Have you been avoiding Algebra Proofs until now?
Don't worry, when it comes to the Proof by Induction, Contradiction, Limits, Inequalities, we have you covered.
This blog post highlights some tips and advice on how to approach revision of Algebra Proofs.
One important thing to note: It's due to come up (see tip #1), so ignore it at your peril!
Proof by Induction on Tacit Maths
The Tacit Maths module on Proof by Induction gives you:
A notes worksheet with examples and exercises of Proof By Induction, Proof by Contradiction, Limits, Abstract Inequalities, Divisibility Proofs.
17 Exam Questions with all the times Proof by Contradiction appeared in LCHL since 2006.
Powerpoint Solutions to ALL of the above.
Marking schemes for all of the Exam Questions, to show you where the marks are going.
Visit the module by clicking the following link:
What are Algebra Proofs?
Have a look at the proof in the image above.
All the values for n shown give you an answer that is divisible by 3.
But how many examples do you need to conclusively prove it is true?
In Maths, we need to be able to prove it's true for EVERY example.
That's where the Algebra Proofs come in handy.
Different Types
On the Leaving Cert course, the most popular Algebra Proof is Proof by Induction.
There are a few standard Proof by Induction questions (see LC 2014 or LC 2020).
Then you have the "divisiblity" proofs, which follow the same process.
You can then have Proof by Induction linked to other topics on your course, like De Moivre's Theorem (shown above), or the Geometric Series formula (see LC 2012).
Next, you have Proof by Contradiction, which came up in 2023, but also, interestingly, in 2022 in Paper II with a circle theorem...
Limits aren't exactly an Algebra proof, but they do fall under the category of "annoying things in Algebra that you just need to learn off".
Finally, abstract inequality proofs are a lot of fun, and, as you will see in Tip #1 below, it hasn't appeared in a long, long time!
The Bad News #1: Algebra proofs tend to come up most years...
As you can see from this checklist, you can see it comes up most years...
Ignore them at your peril!
If you wish to download this, or other checklists on all the other LCHL topics, then visit or other blog post:
The Bad News #2: It's worth a LOT of marks...
As you can see in the image, Proof by Induction is always worth 15 Marks.
That is almost 7% of your entire Paper 1!
If you plan on leaving it out, maybe think again!
On the plus side, you might notice from the marking scheme that it is actually not that difficult to get 8/15 marks or even 12/15.
You just need to know the steps inside out.
More on the steps in the next section...
The Good News: 80% of Proof by Induction is quite routine...
Proof by Induction is made up of 3 steps (as mentioned in the marking scheme) and the Conclusion.
Step 1: Show for n = 1.
Step 2: Assume true for n = k.
These two steps are quite simple.
Step 3: Prove true for n= k +1.
Step 3 is where the magic happens.
But the good news is once you have the first two steps done, you have already picked up most of your marks.
And don't forget your conclusion!
Prediction: What to expect for LCHL 2024
OK, not to blow our own trumpet or anything, but in the 2023 version of this blog post we posted the following (see image below). Sure enough, what do you think appeared in Q3 of Paper 1??
Leaving Cert 2024: Abstract Inequalities
Abstract Inequalities haven't appeared in the LCHL since 2011.
Before that they appeared in 2006, 2007 and 2010.
The problem is that there is not much material to practice with in terms of exam questions.
To help you out, on the Tacit Maths - Algebra Proofs module there contains:
A walkthough and revision exercise on abstract inequalities.
Exam questions from 2006 to 2011 which featured abstract inequalities.
The PowerPoint solutions to all of the above.
For more Tips and Advice for teachers and students on how to tackle different topics in the Leaving Cert Higher Level Course, visit our blog.
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