### Author Topic: Linear Pairs of Angles  (Read 942 times)

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#### EEVblog

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##### Linear Pairs of Angles
« on: April 01, 2020, 01:13:56 pm »
So 8yo Sagan get these questions in his extension math class.
And I'm like

#### StillTrying

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##### Re: Linear Pairs of Angles
« Reply #1 on: April 01, 2020, 02:33:58 pm »
B2 is
180/2 +/- 24/2  =  90 +/- 12  =  102 and  78

The second B2 is the same working 90 +/- 21

The images are tooo big.

« Last Edit: April 01, 2020, 02:36:07 pm by StillTrying »
CML+  That took much longer than I thought it would.

#### Brumby

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##### Re: Linear Pairs of Angles
« Reply #2 on: April 01, 2020, 02:55:29 pm »
"Linear pairs" ? ? ?  When I was in school they were called supplementary angles.

The English used to describe the actual question in B(1) on sheet 2 isn't all that clear to me.  Seeing the answer makes it obvious, but the phrase used isn't how I would expect such an idea to be presented.  Maybe I just need to spend some time in class and get used to the "new speak".

I am a little surprised at these questions thrown at an 8 y.o. as a couple of them lean into algebraic processes.

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#### bsudbrink

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##### Re: Linear Pairs of Angles
« Reply #3 on: April 01, 2020, 03:23:03 pm »
Here's what I see:

To solve Sheet 2, B2, it is clearest to use simultaneous equations:

a+b=180  given by "In a linear pair of angles"
a+24=b    given by"one angle measures 24 more than the other"

a+a+24=180  substitute  a+24 for b
2a=156          subtract 24 from both sides and combine a
a=78              divide both sides by 2
b=102            subtract 78 from 180

To solve Sheet 1, B2, same method as Sheet 2, B2:

a+b=180   given by "a linear pair of angles"
a-b=42      given by "the difference" ... "is 42"
a=b+42    add b to both sides

b+42+b=180  substitute b+42 for a
2b=138          subtract 42 from both sides and combine b
b=69              divide both sides by 2
a=111            subtract 69 from 180

I assume the red mark on Sheet 1, B3 indicates that the instructor thinks there is an error...
It looks correct to me.  Maybe the instructor is being a stickler about the order???

« Last Edit: April 01, 2020, 04:34:45 pm by bsudbrink »

#### mrflibble

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##### Re: Linear Pairs of Angles
« Reply #4 on: April 01, 2020, 04:06:12 pm »
For a short & clear definition, see http://www.mathwords.com/l/linear_pair_of_angles.htm

For the longer version, see: https://en.wikipedia.org/wiki/Angle#Combining_angle_pairs

Two angles that sum to a straight angle (1/2 turn, 180°, or π radians) are called supplementary angles.

If the two supplementary angles are adjacent (i.e. have a common vertex and share just one side), their non-shared sides form a straight line. Such angles are called a linear pair of angles.

#### NivagSwerdna

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##### Re: Linear Pairs of Angles
« Reply #5 on: April 01, 2020, 04:12:23 pm »
What sort of language is "measure of angle"? used to be called angle in my day?

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#### StillTrying

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##### Re: Linear Pairs of Angles
« Reply #6 on: April 01, 2020, 04:30:49 pm »
The first lot of simultaneous equations have given the wrong answer.
CML+  That took much longer than I thought it would.

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#### bsudbrink

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##### Re: Linear Pairs of Angles
« Reply #7 on: April 01, 2020, 04:32:09 pm »
The first lot of simultaneous equations have given the wrong answer.

Done before coffee.  I used to be able to do simple arithmetic.  Corrected.
« Last Edit: April 01, 2020, 04:35:34 pm by bsudbrink »

#### NivagSwerdna

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##### Re: Linear Pairs of Angles
« Reply #8 on: April 01, 2020, 05:33:46 pm »
A while back I did some 11+ tuition... you find that kids prefer guesstimate/successive approximation to simultaneous equations.

#### janoc

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##### Re: Linear Pairs of Angles
« Reply #9 on: April 01, 2020, 06:11:17 pm »
I am quite surprised that an 8 years old is being asked to solve what amounts to linear equations? Isn't that like second year?
At that level kids can usually barely read and write and are learning multiplication tables! Either Sagan is very smart and is doing some extra math or Australia has some rather interesting math curriculum, because these things are usually taught from 5-6th grade up ...
« Last Edit: April 01, 2020, 06:13:23 pm by janoc »

#### IanB

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##### Re: Linear Pairs of Angles
« Reply #10 on: April 01, 2020, 06:25:57 pm »
I am quite surprised that an 8 years old is being asked to solve what amounts to linear equations? Isn't that like second year?
At that level kids can usually barely read and write and are learning multiplication tables! Either Sagan is very smart and is doing some extra math or Australia has some rather interesting math curriculum, because these things are usually taught from 5-6th grade up ...

There is a distribution of ability in any age group. Schools teach subjects by year group to fit the norm, but that always means some children will find the work too easy and others will find it too hard. Year groups in education are more about social cohesion than about ability--you don't typically want to have widely varying ages in a class.

To be clear, I rather doubt an 8 yo is being asked to do formal algebra. It is more a case of solving by experiment and intuition. They do what amounts to "algebra" in their head, and later on they will learn to formalize it.
I'm not an EE--what am I doing here?

#### IanB

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##### Re: Linear Pairs of Angles
« Reply #11 on: April 01, 2020, 06:29:12 pm »
What sort of language is "measure of angle"? used to be called angle in my day?

https://www.amazon.com/Measurement-Paul-Lockhart/dp/0674284380
I'm not an EE--what am I doing here?

#### maginnovision

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##### Re: Linear Pairs of Angles
« Reply #12 on: April 01, 2020, 07:02:33 pm »
I am quite surprised that an 8 years old is being asked to solve what amounts to linear equations? Isn't that like second year?
At that level kids can usually barely read and write and are learning multiplication tables! Either Sagan is very smart and is doing some extra math or Australia has some rather interesting math curriculum, because these things are usually taught from 5-6th grade up ...

Dave does say it's an extension math class. When I was in grade school(1st - 6th) I was also in an after school program called GATE. When I first started I was 6 and in a group of 4th and fifth graders. It may be a similar situation with it being more and advanced compared to the standard curriculum.

#### SiliconWizard

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##### Re: Linear Pairs of Angles
« Reply #13 on: April 01, 2020, 07:11:08 pm »
https://en.wikipedia.org/wiki/Angle

Quote
Angle is also used to designate the measure of an angle or of a rotation

So whereas the correct pedantic term would be "measure of angle", "angle" alone is also correct. Not that this matters here, the teacher can use either, and explicitely talking about the measure allows not to understand "angle" as merely the geometric figure.

#### SiliconWizard

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##### Re: Linear Pairs of Angles
« Reply #14 on: April 01, 2020, 07:15:43 pm »
Anyway, also surprised this would be for 8yo kids - I would probably see this more for 10yo level or something, but OTOH, I'm happily surprised. Especially after seeing this: https://www.abc.net.au/news/2019-12-03/australia-education-results-maths-reading-science-getting-worse/11760880

#### EEVblog

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##### Re: Linear Pairs of Angles
« Reply #15 on: April 01, 2020, 11:01:14 pm »
I am quite surprised that an 8 years old is being asked to solve what amounts to linear equations? Isn't that like second year?
At that level kids can usually barely read and write and are learning multiplication tables! Either Sagan is very smart and is doing some extra math or Australia has some rather interesting math curriculum, because these things are usually taught from 5-6th grade up ...

Yes, this is "extension" class material for those who are good at maths.

#### Nominal Animal

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##### Re: Linear Pairs of Angles
« Reply #16 on: April 02, 2020, 11:07:34 am »
Sheet 1, B2: Difference in a linear pair of angles is $42^o$.

Using a single variable $x$ to denote the smaller angle:
$$\begin{array}{lrl} ~ & x + (x + 42^o) &= 180^o \\ \iff & 2 x + 42^o &= 180^o \\ \iff & 2 x &= 180^o - 42^o \\ \iff & x &\displaystyle = \frac{180^o - 42^o}{2} \\ \iff & x &= 69^o \\ \end{array}$$
The larger angle is then $x + 42^o = 69^o + 42^o = 111^o$.

Verification: $69^o + 111^o = 180^o$, which is true.

Therefore, the answer is $69^o$ and $111^o$.

Alternate method, using a system of two equations in two variables:
\left\lbrace ~ \begin{aligned} x + y &= 180^o \\ x + 42^o &= y \\ \end{aligned} \right.
Solve the lower one for for $y$, getting $y = x + 42^o$.  Substitute that to the upper, getting
$$x + (42^o + x) = 180^o$$
From this point onwards, the solution is the same as using the single-variable method, and the answer is $x = 69^o$ and $y = 111^o$.

#### Nominal Animal

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##### Re: Linear Pairs of Angles
« Reply #17 on: April 02, 2020, 11:23:19 am »
Sheet 1, B. 3)

Angle $y$ is five times angle $x$, and they form a linear pair.
\left\lbrace ~ \begin{aligned} y &= 5 x \\ x + y &= 180^o \\ \end{aligned}\right .
Solve upper for $y$, getting $y = 5 x$.  Substitute into lower, getting
$$\begin{array}{lrl} ~ & x + ( 5 x ) & = 180^o \\ \iff & 6 x & = 180^o \\ \iff & x & \displaystyle = \frac{180^o}{6} \\ \iff & x & = 30^o \\ \end{array}$$
Substituting $x = 30^o$ back to $y = 5 x$ yields $y = 5 \times 30^o = 150^o$.

Verification: $30^o + 150^o = 180^o$, which is true.

Therefore, the answer of $30^o$ and $150^o$ is correct.  The only reason for the mark can be that the teacher wanted "the one angle" (which is five times "the other angle") first, and "the other angle" second.

Side note:

In many languages, it is easy to confuse "y is five times x" and "y is five times more than x", corresponding to $y = 5 x$ and $y = x + 5 x = 6 x$; i.e. to confuse direct comparison and the comparison of the increase.  In this case, the opposite error -- treating the equation as if $y = 4 x$ -- yields an answer of $36^o$ and $144^o$.  Teachers can make mistakes, too, and when you need them to acknowledge an error, it often helps to understand how/why they made the error in the first place. (And you can then just note that "it is easy to accidentally do Z here, which would be an error, because ...", which is more palatable to fragile-egoed people than "you made an error ...".)

#### Nominal Animal

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##### Re: Linear Pairs of Angles
« Reply #18 on: April 02, 2020, 11:32:38 am »
Sheet 2, B. 1) As above, $144^o$ and $36^o$; correct.

Sheet 2, B. 2) Linear pair of angles, with one angle $24^o$ more than the other.

Similar to Sheet 1, B. 2), but the difference is now $24^o$ instead of $42^o$.  So, we have
\left\lbrace ~ \begin{aligned} y &= x + 24^o \\ x + y &= 180^o \\ \end{aligned} \right.
Substituting $y$ into the lower gives us
$$\begin{array}{lrl} ~ & x + ( x + 24^o) & = 180^o \\ \iff & 2 x + 24^o & = 180^o \\ \iff & 2 x & = 180^o - 24^o \\ \iff & x & \displaystyle = \frac{180^o - 24^o}{2} \\ \iff & x & = 78^o \\ \end{array}$$
Substituting $x = 78^o$ back to $y = x + 24^o$ yields $y = 78^o + 24^o = 102^o$.

Verification: $102^o + 78^o = 180^o$, which is true.

Therefore, the angles are $102^o$ and $78^o$.
« Last Edit: April 02, 2020, 11:34:48 am by Nominal Animal »

#### Nominal Animal

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##### Re: Linear Pairs of Angles
« Reply #19 on: April 02, 2020, 11:39:04 am »
I really don't like homework (or questions in general!) that only asks for the answer; the proper way is to see how the answer is worked out.  The answer itself is irrelevant, the procedure/algorithm is the interesting and important thing.

That's why I wrote the above answers: to show how one should work them out.  No insult/offense intended for the other members here.

The question I anticipate any learner to ask, is "how did you choose which one to solve first in the systems of two equations?".

The answer is, "whichever looks simpler".  Choosing either one (in a set of two equations with two unknowns) will work, and you will get the same answers.  The only thing that varies is the amount of work needed.  If one of the equations already isolates one of the variables (like $x + 24^o = y$ does, it gives us $y = x + 24^o$), it is always a good one to start.  Just remember to use the other equation for the rest, because if you reuse the same equation, you'll end up with something non-useful like $0 = 0$.

(I so wish StackOverflow hadn't gone full SJW retard; I miss doing this at math.stackexchange.com.)
« Last Edit: April 02, 2020, 11:45:12 am by Nominal Animal »

#### IanB

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##### Re: Linear Pairs of Angles
« Reply #20 on: April 02, 2020, 04:48:56 pm »
That's why I wrote the above answers: to show how one should work them out.  No insult/offense intended for the other members here.

Did you miss the information that the student is 8 years old? What you wrote is not how an 8 yo should or would solve those problems (at least not that formally using x and y variables and algebra).

It would be interesting if you forgot you ever learned algebra and then think about how you would solve them with only a basic knowledge of arithmetic? Would you draw sketches? Would you work in terms of "smaller angle" and "bigger angle"? Use language like "if one part is four times the other part, then there are five small parts in total", then "let's divide 180 into five small parts", and so on?
I'm not an EE--what am I doing here?

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#### bsudbrink

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##### Re: Linear Pairs of Angles
« Reply #21 on: April 02, 2020, 04:58:26 pm »
In my reply (above), I assumed that I was "talking" to Mr. Jones but I was trying to model the way I think you should approach teaching the method.  Basically, step one is to translate the English into mathematics.  Then step two is to translate the resulting mathematics into the answer.

#### janoc

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##### Re: Linear Pairs of Angles
« Reply #22 on: April 02, 2020, 06:39:19 pm »
I am quite surprised that an 8 years old is being asked to solve what amounts to linear equations? Isn't that like second year?
At that level kids can usually barely read and write and are learning multiplication tables! Either Sagan is very smart and is doing some extra math or Australia has some rather interesting math curriculum, because these things are usually taught from 5-6th grade up ...

There is a distribution of ability in any age group. Schools teach subjects by year group to fit the norm, but that always means some children will find the work too easy and others will find it too hard. Year groups in education are more about social cohesion than about ability--you don't typically want to have widely varying ages in a class.

To be clear, I rather doubt an 8 yo is being asked to do formal algebra. It is more a case of solving by experiment and intuition. They do what amounts to "algebra" in their head, and later on they will learn to formalize it.

I doubt it as well, but regardless of ability distribution, if you are asking the kid to answer such questions then someone had to explain it to them first, no? And this is quite advanced stuff to teach in the second year. Even the angles themselves and their relationships in a triangle are not normally taught this early.
« Last Edit: April 02, 2020, 06:43:45 pm by janoc »

#### janoc

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##### Re: Linear Pairs of Angles
« Reply #23 on: April 02, 2020, 06:41:35 pm »
I am quite surprised that an 8 years old is being asked to solve what amounts to linear equations? Isn't that like second year?
At that level kids can usually barely read and write and are learning multiplication tables! Either Sagan is very smart and is doing some extra math or Australia has some rather interesting math curriculum, because these things are usually taught from 5-6th grade up ...

Yes, this is "extension" class material for those who are good at maths.

Ah okay. That makes more sense. Still some pretty tough questions for an 8 years old! Hats off to the little dude!

#### Nominal Animal

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##### Re: Linear Pairs of Angles
« Reply #24 on: April 02, 2020, 06:58:33 pm »
Did you miss the information that the student is 8 years old?
No, I wrote it for Dave -- who asked the question -- to read and understand, step by step, then translate to whatever form is appropriate and best for the 8 year old.

The only language I know of how to do that for math, is math; it is a precise language for exactly that purpose.
The form I used above is the form I still use when grasping new mathematical techniques, and it works.  And I do computational material physics, and often play with classical potential models and stuff; and am definitely not a mathematician.

I would have never guessed that being precise would be frowned upon here, of all places.

It would be interesting if you forgot you ever learned algebra and then think about how you would solve them with only a basic knowledge of arithmetic? Would you draw sketches? Would you work in terms of "smaller angle" and "bigger angle"? Use language like "if one part is four times the other part, then there are five small parts in total", then "let's divide 180 into five small parts", and so on?
Whatever you do, never underestimate kids capability of understanding things.  Just because they are immature, does not mean they cannot grasp logic and rules.  Heck, they are masters at working out working rule sets from first principles, when inventing new games they play with friends.

Elementary algebra, for equations, is really a small extension to basic arithmetic:
• = is the equals sign.  It means the value or expression on its left side is equal to the one on its right side.
• You can add or substract any value from both sides of the equals sign.
• You can multiply or divide both sides of the equals sign by any value except zero.
• When a value is unknown, you can use a variable to represent it.  Typically, we use single letters like x for variables.
• When you substitute a variable with a known value or expression, you need to put the value or expression in parenthesis to not accidentally break operator precedence rules.
• To find out the unknown value, you use the above operations to isolate the variable to one side; that gives you its value on the other.

Elementary linear algebra, or pairs or systems of equations, add a couple more points:
• You need exactly as many independent equations as there are unknowns.
(Independent means you cannot "cheat" by just modify an existing one to get a new one.  Technically, the proper definition is linear independence, but you don't generally need to care until you do calculus or linear algebra.)
• Simple systems can be solved by isolating each unknown in one equation, then substituting it into the next one, in any order you see fit, until you find the actual value for the (last) unknown.  Then, you work backwards, substituting that value to the unknown in the previous equation, one by one, until you have defined the value for each of them.
• More complex systems require linear algebra to solve.  There, you find new tools, vectors and matrices, that make solving these much easier.

Would I explain the above to an 8 year old?  I probably would, but not to expect them to learn it then and there; more to give their subconscious a gentle nod in the correct direction.

(In case anyone "adult" wonders, "simple system" above means one that can be described using a diagonal, bidiagonal, or tridiagonal matrix, in matrix form.)

Some people are visually oriented, some are more quantitatively oriented.  So, it would definitely depend on the person whether I'd use visual sketches or countable tangible items to describe how to solve these problems.

As an example, round or cylindrical-shaped objects, like Lego figurine heads or axle nuts, can be used to form a half-circle.  If you work out how many of them you need, you can put them on a half-circle arc (the radius depending on how many of them you have), and grasp the solution-finding process that way: then, each object represents a part of the arc -- if you have N objects, each one of them represents one Nth of 180 degrees.

Some people are more visually oriented (in the sense that they gain more insight from drawing descriptions of the problem than handling countable items describing the problem), but the problem here is that dividing a half-circle into a number of sectors is nontrivial.  That is, this particular kind of problem is a lot of work to visualize when solving it; the visualization itself becomes a problem, and can easily frustrate a learner.

Using a computer, for example a simple application that shows you a half circle, and lets you divide it into any number of equal segments, showing the angle in degrees as well, can actually hinder the learning process.  The learner can skip the entire solution method, and just find the solution via brute-force trial-and-error.  It becomes easy repetitive mechanical work, and does nothing to learn the solution method at all.

In my reply (above), I assumed that I was "talking" to Mr. Jones but I was trying to model the way I think you should approach teaching the method.  Basically, step one is to translate the English into mathematics.  Then step two is to translate the resulting mathematics into the answer.
I agree, except that step zero is to make sure Mr. Jones understands the exact methods these types of problems can be solved with.

(You could argue that isn't that step 1.5 instead, but thing is, each problem can be described in many equally valid ways.  Just compare Snell's law and Fermat's principle, which describe the exact same thing using completely different mathematical tools.  So, if you want to solve a problem, you should make sure you translate the problem into mathematical expressions you theoretically know how to solve.)
« Last Edit: April 02, 2020, 07:03:14 pm by Nominal Animal »

Smf