Chain Rule for Differentiation made easy

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.

Chain Rule Visual 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).

You will find this and lots more in the downloadable PDF file “Making the most of the Formulae and Tables Booklet” which can be purchased at https://sellfy.com/p/qR3F/

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Polynomial Long Division Calculator

Found a nice little calculator for polynomial long division at http://www.emathhelp.net/calculators/algebra-1/polynomial-long-division-calculator/.

Polynomial long division calculator

Simply enter the polynomial to be divided into the dividend box.

Enter the number you want to divide by in the divisor box.

Check the “Show steps” box if you want the steps shown. Leave it empty if you just want the answer without an explanation.

It is fairly self explanatory but if you need guidance on how to enter the numbers using acceptable notation click the “show instructions” tab.

Then click the “calculate” tab to set it to work.

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Another Year of Excellent Results

CAO points comparison chart

Proud and delighted with the results obtained  by Galway Maths Grinds students in their Leaving Cert again this year.

For example James got a H1 in each of higher level maths, physics and applied maths and is about to start Medicine in NUIG.

His mother gets an equally impressive mark for the cake that arrived to my door freshly baked and piping hot after the results came out.

 

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Making the most of your Casio Scientific Calculator

cover for calculator pdf

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.

To buy now click on the link.

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A little help can make all the difference.

I 8 some pie maths humour

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.

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Strong enough to compete, and still care.

Price before Easter Holidays:

Dunnes Stores   11.99          Galway Maths Grinds  €9.99

Price After Easter Holidays:

Dunnes Stores   13.99          Galway Maths Grinds  €9.99

GMG do not increase prices just because exams are near. If you find it cheaper anywhere else let us know.

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Happy Easter

Happy Easter

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.

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Happy St. Patrick’s Day

St. Patrick's Day Prayer

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Visual Mathematics

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.

visual maths multiplication

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 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:

visual mathsBoth 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.

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Pascal’s Triangle and the Maths of Christmas

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.

Adding: 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 = 364 gifts.

We are adding the triangular numbers. The triangular numbers can be found in Pascal’s triangle, so we can use Pascal’s triangle.

Pascal's Triangle

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