Modelling sans Algebra

Singapore PSLE Modelling Math (for 11 year-old pupils)

Problem:

1) 20% of John saving is $120 more than 30% of Brian savings.

2) After John spent 5/6 of his saving and Brian spent 1/2 of his saving ,
John has $20 more than Brian.

Find the saving of John at first ?

Solution:

(Technique: work backwards from the End.)

Before 2nd condition:

J: 60 units+ $120

[1][2].....[9][10] [$20]

[1][2].....[9][10] [$20]

[1][2].....[9][10] [$20]

[1][2].....[9][10] [$20]

[1][2].....[9][10] [$20]

[1][2].....[9][10] [$20]

B: 20 units

[1][2].....[9][10]

[1][2].....[9][10]

After 2nd condition:

J: [1][2].....[9][10] [$20]

B: [1][2].....[9][10]

20% of J

= 0.2 x 60 units + 0.2 x $120

= 12 units + $24

30% of B

= 0.3 x 20 units

= 6 units

From 1st condition:

(12 units+$24)- 6 units

->$120

6 units +$24 ->$120

6 units ->$96

1 unit ->$16

J: 60 units+ $120

J= 60x$16+ $120= $1080

[QED]

4-level MathThinking

4 Levels:

L1. S&T (See & Touch) Concrete: 1 apple, 2 oranges...
e.g. Math Modeling: visualise the problem [Primary School]

L2. S~T (See, no Touch but can guess):
e.g. Guess x,y for 2x+3y=8 ? [Secondary School]
e.g. Chimpanzees can guess where you hide the banana.

L3. ~S~T&I (no See, no Touch but Imagine):
e.g. Complex i = [Junior College].

L4. ~S~T~I (no See, no Touch, no imagine)
e.g. Abstract Math: Galois Group, ε-δ Analysis, Ring, Field, etc. [University]

Solution 3 (Modelling): Monkeys & Coconuts

Let x the min number of coconuts initially.

1st monkey took "a" coconuts away, 2nd monkey "b" coconuts....5th monkey took "e" coconuts.

[a][a][a][a][a] + 1 =x

Loan 4 coconuts to the initial pool of x coconuts to divide by 5 evenly at each monkey.

[a][a][a][a][a] + 1 + 4 = [a][a][a][a][a] + 5 = x+4 = X (inflated x by 4 )
1st Monkey: [a’][a’][a’][a’][a’] = X
a’ = $Latex \frac {1}{5} X$

... Left 4a’= $Latex \frac {4}{5} X$
[a’][a’][a’][a’] => [b][b][b][b][b]

b= $latex \frac{1}{5} $ .4a’ = $latex \frac{4}{25}X$

... Left 4b= $Latex \frac {16}{25}X$
[b][b][b][b] => [c][c][c][c][c]
c=$Latex \frac {1}{5}$ .4b= $Latex \frac {16}{125}X$
... Left 4c= $Latex \frac {64}{125}X$
[c][c][c][c] => [d][d][d]d][d]
d= $Latex \frac {1}{5}$.4c= $Latex \frac {64}{625}X$
... Left 4d= $Latex \frac {256}{625}X$
[d][d][d]d] => [e][e][e][e][e]
e= $Latex \frac {1}{5}$.4d= 256.(X/3125)
Since e is integer
=> X = 3125 or multiples of 3125
Minimum X=3125
x+4 = 3125
x= 3121 = minimum Coconut initially.

Note: This solution used the Singapore Modelling Math taught in all Primary Schools for 11-year-old pupils.

Singapore Modelling Math

Modelling Math versus Usual Method

Marie had 20 m of cloth. She used 3/5 of it to make some dresses. How many meters of cloth did she use?
1. Model method:

【】【】【】〖〗〖〗
5 units = 20 m
1 unit = 20 / 5= 4 m
3 units= 3 x 4 = 12 m

2. Usual method:

3/5 x 20 = 12 m

The 2nd method is too abstract for 8 or 9 yr-old kids who will resort to learn fraction “x” or “/” formula by rote.
Models appear 'silly' to adults who prefer abstract concept '3/5 x 20';

Note 1:  Singapore Primary School Modelling Math

Adopted since 1990 from Polya’s “How to Solve It” (1945) 4-Step Problem Solving Process

1. Understand the problem in English Word Math.
2. Devise a plan: draw models
3. Carry out the plan: solve
4. Look back: verify

The Concrete-Pictorial-Abstract Math approach is developed by Dr. Kho Tek Hong (retired in 2007) & his MOE team @ 1980s.

Note 2: Models can also help visualize & 'concretize' Abstract Algebra (Set, Group, Coset, subgroup, kernel, isomorphism ...Galois theory). Feynman invented 'model' (Feynman diagram) to visualize quantum theory.