Sunday, May 17, 2015

[S1_20150516AELI] Apples and Pears on a Table?

Question
Michael has enough money to buy either 12 pears or 36 apples. if he intends to buy equal number of pears and apple, how many of each fruit can he buy with the money

Introduction
     This is an interesting algebra problem meant for secondary 1 students (» grade 7) that has the potential to lead to a system of complicated simultaneous equations in two variables.  Fortunately, there are some interesting and simpler approaches.  I present three of them below.

Solution 1 (using a Table and Unit Costs)


Solution 2 (Insight from proportion)
     This is perhaps the intended algebraic approach.  Observe that since  12 pears are as expensive as  36  apples,  1  pear is “equivalent” to  3  apples.  Michael’s budget is  36  apples-worth of money.  If there are  n  pears and  n  apples,  the  n  pears can be exchanged for  3n  apples.  We arrive at an equation as in solution  1.
  
Solution 3 (Acting it Out)
     You can role-play this with your friend using toy-fruits.  Your friend is the fruit-seller and you are the buyer.  No toy-fruits?  Well, use some counters, bottle caps, Lego bricks, ... whatever to represent the apples and pears.  If you are a lonely person, or if your friend is too busy, or your mother has thrown away all your toys since you are (sort of) more grown up already, maybe just do a thought experiment.  Imagine, at first, you took  36  apples to the check-out counter.  Then you saw some pears, and you grabbed  12 of them, ditching the apples.  You figure out that  1  pear is equivalent to  3  apples.  Just then, you change your mind.  You decide that you want an equal number of apple and pears.  OK, so you replace  1  pear with  3 apples.  You get  11  pears,  3  apples.  Keep on swapping pears for apples.   You get  10  pears,  6  apples.  Then  9  pears,  9 apples.  Bingo!

Final Remarks
     There is no magic potion for mathematics (although heuristics is a start).  There is also no one fixed method for you to memorise to solve problems.  As they say, you have more than one way to skin the cat.

H01. Act it out
H02. Use a diagram / model
H03. Make a systematic list
H04. Look for pattern(s)
H05. Work backwards
H06. Use before-after concept
H07. Use guess and check
H08. Make suppositions
H09. Restate the problem in another way
H13* Use Equation / write a Mathematical Sentence

Suitable Levels
Lower Secondary Mathematics (Secondary 1)
GCE ‘O’ Level “Elementary” Mathematics (revision)
* other syllabuses that involve ratios or algebra

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