learn­ing has pupils watch­ing tuto­r­ial videos for home­work and then tack­ling ques­tions in the class­room. This has a num­ber of advan­tages; pupils can learn at their own pace, quick stu­dents are not held back, nobody is left behind and, per­haps most impor­tantly, teach­ers spend less time mark­ing and “lec­tur­ing” the whole class and far more time one to one with pupils help­ing them as they work in the class­room. The tech­nol­ogy also allows teach­ers to eas­ily track the progress of each pupil and see where they are “stuck”.  Ulti­mately this approach may enable teach­ers and pupils to spend less time wor­ry­ing about exams and more time on chal­leng­ing activ­i­ties to develop math­e­mat­i­cal think­ing and prob­lem solv­ing skills.

Check­list

This is an inno­v­a­tive fea­ture. You can reg­is­ter with Hegarty Maths (as straight­for­ward as you might expect) and select a course or courses to fol­low. This gen­er­ates check­lists which are linked to the rel­e­vant video tuto­ri­als. As you com­plete each tuto­r­ial you indi­cate whether you fully under­stand, par­tially under­stand or do not under­stand at all. This infor­ma­tion is pre­sented in a traf­fic light pie chart– each topic in your course(s) is colour coded as green (100% under­stand), amber/yellow (need more work) or red (don’t under­stand). At the time of writ­ing there are check­lists for:

  • Maths GCSE (Edex­cel) — Higher
  • Maths GCSE (Edex­cel) — Secure Grade C or B
  • Maths A Level (Edex­cel) — Core 1,2,3 & 4
  • Maths A Level (Edex­cel) — Sta­tis­tics 1
  • Maths A Level (Edex­cel) — Deci­sion 1
  • Maths A Level (Edex­cel) — Fur­ther Pure 1 & 2

Live

Another new fea­ture is live online tuto­ri­als. The first tri­als were one hour revi­sion ses­sions in April 2014 cov­er­ing spe­cific top­ics. Appar­ently they were a suc­cess and more ses­sions are “com­ing soon”.

Future

It’s clear that Messrs. Hegarty and Arnold are admir­ers of Salman Khan and his Acad­emy. They real­ized that there was a need for some­thing sim­i­lar in the UK but (to para­phrase Mr. Arnold) “tai­lored to use the same lan­guage, ter­mi­nol­ogy and math­e­mat­i­cal sym­bols that GCSE and A-Level stu­dents study”. I won­der if Hegarty Maths might be devel­oped to include (like the Khan Acad­emy) even more stats and graphs to track progress and coach­ing fea­tures to facil­i­tate the teach­ing of classes/groups.

Con­clu­sion

In my opin­ion Hegarty Maths is an excel­lent resource for pupils (at school, home tutored and mature) par­ents and teach­ers. The web­site caters for a wide range of courses and abil­i­ties and I think this helps to make it acces­si­ble to all. I’d say whether you’re a pupil, mature stu­dent, par­ent or teacher, just jump in and check it out.

Posted in Maths Websites | Leave a comment

Top Ten Maths Websites

Best Ten Maths Websites.

Please also check out my guide (only my per­sonal views) on “How to make the most of maths web­sites” which fol­lows this list of top ten websites.

I couldn’t order these one to ten, they all have dif­fer­ent strengths, so I chick­ened out and here they are in no par­tic­u­lar order:-

NRICH http://nrich.maths.org

NRICH Homepage

NRICH Home­page

This site is unique. It’s run by Cam­bridge Uni­ver­sity and the aim is to “enrich the math­e­mat­i­cal expe­ri­ences of all learn­ers”. There are sec­tions for teach­ers and stu­dents and cov­ers Key Stage 1 right through to A Level. If you’re look­ing to be spoon fed a syl­labus this is not the site for you. If you’re look­ing to be chal­lenged and stim­u­lated this site sets the standard.

NRICH (askN­RICH) has an excel­lent forum. Stu­dents of all ages may ask ques­tions. Quot­ing the forum’s blurb:-

If you have a ques­tion about a par­tic­u­lar math­e­mat­i­cal prob­lem, or about math­e­mat­ics in gen­eral, Ask NRICH is the place to be.

Our team will do their best to help you. They are not here to give you answers to ques­tions, but they will help you to make sense of the math­e­mat­ics involved and to use what you know to think your way through the prob­lems yourself.

They are very quick and friendly! Most ques­tions get a use­ful reply within the day.”

BBC Bite­size http://www.bbc.co.uk/bitesize/

This is prob­a­bly the most well known resource but it’s still very good. There are sep­a­rate sec­tions for KS1, KS2, KS3, GCSE and higher maths. The web­site is well designed, easy to fol­low and makes good use of graph­ics. At the Maths GCSE level, each topic has a revi­sion, activ­ity and test section.

Brain Cells http://www.brain-cells.co.uk/gcse.html

This is mainly a pay site but there are many free Maths GCSE resources:- Revi­sion Pre­sen­ta­tions, Revi­sion Quizzes and Revi­sion Sheets. I like these mate­ri­als because they are very clear and require inter­ac­tion to complete.

Cor­bett Maths http://corbettmaths.com/

This site is free to use and the main focus is Maths GCSE. There are many videos, prac­tice ques­tions and revi­sion cards. There are two things I par­tic­u­larly like; the sim­ple but very effec­tive pre­sen­ta­tion and the use of Symbaloo.com to group the videos by GCSE grade ( for exam­ple these are all the GCSE Grade C videos - http://www.symbaloo.com/mix/gcsemathsgradec)

Exam Solu­tions http://www.examsolutions.net/

Another free site. There are many (over 2,800!) Maths GCSE and A Level videos plus exam ques­tions with thor­oughly worked solu­tions. I think it would be fair to say that it’s aimed at stu­dents who would expect to get grade B or higher at Maths GCSE.

Hegarty Maths http://www.hegartymaths.com/

Nearly 1,000 videos cov­er­ing KS3, GCSE and A Level Maths. This site is main­tained by two teach­ers with help from two young web developers/designers. The videos are clearly used by the site’s own­ers to help their stu­dents. Per­haps because they cover every­thing from KS3 to A Level, I think the videos have a friendly and acces­si­ble feel to them. I think this might be my favourite maths web­site– a full, detailed review to fol­low– watch this space! Here’s my full review of Hegarty Maths.

Khan Acad­emy http://www.khanacademy.org/math

Prob­a­bly the most famous tuto­r­ial videos. The acad­emy is per­fect for USA stu­dents but for UK stu­dents there’s no easy way to map the videos to KS1, KS2, KS3, GCSE etc. It’s still a pio­neer­ing, fan­tas­tic resource but I would sug­gest (if you’re study­ing out­side the USA) you need to have a firm under­stand­ing of what you need to learn (i.e. your detailed syl­labus) to get max­i­mum benefit.

Mr Barton’s Maths http://www.mrbartonmaths.com

Yet another labour of love by a UK Maths teacher. Mr. Bar­ton is the cura­tor of UK Maths online resources. He has sec­tions for teach­ers, par­ents and stu­dents. Mr. Bar­ton is also the Sec­ondary Maths Advi­sor for the TES (Times Edu­ca­tion Sup­ple­ment) - www.tes.co.uk

MyMaths - http://www.mymaths.co.uk

I know this web­site is not free but it is very widely used by schools. Typ­i­cally it’s paid for by schools to sup­port pupils– so for many chil­dren (includ­ing mine) it works out to be free. The site is pro­fes­sional and I’m sure a great help to teach­ers, par­ents and pupils. In my opin­ion the only thing it lacks is a per­son­al­ity– the sort of inter­est and iden­tity pro­vided by the use of videos on some of the other web­sites listed here.

The Maths Teacher http://www.themathsteacher.com/

David Smith is yet another exam­ple of a Maths teacher pro­vid­ing a superb free ser­vice. The site cov­ers GCSE and A level maths and, in my opin­ion, is the eas­i­est maths web­site to nav­i­gate.  For exam­ple, Maths GCSE is spilt between Foun­da­tion and Higher Sec­tions and for each topic there is a video, les­son notes (pdf to com­ple­ment the video) and an exer­cise sec­tion which has about 10 ques­tions and fully worked answers. Log­i­cal, well pre­sented and thor­ough. Then there are exam ques­tions and the same for­mat is fol­lowed; a video and pdf’s with notes and tran­script of ques­tions and answers.

What do you think?

Have I cap­tured the top ten maths web­sites? Are there any glar­ing omis­sions from my list? Please reply below to let me know about your favourite websites.

How to Make the Most of Maths Websites

When I started inves­ti­gat­ing how I could help my chil­dren with their Maths GCSE’s I was amazed to find so many great free maths web­sites. Some of the web­sites are writ­ten by full-time Maths teach­ers and it’s clear that they are real labours of love. The best web­sites have well pro­duced videos and sup­port mate­ri­als such as notes and test questions.

I could have included a num­ber of other web­sites, choos­ing the “best” web­sites is, of course, highly sub­jec­tive. I con­sid­ered the fol­low­ing fac­tors to make my choice:-

- Good qual­ity, easy to under­stand videos

- A clear struc­ture to make it easy to under­stand the con­tent of the website.

- Up to date mate­r­ial, well maintained.

- Author­ity and rel­e­vant expe­ri­ence of the web­site owner.

- Free! I’ve only included web­sites that are wholly free to use OR where sub­stan­tial sec­tions are free to use.

Using Maths Websites

Because there are so many free resources, it’s tempt­ing to think you could rely on them to get you through your exam. In my opin­ion that would be a big mis­take. As far as I can see none of these web­sites clearly map their mate­r­ial to give total con­fi­dence that all the top­ics required for any spe­cific exam board are 100% cov­ered. For exam­ple, a web­site may have excel­lent Maths GCSE mate­r­ial but if you are due to sit the AQA Maths GCSE, how do you know that web­site cov­ers all the top­ics required by AQA? In addi­tion you may be tar­get­ing a cer­tain grade– any­thing up to A star, how do you know where to focus?

Use Your Teacher

There is no sub­sti­tute for a teacher to guide you through your stud­ies. A teacher will help you to under­stand your whole syl­labus, which top­ics you need to study and which top­ics you need to focus on to get your tar­get grade. If you have access to a teacher make sure you get your money’s worth!

Find a Teacher?

What if you don’t have a teacher? Per­haps you’re a mature stu­dent. You might be sur­prised by the help that’s avail­able. Maths and Eng­lish are known as “Skills for Life” and the gov­ern­ment is keen to ensure that as many peo­ple as pos­si­ble have these skills (up to and includ­ing Level 2 or GCSE) so there may be help avail­able. Click here to find out more about gov­ern­ment help with numer­acy and maths GCSE.

Use a Textbook

If you don’t have a teacher, in my opin­ion you should use a text­book which is recog­nised by your exam board. So if you’re going to sit AQA Maths GCSE you need to get a text­book  which specif­i­cally cov­ers the AQA Maths GCSE. You also need to make sure that the text­book is up to date and cov­ers the cur­rent syllabus.

Shop Around, Mix and Match

Once you under­stand what you need to learn (either from your teacher or from your recog­nised text­book, see above) you can seek out the best web­sites that work for you. The fol­low­ing list of my per­sonal favourite ten maths web­sites only scratches the sur­face. There are many to choose from and it’s an ever grow­ing list. Obvi­ously you don’t have to stick to one web­site, you can mix and match. You choose videos at one web­site, pdf down­loads at another and  a forum at yet another. All web­sites have their strengths and weaknesses.

Inter­ac­tive and Procative

I’m sure that most peo­ple learn Maths most effi­ciently by actu­ally solv­ing Maths prob­lems. Pause videos and try ques­tions your­self rather than just watch­ing them all the way through. Some web­sites pro­vide forums to ask ques­tions, for exam­ple NRICH (see above) has an “Ask A Math­e­mati­cian” ser­vice. As NRICH is run by Cam­bridge Uni­ver­sity, I’m sure you’re likely to get some help­ful answers!

Sum­mary of How to Use Maths Websites

1. Under­stand what you need to learn, prefer­ably with a teacher’s guid­ance but, if not, by using a text­book approved by your exam board.

2. Shop around and mix and match. Find web­sites that work for you. Use dif­fer­ent web­sites for dif­fer­ent top­ics and services.

3. Be proac­tive and inter­ac­tive. Try to solve maths prob­lems before you’re shown the solu­tion. Join forums and ask questions.

Posted in Maths Websites | 2 Comments

Triangles, Polygons and Constructions

Inte­rior and Exte­rior Angles of a Triangle

First a cou­ple of rules using this triangle:-

Interior and Exterior Angles of a Triangle

Rule 1 — The sum of the angles of a tri­an­gle (x + y + z) equals 180°

Rule 2 — The exte­rior angle (angle w, above) of a tri­an­gle is equal to the sum of the two oppo­site angles (x and y, above)

Note that an exte­rior angle lies out­side the tri­an­gle and is made by extend­ing one sides of the triangle.

This video shows these 2 rules in action

Con­struct­ing Triangles

SAS and ASA

SAS — if you know the length of 2 sides and one angle you can con­struct a full triangle.

ASA - if you know 2 angles and the length of one side between those angles you can com­plete a triangle.

Here’s a video show­ing the con­struc­tion of a tri­an­gle based on SAS:-

Here’s a video show­ing the con­struc­tion of a tri­an­gle using ASA:-

Quadri­lat­er­als

A quadri­lat­eral is a shape formed by 4 straight lines. These are all types of quadrilateral:-

Special Quadrilaterals

The above were some spe­cial, named quadri­lat­er­als. All quadri­lat­er­als have cer­tain attributes:-

- Quadri­lat­er­als are 2D shapes bounded by four straight lines.

- The diag­o­nal — a straight line join­ing two oppo­site ver­tices or cor­ners– divides the quadri­lat­eral into two triangles.

We know that the sum of a triangle’s inte­rior angles adds up to 180° so it fol­lows that if  quadri­lat­er­als can be divided into 2 tri­an­gles then the sum of a quadrilateral’s inte­ri­ors angles add up to 180° plus 180° = 360°

Poly­gons

There are a num­ber of rules about ploy­gons. Quite often ques­tions will ask you to use these rules to cal­cu­late inte­rior or exte­rior angles.

A poly­gon is a two dimen­sional shape bounded by straight lines, so a tri­an­gle is a polygon.

Reg­u­lar poly­gons have sides that are the same length and all the angles in a reg­u­lar poly­gon are the same size. These are all reg­u­lar polygons:-

3 sides — Equi­lat­eral triangle

4 sides — Square

5 sides — Reg­u­lar Pentagon

6 sides — Reg­u­lar Hexagon

7 sides — Reg­u­lar Heptagon

8 sides — Reg­u­lar Octagon

The sum of the exte­rior angles of any poly­gon is 360°

In a reg­u­lar ploy­gon all the exte­rior angles are the same size and can be cal­cu­lated using this rule:-

Exte­rior angle of a reg­u­lar poly­gon = 360°/number of sides

So for a reg­u­lar pen­ta­gon (5 sides) each exte­rior angle will be:-

360°/5 = 72°

Quadri­lat­er­als (see above) are four-sided poly­gons. We saw that a quadri­lat­eral could be divided into 2 tri­an­gles by draw­ing a line from one ver­tex (cor­ner) to the oppo­site ver­tex and that this proved that the inte­rior angles of a quadri­lat­eral were 360° (180° + 180°).

In a sim­i­lar way any poly­gon can be divided into a num­ber of tri­an­gles. A quadri­lat­eral (4 sides) can be divided into 2 tri­an­gles, a pen­ta­gon (5 sides) can be divided into 3 tri­an­gles and hexa­gon can be divided into 4 tri­an­gles. Can you see the pat­tern or rules that’s emerging? -

The sum of the inte­rior angles of a poly­gon is (num­ber of sides — 2) x 180°

When you con­sider a polygon’s vor­tex and it’s inte­rior and exte­rior angle, note that the inte­rior and exte­rior angles lie on a straight line so:

Inte­rior angle + Exte­rior angle = 180°

OR

Inte­rior angle = 180° — Exte­rior angle

OR

Exte­rior angle = 180° — Inte­rior angle.

 

Posted in 23. Triangles Polygons and Constructions | Tagged , , | Leave a comment

Angles, Parallel Lines and Bearings

 

Angles In Par­al­lel Lines

We use arrow­heads to indi­cate that 2 lines are parallel:-

Arrowheads to indicate parallel lines

 

 

 

 

 

When a line crosses 2 par­al­lel lines it cre­ates pairs of equal angles

Inte­rior Angles

In the dia­gram below x and y are inte­rior angles.

Inte­rior angles add up to 180°

so x + y = 180°

Interior angles add up to 180 degrees

 

 

 

 

 

Here’s a video that explains inte­rior (or co-interior angles) very clearly:-

Cor­re­spond­ing Angles

What are cor­re­spond­ing angles? That’s best explained by a cou­ple of exam­ples. In the fol­low­ing dia­grams; a and b and c and d are cor­re­spond­ing angles. Note that the lines that join up cor­re­spond­ing angles make an ‘F’ shape:-

Corresponding angles are equal

 

 

 

 

 

Here’s a video that explains cor­re­spond­ing angles clearly and in detail:-

Alter­nate Angles

This dia­grams shows 2 pairs of alter­nate angles.

Alter­nate angles are equal. Notice how the lines make a ‘Z’ shape.

Alternate angles on parallel lines are equal

Here’s a video to explain alter­nate angles in par­al­lel lines:-

Bear­ings

Bear­ings are mea­sured clock­wise from the North:-

Bearings- example of 60 degrees

 

 

 

 

 

 

To get a fuller under­stand­ing of all bear­ings, con­sider the com­pass. Can you work out each direc­tion shown here as a bearing?

Compass bearings

 

 

 

 

 

 

 

 

NE = 045

E = 090

SE = 135

S = 180

SW = 225

W = 270

NW = 315

Here’s a video that explains bear­ings in some detail:-

Here’s another video that walks through some Bear­ings exam questions:-

Par­el­lel Lines and Bearings

You can use the fact that North lines are always par­al­lel and knowl­edge of angle facts to work out bearings.

This is best illus­trated with an exam­ple. In the fol­low­ing dia­gram, work out the bear­ing of B from A.

Bearings question using North , parallel lines and angle rules

 

 

 

 

 

 

 

 

There are 2 pos­si­ble approaches to answer­ing this ques­tion. Both start with draw­ing the North line from A so that we have 2 par­al­lel lines point­ing North.

Draw the parallel North line

 

 

 

 

 

 

 

 

The first method we can use is based on the fact that co-interior angles add up to 180°

There­fore we know the angle below marked in red must be 180° — 135° = 45°

Interior Angles add up to 180 degrees

 

 

 

 

 

 

 

 

But remem­ber bear­ings are mea­sured clock­wise from the North. So the bear­ing of B from A = 360 — 45 = 315 see the dia­gram below:-

Co-Interior Angle Solution to Bearing Question

 

 

 

 

 

 

 

 

 

Here’s an alter­nate solu­tion based on.….alternate angles. Again we start by draw­ing the North line from A:-

Draw the parallel North line

 

 

 

 

 

 

 

 

Then we find the alter­nate angles (it helps to extend the line below A and high­light the ‘Z’ shape we’re look­ing for):-

Deduce Bearings using alternate angles

 

 

 

 

 

 

 

 

 

 

 

Finally, remem­ber­ing that bear­ings are always mea­sured clock­wise from the North we just have to cal­cu­late the answer:-

Use alternate angles to deduce bearing- answer

Posted in 22. Angles Parallel Lines and Bearings | Leave a comment

Further Algebra

This another sec­tion where all the ques­tions are in the B to A* range. I think that algebra’s rep­u­ta­tion as being very dif­fi­cult is exag­ger­ated. Like all maths top­ics if you really get to grips with the basics, you’ll be able to tackle the more chal­leng­ing stuff. If you’re strug­gling step back to a lower level and when you’re more con­fi­dent come back to the higher level. A lit­tle rep­e­ti­tion and prac­tice goes a long way. You might not get it the first time but if you keep com­ing back to it, you’re more likely to succeed.

Fur­ther Simul­ta­ne­ous Equations

First some key facts:

When 2 straight lines cross or inter­sect they only do so in one point.

When you plot a qua­dratic and lin­ear func­tion on the same graph, there are poten­tial outcomes:-

  • The lin­ear graph crosses or inter­sects the qua­dratic graph at 2 points
  • The lin­ear graph touches the qua­dratic graph at one point
  • The lin­ear graph and the qua­dratic graph do not intersect

Here’s the graph of y = x² — 4x + 1 and y = x + 1

Graph of quadratic function y = x² -4x + 1 and linear function y = x + 1

 

 

 

 

 

 

 

 

 

 

The lines cross or inter­sect at 2 points, (0, 1) and (5, 6)

So you can solve these simul­ta­ne­ous equa­tions by plot­ting the graphs but you also use algebra:-

1. Start with the lin­ear equa­tion and, if you need to, make x or y the subject.

So we have:-

y = x + 1

and

y = x² — 4x + 1

2. Next sub­sti­tute the expres­sion for “y” from the lin­ear equa­tion into the qua­dratic equation:-

x + 1 = x² — 4x + 1

3. Adjust so that you move all the terms to the left hand side:-

x + 1 = x² — 4x + 1

x² — 4x + 1 = x + 1

x² — 5x = 0

x(x — 5) = 0

So x = 0 or 5 

i.e. when x is 0 or 5, then x(x — 5) does equal zero. 

to be continued…

Posted in 21. Further Algebra | Leave a comment

20. Quadratic Equations

The whole of this topic is in the range grade B to grade A*. I sus­pect that the mere men­tion of “qua­dratic equa­tions” is enough to put off many peo­ple. There is no doubt that some of the equa­tions look scary but if you work through them and prac­tice them, you’ll see that there’s no need to be afraid.

A qua­dratic expres­sion is where the high­est power of x is x².

Exam­ples of qua­dratic expressions:-

4x² + 3x + 7             x² — 4x — 3                 6x² +9             16x² — 4x

There are cer­tain types of expres­sion which are one square num­ber sub­tracted from another square num­ber. Note in the fol­low­ing exam­ples all the num­bers are square numbers:-

x² — 9                    a² — 25                  16c² — 36

Where you have this type of expres­sion in the form a² — b², where a and b are either num­bers or alge­braic terms, it’s known as the dif­fer­ence of two squares.

You need to remem­ber that a² — b² = (a — b)(a + b)

Check:  (a-b)(a + b) = a² + ab — ba — b² = a² — b²

Fac­tor­ize x² — 16

x² — 16  =  x² — 4²  = (x — 4)(x + 4)

Fac­toris­ing qua­drat­ics x² + bx + c

to be continued…

Posted in 20. Quadratic Equations | Leave a comment

Maths News Headlines

Links to any maths news that catches my eye. Could be zero or twenty sto­ries on any par­tic­u­lar day.

Fri­day Feb­ru­ary 1st 2013

1. Elite Maths School Pro­posed for Exeter

New Free school to be estab­lished in Exeter will be a cen­tre of excel­lence for Maths. This has been sup­ported by local Labour MP Ben Brad­shaw but opposed by teach­ing unions.

In my opin­ion this is mis­sion creep from the orig­i­nal Free School prin­ci­ples. It will be an elit­ist school and only pupils with excep­tional maths skills will be eli­gi­ble. If this was repli­cated across other sub­jects and spread to all parts of the coun­try, wouldn’t this amount to selec­tion (of the type we saw when we had gram­mar and sec­ondary schools)  by the back door? Read more here.

Thurs­day Jan­u­ary 31st 2013

1. OCF joins new ini­tia­tive to improve adults’ maths skills

OCF (Online Cen­tres Foun­da­tion) is join­ing Maths4us along with over 20 other orga­ni­za­tions to boost adults’ maths skills. An action plan will be devel­oped over the com­ing weeks. Read more here.

At the moment the details seem vague but I guess these will become clear as the action plan is devel­oped. In the mean­time Maths4us has a web­site which has some use­ful infor­ma­tion for all adults want­ing to improve their maths skills and for all maths and numer­acy teachers.

2. UK’s Games Indus­try Calls for Improved Stan­dards in Maths and Sci­ence Education

TIGA (The Inde­pen­dent Game Developer’s Asso­ci­a­tion) has responded to Sir Tim’s Berner-Lee’s com­ments about com­puter sci­ence edu­ca­tion. Read full press release here.

3. Maths Should Be Cool Not Scary

Ok this is not really news but it’s an inter­est­ing Guardian blog about the way that maths is per­ceived by chil­dren and soci­ety at large. It’s all the more com­pelling because it’s writ­ten by an Eng­lish teacher! The com­ments are also a good read. Read arti­cle here.

Posted in Maths News Headlines | Tagged | Leave a comment