Substitution - do you know what it is yet?

 

Substitution: Do you know what it is yet?

 

Step 1

A bit of preparation, but once done it is a resource that can be used time and time again. Each pupil needs a set of cards numbered 0 to 9. Make them big get them laminated, held together with a elastic band and stored in a cheap envelope.



Step2

Write an expression on the board such as

3a

 

Emphasise that this is an expression not an equation. You may have to tell them what the difference is.



Step 3

Now tell them a = 4, using the cards show the answer. Hopefully the will all demonstrate they know the answer is 12, if not you have an opportunity to rectify any misconceptions. You can now progress depending as always upon the answers you receive, so for example you might want to put up the expression

2a – 1

 

You can continue with this exercise dependent upon the age and stage of the class.

Once again the same process is followed you can ask the class show me the answer. Where you go from here is up to you. You can continue with this exercise dependent upon the age and stage of the class. For example you can take this forward when introducing brackets, what is the answer to 2(x-1)? Or when you need to tackle the big problem that we all face of when many pupils become totally confused with the difference between x² and 2x?

 

This is an excellent activity for an instant assessment of what they have learnt. It ensures every child participates in the class and if you are in the UK it will satisfy some of the demands of the OFSTED inspection regime.

This activity was not thought up by me. I found it in an excellent book which is full of ideas, starters and worksheets that can be photocopied. It is not as dry as the normal text books or dare I say it teacher produced worksheet. The title is ‘Activities for Teaching Numeracy’ Year 7 algebra. The whole series is excellent and I have bought numerous copies over the years for departments I have run, oftenI have paid for them out of my own money, because I believe them to be so good.

 

 

Area, equations and that frog again

Those of you who read my post about how to use a frog to solve equations might want to use this before that lesson. It can also be used independently before starting to find the areas of complex shapes. It is deceptively simple, but powerful.

 

All you need to do I draw two lines and on one put its length, say 10cm, and on the other x cm and 8cm. Explain that the two lines are the same length. Two frogs have a jumping competition, they agree to jump over the same course of 10 cm. The first jumps the full 10 cm. Fred the frog jumps but only a distance of  x cm but then covers the rest of the course, which is 8 cm. How far is x cm?

As you know this is really x + 8 = 10. Depending on the level of pupils you can continue with further examples, make the link to algebra or move on.

 

The next step (or jump if you are a frog) is to draw similar lines  such as one which is 10 cm and the other which is x cm, x cm and 3 cm or whatever appeals to you. Again you can use the story of the two frogs one jumping the full 10 cm the other Fred covering x, cm then x cm then 3 cm. Emphasise that both x cm are the exactly  the same distance. Then ask how far is x cm?

 

Of course the pupils are solving 2x + 3 = 10. You can judge how far to take this idea it really does depend on the class or pupil. Use in its simplest form before doing complex shapes as pupils often fail to grasp how to find a missing dimension.

 

If you are wondering about previous mentions of Fred the frog see my previous post ‘solving equations with a frog’.

 

 

If anyone has used the ideas that I have given you over these last few blogs how did it go? Were they successful? How did you improve them? (I'm sure you can). Please leave a comment.

An algebra starter - money in my pocket

Algebra starter - just for a change

Want to revise basic algebra with your class? Want a quick starter that doesn't involve equipment, not even a worksheet?  This is a starter activity for you.

Use of algebra

 This activity is fantastic for starting algebra or revising basic concepts. You need to have some coins in your pocket or purse, preferably of a low denomination. Tell the class, as you jangle the coins (that you have out of sight) that you have some money but you don’t know how much. Ask them what shall we call the amount? A variety of answers will be offered, many quiet amusing, such as ‘cash’ ‘wonga’, ‘dosh’ or whatever the current vogue word is for money. Once you have arrived at a consensus for the change that you have in your pocket guide them to some algebra by saying that you will use one of their chosen words but you'll only use the first letter. Establish that the unknown amount of money in your pocket is m for example if you are using the word money.

The generous Maths teacher

Now in a fit of generosity, well all Maths teachers are kind, caring warm and generous, give someone in the class 1p. Ask how much you have in your pocket now. Again the answers will be very revealing, as they struggle with algebra, their understanding of the concept of an unknown and how it is represented, eventually the discussion should led to m – 1. Retrieve your 1p and now ask them how much now do you have in your pocket? Hopefully they will get to m.

The generous Government

Next explain that the Government has awarded all teachers a bonus of 5p which has just magically arrived in your pocket. I did have m how much do I have now in my pocket? n will be a popular response, this really does demonstrate their misunderstanding and misconceptions about algebra. That is why this exercise is so important. In a short time you should get to m + 5. Now ask how much you have if you give away 3p reminding them that you now have m + 5. Again the responses will be very illuminating, highlighting understanding, or not as the case may be. Keep asking questions of this nature. It is often useful to record your answers on the board. Vary the questions depending upon the responses.

Further work

This oral question and answer session is very good for highlighting misconceptions or building understanding. Further question could be ‘I have m in my pocket what do you have in yours?’ If they say m you say ‘so if we put our money together what do we both have? Or if they say c what do we both have? If I lose 2p what do I have? After I have lost 2p I double my money what do we have now? The variations are endless.

For another esy starer why no see Stand up/Sit down.

 

I suggest that you repeat this exercise at regular intervals just to keep the class on the ball. we all know how knowledge slips way if it is not used.  Repeating this activity will give your class a head start when it comes to basic algebra.

 

An excellent book for mathematical starters is 101 red hot starters.

Letts Red Hot Starters - Maths (Letts 101 Red Hot Starters)

 

Why not follow me on Twitter at croftsr1

 

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Solving equations

When planning to teach a class how to solve equations I would normally follow the steps outlined in my previous blogs which are

1. Solving equations, a staring activity

2. Solving equations with a frog.

The class should now have all the skills and knowledge to be able to solve equations. I reintroduce the frog story but this time I show them we cn write it as 3x +1 = 16. It is important to now make it clear what your objective is, to find x. Link it back to your diagram by asking 'what did you do first when trying to find how far Fred jumps?' Hopefully they will repond subtract 1, which you now show them.



Once again link back to the Fred the frog story, reminding them about the three jumps covering 15 metres and ask 'so how long was one jump?' To which they should answer 5. 'How do you know?' This leads on to division by 3. You then show them usual notation used with the arrows by the side.

I am confident that hould you follow the three stages I have outlined in thee blogs your success rate in teaching solving equtions will soar. Don't forget to use a different colour for the arrows and operations as this help pupils to distinguish what is happening to the equations form the equation itself.



Solving equations - life from both sides

Equation problem

Solving equations can be taught time and time again, but what percentage of a class can perform the complex operations involved. What percentage could repeat what they have done a day later? To us it seems obvious that 'what we do to one side we to the other', how frustrating then that this simple procedure seems to be so difficult for many students to grasp.

I have often been bewildered by students who could solve equations one day but not have a clue the next. I would ask myself what was wrong with them? Followed by what was wrong with my teaching? What else could I do? As a young teacher I asked my colleagues what they did they reassured me the same as me and not to worry most kids don't understand equations.

The root of the problem

It is not so much being able to reach an answer that causes the problem more it is the lack of appreciation that the equation has two distinct sides. They are also equal. Once this fact is conquered the next issue is the. Realisation that what you do to one side you do to the other.

The solution

This article is about tackling the both sides problem. When this was first suggested to me I was very cynical. Why should I try this I've always taught solving equations the same way, with the same amount of success as everyone else. But I decided to give it a try. Many pupils have a problem balancing an equation. Many Maths books have  pictures of weighing scales, some letters and numbers are placed in either tray and successive numbers and letters are removed until a satisfactory result is arrived at. But it doesn't mean much to pupils, how any of them have seen scales in this digital age?. This analogy doesn't seem to help much. This is what I tried and it was really successful.


First I wrote a simple equation on the board like this.

Most pupils will be able to tell you that x is 2. But that is not the object of the exercise, you want to get them to realise that what ever you do to one side you do to the other. I know you can tell them that but it doesn't not always register. I now ask them for a number any number which I then add to both sides. For xample lets say they chose 5 I would write

Putting the arrows on and in different colours has proved to me to be very important. It places the emphasis on what we are doing to both sides not on why we are doing it, that will come later. I then ask for another number. Lets ssume someone says 100.

I quite happily write +100 on both sides and continue. I might then suggest or it could come from the students subtracting. I then give an example of subtraction.

It depends upon the class how long you continue this. You could demonstrate a few more examples on the board, or put a starting point and get pupils to come up to the board and take suggestions from the class and develop the equations that way. It is not important to find x your objective is to get them to understand how to balance equations. I would follow this up with individual work, giving them starting points and letting their creative juices flow. 

 

Why not read the accompanying post

How to solve an equations with a frog

 

 

 

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Solving equations with a frog




Fred, ready to solve an equation

 

 Solving equations with a frog

Solving equation can be a slow and painful experience for many pupils and teachers. It just seems to be a complete mystery for far too many kids despite our best efforts that is why I have found using Fred the Frog such a powerful tool, you don’t even have to teach them about equations.

I start by telling them they are going to solve an equation butt I’m not going to tell them how to do it because they already know. This causes quite a few puzzled looks. Here is the story.

‘Fred the frog is resting on his Lilly pad not having had breakfast and feeling quite hungry when he spotted a bug in the distance his favourite food.’

 I then draw a picture of Fred and the bug. As you can see no special artistic skills are needed for this.



‘Fred jumps a distance we don’t know how far. Let’s give it a letter, what letter would you like?’

‘d’ says our enthusiastic pupil. I then draw d on the diagram like this.



I then say, ‘Fred jumps again and being an exceptional frog he jumps exactly the same distance again’. Further drawing now takes place.

 ‘Surprise, surprise he jumps the same distance again.’ More drawing takes place.

Finally he does one more jump but this time we know it is exactly one metre and Fred has breakfast  he eats the bug. Yet more drawing.





I then explain that Fred is truly exceptional, not only can he jump the same distance he can also read. ‘He looks up from his breakfast and sees a sign which says he is 16 metres from where he started.

Now ask ‘How far is one jump?’ Very quickly you will get the answer 5 metres. Now the killer question to ask ‘how do you know?’ Pupils will tell you to take 1 off the 16 so you are the last d that make 15 . Because there are 3 ds them you divide 15 by 3 to get 5. Its now up to you how you develop this. You can tell more Fed the frog stories with different numbers or you can go straight into ‘you’ve just solved 3d + 1 = 16’ and show them what they have done using traditional algebra.  I taught one girl who loved Fred so much she solved all her equations using this technique, never progressed to conventional notation and passed all her maths exams. You can also extend it to solving simultaneous equations, but that’s another story.

Collecting like terms a story of cards and beer

How to have success with Algebra

Collecting like terms

Have you taught collecting like terms, thought most pupils understood it, only to discover the next lesson your back at square one?  

Use this idea and you'll find far greater success. It is simple and easy to use.

I like most maths teachers used the fruit analogy. You know  what I mean, let 'a for apple and b for banana', then expected pupils to add a + 4b +2a + 2b. 

Virtually impossible to take that amount of fruit into the classroom as a visual aid. Reduce the stress in your life by using this idea. The visual aid you'll need? You can carry it around in your pocket.

A story of how cards and beer solved an algebra problem.

Whilst having a chat with my friend Dave he confided in me that his daughter Emily was very worried about Maths. Dave being an intelligent and perceptive fellow had come to the right man. 

Having taught Maths since 1975 I had seen many a youngster through the years struggle with what is thought to be  difficult subject and during this time I had even picked up some good ideas on how to overcome some fundamental problems.

Dave's problem

‘What’s the problem Dave?’

‘It’s this Algebra thing, you know when you put all those a b and c’s together.’ He replied with a worried look and furrowed brow. ‘It was something like 4x + 3y then you had to add it to
x + 5y.’

Collecting like terms is what we call it in the trade. All I need is a beer, a pack of cards, pen and paper and your undivided attention and I’ll show you how to teach Emily to collect like terms.’

Pack of cards

Dave intrigued provided me with a pack of cards, a beer, pen and paper and his undivided attention. I dealt him ten cards face down and asked him to sort them into their respective suits. He had 4 spades, 2 clubs, 1 diamond and 3 hearts.

The first set of terms

‘OK Dave write down what you’ve got but don’t bother with the full names of the suits, lets have a code for each, what do you suggest?’ 

Quick as a flash he came up with s, c, d and h, no fool this boy I thought. I suggested also using + instead of ‘and’ because I knew his spelling wasn’t up to much. He then wrote down

Algebra begins

4s + 2c + d + 3h

I then dealt him another 10 cards. The results were 3 spades, 2 clubs 2 diamonds and 3 hearts, I encouraged him to write the results underneath the original using our code or notation as we Mathematicians like to call it. His paper now looked like this

The second set of terms

4s + 2c + d + 3h

3s + 2c + 2d +3h

I drew a line underneath and said add them, his results were

Collecting like terms

7s + 4c + 3d + 6h

‘You’ve just done algebra, collecting like terms to be precise’ I explained. ‘So if you had 4h + 3s and added it to 7h what would you have?’ ‘11h + 3s’.

‘Now if I changed the h to x and s to y’?

‘11x + 3y. So just treat them as if they are suits in a pack of cards, how many of each do you have. Is that right?’ ‘Sure is'I replied. ‘Oh boy 11x + 3y, I can do algebra, you wait until I show Emily.  

Then 4x + 3y  added to x + 5y is 5x + 8y. Wow it’s easy, there is only one thing I don’t understand where does the beer come into it?' 

‘It’s for me, cheers!’

You might also like to read my post Solving equations with a frog

 A great collection of starters for a classroom is '101 Red Hot Starters.'

'Start your lessons with a bang! Letts Red Hot Starters contains a bank of snappy, interactive starter activities. Each starter consists of a simple, effective activity involving minimal preparation, answers and suggestions for differentiation.'



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