Substitution: so what does it all mean?

Substitution: so what does it mean?

 

Substitution is taught every year and yet it is always causing problems. In particular when we ask a student to substitute a number into 2a then a².It is not the act of substituting that causes the problem but the understanding of algebra. Common misconceptions when substituting into a term happens time and time again. This activity helps to identify those that do, and do not understand, algebra particularly when those difficult areas of powers and brackets are used.

 

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 an elastic band and stored in a cheap envelope.

 

Step2

Hand out a set of cards to each pupil or give them a set between two if you want a more collaborative approach. Write an expression on the board such as

 

2a

 

Emphasise that this is an expression not an equation. You may have to tell them what the difference is, again another big problem what is an expression, what is an equation.

 

Step 3

Now tell them a = 4, using the cards show the answer. Hopefully they will all demonstrate they know the answer is 8, if not you have an opportunity to rectify any misconceptions. Now put the following on the board

 

a²

 

Once again the same process is followed you can ask the class show me the answer. It is now some confusion occurs, some will give you the answer 16, others 8 again. Now is the opportunity to find out who understands and who doesn’t. Dependent on the answers it is up to you how you progress. Try other numbers and other powers. This quick simple exercise will inform your assessment and guide future lessons.

 

Extension

Once you have successfully rectified any problem with these issues, and it is a major problem to be solved if you want them to progress in algebra, you can now try something like these more challenging terms 3a² and (3a)². This starting activity will provide the basis for much useful discussion.

 

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.

 

 

Bingo in the classroom

Maths Games - and now for something completely different

It is always a struggle to find innovative or new ways to practise multiplication tables. I know there are plenty of computer games but it is not always possible to get organise and sometimes you may just want a brief recap before tackling something else

Kids love playing games, even maths games, and this activity has endless variations on the game of bingo, the only limit is your imagination. It can be used to reinforce a previous lesson, warm up the class before a new lesson or just bit of fun if the lesson is flagging or boredom and irritability sets in on a wet Friday afternoon.

Click on the link below for a great source of ideas for starter lessons '101 red hot starters', take a look. This book will give you a source of ideas for years to come; a snappy start to any lesson. As one reviewer said,

 'This book is an excellent start for developing your own collection of maths starters.'

Virtually no preparation is required to start using this in the classroom. It can be bought for as little as 1p from Amazon - if that is not a bargain I don't know what is.

Maths starters

Step 1

I would ask the pupils to draw a grid in the back of their books 4 x 4, leaving a lot of space in each cell. Looking at multiplication tables I would ask them to fill in numbers bewteen say 10 and 40 it is their choice. No repeats allowed. Alternatively for example, we were practising identifying different types of triangles I would ask them for the names of the triangles and get them to recall their properties. So in random cells they would write I (Isosceles), E (Equilateral), S (scalene).

Step2

I would then tell them we are playing times table bingo or shape bingo, the winner is the one who has a complete line or get a full house, whatever suits your purpose. Ask a multiplication question or  read out a description of the triangle in question, for example, ‘two equal sides and the base angles are equal’ or you could show them a picture of the triangle. They have to cross out the answer or what they think is the name of the triangle. Keep a record of what you have done.

I found it best to have 6 cards with either the description or picture on, I would shuffle the cards and use them to pick the question. Once all six have been used reshuffle and start again until a winner has been found.

This is so popular that often I have walked into a classroom and they have asked for this as a starter, not often do you get kids begging for Maths questions. Just like Oliver Twist 'More please Sir'

Yo: A Math Teacher's Blog: Is this going to be on the test?

Yo: A Math Teacher's Blog: Is this going to be on the test?

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.