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Level Four > Number and Algebra

Free Cell

Achievement Objectives:

Achievement Objective: NA4-3: Find fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals.
AO elaboration and other teaching resources
Achievement Objective: NA4-5: Know the equivalent decimal and percentage forms for everyday fractions.
AO elaboration and other teaching resources
Achievement Objective: NA4-7: Form and solve simple linear equations.
AO elaboration and other teaching resources

Specific Learning Outcomes: 

Solve problems involving fractions and percentages

Devise and use problem solving strategies (guess and check, be systematic, look for a pattern, make a table)

Description of mathematics: 

To do this problem the students need to have a good look at what they are given and see precisely what that information tells them. Then they have to put the two pieces of information together.

 This problem can be made considerably simpler if the students know how to use algebra, though the algebraic approach is probably not one that they could handle at this Level. Nevertheless, algebra is a very powerful tool and becomes increasingly so the further on they go in mathematics.

We include an algebraic solution here for your information. If you have a bright student you might like to show them how it works.

Required Resource Materials: 
Calculator.
Copymaster of the problem (English)
Copymaster of the problem (Maori)
Activity: 

The Problem

I’m addicted to Free Cell a game on our computer. At the moment I’ve won 3300 games, which is 60% of the games I've played. I’d like to be able to say that I had won two-thirds of the games that I had played. How many games would I have to win in a row to get to the two-thirds winning mark?

Teaching sequence

  1. Write 2/3 on the board and ask the students to tell you all they know about it. (For example: it’s a fraction; it means 2 out of 3 parts; it’s the same as 4/6; …)
  2. Pose the problem to the class. Ask them to retell the problem using their own words to ensure that they all understand what is required.
  3. As the students solve the problem ask questions that focus on their understanding of percentages and fractions:
    What is a percentage?
    Would you rather have 2/3 or 60% of a chocolate bar? Why?
    Show me how you calculate 2/3 of 120 using the calculator.
    Tell me how you started the problem. Why did you start in that way?
    Are you convinced that you have the correct answer? Why?
  4. Tell the students that they need to record their solutions for display.
  5. Share solutions.

Other Contexts

The computer game could be replaced by any other game.

Solution

What information do I have? First I have won 60% of my games. This number is actually 3300 so I can find the number of games that I have played. This is because 3300/games played = 60/100. So games played = (3300 x 100)/60 = 5500. Where does the two-thirds come in? Well, I’m going to play some more games and I’m going to win them all. So 2/3 = (3300 + more games) / (5500 + more games). In the table below, I’ll let the fraction on the right be F. This is now set up for a classical ‘guess and improve’ strategy and the use of a table.

More games played F Compared to 2/3
100 0.6071 too small – increase games played
1000 0.6615 too small – increase games played
2000 0.7067 too large – decrease games played
1500 0.6857 too large – decrease games played
1300 0.6765 too large – decrease games played
1200 0.6716 too large – decrease games played
1100 0.6667 Bang on!

So if I play another 1100 games and win them all, then I shall have won two-thirds of all the games I have played. (I’m afraid that there is not much chance of that!)
Algebraic approach: We already know that 2/3 = (3300 + more games) / (5500 + more games). So let m = more games and we then have 2/3 = (330 + m)/ (5500 + m). Multiplying both sides by 3 and 5500 + m gives 2(5500 + m) = 3(3300 + m),
so 11000 + 2m = 9900 + 3m,
or m = 1100.

AttachmentSize
FreeCell.pdf61.14 KB
FreeCellMaori.pdf76.5 KB