Download Fractions diagnostic test June 08

Fractions Diagnostic Test
Name: __________________________ Class: _______ Date:__________
1.What fraction of this
set of counters is shaded black?
3 sevenths,
7
,
3
3 out of 4,
9. Jacob spends
7 thirds
2. Shade 5 eighths of this picture
1
of his pocket money and Sara
4
1
of her pocket money. Is it possible for
2
Jacob to have spent more than Sara?
YES or NO (circle)
spends
10. Work out 1 whole – 17 twentieths of a whole
Answer: ____________________
11. Find two fifths of 25 __________________
3. Shade two thirds of this
Shape
12. Find
13. Shade this diagram to find what share each
person receives when 3 pizzas are shared among
4 people
4. Circle the largest number
1 third,
1 quarter,
1 sixth,
1 fifth
Answer: _______________
5. Circle the largest number
1
,
6
1
,
3
2
of 21 ___________________
3
1
,
5
1
4
14. Complete this statement with a fraction word
7 ÷11 = 7 _____________ of one whole
6. Reuban says that this
picture shows one third has
been shaded.
Is this true
YES or NO
(circle)
15. Put a circle around the bigger fraction in
each of these pairs
a)
7. Mele puts an arrow on this number line. Does
2
it show
YES or NO (circle)
3
0
1
2
3
8. Which is the smallest number
3 seventh, 3 ninths, 3 elevenths
Answer: ____________________
3
5
3
4
b)
3
7
5
7
c)
1
5
16. write down a fraction between
3
10
1
2
and
2
3
________________
17. I am putting tiles on the floor; they are
shaded. What fraction of the floor has been
tiled?
_____________
Fractions Diagnostic Test – Teacher Notes
1.
What fraction of this set of counters is shaded black?
3 sevenths,
3 out of 4,
7
,
3
7 thirds
Only correct answer involves fraction words. It helps to detect those students who view
fractions as “out of”, which is inadequate when trying to get students to understand number
operations with fractions. i.e ¾ of 20 as ‘3 out of 4 of 20’ !!
2.
Shade 5 eighths of this picture
3. Shade two thirds of this shape
5
2
and ), again to detect students who regard
8
3
a fraction as ‘out of’. Question 3 requires the student to deal with more pieces than the
wording of the question implies.
Fraction deliberately written in words (not
4.
Circle the largest number
5. Circle the largest number
1
,
6
1 third, 1 quarter, 1 sixth, 1 fifth
1
,
3
1
,
5
1
4
Designed to test that students understand that there are two equivalent ways of writing a
fraction. One or both wrong indicate a student has a problem with fundamental
understanding of what a fraction is.
6.
Reuban says that this picture shows one third has been shaded.
Is this true
YES or NO
A ‘Yes’ indicates student does not understand that fractions deal with equal parts of whole.
7.
Mele puts an arrow on this number line.
2
Does it show
YES or NO (circle)
3
0
1
2
3
A ‘yes’ answer suggest that the student sees fractions as ‘out of’ i.e 2 out of 3 rather than 2
thirds of one whole
8.
Which is the smallest number - 3 seventh, 3 ninths, 3 elevenths
By expressing the denominators as words rather than numbers, this question prevents
students applying a rote rule about comparing fractions with equal denominators.
9
1
1
of his pocket money and Sara spends
of her pocket money. Is it
4
2
possible for Jacob to have spent more than Sara? YES or NO (circle)
Jacob spends
If Julie’s pocket money is at least double Sam’s pocket money then Juile spends more. This
problem is designed to see whether students realize that fractions are always of something –
an object or a number. So a small fraction of a large amount may well be more than a
larger fraction of a smaller amount.
10.
Work out 1 whole – 17 twentieths of a whole
This item is designed to see whether the student understands that 1 whole has an infinite
number of fraction names. In this example, 1 whole has been split into 20 twentieths.
2
of 21
3
Superficially these two questions look like the same kind of problem. Yet there are many
students who will get #11 right but #12 wrong. In #11 the word “fifths” guides students into
working out one fifth of 25 is 5, so 2 fifths is 2 x 5 = 10. Whereas in #12 “times’ is a
different concept from “of”. It is likely that knowledgeable students will turn it into 2 thirds
of 21, which leads easily to the answer.
11.
Find two fifths of 25
12.
13.
Shade this diagram to find what share
each person receives when 3 pizzas are
shared among 4 people
Find
12 quarters are shown. 12 ÷ 4 = 3, so each person receives a 3 quarters of one apple. A
very common error made by students, who conceive of fractions as “out of”, is 3 out of 12,
which they write as 3/12. This highlights why it is so important for teachers not to refer to
fractions as “out of”.
14.
Complete this statement with a fraction word 7 ÷11 = 7 _____________ of one whole
This item is designed to highlight a wide spread misunderstanding that, in the standard
short form of a fraction, the line means division. It does not. Division is an operation whose
answer is a fraction. This question highlights whether a student understands how to divide
whole numbers that lead to a fraction answer. Here imaging, say, 7 apples shared among 11
people, each apple is cut into 11 pieces. Each piece is one eleventh of a whole apple. Each
person receives one piece per apple i.e 7 pieces which is 7 elevenths of one whole.
15.
Put a circle around the bigger fraction in each of these pairs
a)
3
5
3
4
b)
3
7
5
7
c)
1
5
3
10
Parts a) and b) have either the numerator common, or the denominator, so the questions
are essentially using ‘more of the same size pieces’ or the ‘same number of smaller pieces’.
Part c) requires use of equivalent fractions – converting 1/5 to tenths
Fractions – Common Misconceptions
Question
Possible Student
Answers
Reuban says that this picture shows one third
has been shaded.
Is this true
YES or NO
Shade this diagram to find what share each
person receives when 3 pizzas are shared
among 4 people
Circle the largest number
1 third, 1 quarter, 1 sixth, 1 fifth
Also
1
,
6
1
,
3
1
,
5
1
4
Mele puts an arrow on this number line.
Does it show
0
2
YES or NO (circle)
3
1
2
3
Work out
3
2
+
4
5
Jacob spends one quarter of his pocket money
and Sara spends a half of her pocket money.
Is it possible for Jacob to have spent more
than Sara? YES or NO (circle)
Possible Misconceptions
Possible Teaching Responses
Fractions Diagnostic Test
Do Not write on these sheets. Write all answers on the answer sheet provided
SECTION A
1.
5 eggs in a box of 12 are found to be cracked
(a)
What fraction of the box of eggs is cracked?
(b)
What fraction of the box of eggs is not cracked?
2.
What fraction of each of the shapes below is shaded?
(a)
3.
(b)
Write down the bigger fraction in each of the pairs below.
(a)
4.
(c)
3
5
3
4
(b)
3
7
5
7
(c)
1
5
3
10
Write down the missing number in ease case:
(a)
1 2
=
3 ?
(b)
6 3
=
8 ?
3
2
of the flour is used for bread and of the flour is used for cakes. What
8
8
fraction of the flour has been used?
5.
In a bakers shop
6.
John pays
3
of his wages in tax. What fraction of his wages is left?
5
SECTION B
1.
Put these fractions in order, starting with the smallest:
1
4
1
2
1
100
1
3
2.
Write down the missing number in each case:
2 ?
4 1
=
=
(a)
(b)
3 15
12 ?
3.
A relay race is run in stages of
4.
John saves
5.
The area of this rectangle is 10cm2. What is the
width of the rectangle?
1
km each. Each runner runs one stage. How many runners
8
3
would be required to run a total distance of
km?
4
1
1
of his own pay. Paul earns $6 and saves of it. How much must John earn to
4
3
save as much as Paul?
width
4cm
SECTION C
1.
What number goes in the box (
)?
2
?
10
=
=
7
15
2.
I am putting tiles on the floor; they are shown shaded. What fraction of the floor has been
tiled?
3.
Mele pays
2
1
of his wages in tax and
of his wages on rent. What fraction of his total
5
10
wages does he spend on tax and rent?
SECTION D
1
The area of this rectangle is
1 2
cm . What is the
3
2
cm
5
length of the rectangle?
length
2.
Rangi is on holiday in the UK, where distances are given in miles. 1km is the same length as
2
5
mile. His hotel is km from the petrol station. What distance is this in miles?
8
3
3.
What fraction of the square metre is shaded?
1m
1m
5
m
8
2
m
9
4.
(a)
An Australian in London converts pounds to Australian dollars when he wishes to
3
5
buy something. £1 is equivalent to 1 Australian dollars. If he pays £45.50 for a coat, how
much is this in Australian dollars?
(b)
Sometimes he wants to convert the currency the other way. He knows a suit in
Australia costs 65 dollars. To change this into pounds which sum would he do? Write down
the one you think is correct?
i) 65 x 1
5.
3
5
ii) 65 ÷ 1
3
5
iii) 1
3
÷ 65
5
a, b and d are positive whole numbers. When is
a
a
is less than ?
b
d