Linear Simultaneous Equations

Solving linear simultaneous equations is rather satisfying. In this postal service I advise approximately techniques too resources for pedagogy this topic.

Equality too 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 matter are also equal to ane another’. Influenza A virus subtype H5N1 well-made betoken inward a give-and-take nearly racial equality, but also an interesting claw into a method for solving simultaneous equations such as

y = 4x + 6

y = 2x + 3

Both 4x + six too 2x + three are equal to y, thus logically nosotros convey 4x + six = 2x + three (this is essentially merely a exchange method). Why non present this this blog gives a adept explanation of Lincoln's point)

Elimination too 'SSS'
When solving simultaneous equations yesteryear elimination, my pupils abide by it really slow to recall the ‘SSS’ dominion ie ‘Same Sign Subtract’. For example, inward the equations 3x + 5y = 59; 2x + 5y = 56, this dominion says that because the 5y inward each equation is positive (‘same sign’), we subtract one equation from the other.

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

Graphical Solutions
It’s then of import for pupils sympathise that whenever they solve linear simultaneous equations, they are inward fact finding the betoken of intersection of 2 direct lines. It’s essential to position a query involving parallel lines (eg solve 3x + 4y = 10; 6x + 8y = 15) too perish them thinking nearly why at that spot are no solutions.  I similar to present them this cute picture too.

Wordy Equations
This is ane of my favourite simultaneous equation questions: “A zoo has several ostriches too several giraffes. In full they convey xxx eyes too 44 legs. How many ostriches too how many giraffes are inward the zoo?”. Make certain you lot 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 make of fantastic questions too activities.
• TeachitMaths has a swell ‘wordy question’ worksheet too clear slides on elimination too 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 adept practise inward solving simultaneous equations graphically amongst ICT.
• Peter Bland and JustMaths both render assay questions on this topic.
• MEI has a overnice extension activity. 

5 Maths Gems #17

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

1. Balancing

Do you lot recollect inward Gems 13 when I featured the bright 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 like problems for purpose on an interactive whiteboard, tablet or computer. The website, from the Education Development Center, allows users to both create as well as solve mobiles. The website says that 'the logic of these puzzles is the same equally the logic behind the solving of equations as well as systems of equations, but the visual format makes these puzzles, as well as the logic they develop, accessible as well as appealing...'. 

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

2. Blanks

Sarah Aldous (@mathsfeedback) shared a overnice stance for instruction logs that could live on used for a publish of topics. The stance is that you lot lay questions upwards only about the room as well as students respond on a post-it, but their respond must differ from the answers already given.

Here's my examples of like questions for fractions:

as well as indices:

You larn the idea. Students brand suggestions for the blanks as well as inward doing hence they demonstrate their agreement of the topic. Plus, they receive got to intend creatively if their stance 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 amongst examples as well as interactivity. Here's the definition of the news following (check out the website yourself to play amongst 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 as well as perpendicular.

4. Simultaneous Equations

I dearest this Linear Simultaneous Equations.

5. Whiteboards (again!)

I've written close vertical whiteboards as well as acrylic sheets earlier (see Gems 7) - I dearest these ideas as well as promise that 1 twenty-four hours I receive got the budget to practise both.  Andrew Stadel (blog post this calendar week close his sense of using large whiteboards - it's good worth a read. One stance is 'Placemat':

If students are working on a business together, they operate individually on split upwards sections of a large whiteboard, as well as hence purpose the centre to tape their terminal answer. This approach helps students compare their operate as well as back upwards each other. It's too a proficient means for the instructor to assess private agreement as well as banking concern gibe that everyone is contributing to grouping discussion.

Another stance is Brain Dump:

Here nosotros receive got an icon inward the middle as well as students write downwards everything they know close it inward their private sections, earlier they compare as well as hash out their answers.

And nosotros receive got Number Talks:

In this trial a publish is written inward the middle as well as students operate inward their private sections to write downwards equally many ways they tin strength out intend of to brand that number. 

Do read Andrew's ship service for the total details as well as links to lots to a greater extent than ideas. If your region doesn't receive got the budget to purchase large whiteboards or acrylic sheets as well as hence you lot could purpose flipchart newspaper for these activities. Whiteboards are much ameliorate though, because of the power of the eraser ('because the markings tin strength out live on easily erased, students are straight off inclined to write without hesitation') - you lot tin strength out read close this inward the fantabulous post Whiteboards vs. Chart Paper by Thomas Ro (@MrT_Ro). 

That's it for this week. I'll locomote out you lot amongst a nifty occupation for your students from UKMT, shared past times Matt Smith (@mathsmatt). Find the angles inward the rhombus, assuming there's 1 equilateral as well as iii isosceles triangles.

Linear Simultaneous Equations

Solving linear simultaneous equations is rather satisfying. In this postal service I advise approximately techniques too resources for pedagogy this topic.

Equality too 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 matter are also equal to ane another’. Influenza A virus subtype H5N1 well-made betoken inward a give-and-take nearly racial equality, but also an interesting claw into a method for solving simultaneous equations such as

y = 4x + 6

y = 2x + 3

Both 4x + six too 2x + three are equal to y, thus logically nosotros convey 4x + six = 2x + three (this is essentially merely a exchange method). Why non present this this blog gives a adept explanation of Lincoln's point)

Elimination too 'SSS'
When solving simultaneous equations yesteryear elimination, my pupils abide by it really slow to recall the ‘SSS’ dominion ie ‘Same Sign Subtract’. For example, inward the equations 3x + 5y = 59; 2x + 5y = 56, this dominion says that because the 5y inward each equation is positive (‘same sign’), we subtract one equation from the other.

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

Graphical Solutions
It’s then of import for pupils sympathise that whenever they solve linear simultaneous equations, they are inward fact finding the betoken of intersection of 2 direct lines. It’s essential to position a query involving parallel lines (eg solve 3x + 4y = 10; 6x + 8y = 15) too perish them thinking nearly why at that spot are no solutions.  I similar to present them this cute picture too.

Wordy Equations
This is ane of my favourite simultaneous equation questions: “A zoo has several ostriches too several giraffes. In full they convey xxx eyes too 44 legs. How many ostriches too how many giraffes are inward the zoo?”. Make certain you lot 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 make of fantastic questions too activities.
• TeachitMaths has a swell ‘wordy question’ worksheet too clear slides on elimination too 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 adept practise inward solving simultaneous equations graphically amongst ICT.
• Peter Bland and JustMaths both render assay questions on this topic.
• MEI has a overnice extension activity. 

5 Maths Gems #17

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

1. Balancing

Do you lot recollect inward Gems 13 when I featured the bright 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 like problems for purpose on an interactive whiteboard, tablet or computer. The website, from the Education Development Center, allows users to both create as well as solve mobiles. The website says that 'the logic of these puzzles is the same equally the logic behind the solving of equations as well as systems of equations, but the visual format makes these puzzles, as well as the logic they develop, accessible as well as appealing...'. 

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

2. Blanks

Sarah Aldous (@mathsfeedback) shared a overnice stance for instruction logs that could live on used for a publish of topics. The stance is that you lot lay questions upwards only about the room as well as students respond on a post-it, but their respond must differ from the answers already given.

Here's my examples of like questions for fractions:

as well as indices:

You larn the idea. Students brand suggestions for the blanks as well as inward doing hence they demonstrate their agreement of the topic. Plus, they receive got to intend creatively if their stance 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 amongst examples as well as interactivity. Here's the definition of the news following (check out the website yourself to play amongst 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 as well as perpendicular.

4. Simultaneous Equations

I dearest this Linear Simultaneous Equations.

5. Whiteboards (again!)

I've written close vertical whiteboards as well as acrylic sheets earlier (see Gems 7) - I dearest these ideas as well as promise that 1 twenty-four hours I receive got the budget to practise both.  Andrew Stadel (blog post this calendar week close his sense of using large whiteboards - it's good worth a read. One stance is 'Placemat':

If students are working on a business together, they operate individually on split upwards sections of a large whiteboard, as well as hence purpose the centre to tape their terminal answer. This approach helps students compare their operate as well as back upwards each other. It's too a proficient means for the instructor to assess private agreement as well as banking concern gibe that everyone is contributing to grouping discussion.

Another stance is Brain Dump:

Here nosotros receive got an icon inward the middle as well as students write downwards everything they know close it inward their private sections, earlier they compare as well as hash out their answers.

And nosotros receive got Number Talks:

In this trial a publish is written inward the middle as well as students operate inward their private sections to write downwards equally many ways they tin strength out intend of to brand that number. 

Do read Andrew's ship service for the total details as well as links to lots to a greater extent than ideas. If your region doesn't receive got the budget to purchase large whiteboards or acrylic sheets as well as hence you lot could purpose flipchart newspaper for these activities. Whiteboards are much ameliorate though, because of the power of the eraser ('because the markings tin strength out live on easily erased, students are straight off inclined to write without hesitation') - you lot tin strength out read close this inward the fantabulous post Whiteboards vs. Chart Paper by Thomas Ro (@MrT_Ro). 

That's it for this week. I'll locomote out you lot amongst a nifty occupation for your students from UKMT, shared past times Matt Smith (@mathsmatt). Find the angles inward the rhombus, assuming there's 1 equilateral as well as iii isosceles triangles.