how to write equations and expressions, mathematical grammars

1. Author: Mr Lam

2. Novelist communicates their passion and
imagination to his/her readers by alphabets.
Mathematician shows their ideas by
symbols.
An English article has its standard formats or
we called this “grammars”. Grammars can
help the reader easier to read

3. If a piece of mathematic work is written properly, readers from
any country can understand it. In some sense, Mathematics
has it own language for communicating ideas.
Similar to English, Mathematics also has “grammar”
Correct use of mathematical grammars can make the reader
easier to understand the author.
Moreover, it can show the author’s understanding of the
topics. Usually, an unintelligible mathematical work
associated with unclear mathematical minds and ability

4. These mathematical grammars are the ones
commonly ignored by the students, not
explicitly showed in the textbook, or not
taught properly by their teachers.
This presentation is trying to address some
of these common mathematical grammars in
expressions and equations.

5. Many students cannot tell the difference
between an expression and an equation.
Equation has an equal sign
e.g. 5푥 + 3푦 − 12 = 7
Expression does not have an equal sign e.g.
5푥 + 3푦 − 12 − 7

6. Consider an expression: “7푥 + 2푦 − 5”
"7푥", "2푦" and “5” are the terms of this expression
“푥” and “푦” are the variables
“7” is the coefficient of 푥
“2” is the coefficient of 푦
“-5” is the constant term or just simply “constant”
Consider another expression: “103푥푦(5 − 푤)"
“푥” , “푦” and " 5 − 푤 ” are the factors of the above
expression
Technically 103 is also a factor, but more specifically we
call it coefficient

7. We usually write the coefficients in front of
the variables, e.g. 7푥 instead of 푥7
There is no need to write “1” in front of a
variable, e.g. “푥” instead of “1푥”, “푦” instead
of “1푦”

8. There are different ways to express multiplication.
“9 × 3”, “6 ∙ 5”, “5푘“
Note: The 1st expression is called a cross product.
The 2nd expression is called a dot product. The dot
is in the middle and do not confused it with 6.5 or
(6
1
2
) The later is a decimal point, which is at the
bottom between the numbers.
There are some differences between these symbols
in advanced mathematics (vector). But in junior
secondary school mathematics, they all means the
same.

9. Solve for 푥
2푥 = 24
The correct answer is 12.
If your answer is 4, then revise this power point
again.

10. Express the following expressions in fraction
with denominator = 10
푎) 2.3
푏) 5 ∙ 3
Answers:
푎)
23
10
푏)
150
10

11. Because in algebra, we use “푥” a lot as our
variable, we prefer to not use the cross product
to avoid confusion.
푦 × 푥 × 6 × 푘 × 푥 × 푚 × 2 can be just simply
expressed as 12푦푥2푘푚
Some teachers may insist you need put the
variables in alphabetic order 12푘푚푥2푦
In my opinion, it is not totally necessary as long
as you are not missing any variables and
putting the coefficient in the front.

12. Bad Good
13 = 6푥 + 1
13 − 1 = 6푥
12 = 6푥
12
6
= 푥
2 = 푥
13 = 6푥 + 1
13 − 1 = 6푥
12 = 6푥
12
6
= 푥
2 = 푥
푥 = 2

13. There is some very subtle difference
between 푥 = 2 푎푛푑 2 = 푥.
I can illustrate the difference in English as:
Crocodile is reptile but reptile is not crocodile.
Therefore, it is better to express your answer
as
푥 = 푠표푚푒푡ℎ푖푛푔
Instead of
푠표푚푒푡ℎ푖푛푔 = 푥

14. Working on equations, it is best to align all
the equal signs on the same column
In general, no two equal signs should be
occurred in one line.
Bad Good
푦 + 1 + 푥 = 2푥 + 3
푦 + 1 = 푥 + 3
푦 = 푥 + 2
푦 + 1 + 푥 = 2푥 + 3
푦 + 1 = 푥 + 3
푦 = 푥 + 2

15. There are two formats for simplifying
expression. Both are good as long as the
equal signs are aligned in the same column
ퟔ풙 + ퟏ + ퟏퟗ풙 − ퟏퟒ
= ퟔ풙 + ퟏퟗ풙 + ퟏ − ퟏퟒ
= ퟐퟓ풙 − ퟏퟑ
ퟔ풙 + ퟏ − ퟏퟗ풙 − ퟏퟒ = ퟔ풙 + ퟏퟗ풙 + ퟏ − ퟏퟒ
= ퟐퟓ풙 − ퟏퟑ

16. Think about this English paragraph:
Tom lives with his parents. Tom has a younger
sister. Tom goes to see his friend very day.
Tom loves chocolate cake. Tom’s mother is an
accountant.
Do you feel the above paragraph clumsy? Can
you improve it?
In mathematics, sometimes we purposely omit
the terms on the left hand side if they are not
changing. This is to make the workout much
neater.

17. Good Better (neater)
9푥 + 1 = 19푥 + 7 + 6푥 − 2
9푥 + 1 = 19푥 − 6푥 + 7 − 2
9푥 + 1 = 13푥 + 5
9푥 − 13푥 = 5 − 1
−4푥 = 5 − 1
−4푥 = 4
푥 =
4
−4
푥 = −1
9푥 + 1 = 19푥 + 7 + 6푥 − 2
= 19푥 − 6푥 + 7 − 2
= 13푥 + 5
9푥 − 13푥 = 5 − 1
−4푥 = 5 − 1
= 4
푥 =
4
−4
= −1
Although the work out in the first column including the answer is absolutely
correct, but it looks clumsy, especially, if the equations is a complicated
one. Also note: omission can only use for left hand side of the equations.
If the right hand side does not change, we still need to rewrite the right
hand side. See the highlighted red text in the above example.

18. Some teachers /schools insist students to
write their “x” in this form:
I myself am not too fussy about this as long
as you can make a consistent difference
between “x” and “×” (the multiplication sign)
e.g.

19. Not to confuse z with 2
Not to confuse l with 1
Not to confuse 6 with b Not to confuse g with y
These are only a few examples. You can name more,
e.g. h and b, x and y, 5 with S, K with R, q with 9 and H
with 1-1