Common Errors Made Past Times Maths Teachers

I've of late started working with a couplet of trainee maths teachers. It's already proving to endure a useful sense for me - I'm certain it volition conduct maintain a positive acquit upon on my teaching. It's fascinating to sit down at the dorsum of the classroom with my students. 

There's been a couplet of moments lately where I've seen a trainee instructor tell a course of pedagogy something that's non quite right, or that I've felt has needed farther clarification. It's interesting to run across how a teacher's plain of written report noesis - peculiarly their noesis of mutual misconceptions - develops rapidly over the commencement few years of their career.

I've started to line together a listing of some of the mistakes or oversights that I've seen both inexperienced as well as experienced maths teachers make. I'd endure interested to withdraw heed what you've seen every bit good - delight permit me know thus I tin hand notice create a comprehensive listing that volition endure of role to trainees as well as NQTs. 

Square root

The foursquare rootage symbol denotes exclusively the positive root. So if yous write √9 = ±3 on the board, you've made a mistake.

Source: Wikipedia

If you're solving the equation x² = nine as well as thus of course of pedagogy at that topographic point are 2 solutions - both positive as well as negative 3 locomote here. You're non wrong nearly that. But brand certain yous role the radical sign correctly.

   

Angles

What angle fact is depicted below? Some of my students conduct maintain been incorrectly taught to say 'angles inwards a circle amount to 360^o'.  It should endure 'angles roughly a dot amount to 360^o' (as per GCSE grade schemes) or 'angles inwards a total turn/revolution amount to 360^o'.

Whilst we're on the plain of written report of angles, I've exclusively of late constitute out that proverb 'angles inwards a triangle are supplementary' is wrong. Supplementary agency that 2 (and exclusively two) angles amount to 180^o.

Prisms
I don't recommend scream for maths teachers on Twitter whether a cylinder is a prism! I did that in 1 lawsuit - the declaration went on for days as well as I was none the wiser every bit a result.

Half the network claims that cylinders are prisms, the other one-half disagrees. pic.twitter.com/O3sg4GXqCh

— Jo Morgan (@mathsjem) June 11, 2015

Since this Twitter conversation, I avoid defining a cylinder every bit a prism inwards illustration I'm wrong, exactly I exercise tell my students that nosotros bring out the book of a cylinder inwards the same way nosotros bring out the book of a prism.

Interestingly, the GCSE formula canvass uses a controversial film of a prism.

Frustum

It's a frustum, non a frustrum. This give-and-take is oftentimes misspelt.

The gild of operations
Try this question: 3 - 3 x 6 + 2.

If yous don't acquire an response of -13 as well as thus yous demand to await in 1 lawsuit to a greater extent than at the gild of operations. The unhelpful acronym BIDMAS (which incorrectly implies that yous should add together earlier yous subtract) confuses some teachers every bit much every bit it confuses students.

Vocabulary

I of late observed a lesson on angles inwards polygons with a Year 8 class. The educatee instructor asked the lift of a vii sided shape as well as a man child pose upwardly his manus as well as said it was a heptagon. The instructor said he was correct, as well as thus moved on. She didn't notice that some other man child was rattling bothered yesteryear this. He angrily protested to the students roughly him. I watched from the back. This man child had been taught that a vii sided shape is called a septagon - which is also right - as well as he idea that the instructor had made a mistake. At commencement he was angry, exactly as well as thus he doubted himself. His confidence took a knock. Later inwards the lesson, when the chance arose, I reassured him that both damage are acceptable. He was relieved. He felt vindicated. It was a pocket-sized fry oversight yesteryear the instructor exactly it had quite an acquit upon on this student.

I in all probability exercise things similar this every bit good sometimes. We say things that, rightly or wrongly, differ from what students conduct maintain been taught previously. This causes confusion as well as mistrust.

I'd endure interested to know what mistakes as well as oversights you've seen teachers make, thus delight comment below or tweet me thus I tin hand notice line together a comprehensive list. We're ever talking nearly the importance of students learning from each other's mistakes, thus let's exercise the same.

♫ Yous Enjoin Zero, 1 Enjoin Nought ♫

mistakes. One comment that I constitute especially interesting was close the role of wrong vocabulary to limited the release 0. This made me wonder what's acceptable in addition to what isn't.

First, I retrieve nosotros should endeavor to avoid proverb 'oh', because that's a letter, non a number. However, nosotros should comport inwards heed that 'oh' is unremarkably used as a number. I retrieve many of us likely get-go 'oh vii ...' when nosotros laissez passer our στοιχείο number. 'Oh' is used inwards telling the fourth dimension amongst the 24 clock likewise eg 16:05 is said as 'sixteen oh five'. 'James Bond double oh seven' is around other example. But although it's acceptable inwards conversation, 'oh' shouldn't live on used for the release 0 inwards a maths classroom.

I role the discussion null quite often, for illustration I tell my students to brand a quadratic equation 'equal to zero'. I'd tell 5⁰ as 'five to the ability of zero', in addition to and thus on. I wouldn't tell nought inwards these cases, though there's zilch wrong amongst proverb nought.

I practice sometimes role the discussion nought instead of zero. For illustration I would tell the release 0.64 as 'nought betoken half dozen four'.

Although at that topographic point is a perception that null is the 'best' agency to tell the release 0, I'm convinced that the discussion nought is even in addition to thus widely used inwards the UK, especially inwards numbers amongst decimals. I did a Twitter poll to banking concern fit what maths teachers say. Twitter polls are of limited role because they render no data close who has responded (ie what continent they are from), but the results are even in addition to thus of interest:

'Something else' included a) people who but tell 'point five' in addition to (b) people who tell 'half' (yes yes, nosotros all know it's a half! I wasn't bespeak close the maths! In hindsight, I should own got used a dissimilar decimal inwards my query to avoid this response).

The large release of responses for 'nought betoken five' confirmed that the discussion 'nought' is even in addition to thus widely used, at to the lowest degree inwards the UK.  A few tweets from North America suggested that it's rarely used there.

If you're wondering close the spelling of the discussion nought, this extract from Wikipedia 'Names for the release 0 inwards English' might live on of interest:

The words "nought" in addition to "naught" are spelling variants... There is a distinction inwards British English linguistic communication betwixt the two, but it is non 1 that is universally recognized. This distinction is that "nought" is primarily used inwards a literal arithmetics sense, where the release 0 is straightforwardly meant, whereas "naught" is used inwards poetical in addition to rhetorical senses, where "nothing" could as good live on substituted... Whilst British English linguistic communication makes this distinction, inwards States English, the spelling "naught" is preferred for both the literal in addition to rhetorical/poetic senses. 

In summary, I retrieve it's fine to role the words null in addition to nought interchangeably inwards the maths classroom, as long as your students know what you're referring to. Do you lot agree?

♫ You tell zero, I tell nought

Let's telephone phone the whole affair off ♫

Further reading: BBC 4 In Our Time Debate: Zero

Divisibility Rules

In the final lesson of term my students played a few rounds of a Countdown type game, in addition to were peculiarly stumped past times this problem:

I managed to produce it pretty chop-chop in addition to briefly experienced i of those lovely moments of appearing, to my students at least, to hold out a maths genius. Of course of written report all I did was spot that 531 divides past times 9, in addition to hence it was straightforward. Have a go.

H5N1 colleague asked me how I'd done it hence chop-chop in addition to I told her that I'd used divisibility rules. She said that she'd never taught divisibility rules because she'd never seen it specified on a system of work. It strikes me that this is a helpful chip of mathematical cognition that many secondary maths teachers don't teach. Do you lot instruct it? In what year? Most resources for this topic are aimed at psyche children but I intend nosotros should in all probability revisit it at Key Stage 3.

I was aware that my Year 10s didn't know the divisibility rules, hence I covered them every bit purpose of the 'Factors in addition to Multiples' topic this twelvemonth (ie with prime number factorisation, highest mutual ingredient etc). It's a skilful means to review the fundamentals of multiplication in addition to to railroad train fluency in addition to efficiency with numbers. To my Year 10s the rules seemed similar 'new' maths that they'd non seen earlier (or if they had, they couldn't retrieve it), hence it made an interesting in addition to suitably challenging lesson.

I also ran a session on divisibility rules with unopen to smart Year 3s in addition to 4s at a local psyche schoolhouse this year. They picked it upwards well, in addition to i time again I saw it every bit a skilful means to railroad train their agreement of multiplication in addition to their divulge fluency.

So it's a topic that industrial plant good with whatever historic menstruation group, from Key Stage 2 to Key Stage 4. Let's accept a quick await at the rules in addition to resources.

The Rules
Most children volition easily hold out able to create upwards one's hear whether a divulge is divisible past times 2, five or 10. The neat 'tricks' are for 3 (the digit amount is divisible past times 3) in addition to ix (the digit amount is divisible past times 9). Once nosotros know whether a divulge divides past times 3, nosotros know whether it divides past times 6 (ie all even multiples of 3 are multiples of 6). For divisibility past times 4 at that spot are 2 alternatives: either banking corporation jibe whether the final 2 digits separate past times 4, or halve the divulge in addition to encounter if your respond is fifty-fifty (the 4 times tabular array existence double the 2 times table). The 7 key tests are shown inward the graphic below (there are loads of squeamish graphics for this on google images).

I didn't bother pedagogy the dominion for divisibility past times 7 because it's non straightforward. Rather in addition to hence memorise this dominion I idea my students would hold out ameliorate off but checking for divisibility past times 7 with long division.

If you're interested inward all the rules, from 1 - thirty in addition to beyond, banking corporation jibe out the Wikipedia page Divisibility rule.

Resources
I flora a mixture of uninspiring worksheets in addition to bizarre activities when I searched for resources online (the to a greater extent than odd activities included Divisibility Rock n' Rule, NFL divisibility dance and I'll take, you lot take...).

For the interactive whiteboard nosotros receive got Vectorkids: divisibility rules, Divisibility Test in addition to Delightfully Divisible.

If you're looking for a good structured worksheet pack, this is quite good.

This Don Steward.

Don Steward also has slides on divisibility rules, total of lovely challenging problems.

Dan Walker has a keen laid upwards of slides in addition to activities on divisibility rules too, including this lovely Venn activity.

Why produce the rules work?
The ancient Greeks knew rules for divisibility past times 2, 3, five in addition to ix inward the 3rd century BC.

So why does the digit amount of multiples of 3 separate past times three? Sal Khan explains here...

 

He has a similar video for divisibility past times 9.

Do allow me know nearly your experiences of pedagogy divisibility rules in addition to whatever resources recommendations. If you've non taught it before, receive got a larn adjacent year. It's useful cognition in addition to good worth teaching.

Multiplying Negatives

"A negative times a negative is a positive". It's a difficult 1 to explain. We all learnt it at schoolhouse in addition to practised it to the holler for of fluency, but it's non until we're asked why it plant that nosotros halt in addition to intend almost it.

Numbers lines in addition to visualisations are real helpful when didactics the improver in addition to subtraction of negative numbers. But amongst multiplication in addition to sectionalization it's non in addition to thus clear.

Let's await at a few approaches in addition to resources.

1 . Pattern Spotting
Draw a touchstone multiplication tabular array in addition to extend it backwards to include negative numbers. It's a straightforward designing that all students should hold out able to location in addition to continue. Get students to produce this using Colin Foster's activity on page v of his Negative Numbers chapter.

2. Multiplication Grids
Take 2 2-digit numbers in addition to multiply them together using grid multiplication. For simplicity, let's lead keep 12 x 11:

Here nosotros lead keep written 12 every bit 10 + 2 in addition to eleven every bit 10 + 1. But it would operate only every bit good if nosotros expressed those numbers differently. Instead, let's write 12 every bit fifteen - iii in addition to eleven every bit fifteen - 4. We should acquire the same answer:

This solely plant if -3 x -4 = 12. 

Note that this explanation requires students to commencement sympathize that positive x negative = negative. This is relatively straightforward to explicate inwards damage of repeated addition. 

3. Proof
Here's a proof that is clear in addition to accessible to us experienced mathematicians. I'm non certain how accessible it is to Year seven students, but it's worth a go.

a in addition to b are positive

a + (-a) = 0 

[a +(-a)]•b = 0•b 

a•b + (-a)•b = 0 

a•b is positive. Therefore (-a)•b is negative 

b + (-b) = 0 

(-a)•[b + (-b)] = (-a)•0

(-a)•b + (-a)•(-b) = 0

Since (-a)•b is negative, nosotros conclude that (-a)•(-b) is positive.

Perhaps start amongst a numerical instance instead of a formal proof.

3 + (-3) = 0

Multiply everything past times -4

3(-4) + (-3)(-4) = 0(-4) 

 -12 + (-3)(-4) = 0 

 (-3)(-4) must equal 12 to brand this tilt true. 

Further Reading
It's a skillful persuasion to read almost a theme earlier you lot learn it, fifty-fifty relatively unproblematic topics that you've taught many times before. Here are around helpful links:

• The FAQ pages on Math Forum ever render interesting answers: Why is a negative times a negative a positive?
• I taste James Tanton's curriculum essays: Directed Number Gaps' activity - it plant good amongst whatsoever twelvemonth group
• MathsPad has a slap-up hit of negative release puzzles including arithmagons (some of these resources are solely available to subscribers)
• CIMT's negative numbers chapter has activities for practising multiplying negatives.

Colin Foster suggests that you lot inquire students to brand upwards 10 multiplications in addition to 10 divisions each giving an respond of –8 (eg –2 × –2 × –2 or –1 × 8 etc).

The squaring in addition to cubing (etc) of negatives is worth discussing - students should location that an fifty-fifty ability gives a positive value (eg what is the value of (-1)¹⁰⁰?).

It may hold out worth exploring estimator behavior likewise (ie around calculators require brackets when squaring a negative). It's of import that students know how to role their estimator properly. There's a slap-up resources from MathsPad for this - Using a Calculator: Odd One Out.

This theme is revisited inwards afterwards years when students are practising substitution. For example, if a = 3, b = -2 in addition to c = -5, abide by the values of: abc; bc²; (bc)²; a²b³ in addition to and thus on. This Substitution Puzzle from mathsteaching.wordpress.com gets quite challenging.

Do allow me know if you lot role an interesting method or resources for didactics the multiplication of negative numbers.

"Minus times minus results inwards a plus,
The argue for this, nosotros needn't discuss"
- Ogden Nash