Page 25 of the Formulae and Tables booklet gives a table with some standard functions of x and their derivatives as shown below left. Below right I have added a version that shows how these can be applied to functions of functions of x. In these cases we need to use the chain rule. In the blank box in entries on the table on the right the inner function is placed. The shaded box represents the derivative of what is placed in the blank box. Some students might find this more visual representation of the chain rule useful.

For example to get the derivative of cos(3x+5), the outer function is cos ( ) which has as derivative -sin( ). The inner function is (3x+5) which has derivative 3. Using the chain rule the derivative of cos(3x+5) is -3sin(3x+5).

Apart from what is in your head the only resources you can bring into your Junior Cert and Leaving Cert exams are the Formulae and Tables booklet and your calculator.

Your calculator is an invaluable and even necessary tool when doing maths. As a student you need to become proficient in its use.

Reducing the time spent in entering calculations into your calculator means more time is available for actually answering questions in your exam. Entering them in an efficient manner also means less mental effort is required to combine various sub calculations correctly.

I have finally gotten around to producing the document on the use of the scientific calculator. The aim of this booklet is to help you to be more productive in your studies and more successful in your exams by pointing out ways in which to use your scientific calculator both efficiently and effectively, and to help you avoid pitfalls by pointing out its limitations.

This document uses the Casio fx-83GT PLUS or fx-85GT PLUS as an example because the vast majority of my students use these models.

Just a few weeks left. Serious effort should be going in now, but keep things in perspective. A little humour is no harm. And if you need help the sooner you get it the better.

Happy Easter to all from Galway Maths Grinds. Enjoy the break and take the oppotunity to relax, before the final seven or eight weeks until the exams are over. It could be a good time to get those run around jobs out of the way. Do you need to get a new calculator or maths set? Have you all the stationary you need for the last few weeks of revision.

The most powerful learning occurs when we use different areas of the brain together to solve problems. When students work with symbols, such as numbers, they are using a different area of the brain than when they work with visual and spatial information, such as an array of dots. Learning and performance is optimised when the two areas of the brain are communicating. Training students through visual representations may improve their maths performance significantly, even on numerical maths.

Students can be excited and inspired when they see mathematics as pictures, not just symbols. For example, consider how you might solve 18 x 5, and ask others how they would solve 18 x 5. Here are some different visual solutions of this problem.

Each of these visuals highlights the mathematics inside the problem and helps students develop understanding of multiplication. Pictures help students see mathematical ideas, which aids understanding. Visual mathematics also facilitates higher-level thinking, enables communication and helps people see the creativity in mathematics.

Mathematics is a subject that involves precise thinking. But it also involves creativity, openness to new ways of seeing things, visualisation, and flexibility in approach.

Nice neat formulae and procedures are used to solve familiar type questions posed in a familiar way. But understanding is required to solve them when they are presented in an unfamiliar manner. Students should be challenged to discover new ways in which to see and solve problems.

Take the following example.

A man is on a diet and goes into a shop to buy some ham slices. He is given 3 slices which together weigh ⅓ of a pound but his diet says that he is allowed to eat only ^{1}/_{4}of a pound. How much of the 3 slices he bought can he eat while keeping to his diet?

One approach would be to use ratios and algebra as follows:

3: ^{1}/_{3} = x: ¼

9:1 = 4x:1

4x = 9

x = 9/4 =2 ¼

Another would be to solve it visually as follows:

Both methods are equally valid. Both result in the same correct solution. Students who have difficulty with one approach may find success comes easily with the other. Don’t be afraid to experiment. For some doodling on scraps of paper may be an important part of learning and thinking their way through problems to the solutions.

In the Christmas song, “The 12 Days of Christmas”, how many total gifts does my true love give to me? The gifts are:

A partridge in a pear tree,
Two turtle doves,
Three french hens,
Four calling birds,
Five gold rings,
Six geese a-laying
Seven swans a-swimming,
Eight maids a-milking,
Nine ladies dancing,
Ten lords a-leaping,
Eleven pipers piping,
Twelve drummers drumming.

There are two ways that people usually do this problem:

We can count the total number of each gift.

I get 1 partridge in a pear tree on each of the 12 days. 1 x 12 = 12

2 turtle doves on the last 11 days. 2 x 11 = 22

3 french hens on the last 10 days. 3 x 10 = 30

4 calling birds on the last 9 days. 4 x 9 = 36

5 gold rings on the last 8 days. 5 x 8 = 40

6 geese a-laying on the last 7 days. 6 x 7 = 42

7 swans a-swimming on the last 6 days. 7 x 6 = 42

8 maids a-milking on the last 5 days. 8 x 5 = 40

9 ladies dancing on the last 4 days. 9 x 4 = 36

10 lords a-leaping on the last 3 days. 10 x 3 = 30

11 pipers piping on the last 2 days. 11 x 2 = 22

12 drummers drumming on the last day. 12 x 1 = 12

That is a total of 364 gifts.

Or We could count the number of gifts I get on each day.

On the first day I get 1 gift

On the second day 1 + 2 = 3gifts.

On the third day 1 + 2 + 3 = 6 gifts.

On the fourth day 1 + 2 + 3+ 4 = 10 gifts.

On the fifth day 1 + 2 + 3+ 4 + 5 = 15 gifts.

On the sixth day 1 + 2 + 3+ 4 + 5 + 6 = 21 gifts.

On the seventh day 1 + 2 + 3+ 4 + 5 + 6 + 7 = 28 gifts.

On the eighth day 1 + 2 + 3+ 4 + 5 + 6 + 7 + 8 = 36 gifts.

On the ninth day 1 + 2 + 3+ 4 + 5 + 6 + 7 + 8 + 9 = 45 gifts.

On the tenth day 1 + 2 + 3+ 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55 gifts.

On the eleventh day 1 + 2 + 3+ 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 66 gifts.

On the twelfth day 1 + 2 + 3+ 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11+ 12 = 78 gifts.