Transcript Misconceptions and Common Mistakes

Math Department
Endeavour Primary School
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Purpose of workshop
Misconceptions and mistakes by topic:
◦ Whole Numbers (P5)
◦ Fractions (P5 and P6)
◦ Ratio (P5 and P6)
◦ Percentage (P5 and P6)
◦ Geometry (P5 and P6)
◦ Mensuration (P5 and P6)
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Equip parents with knowledge of
common misconceptions and mistakes
pupils make.
Help parents to help pupils better.
Consistent between what is taught in
school and support from home.
Provides a good support structure that
can reduce stress in pupils.
If pupils can overcome these
misconceptions, they can fare much
better.
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What is a misconception?
It is defined as – ‘a view or opinion that is incorrect
because it is based on wrong thinking or
understanding.’
Misconception 1: When you divide a number, the
answer will be smaller and when you multiply a
number, the answer becomes greater.
Misconception 2: When you see the word ‘more’ in
the sentence, you always add.
1. Writing in numerals or in words
Write in numerals:
Three hundred and four thousand and sixtyfive
Correct answer : 304 065
Common mistakes: 300465, 4365
300 + 4000 + 65
Three hundred and four thousand and sixtyfive
304 065
Write in words:
217 389 vs 217 089
Misconception: can only use ‘and’ once!
2. Order of Operations
Do the following sum:
a) 8 + 3 – 1 = ?
Ans: 10 and 6
Did you get them right?
b) 8 – 3 + 1 = ?
Now do these:
c) 5 x (8 – 4 ÷ 2) ÷10 =
Is your answer 1 or 3? Or do you get a totally
different answer?
Why do pupils get the wrong answer?
BODMAS
It is not true that you have to follow the order as
shown, e.g. D before M and A before S
BODMAS is a misconception
In Primary school, we seldom encounter ‘of’ in
the order of operations.
Pupils generally only see B, D, M, A, S
Use hierarchy structure
First priority
B
Then,
D M or M D
Lastly,
A S or S A
from left to right
from left to right
Let’s try again:
5 x (8 – 4 ÷ 2) ÷10
=
5 x (8 – 2) ÷10
=
5 x 6 ÷10
=
30 ÷10
=
3
1. Fraction as division
5 =?
8
Pupils work out 8 ÷ 5 instead of 5 ÷ 8. Why?
Misconception: Only a larger number can be
divided by a smaller number.
Show counter example: 2 pizzas can be shared
with 4 people.
2. Answering ‘Fraction of …’ qns:
John has 7 books, Mary has 5 books.
Express the number of Mary’s books as a
fraction of the number of John’s books.
5
7
Express the number of John’s books as a
fraction of the number of Mary’s books.
7
5
Pupils may give the first answer as they believe the numerator
has to be smaller than the denominator
Misconception 1: First number is the
numerator, second number is the denominator,
Misconception 2: Larger number is the
denominator.
Consider another question:
John has 7 books, Mary has 5 books.
What fraction of the number of Mary’s books is
the number of John’s books?
5
7
7
or
5
Rule: the first number after ‘fraction of’ is the
denominator
3. Multiplication of fractions (cancellation)
22 x 14 x 1
7
1
2
What are the common mistakes usually found
here?
a) Cancellation between 2 numerators or 2
denominators
b) Double cancellation: 1 denominator with 2
numerators
4. Calculator error
Use your calculator to do this:
1
1 x2=?
2
Pupils sometimes did not use mixed number
key but used fraction key instead.
Pupils who pressed ‘1’ first, then the fraction
key to enter half gets a wrong answer. Try
doing this?
1) Press the shift button
2) Press the fraction button
5. Dealing with remainder
Compare the two questions:
1)
1
1
Sarah spent 4 of her money on a bag and
4
2)
1
1
Sarah spent
of her money on a bag and
4
4
of it on a purse.
of the remainder on a purse.
Tendency to missed out the ‘remainder’.
1
7
6. Mrs Tan had
kg of flour. She used
2
8
of it to make some cookies. How much flour
had she left?
7 1

8 2
or
7 1
x
8 2
Pupils need to be alert on the presence or
absence of units.
1. Simplest form
All ratio answers need to be in the simplest
form unless specified.
Do not leave ratio in decimal notation.
e.g.
3.5 : 4 : 2 = 7 : 8 : 4
2. Not alert on ratio requested.
Sam has $34 and Frank has $35.
What is the ratio of Frank’s money to Sam’s
money?
What is the ratio of Frank’s money to the total
amount of money?
Tendency to give the ratio answer based on
order the numbers appear.
3. Answering in the wrong format
The ratio of the number of boys to the number
of girls in the school in 2 : 3. What fraction of
the total number of pupils are girls?
Pupils answer in ratio instead of fraction.
4. Unable to identify standard ratio type
1)
2)
3)
The ratio of Ali’s stamps to John’s stamps is
3 : 1. Ali uses 12 stamps and the ratio
1 Quantity unchanged - John
becomes 3 : 2.
The ratio of Ali’s stamps to John’s stamps is
3 : 1. After Ali gave John 12 stamps, the
ratio becomes 2 : 1. Total remain unchanged
The ratio of Ali’s stamps to John’s stamps is
3 : 1. If they both buy 12 stamps each, the
ratio becomes 2 : 1. Difference is unchanged
5. Using wrong original ratio
E.g. The ratio of Ali’s stamps to John’s stamps
is 3 : 1. Ali uses 12 stamps and the ratio
becomes 3 : 2. How many stamps do they have
altogether in the beginning?
1 Quantity unchanged - John
Before
Ali : John
3 : 1
After
Ali : John
3 : 2
Before
Ali : John
3 : 1
6 : 2
Ali – 12
After
Ali : John
3 : 2 1 Quantity unchanged - John
3u ---- 12
1u ---- 4
4u ---- 4 x 4 = 16
Should have solved for 8u instead of 4u.
Good practice: Cancel out the original ratio.
1. Wrong mathematical sentence
Do this: Change to percentage
12
a)
25
7
b)
8
Method 1 is to convert denominator to 100.
Method 2 is to multiply fraction with 100%
Method 2 is to multiply fraction with 100%
12
x 100 = 48% (incorrect statement)
25
12
x 100% = 48% (Correct statement)
25
However, try doing both using the calculator.
What do you notice?
When using calculator, do not press the % key.
2. Using wrong base
Mr Jahan’s mass was 70kg in 2004. His mass
increased to 84 kg in 2014 and reduced after
much dieting to 67.2kg in 2016.
a)
b)
What is the percentage increase in his mass
in 2014?
What is the percentage decrease in his mass
in 2016?
a) Increase: 84 – 70 = 14
14 x 100% = 20%
70
b) Decrease: 84 – 67.2 = 16.8
16.8
x 100% = 20%
84
Common mistake: use wrong
denominator
Mr Jahan’s mass was
70kg in 2004. His
mass increased to
84 kg in 2014 and
reduced after much
dieting to 67.2kg in
2016.
a)
What is the
percentage
increase in his
mass in 2014?
b)
What is the
percentage
decrease in his
mass in 2016?
1. Unable to identify parallel lines and angles
within
2. Not labeling angles
180° – 132° = 48°
48° – 23° = 25°
180° – 90° – 25° =
65°
132° – 65° = 67°
No way of knowing what the working done are
for.
3. Unable to see isosceles triangles within a
rhombus
By adding the equal
lines, pupils can
see the isosceles
triangles better.
4. Wrong naming of type of angles
180° - 23° - 132° = 25°
(alternate angle)
(vertically opposite angle)
1. Forgetting to multiply the half
2. Not able to identify the correct base and
height pair
2. Not able to identify the correct base and
height pair
3. Difference of areas mistake
1
x 42 x 3
2
1. Wrong use of formula
Area of circle
π x r x r (encouraged)
π x r2
Circumference of circle
2xπxr
π x d (encouraged)
Encouraged to distinguish the formulas better.
2. Calculate for a circle instead of part of a
circle
3. Identifying the correct radius and diameter
Small semicircle:
Radius = 5cm
Diameter = 10cm
Big quadrant:
Radius = 10cm
Diameter = 20cm
4. Incomplete sides when finding perimeter
Pupils failed to add
the two straight
lines.
4. Incomplete sides when finding perimeter
Pupils failed
to add the
two straight
lines.
5. Mathematically wrong statement
Correct: 2u
60 or 2u = 60
Incorrect: 2
60
2 = 60
Statement says area of circle but pupils calculated for
a quadrant instead.
Missing π in the working answer
6. Doing things the hard way
Find area of quadrant
Answer x 2
Find area of square –
Area of quadrant
Answer x 2
Add both answers
Shortcut:
Area of square x 2
6. Doing things the hard way
Shortcut:
Area of square –
Area of semicircle
Can you see it?
7. The value of pi
Usually, we use 4 values of pi
1)
2)
3)
4)
3.14
In terms of π
Calculator value
22
7
Pupils must use the value of π as mentioned in
the question.
Some pupils used the wrong value.
What is the value of π in the calculator?
3.141592654
So how do pupils use this value?
Imagine needing to find perimeter of quadrant
Pupils tend to round off first and that will result in
inaccurate answer.
Pupils are advised to use the symbol π until the last
step, then press the value of π into the final answer.
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When pupils do not get the marks they are
supposed to get, there are usually 3 reasons:
a) Do not understand the question.
b) I understand but I do not know how to do.
c) I understand and I know how to do but I am
not alert and careless.
Common mistakes made in exams may cause
pupils to lose enough marks to cause a drop of
1 grade and in some cases, even two grades.