Angles In Parallel Lines

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

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°

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:-

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.

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

Bearings

Bearings are measured clockwise from the North:-

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

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.

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.

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°

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

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

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

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

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

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…

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…

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.

Line Segments and Mid-points

A graph paints a thousand words:

In examples like the one above you can just use common sense to find the mid-point of a line segment like the one above but you can calculate the mid-point from just the co-ordinates of a line segment with this formula:-

So, for example a line segment AB has coordinates

A: (2, 10)
B: (10, 16)

What is the mid=point?

Using the formula from above, we have

Midpoint = ((2+10)/2), ((10+16)/2)

= (6, 13)

You can see this on a graph:-

Plotting Straight-Line Graphs

When you plot a linear function on a graph you get a straight line. A linear function is usually expressed in the format y= mx + b.

You can plot a linear function by feeding in values of ‘X’ and calculating ‘Y’. In theory, because a linear function will result in a straight line you only need two sets of coordinates. Normally you will be asked to calculate a few sets of coordinates in a table and then plot the line.

For example; draw the graph of y = 3x + 3 for values of x from –3 to +3

Just 3 simple steps; draw up a table, calculate the values of y and plot the graph:

Table of Values and Graph of Linear Function y = 3x + 3

To be continued…

Percentage Increase and Decrease

Here you are given a start number and you have to increase or decrease that number by a certain percentage.

There are two ways to solve these type of problems.

The first way is normally how you would do it in your head or on paper. For example: Fred earns £110 per week and gets a 5% increase, how much will his new weekly wage be? In your head you would work out 5% of £110 and add it to the original sum. 5% of £110 is £5.50, add this to £110 to give the answer of £115.50.

Alternatively, if you were using a calculator you would add the percentage increase to 100% to give 105%, convert this to a decimal to give 1.05 and then multiply it by the original wage, 1.05 x £110 = £115.50.

Percentage Profit or Loss

Is the percentage profit or loss of the cost price.

So if the cost price is £10 and the sales price is £20, then the profit is £10 (£20 — £10) and the percentage profit = £10/£10 x 100% = 100%

In words:-

Percentage Profit (or loss) = actual profit (or loss)/cost price X 100%

Repeated Percentage Change

This is most often used in finance. Usually when you invest money, interest is calculated using compound interest. This is where the interest is paid on the initial investment plus on any interest already earned. Sounds more complicated than it is! Best to look at an actual example:

John invests £1000 in a savers account which earns 5% interest annually compound  interest. Assuming he makes no withdrawals, how much will he have in his account;

a) After 1 year

b) After 2 years

Answer

a) £1000 x 5% = £50, so after one year he’ll have £1,000 + £50 = £1,050

b) After 2 years he’ll have £1,050 (from the first year) + 5% interest on £1,050=

£1,050 + (£1,050 x 0.05) = £1,050 + £52.50* = £1,102.50

*1% of £1,050 = £10.50, so 5% of £1,050 = 5 x £10.50 = £52.50

Reverse Percentages

This is another case where the two different methods, one without a calculator and one with. Best explained with an example.

A shirt is on sale at £60 and the ticket says that it has been reduced by 20%. What was the original price?

If you don’t have a calculator, follow these two steps:

a) Work out what percentage the new price represents of the original price:-

In this case 100% — 20% = 80%

b) Multiply the revised price by 100/(your answer from (a))

In this case £60 x 100/80

At this point could divide by 80 and multiple by 100 BUT it’s often a lot simpler to use your knowledge of fractions:-

so £60 x 100/80 = £60 x 10/8 = £60 x 5/4 = £60 x 1.25 (we know that 0.25 = a quarter)

so £60 x 1.25 = £60 + a quarter of £60 = £60 + £15 = £75.

If you do have a calculator follow these steps (note the first step is the same as the “without a calculator method” shown above):

a) Work out what percentage the new price represents of the original price:-

100% — 20% = 80%

b) Divide the answer by 100 to get a “multiplier”:-

80/100 = 0.8

c) Divide the new price by the multiplier to get the original price:-

£60/0.8 = £75

So, the basics:-

A sequence is a list of numbers in a given order

Consecutive numbers are numbers next to each other

We call the numbers in a sequence, terms

This leads us to the most basic sequence:-

A liner sequence increases or decreases in equal sized steps.

For example:-   2, 4, 6, 8

In this sequence the terms increase by 2 in each step.

We can say:  The nth term = 2n (i.e the 1st term = 1 x 2 = 2, the 2nd term = 2 x 2 = 4, the 3rd term = 3 x 2 = 6 and the 4th term = 4 x 2 = 8).

When you’re trying to understand a sequence it’s very useful to write the differences between each term under the sequence, like this:-

When you have a linear sequence if you look at differences between the terms to start to define the sequence, if the differences are all 2 then the sequence definition will include 2n, if the differences are all 3 then the sequence definition will include the term 3n, if the differences are all 4 then the the sequence definition will include the term 4n etc., etc.

This is best explained with a couple of examples:-

Find the nth term of the sequence 4, 6, 8, 10,.…

First look for the difference between each term:-

You now know that the sequence includes 2n. Set the information in a table including 2n:-

This method might seem over the top for simple linear sequences, but it shows its worth with more complicated examples:-

Find the nth term of the sequence 1, 9, 17, 25

So start by finding the difference between each term and then complete the grid as shown above:-

Most questions ask you to find the nth term but sometimes this is turned on its head. For example what is the 10th term if the nth term = 8n — 7? Just feed the actual number into the statement:

10th term = (8 x 10) — 7 = 73 

Alternatively you might be asked to find out if a particular number is a term in the sequence, for example is 155 a term in the above sequence? Again just feed the number into the sequence definition.

If 155 is in the sequence, then 8n — 7 = 155

and 8n = 162

n = 162/8 = 20.25

So “n” is not a whole number and 155 is not in the sequence.

Another example using the same sequence, is 177 a term in the sequence?

If 177 is in the sequence then 8n — 7 = 177

and 8n = 184

n = 184/8 = 23

So “n” is a whole number and 177 is in the sequence.

Quadratic Sequences

The nth term of a quadratic includes an “n²”. To find out if a sequence is quadratic you check the second differences. So what is a second difference? We used the first differences between terms when we looked at linear sequences above. The second difference is the difference between the differences. This is best understood by using an example. Let’s invent our own quadratic sequence, n² + 1. So looking at the first few terms and the differences and the second differences we have:-

You may get a question where the answer is a multiple or fraction of n². In these cases the second difference will not be 2, but it will be constant. For example, find the nth term of this sequence:-    0.25, 1, 2.25, 4, 6.25

Proofs and Proof using algebra to follow.

The lattice method has a number of different names, including: Gelosia, Sieve, Shabakh, Venetian Squares, Hindu Lattice. I discovered it the other day when my son was struggling to multiply decimals using the grid method. My daughter saw him getting frustrated and me struggling to explain where he had gone wrong, and said “Why don’t you just use the lattice method, it’s really easy”. I was skeptical but after she showed me how quickly and confidently she could do long multiplication, with or without decimals, I’m convinced it’s a valid and easy to understand method.

Example of the Lattice Method of Multiplication

The method can only really be explained by using an example:

723 x 63

Step 1 Draw the Blank Lattice

This example involves the multiplication of a 3 digit number by a 2 digit number. So we need a lattice with 3 columns and 2 rows (you will see that you’ll also need space around the lattice). This looks like a grid but as we progress it’s transformed into a lattice.

Step 2 Add the numbers to be multiplied:

Step 3 Draw Diagonals to form the Lattice

Notice how the diagonals project beyond the initial columns and rows. This gives space to write the answer.

Step 4 Multiply the Lattice

Notice how the multiplication results are entered into the lattice. There are 6 multiplications:-

7 x 6 = 42

2 x 6 = 12

3 x 6 = 18

7 x 3 = 21

2 x 3 = 6

3 x 3 = 9

Where there is a 2 digit answer, the first digit is entered to the left of the diagonal and the second digit is entered to the right of the diagonal. Where there is a one digit answer, zero is entered to left of the diagonal and the single digit entered to the right of the diagonal.

Step 5 Add up the Diagonals

Working from the right-hand side we have:-

9 = 9

8 + 6 =14 NB when you have a 2 digit answer, write down the second digit and carry the first digit over to the next diagonal. In this case we write down “4” and carry over “1” (see  the number one written in green)

1 (the one carried over, see above) + 1 + 2 + 1 =5

1 + 2 + 2 = 5

4 = 4

Step 6 Read off the final answer

Final Answer 723 x 63 = 45,549

Advantages of the Lattice Method of Multiplication

1. You only need to know the times tables from 1 x 1 through to 9 x 9 to be able to complete any multiplication question.  Check out “How to Learn Your Times Tables Fast”  to become more confident with your times tables.

2. The method is step by step and relatively easy to follow.

3. You multiply and then add in 2 distinct separate steps.

4. As long as you take care drawing the diagonals you should not muddle units, tens, hundreds, thousands etc.

5. You can use this method to multiply decimals (separate article to follow).

Disadvantages of the Lattice Method of Multiplication

1. If you’re careless or untidy in preparing the lattice you are likely to make mistakes.

2. It’s possible to learn this method by rote without understanding why it works.

7.23 x 6.3

My son is ok at Maths, in fact he’s quite good, he was one of only a few at his primary school that achieved “Level 6″. However, he was really struggling with this question. He was using the grid method of multiplication. The grid method was never used when I was at school but now it’s commonly used as a stepping stone to the traditional method of long multiplication.

The Grid Method of Multiplication

The grid method can only really be explained by using an example.

So let’s use:-

16 x 23

First you draw up grid. In this example which is multiplying a two digit by a two digit number, we need 2 columns and two rows. Next we split the numbers into tens and digits. So 16 becomes 10 and 6 and 23 becomes 20 and 3 and enter as below. Then multiply out (refer the grid below) 20 x 10 = 200, 20 x 6 = 120, 3 x 10 = 30 and 3 x 6 =18. Then add up each column 200 + 30 = 230 and 120 + 18 = 138. Finally (see the sum beneath the grid) just add 230 +138 = 368.

Here’s a video that takes you through this example, step-by-step:-

I can see the advantages of using the grid method. It is highly visual, contrast how difficult it was to follow my written explanation compared to how easy it was to just look at the actual grid! The other advantage is that it clearly separates tens and units (and hundreds and thousands etc. for larger numbers). In my view this helps children to understand how it works.

Multiplying Decimals Using The Grid Method

As my son now realises, you have to be careful when you use the grid method to multiply decimals. As I mentioned above he had to solve this multiplication question:-

7.23 x 6.3

This was roughly how he set out his grid to answer this question (THIS IS AN EXAMPLE OF HOW NOT TO DO IT!):-

The cells in the grid above are correct. My son, using pen and paper and the typically less than neat presentation skills of an eleven year old, managed to get the decimal point in the wrong place in more than one of the cells. The trick here is to eliminate the decimal point when you use the grid and use a simple rule to introduce it back after you’ve used the grid. So we have:

7.23 x 6.3

Step 1 — Eliminate the decimal points;

7.23 x 6.3 becomes 723 x 63

Step 2 — Use the grid method;

Step 3: Reintroduce the decimal point using this simple method.

Count the number of digits in the original question that are after the decimal point and then alter the answer from the grid so that there are that number of digits after the decimal place. Hmmm… that’s not easy to put into words! Best look at our example:-

The original question was 7.23 (2 digits after the decimal place) x 6.3 (1 digit after the decimal place), so in this case there are 3 digits after the decimal place. So we need to take our answer from the grid:- 45549 and alter it so that there are 3 digits after the decimal place = 45.549

Final Answer 7.23 x 6.3 = 45.549

You don’t need to know this, but the following proof explains why flipping the dividing fraction and then multiplying works. It’s not difficult to follow and if you understand it, you’ll never forget how to divide by a fraction. It’s also, I think, more satisfying to know why something works rather than just know how to do it. The proof is also quite elegant (mathematicians sometimes say that proofs are “beautiful”, that’s a bit over the top). Anyway, here’s the proof:-

Proof: To Divide by a Fraction, Just Flip it and Multiply

Proof to Divide a Fraction, Flip and Multiply

Just as saw when we were multiplying fractions, you will probably have to deal with mixed numbers and improper fractions. This video walks through the five simple steps that will cover all possibilites:-

The video looks at the following division:-

These are the five steps that are shown in the video and that can be used to answer divide by a fraction question:-

Step 1:-  Convert any Mixed Numbers to Improper Fractions

A mixed number is a combination of an integer (or whole number) and a fraction. Any mixed numbers need to be converted to an improper fraction, a fraction where the top number (numerator) is greater then the bottom number (denominator):-

Step 2:- Take the dividing number and switch or flip its numerator and denominator.

The proof shown above shows why this works. Don’t forget to change the divide sign into the multiply sign:-

Step 3:- If it’s possible simplify any of the fractions

Again, this is not always possible, but if you can simplify it makes the remaining steps easier:-

Step 4:- Multiply

Now we’re ready to just multiply the numerators and the denominators:-

Step 5:- If Required Change your answer from an Improper Fraction to a Mixed Number

Your initial answer may or may not be an improper fraction (numerator greater than the denominator) but if it is you should convert it to a mixed number. You may lose marks (or to be more positive gain extra marks if you do) if you don’t complete this final step:-

Re-cap: How to Divide by a Fraction in Five Simple Steps

Here’s all five steps summarized:-