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What Makes A Good Resource - Simultaneous Biscuits


This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 08 September 2009 by ncetm_administrator
Updated on 03 September 2010 by ncetm_administrator

What Makes a Good Resource
 

Simultaneous Biscuits

Teacher comments:
I really dislike the way that simultaneous equations are separated into lots of different types, and students expected to remember the methods for all the cases. 

I decided to teach how to play with equations, developing confidence and strategy, so that ultimately the kids will teach themselves the method of elimination for solving simultaneous equations.

I’d worked with year 7s on solving equations using the context of x= number of biscuits in a packet.  I had to do simultaneous equations with a couple of kids the next day, so I tried the metaphor of two sizes of biscuit packs.  It worked well, and now I use it with full classes as an introduction to simultaneous equations.

 

What I did:
“My favourite biscuit comes in two different packs, red and blue”

“Yesterday, I bought 7 red packs and 3 blue packs, when I got home I opened them up and counted – I had 41 biscuits altogether."
7r+3b=41

At the weekend I bought 5 red packs and 10 blue packs – and it was 45 biscuits.”
5r+10b=45

“Can anyone tell me another shopping list and tell me how many biscuits it would buy?  Only say it if you are SURE”

Common responses are:
10r+20b=90 (double)
70r+30b=410 (other multiple) etc

I keep accepting suggestions until there are about ten valid equations, and the class look confident.  If no-one thinks up sums or differences, I’ll leave introducing that for the moment, if they do mention them, then we start acting out the two scenarios below.

I let them loose to work out as many different shopping lists as possible – I’ll usually interrupt to demonstrate sum/difference after 5 mins, then let them work for another 10 mins.

Sum and difference
Enact one student buying 7r and 3b, and another buying 5r and 10b    They come together and combine their shopping:  12r+13b=86

Then we imagine having one person buying 10r+20b, and then giving away 7r+3b to a friend:  3r+17b=49

“So now there are lots of valid ways of combining equations to make new true equations.  Your task this lesson is to make as many valid equations as possible.”

While they work, I wander round muttering distractedly “you can add equations or take equations, or you can times an equation by a number”

Suddenly, someone will say r=5

I stop everyone, and invite him/her to demonstrate how they made their breakthrough.  I usually have a spare example, if needed, for kids to work on if the denouement comes too soon. 

 
What happens next
I have a couple more equations on the board, and ask them to create more equations from them.  “It ought to be possible to get r= or b=, so tell me if you happen to get one of these”

I go round the room looking for kids to say things like:
  • “I’ve found out r”
  • “You have to get them to match”
  • “If you’ve found r then you can get b too”
  • I repeat these insights, to anyone who wants to know “Joe noticed that...”  “Fiona says it’s best to...” Why don’t you try that? 
  • “so how will you find the other variable?”
  • “how can you check if your answers are correct?”

We usually end up with a method like this which works for all simultaneous equations

  1. multiply/divide until two terms match
  2. add or subtract the equations to kill one variable
  3. solve a variable
  4. substitute this answer to simplify one equation
  5. solve for the second variable
  6. substitute both values into original equations to check
  7. draw a smiley face in triumph

 

Reflections:
Classes seem to be more confident in approaching simultaneous equations, they tend to approach each question as a puzzle to crack, everyone has access to the basic moves, and they tend to ask each other for help spotting what to do next and where to go.  I love teaching any maths where the kids can know for sure that they have solved it perfectly before they move on.  Checking and drawing smiles seems to relieve frustration/demonstrate triumph/give a sense of achievement.  It also forces people to confer and ask for help when they need it.
 
 
 
Secondary Resources
 
What makes a good resources - using the materials


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Comments

 


10 April 2013 23:04
Looks great. I've got OFSTED in tomorrow and I'll give it a go. I'll let you know. :)
22 January 2013 22:37
Love it. Especially muttering distractedly. Will try this next time sim eqns comes up... Thanks!
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