Transcript Division of Fractions 1

TEACHING FRACTIONS FOR
MASTERY IN KS3
A lesson given in November 2015 by a
Shanghai teacher to a year 8 class at a
secondary comprehensive school in
Devon
分数的除法（1）
------倒数
Exercise 1:Calculate
1
(1)4 ´ =
4
5 12
(2) ´ =
12 5
8 3
(3) ´ =
3 8
p q
(4)  ( p  0, q  0)
q p
Exercise 1:Calculate
1
(1)4 ´ =
4
p q
(4)  ( p  0, q  0)
q p
8 3
(3) ´ =
3 8
5 12
(2) ´ =
12 5
Exercise 2:Fill in the blanks
A
B
A´ B
2
3
3
2
9
4
4
9
5
2
p
 p  0, q  0 
q
1
5
1
2
q
p
1
1
1
1
1
Observation
&
Conclusion
Please observe the table, what
conclusion can you get?

If the numerator and
denominator of two
fractions are inverted,
then their product is 1.
p q
´ =1 p ¹ 0,q ¹ 0
q p
(
)
Reciprocal倒数
Definition
If the product of number A and B is 1, then
A is the reciprocal of B.
Exercise 3:Draw a line to connect the reciprocal from
the left to the right.
2
5
4
3
1
1
6
7
12
12
7
6
1
3
4
5
2
Exercise 4:
（ 1 ） Find the reciprocal of the following fractions and conclude the
solution .
Reverse the
3
4
7
2
5
6
4
11
numerator and
denominator
（2） Find the reciprocal of integers and conclude the
solution
3
4
15
1
0
(3) Find the reciprocal of mixed fractions and
conclude the solution
3
1
4
1
3
2
5
1
6
4
2
11
Change integer a
to a/1 first
Change a mixed
number to
improper fraction
Game time
Exercise 5: Say a number and pick a friend
to say the reciprocal of the number.
Exercise 6:True or false.
(1)If the product of number A and B is 1, then A is the
reciprocal of B.
(2)If C is the reciprocal of D, then the product of C
and D is 1.
5
3
(3)The reciprocal of 2 is 2
.
3
5
(4)Each number has a reciprocal.
(5)No number has a reciprocal which is exactly itself.
(
)
(
)
(
)
(
)
(
)
Conclusion


1
The reciprocal of x ( x ¹ 0) is .
x
q
p
The reciprocal of  p  0, q  0  is p .
q

Zero does not have a reciprocal because no real number
multiplied by 0 produces 1.(the product of any number with zero
is zero)

In positive numbers, only one number has a reciprocal which is
exactly itself, 1.
Reciprocal & Division
1
1
1  4  ,1   4
4
4
1
4  =1
4
Exercise 7:Change the multiplication to division.
5 12
12 5
 , 1 
12 5
5 12
5 12
(1) ´ =1
12 5
1
8 3
(2) ´ =1
3 8
8 3
3 8
1   ,1  
3 8
8 3
p q
(3) ´ =1 p ¹ 0,q ¹ 0
q p
(
)
1
p q
q p
 ,1  
q p
p q
0 the quotient is the reciprocal
If 1 is divided by a number（ ¹ ）,
of this number.
Summary

Reciprocal
• Definition
• Reciprocal of a fraction
• Reciprocal of an integer
• Reciprocal of a mixed fraction
• Some other conclusions

Relationship between Reciprocal & Division
Exercise 8:Calculate
5 3
(1) ´
9 10
5
(2)5 ´
12
2 3
(3)4 ´
3 7
1
(4)6 ´ 7
14
End of lesson
I did not see the next lesson, but the next slide shows
how this knowledge of reciprocals and of multiplying
fractions is developed into dividing by fractions in the
Shanghai textbook.
2 3
 x
5 4
3
2
 x
4
5
4 3
2 4
  x 
3 4
5 3
2 4
x 
5 3
8
x
15
2 3 2 4 8
    
5 4 5 3 15
The result would then be generalised to:
a p a q aq
   
b q b p bp
b  0, p  0, q  0
Division by fractions is taught not as an arbitrary rule:
‘When dividing by a fraction, turn it upside down and
multiply.’
Instead, it is derived as a logical consequence of pupils’
understanding of multiplication, division and reciprocals,
systematically developed in earlier lessons.