A very elegant piece of algebra

I set this problem today, during a lesson in which we had been looking at simultaneous equations with 2 unknowns:

A + B = 92

A + c = 57

B + C = 59

M came up with this answer:

POSTSCRIPT –

M then set me this problem:

A-B = 25

B+C = 151

A+C = 176

which turned out to have an unexpected twist in it….. Have fun!

Getting used to negative numbers Part I – a simple game

The Philosophy of the Game

The learner will be more confident with negative numbers once they can sketch a number line like this in their heads. Helping to make one by numbering the chips is a great way to get to know how the line works. During play, the colour of the card will tell you whether to move towards the losers’ end (boo, red, negative) or the winners’ end (hooray, black, positive). This emotional response to the DIFFERENCE between plus and minus is really important later, especially in algebra when kids may not see the minus signs. Confidence subtracting EVEN WHEN THE ANSWER IS NEGATIVE takes time to learn, but this game is ideal, treating the minus numbers as just more places on the number line after you count 5,4,3,2,1,0….

You will need:

• A pack of playing cards, use just the Ace,2,3,4,5 of all 4 suits.
• A piece of A4 paper, scissors, a ruler, sellotape and a pen
• 2 different small counters. A 1p coin and a 5p coin would do fine.

To prepare the game board

• Cut the paper in half legthways and sellotape the two peices together to make a long thin shape. Rule a line across it and mark on it a number line with 0 in the centre, the negative numbers on the left, and the positive numbers going right. The scale must be regular (use the ruler to mark out 2cm-apart chips). It should look like this:
• Place both the counters on the 0 (one below and one above, will avoid collsisions!

To Play:

• The BLACK cards are positive values 1 to 5, and the reds are negative 1 to 5.
• Shuffle the cards and take turns to choose one, moving either right (black) or left (red) the right number of chips.
• The winner is the first one to go over the 11 (ie score 12), OR the loser is the one who falls off the left hand end by scoring -12.
• If noone falls of the end, declare the winner after a timed period, say 5 minutes.
• To make it more challenging, pick 1,2 or 3 cards each. The player decides how many cards to take and then takes them.

Next Time you play:

• Can the child make the board? With a bit of help, perhaps?
• Can they total their 2 or 3 cards before they move, or do they prefer to make the moves for each of their cards in turn?

Next Time:

• Once the number line is a simple concept for them, you can play with all the cards from Ace to 10, and just keep a score on paper without counting any counters up and down the number line. Winner is the one with the highest (or maybe the least negative) score after say 15 rounds. Players can choose up to 10 cards in one round, and some interesting strategies develop for totalling their hands. More in the next post.

Substitution into a formula (calculator method)

Using formulae to solve problems can be fraught with difficulties if some of the numbers are negative. If the calculator is given the values of the letters, then the formulae is keyed in using those letters, accurate results can be achieved every time. The work is also easier to check.

Click here to see a film of the calculator being used to solve this problem.

To set the letter A to the value 12 , B to -3.6 and C to 15

• 2   Shift   RCL(Sto)   (-)A
• 7  Shift   RCL(Sto)   ..”’ B
• -12 Shift   RCL(Sto)   hyp C

Once that is done you can type entire algebra phrases into the calculator and find their values.  Here is an example of using the calculator to use a formula. You will see that the formula is written with capital letters to match the way the calculator sees the alphabet.

Trial and Improvement for George

George was guessing answers today and got it down to a fine art – usually getting it right in 3 or 4 tries. We were working really fast because we had set our calculators up to give the answers exactly the way we wanted to see them, ie to one decimal place, and as a number not as a square root (or Surd). Here’s how:

• Shift    Mode   6:Fix    1      gives the answers Fixed to 1 decimal place
• Shift    Mode   1:MathIO    2:LineO     gives you nice mathematical INPUT but just a decimal OUTPUT which is perfect.

With Guessing you can go round in circles and forget what you tried…. so we do Trial and Improvement which sounds really posh but it just means we jot down the guess each time and the answer it gives. It’s nice to see the list getting shorter, as you get better and better at making a good first guess.

*** When you have finished this work, you can put your calculator back into its normal mode by pressing:

• Shift 9(Clr)  3(All)  =(to confirm) AC(to start again)