Wednesday, April 26, 2017
Problem / Question
This problem for primary 5 from one of my acquaintances on Facebook, considered to be of intermediate level difficulty (in
). But it looks rather challenging to draw all
those bar diagrams, doesn’t it? Singapore
Here is my quickie solution without explicit algebra and without bar diagrams.
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.
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”.
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
* Primary School / Elementary School Mathematics
* any precocious or independent learner who is interested
Sunday, January 1, 2017
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.
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?