[Pri5 20170426FEM] Baking Éclairs and Macaroons

Problem / Question

     This problem for primary 5 from one of my acquaintances on Facebook, considered to be of intermediate level difficulty (in Singapore).  But it looks rather challenging to draw all those bar diagrams, doesn’t it?

     Here is my quickie solution without explicit algebra and without bar diagrams.

Solution

     For convenience, we use 6 circle units for Eclairs and 6 square units for Macaroons.

Suppose there were half as many Eclairs and Macaroons, then there would be 15 more Eclairs.  So 3 square units add 15 can be changed to 3 circle units.

     Add 15 to the 17 and change 3 square units to 2 circle units.  We deduce that 5 circle units is the same as 85.  From here we can easily figure out the rest.

Ans: 102 éclairs.

Comment

     The problem can be solved by bar diagrams.  However, there are many ways to skin the cat.  For more good stuff, please join my Facebook group “Effective and Elegant Mathematics”.

Heuristics Used

H02. Use a diagram / model

H05. Work backwards

H06. Use before-after concept

H08. Make suppositions

H09. Restate the problem in another way

H11. Solve part of the problem

Suitable Levels

* Primary School / Elementary School Mathematics

* any precocious or independent learner who is interested

[Enrich20170101SQT] Calculating Square Roots by Hand

Introduction

          Happy New Year to our readers!  I wish this year will be a fruitful one for everybody.

          Today, I will illustrate how to calculate square roots by hand, using  54 756  as an example.  It is similar to long division, but has some modifications.

Solution

          Starting from the right, pair up the digits.

          2×2 = 4  is the nearest perfect square to  5.  Subtract and bring down the next two digits, giving  147.

          Double the digit  2  to get  4.  Think:  ? × 4?  gives  147  or nearest possible value.  We have 3×43 = 129.

          Subtracting and bringing down the next two digits gives  1856.  Replicate the digit  4  on the left and double the digit  3,  giving  46.

          Now think:  ? × 46?  gives  1856  or nearest possible value.  It turns out that  4 × 464 gives exactly  1856.  We are done!  The square root of  54 756  is  234.

How does it work?

          This relies on the algebraic identity  (10a + b)² = 100a² + 20ab + b², the right-hand expression is equal to   100a² + (20a + b)b.  For example, at stage 4, we have  a = 23,  b = 4  and  (20a + b) = 464.

          Did you learn something today?