Linear Simultaneous Equations

Solving linear simultaneous equations is rather satisfying. In this postal service I advise unopen to techniques as well as resources for pedagogy this topic.

Equality as well as Lincoln
When I saw the cinema Lincoln I was struck yesteryear Abraham Lincoln’s reference to Euclid’s First Axiom ‘Things which are equal to the same affair are also equal to 1 another’. Influenza A virus subtype H5N1 well-made indicate inwards a tidings nigh racial equality, but also an interesting claw into a method for solving simultaneous equations such as

y = 4x + 6

y = 2x + 3

Both 4x + vi as well as 2x + three are equal to y, hence logically nosotros convey 4x + vi = 2x + three (this is essentially simply a exchange method). Why non demo this this blog gives a proficient explanation of Lincoln's point)

Elimination as well as 'SSS'
When solving simultaneous equations yesteryear elimination, my pupils uncovering it real tardily to shout out back the ‘SSS’ dominion ie ‘Same Sign Subtract’. For example, inwards the equations 3x + 5y = 59; 2x + 5y = 56, this dominion says that because the 5y inwards each equation is positive (‘same sign’), we subtract one equation from the other.

However, I noticed that when solving simultaneous equations such every 2nd -3x + 5y = 41; 2x + 5y = 56, unopen to pupils saw that the -3x as well as the 2x were different signs as well as incorrectly decided that this meant they should add the 2 equations. So it’s worth clarifying which signs nosotros are referring to inwards the ‘Same Sign Subtract’/’Different Sign Add’ approach.

Graphical Solutions
It’s as well as then of import for pupils empathise that whenever they solve linear simultaneous equations, they are inwards fact finding the indicate of intersection of 2 conduct lines. It’s essential to put a query involving parallel lines (eg solve 3x + 4y = 10; 6x + 8y = 15) as well as decease them thinking nigh why at that spot are no solutions.  I similar to demo them this cute picture too.

Wordy Equations
This is 1 of my favourite simultaneous equation questions: “A zoo has several ostriches as well as several giraffes. In full they convey xxx eyes as well as 44 legs. How many ostriches as well as how many giraffes are inwards the zoo?”. Make certain yous discourage pupils from using the missive of the alphabet O for ostrich, because it looks similar a zero.

Resources
There's lots of resources online – here’s a few I’ve picked out:

• Don Steward’s Median blog has a huge arrive at of fantastic questions as well as activities.
• TeachitMaths has a slap-up ‘wordy question’ worksheet as well as clear slides on elimination as well as substitution.
• UEA has a worksheet on linear simultaneous equations amongst a overnice progression of questions.
• There is a useful worksheet on Solving Simultaneous Equations Graphically yesteryear Tristan Jones on TES. 
• This Geogebra worksheet yesteryear themathsteacher.com is a proficient practice inwards solving simultaneous equations graphically amongst ICT.
• Peter Bland and JustMaths both supply examination questions on this topic.
• MEI has a overnice extension activity. 

5 Maths Gems #17

Ho-ho-hello together with welcome to my seventeenth weekly gems post. This is where I circular upwards some of the best maths didactics ideas I’ve seen on Twitter. I’ve got some crackers for you lot today. 

1. Balancing

Do you lot retrieve inward Gems 13 when I featured the vivid balancing problems from Don Steward? My readers loved them! Well this calendar week Damian Watson (@dwatson802) shared the visual algebra puzzle website SolveMe Mobiles which has similar problems for purpose on an interactive whiteboard, tablet or computer. The website, from the Education Development Center, allows users to both create together with solve mobiles. The website says that 'the logic of these puzzles is the same every bit the logic behind the solving of equations together with systems of equations, exactly the visual format makes these puzzles, together with the logic they develop, accessible together with appealing...'. 

And there's already been some skilful feedback on Twitter:

2. Blanks

Sarah Aldous (@mathsfeedback) shared a prissy sentiment for didactics logs that could last used for a publish of topics. The sentiment is that you lot position questions upwards around the room together with students reply on a post-it, exactly their reply must differ from the answers already given.

Here's my examples of similar questions for fractions:

together with indices:

You learn the idea. Students brand suggestions for the blanks together with inward doing hence they demonstrate their agreement of the topic. Plus, they receive got to intend creatively if their sentiment has already been taken.

3. Dictionary
Steven W. Anderson (maths lexicon for kids. This lexicon from Jenny Eather is actually lovely - it's the best maths lexicon I've seen for students. Definitions come upwards alongside examples together with interactivity. Here's the definition of the intelligence following (check out the website yourself to play alongside the interactive features).

And here's the definition of sector. Again, it's interactive.

And here's 2 to a greater extent than examples - palindromic together with perpendicular.

4. Simultaneous Equations

I honey this Linear Simultaneous Equations.

5. Whiteboards (again!)

I've written well-nigh vertical whiteboards together with acrylic sheets earlier (see Gems 7) - I honey these ideas together with promise that 1 twenty-four hr current I receive got the budget to practice both.  Andrew Stadel (blog post this calendar week well-nigh his sense of using large whiteboards - it's good worth a read. One sentiment is 'Placemat':

If students are working on a chore together, they operate individually on split sections of a large whiteboard, hence purpose the centre to tape their terminal answer. This approach helps students compare their operate together with back upwards each other. It's likewise a skilful means for the instructor to assess private agreement together with depository fiscal establishment lucifer that everyone is contributing to grouping discussion.

Another sentiment is Brain Dump:

Here nosotros receive got an picture inward the middle together with students write downward everything they know well-nigh it inward their private sections, earlier they compare together with hash out their answers.

And nosotros receive got Number Talks:

In this instance a publish is written inward the middle together with students operate inward their private sections to write downward every bit many ways they tin compass the sack intend of to brand that number. 

Do read Andrew's postal service for the total details together with links to lots to a greater extent than ideas. If your division doesn't receive got the budget to purchase large whiteboards or acrylic sheets hence you lot could purpose flipchart newspaper for these activities. Whiteboards are much amend though, because of the power of the eraser ('because the markings tin compass the sack last easily erased, students are at nowadays inclined to write without hesitation') - you lot tin compass the sack read well-nigh this inward the fantabulous post Whiteboards vs. Chart Paper by Thomas Ro (@MrT_Ro). 

That's it for this week. I'll exit you lot alongside a smashing work for your students from UKMT, shared past times Matt Smith (@mathsmatt). Find the angles inward the rhombus, assuming there's 1 equilateral together with 3 isosceles triangles.