Number sense, related concepts and basic number competency
Maths: Class 3
1 Why was the question asked?
Question
2 What did students answer?
7 ones + 9 tens =
A. 26.7%
Options
A
16
B
27
C
79
D 97
P
The concept of place value is one of the most important ones in
Mathematics. This question checks whether students can identify a
number split up in a simple way but presented in an unusual way (usual
way would be 9 tens + 7 ones).
D. 46.1% No. of students
P
9187
C. 23.0%
Only 46.1% answered correctly
3
About 46% of the students answered the question correctly but about
28% of the students chose the wrong option A and about 23%
students chose the wrong option C.
B. 1.1%
Possible reason for choosing A: These students probably didn't
understand what was asked and just added the given numbers by seeing
the addition sign.
Possible reason for choosing B: Very few students chose this option.
These students most probably selected this option randomly.
Possible reason for choosing C: These students probably just saw the digits 7
and 9 and wrote the digits in that order to form the number 79, because the standard
questions are like 9 tens + 7 ones where the students would just need to write the digits
in the same order as given to form the number 97.
Learnings
Many students don't seem to have understood the given expression, probably since the ones and tens are interchanged as
compared to how such questions are asked traditionally.
This indicates that many children are looking for “familiar” question types or patterns and answering based on an observed pattern. These children are not attempting to understand what exactly the question is
asking, or what terms like “tens” and “ones” mean. Most likely, it is teaching practices, the curriculum and “rote” assessments that lead to such an approach among children. While rote practices are important in
certain areas, children should also be encouraged to understand the deeper concepts, and “read” and interpret questions before answering. It is important that teachers know whether children have
understood important concepts like place value or not, which can be diagnosed only if children show the understanding of place value in unfamiliar contexts.
4 How do we handle this?
It is important that students understand the meanings of the terms and expressions, rather than just learning the procedures without understanding. In the case of this question, students need to learn and understand.
• Why 97 is written as 9 tens + 7 ones; Is it the same as writing 7 ones + 9 tens? • What will 79 be written as in terms of tens and ones? • What are the 9 tens and 7 ones representing? • Why is there a '+' in
between? Why are we adding the 2 terms to represent the number? It is only when students are able to answer all the above questions, can we say that they do understand the concept of place value. Expressions
like these are not something to remember, and students should be discouraged from doing so. Developing understanding of these concepts using different activities, involving place value blocks would help
students work with such expressions effectively and prepare them to deal with higher numbers.
One such activity would be - • Give 43 chocolates to a group of children • Ask them to form groups of tens using the chocolates so that they realize that they cannot form more than 4 groups of tens and also that
the remaining are 3 • Then introduce the notation 43 = 4 tens + 3 ones for them to get the representation along with understanding.
Useful resources:
Websites: http://coe.jmu.edu/Mathvids2/videos/plans/ipv/plvl1a_med.mov
http://www.susancanthony.com/Resources/base10ideas.html
Do write to us at ts@ei-india.com, using the feedback format provided to your school. Formats, Useful Resources Links, and other details are available online at http://www.ei-india.com/teacher-sheet/
M3-0708-12
Educational
INITIATIVES
Maths: Class 3
Number sense, related concepts and basic number competency
1 Why was the question asked?
Question
The question checks an understanding of 2 concepts:
Operations – that the expression 4+50–3 is equivalent to each of the
following forms:P
(4+50)–3 P
(4–3)+50 P
4+(50–3)
Place value – The shift of '0' from 50 to 40 is not just a reordering of the
digits in the numbers but is actually decreasing the value of the number
(and the complete expression) by 10.
2 What did students answer?
Which of the following is the same as 4+50–3?
About 35% of the students chose the correct answer D but almost
50% of the students have chosen the wrong option A.
12022
A. 49.4%
Students who chose the correct answer D, might have answered either
by calculating the values of all the expressions and comparing their values
or by applying one of the intelligent strategies below:
a) realizing that the number in the question is 51 (very close to 50)
C. 3.9%
b) realizing that A, B, C are very much “away” from 50, by looking at the shift
in the value of the tens. c) leaving “50” as it is and working with “4–3”, the single
digits in the number to make out that D is also 51.
Only 34.4% answered correctly
Possible reason for choosing A: Students probably saw that the digits 4, 5, 3 are
appearing in the same order as in the question and have decided that only the '0' shifting
from one number to the other doesn't make a difference.
Possible reason for choosing B: These students probably added 4+50 by putting 4 in the tens
place and hence got the answer as 90. This could just be a random guess too.
Possible reason for choosing C: These students probably calculated 50–3 by putting 3 in the tens place and
got the answer as 20. This could just be a random guess too.
A
40+5–3
B
90–3
C
4+20
D 1+50
P
3
B. 5.
Options
7%
D. 34.4% No. of students
P
Learnings
• It can be seen that the expression in the question and the expressions in the options are such that the child could have easily calculated their values but the fact that almost half the students who appeared in the
test have chosen an incorrect answer A, shows that many students didn't even think of doing that.
• From the kinds of answers students are choosing, we can conclude that they could be facing one or both of these difficulties:
P
Understanding place value; that the value of a digit changes according to its placement in the number. P
identifying the same expressions written in different ways, using different combinations.
4 How do we handle this?
It is very important that students understand what they are doing and why certain rules work. For example knowing that 48 + 53 is the same as 43 + 58 but is not the same as say 45 + 83, is important but
students shouldn't make a blind rule that interchanging the unit's digits in the two numbers doesn't alter the sum, they should be able to reason out why it works that way.
It is worthwhile to ask the same question with some more options, say 4+5-30, 4+20+30-3, 4+20-30+3 etc. and see what students are choosing, and have a discussion on why they are choosing particular
answers.
It would also be interesting to investigate what students understand by “same”. i.e. Do they think the sum/ difference should remain the same, or are they attaching some other meaning to “same”?
Useful resources:
Websites: http://books.google.com/books?id=0S9n17SJBMIC&pg=PA308&dq=is+it+important+to+understand+place+value+to+understand+arithmetic+operations&sig=rGqrCRUzScuumXjzhRtzUdNDWCA
Do write to us at ts@ei-india.com, using the feedback format provided to your school. Formats, Useful Resources Links, and other details are available online at http://www.ei-india.com/teacher-sheet/
M3-0708-02
Educational
INITIATIVES
Maths: Class 4
Number sense, related concepts and basic number competency
A . 7.4%
Question
Which of these is 341 CLOSEST to?
Options
A
34 ones
B 34 tens
P
C
43 tens
D
3 hundreds.
1 Why was the question asked?
Students are introduced to grouping/regrouping numbers and to the
concept of place value from Class 2 onwards. This question was designed
to test if students are able to identify the numbers represented in options
as groups of ones, tens or hundreds, and find the one which is closest to
the given number.
2 What did students answer?
Though about 46% of the students were able to answer this question
correctly, about 40% of the students chose the wrong option D.
Possible reason for choosing A: Students selecting this option are
B. 45.7% P
probably just translating 'ones' to '1' and writing 34 ones as 341. Some of
these students might have just made a random guess.
Possible reason for choosing C: Students selecting this option have not
C. 4.5%
understood the question and are most probably making a random guess.
Only 45.7% answered correctly
Possible reason for choosing D: Students selecting this option are probably just
seeing that the given number is three hundred forty one and since the only option in
which they are able to see '3 hundreds' explicitly is D, they are selecting it. They probably
think that a number given as 3 hundreds will be closer to the given number, which is also in
hundreds, than a two-digit number given with ones or tens.
D. 40.5%
No. of students
1166
3
Learnings
As the data indicates, students are perhaps not able to identify numbers given in groups of ones or tens. This could be because two-digit numbers are given along with ones and tens (34 ones, 43 tens) where as they are
used to seeing single-digit numbers like 7 or 5 (7 ones, 5 tens). They might be able to see that 3 hundreds is 300 because it is written in a straightforward manner. But for 34 tens, they need to see that it means 34 groups
of tens which equals 340 ones. Perhaps they fail to see this relationship between the groups of tens and ones. Many students are probably failing at this initial step of identifying the number given in groups.
Many times students try to use shortcuts. In this question, they might be just looking at words like hundreds and comparing it with the given number, without realising that they have to consider the entire term
'34 tens', translate it into the number it represents and work with it.
4
How do we handle this?
Grouping/regrouping numbers is the base of the concept - place value. Students should be able to identify the number given as a group of ones, tens or hundreds.
For this question, let students use some physical objects like straws and find out what number each option represents. For example, give them 34 bundles of straws, each having 10 straws and ask them to open
the bundles and see how many total straws are there. Similarly, they can check this for each and every option and then identify the number that is closest to 341. This kind of activity will give them the feel of the
number – it will help them see how much 34 tens or 43 tens means.
Give them questions that will then test this understanding of grouping. Give them options and ask them to select same numbers from the group. For example, “23 ones, 23 tens, 32 ones, 32 tens, 2 hundreds and
3 ones, 2 hundreds and 30 ones, 2 hundreds and 30 tens, 3 hundreds, 3 hundreds and 2 ones, 3 hundreds and 20 ones. Select pairs of numbers from the above list that are the same.”
Useful resources:
Websites: http://coe.jmu.edu/Mathvids2/videos/videos.html#ghto - A good collection of videos – the ones on grouping
Books: Adding It Up – Helping Children Learn Mathematics (http://www.nap.edu/catalog.php?record_id=9822)
tens and hundreds demonstrate some hands-on activities to teach grouping of objects to students.
A good book published by National Research Council, USA, that addresses certain basic questions regarding school mathematics programs.
Do write to us at ts@ei-india.com, using the feedback format provided to your school. Formats, Useful Resources Links, and other details are available online at http://www.ei-india.com/teacher-sheet/
M4-0708-02
Educational
INITIATIVES
Maths: Class 4
Number sense, related concepts and basic number competency
Question
A
5
B
48
C 50
P
D
91
D. 18.5%
No. of students
1963
2 What did students answer?
About 21% of the students could answer this question correctly.
Around 52% of the students chose the wrong option A.
Possible reason for choosing A: Students selecting this option are just
looking at the two numbers '48' and '43' and finding out the difference
between them.
Possible reason for choosing B: Students selecting this option are most
probably making a random guess.
Only 20.6% answered correctly
Possible reason for choosing D: Students selecting this option are perhaps just
matching the word 'more' and then adding the two numbers '48' and '43' to get 91 as the
answer.
C. 20.6%
P
A. 51.6%
2%
Options
Students start learning about grouping numbers into groups of ones and
tens right from class 2. By class 3 they even start applying the
understanding of place value in the context of operations like addition
and subtraction. This question was designed to test if students are
correctly able to identify the numbers represented as groups of tens
and find the difference between them.
B. 5.
How much more is 48 TENS than 43 TENS ?
1 Why was the question asked?
3
Learnings
For this question, students are expected to 1. Understand what the question is asking: difference between two numbers 2.Recognise the two numbers: 48 tens = 480; 43 tens = 430 3.Find the difference: 480 – 430 = 50
Students who have internalized the relevant concepts solve this easily by subtracting 43 from 48 and since both are groups of tens, see that the answer would be 5 tens - 50.
However, as the data indicates, most students seem to be making a mistake in the second step. Clearly, many students don't understand that 48 tens mean 48 groups of tens which is 480, and so on.
The same students would probably have “procedural” knowledge of how to expand a number like 457 and write it as 4 hundreds + 5 tens + 7 ones. However that's not the same as having a core understanding of the topic.
4
How do we handle this?
1.Use pictorial representations or physical objects like straws or beads and make groups of ones and tens. Give numbers in forms of groups like 4 ones, 5 tens etc. and ask them to represent the same using
straws. The numbers can then be given in different ways - like 112 can be given as 11 tens and 2 ones, 1 hundreds and 12 ones or 112 ones, and they can be asked to represent the same using straws. For this
question, let them find out how many beads will be required to make 48 groups of tens and then for 43 groups of tens. Having done that they, let them check how many more groups of tens are there in 48 tens
than 43 tens. This will help them see how many more beads are there in the 48 tens than 43 tens.
2. Expose students to a variety of simple one line word problems. For example the given question can be asked in a variety of ways – “What is the difference between 48 tens and 43 tens?” “What will be the result
if 43 tens are taken away from 48 tens?” “How much less is 43 tens than 48 tens?” If only certain ways of asking questions, for example, using terms like “taken away” or “subtracted” for subtraction, are
common, such errors are bound to happen.
Useful resources:
Websites: http://coe.jmu.edu/Mathvids2/videos/videos.html#ghto - A good collection of videos – the ones on grouping tens
Books: Adding It Up – Helping Children Learn Mathematics (http://www.nap.edu/catalog.php?record_id=9822)
A good book published by National Research Council, USA, that addresses certain basic questions regarding school mathematics programs.
and hundreds demonstrate some hands-on activities to teach grouping of objects to students.
Do write to us at ts@ei-india.com, using the feedback format provided to your school. Formats, Useful Resources Links, and other details are available online at http://www.ei-india.com/teacher-sheet/
M4-0708-12
Educational
INITIATIVES
Number sense, related concepts and basic number competency
Maths: Class 5
1 Why was the question asked?
Knowing numbers is as important as knowing how to speak a language, as
without that we can't talk about numbers. The question was asked to test
if the students are able to translate the number name into numerals.
A. 4.9%
Question
2 What did students answer?
Which of the following could be written as SIX AND A HALF
THOUSAND?
Options
A
6050
B 6500
P
C
6550
D
6000+ 1
2
About 50% of the students answered the question correctly. But about
44%
of the students chose the wrong option D.
D.43.6%
Possible reason for choosing A: Just about 5% of the students chose
B. 49.0% P
this option. Most probably these students just guessed the answer seeing
that the number should be something in six thousands.
Possible reason for choosing C: Just about 2% of the students chose this
option. These students also most probably guessed the answer using a similar
C.1.6%
logic as the ones choosing A.
Only 49.0% answered correctly
Possible reason for choosing D: Students choosing this option probably couldn't
relate 'half' and 'thousand' and separated 'six thousand' and 'half'.
No. of students
1125
3
Learnings
Many students are not able to understand what 'half thousand' means. They probably have not come across such kind of a
name for a number earlier, and are not able to interpret the number to be six thousand and half thousand. They probably would
understand the meaning of 'six thousand five hundred' and much higher numbers too, if given in the traditional forms they learn in their schools.
The data on this question indicates that many students may not be using numbers in daily life conversations- or if they do, they have not internalized the “common names” of numbers. A probable reason of this is
the exposure that is given in classrooms. Probably there is too much focus on what is given in the textbooks and what is to be learnt 'exactly as per the syllabus' and the students are not exposed sufficiently to
numbers in daily life. Preparing students to use what is learnt in common real-life situations is extremely important.
4
How do we handle this?
Students by this stage are aware of the number concepts like place value, basic operations on numbers, fractions etc. and are also able to do simple problems involving the same. Therefore, while using number
names with the students at this stage, they can be introduced to such non-traditional names as 'one and a half thousand laddoos, two and a half lakh rupees etc. wherever applicable.
The other way of asking different ways of naming a given number would also be a good idea. Activities involving comparison of numbers like 'which of the following are greater than '3 and half thousand' etc.
would also help in dealing with the misconceptions highlighted through this question.
Our experience is that students get varying amounts of exposure to the use of numbers and arithmetic in their daily lives, depending on their home environment and other factors. Students with a fair amount of
real-life exposure would have answered this question correctly. It should be our effort to provide a minimum level of exposure to children in a systematic manner- this could be through classroom activities,
films, field trips or by getting parents to do some activities with children.
Useful resources:
Websites: http://books.nap.edu/openbook.php?record_id=9822&page=17
(This chapter in the book talks about the relation between mathematics and reading)
Do write to us at ts@ei-india.com, using the feedback format provided to your school. Formats, Useful Resources Links, and other details are available online at http://www.ei-india.com/teacher-sheet/
M5-0708-12
Educational
INITIATIVES
Maths: Class 5
Number sense, related concepts and basic number competency
D. 1.5%
Question
21 hundreds, 35 tens and 4 ones are equal to
Options
A
2139
B 2454
P
C
21354
D
21390
A. 9.1%
1 Why was the question asked?
The concept of place value is one of the most important ones to be learnt
and internalised. This question was asked to check whether students are
able to split/combine the groups of tens/hundreds to identify the
number given. Here, they need to understand that 21 hundreds is 20
hundreds and 1 hundreds or 21 hundreds = 21 x 100 and so on.
2 What did students answer?
P
Only about 21% of the students answered the question correctly. And
about 68% of the students chose the wrong option D.
Possible reason for choosing A: Only about 9% of the students chose
this option. These probably just guessed the answer or answered by seeing
C. 67.6%
21 hundreds and 39 (35 + 4), in this option.
Possible reason for choosing C: Students who chose this option probably just
applied the logic of hundreds preceding tens and tens preceding ones in a number
Only 21.2% answered correctly
(when read from left to right) and wrote 21, 35, 4 in that order to form 21354.
Possible reason for choosing D: Only about 2% of the students chose this option.
These students most likely guessed the answer probably without understanding what was being
asked.
No. of students
4044
3
B. 21.2%
Learnings
Many students seem to have missed out splitting 21 hundreds into 2 thousands and 1 hundred and similarly 35 tens into 3 hundreds and 5 tens. Because of this, they hadn't realized that the place value of some
digits is changing and regrouping is required. This indicates that even though the place value concept is taught in lower classes students have still not internalized it fully, even in higher classes.
Many students are probably so used to splitting a number in 'a' particular way, for example, 5235 as 5 thousands + 2 hundreds + 3 tens + 5 ones. Seeing such expressions, students probably try to find a pattern
which they can blindly follow – for example since 5235 is read as 5 thousand 2 hundred thirty five, there should be 3 tens, but they never probably realise that there are 23 tens in 5235!
4
How do we handle this?
Students not having the understanding of place value may face difficulty in understanding many concepts like arithmetic operations.
Various activities related to grouping and splitting and establishing the equivalence between different groupings of the same number, would help a lot in developing a good and proper understanding of place value.
An example of 2 types of groupings of a number is given below:
Activity 1: Ask students to - a. split 160 boys into 1 group of hundred boys and rest of ten boys each b. split the 160 boys into groups of ten boys each
Activity 2: Ask students to - a. split 1670 into groups of only hundreds and tens, groups of only thousands and hundreds b. different students can come up with different types of groupings – two different ways
of splitting 1670 into hundreds and tens are 16 hundreds and 7 tens, 15 hundreds and 170 tens
These activities can be done using place value blocks or any other articles.
Useful resources:
Websites: http://www.garlikov.com/PlaceValue.html
Books. 1. Burns, Marilyn, About teaching mathematics, A K-8 resource (pp. 173-182). Sausalito: CA: Math Solutions Publications. 2. Chapin, Suzanne and Johnson, Art (2000), Understanding the Math You Teach, Grades K-6 (pp. 17-18). Sausalito: CA: Math Solutions Publications.
Do write to us at ts@ei-india.com, using the feedback format provided to your school. Formats, Useful Resources Links, and other details are available online at http://www.ei-india.com/teacher-sheet/
M5-0708-02
Educational
INITIATIVES
Maths: Class 6
Arithmetic Operations, Order of operations, Properties
A. 3.0%
Question
and are two numbers and
What is the value of + ?
+
= 74.
B. 10.5%
A
11
B
47
C 74
P
D
We can’t say
Students start adding single digit numbers even before Class 1 and by the
time they come to Class 6, they are familiar with all the four operations,
their properties and the relationships between them. This question was
designed to test if students understand if the commutative property of
addition – that the result remains the same, in whichever order the
numbers are added.
2 What did students answer?
About 42% of the students were able to answer this question correctly.
Around 43% of the students chose the wrong option D.
Possible reason for choosing A: Students are either making a
random guess or adding the two digits 7 and 4. They don't seem to
C. 42.3% P
understand that each symbol represents a number (and not the digits of
the number 74) which add up to 74.
Possible reason for choosing B: Students selecting this option are probably
thinking
that if the order, in which the numbers (represented by symbols) are
Only 42.3% answered correctly
added, is reversed, the digits of the resulting number will also be reversed. And so for
this addition problem, the number 74 will be reversed to 47 on reversing the order of the
two numbers being added.
Possible reason for choosing D: A large percentage of students have selected this wrong option.
They probably think that the two numbers that are being added should be known to find out the answer.
D. 43.3%
Options
1 Why was the question asked?
No. of students
6655
3
Learnings
Though this is a basic concept that students are expected to know by this age, the data indicates that 43% of Class 6 students are not able to understand the representation in the question, and relate it to the
commutative property of addition.
Students are used to adding numbers and finding out the answer. Most of the students, who chose the wrong option, will probably be able to add two-digit or even three-digit numbers efficiently. But the same
students are not able to see the property of this operation, when symbols are given. Perhaps they feel that the two numbers should be known to find out the answer. They do not realise that the symbols
represent numbers and so whether the first symbol is added to the second one or the second one is added to the first one, the answer remains the same.
4
How do we handle this?
When students start adding numbers, use physical objects like straws or coloured balls. Ask them to put 4 green balls in a jar followed by 5 red balls and see the total number of balls. Then ask
them to do the reverse, put the red balls first and then the green balls and see if the total number of balls change.
This can then be extended to numbers on paper, for example asking them to add 12 and 7 and then 7 and 12 and see the result. Perhaps, simultaneously giving two addition problems using the
same two numbers, just in different order, can help them see this property of addition. Instead of teaching it to them, let them figure out this property. Having done that, they can be moved to
a generalization of this rule, by giving symbols instead of numbers. Give questions using symbols instead of numbers – for example, “If o + ? = 23 + 5, what is ? + o equal to?” “o + ? + o
=525; What will be o + o + ? equal to? 225, 525, 552”
Useful resources:
Websites: http://www.learnnc.org/lessons/WmMKrupicka5232002551 A resource that talks about certain activities to introduce the commutative property of addition in a very early grade.
http://www.mhschool.com/math/2003/teacher/teachres/mathissues/pdfs/basic_facts.pdf An interesting article that talks about the importance of basic facts and strategies of teaching basic operations.
Do write to us at ts@ei-india.com, using the feedback format provided to your school. Formats, Useful Resources Links, and other details are available online at http://www.ei-india.com/teacher-sheet/
M6-0708-12
Educational
INITIATIVES
Number sense, related concepts and basic number competency
Maths: Class 6
1 Why was the question asked?
Question
Which of the following numbers will INCREASE the MOST on
interchanging their digits at the one's and the ten's place?
Options
A
12345
B 24315
P
C
54312
D
55555
A. 14.2%
Students start learning about the place value concept from Class 3
onwards. They are familiar about the different places - ones, tens etc., and
are expected to clearly understand that a digit at a given place will have a
certain value based on its position. This question tests whether children
really understand the impact of place value on the value of a number.
2 What did students answer?
Only about 16% of the students were able to answer this question
correctly. Around 56% of the students chose the wrong option D.
D. 55.7%
Possible reason for choosing A: Students probably choose this option
thinking that the two digits at the ones and the tens place, 5 and 4, are
C. 11.7%
bigger than the other digits of the number.
Possible reason for choosing C: Students selecting this option are most
probably making a random guess.
Only 16.3% answered correctly
Possible reason for choosing Option D: Students probably think that the two
numbers at the ones' and the tens' places are the largest here, and therefore this is the
answer. They do not seem to have understood the question.
No. of students
6655
3
B. 16.3%
P
Learnings
As the data indicate, a large percentage of students are not able to answer this question correctly, and clearly they haven't
internalized the concept of place value.
Students need to first understand what the question is asking. To avoid any kind of careless mistake, keywords like INCREASE and MOST are already given in capital letters. They are not expected to interchange
the digits and mechanically subtract the given number from the resulting number. They are expected to apply their understanding of place value, and just compare the two numbers by interchanging the digits at
the two places. For example, in option A they should see that if the two digits are interchanged, with the other digits remaining unchanged, the number will change from 45 to 54 and the increase will be 11;
whereas for option B, it would be 36 (51 - 15).
It seems that students are not applying this simple understanding. Instead, they seem to be looking at certain keywords like 'increase' or 'most' and following certain blind strategies to solve the question - like
thinking that “increase the most” would mean the biggest number and selecting the biggest of the four given numbers. It is a matter of concern that students don't realise that the question is asking them to
compare the given number with the number obtained by interchanging two of its digits. Even if they aren't exposed to such problems generally, a student with a clear understanding of place value should be able
to answer this correctly.
4
How do we handle this?
It is important that students understand what the question is asking. For this problem, ask them to break down the problem into smaller steps in which they should proceed –
• Interchange the digits at the ones' and the tens' place • Find the difference between the new number and the original number • Find the number for which the difference is the maximum
Having done this, option B will clearly emerge as the answer. They should be able to break down problems like this into simple steps as shown above.
Try and give a variety of problems to check if students have understood the meaning of place value – for example, “which digit has the highest value in the number 132?” “How much less will be the number 256 if
the digit in the tens' place is reduced by 3?”
Useful resources:
Websites: http://coe.jmu.edu/Mathvids2/videos/videos.html#ghto - A good collection of videos – the ones on grouping tens
Books: Adding It Up – Helping Children Learn Mathematics (http://www.nap.edu/catalog.php?record_id=9822)
A good book published by National Research Council, USA, that addresses certain basic questions regarding school mathematics programs.
and hundreds demonstrate some hands-on activities to teach grouping of objects to students.
Do write to us at ts@ei-india.com, using the feedback format provided to your school. Formats, Useful Resources Links, and other details are available online at http://www.ei-india.com/teacher-sheet/
M6-0708-02
Educational
INITIATIVES
Arithmetic Operations, Order of operations, Properties
Maths: Class 7
1 Why was the question asked?
Question
What is 1 + 2 x 3 ÷ 4?
Options
22
1
B 24
1
C 22
P
A
D
25
A . 7.4%
D. 2.0%
Students learn about the four basic operations on whole numbers much
earlier than class 7. By this stage, they are expected to do problems
involving any number of operations and are expected to know the order
in which the operations should be done. The question tries to capture
the same understanding.
2 What did students answer?
C. 31.3%
P
No. of students
Only about 31% of the students answered the question correctly
whereas about 58% of the students chose the wrong option B.
Possible reason for choosing A: Just about 7% of the students chose
B. 57.8%
this option. Most probably these students just made a random guess.
Possible reason for choosing B: These students have performed the
operations in the same order as given in the expression in the question without
Only 31.3% answered correctly
understanding that operations should follow a certain order.
Possible reason for choosing D: Just about 2% of the students chose this option. Most
probably these students just made a random guess.
403
3
Learnings
As indicated by the data, students are going by the order of operations given in this expression rather than following the correct
order of operations. By class 7, probably students have forgotten some of these basic conventions, and therefore make mistakes like these. Even
though, this order does not have a deeper mathematical significance and is only a “convention”, it is still a very important rule, without which there would be a lot of confusion and communication would become
quite difficult.
4
How do we handle this?
Ask children if they know what a convention means. If they don't, explain this through the means of an example- maybe of traffic lights. It is a convention that “red” means stop, and “green” means go. Ask them
what would happen if “red” means go in some other country, and we went there – accidents of course! Explain to students that BODMAS is a similar convention they should learn to avoid mistakes and accidents
in Maths. Help them internalize the rules through the following steps.
Help students practise such problems with parentheses in the beginning. This would help students in understanding how to work with such expressions. For example, giving expressions like
1 + (2 x 4) instead of 1 + 2 x 4, 1 + (3 x 5) – (6 ÷ 3)
might help since students probably don't get confused when brackets are given. Once students are comfortable doing the above, we can go on to giving expressions without brackets stepwise like:
1 + 3 x 5 – (6 ÷ 3), 1 + 3 x 5 + 6 ÷ 3
Students generally tend to make a mistake when a subtraction sign is involved. Therefore, it would be good if brackets are maintained in cases where subtraction is involved and only after it is clear that students
understand such expressions and work well with them, expressions involving subtraction should be introduced.
Useful resources:
Websites: http://mathforum.org/dr.math/faq/faq.order.operations.html
(This link provides information about BODMAS rule and also describes some other names of the same)
Do write to us at ts@ei-india.com, using the feedback format provided to your school. Formats, Useful Resources Links, and other details are available online at http://www.ei-india.com/teacher-sheet/
M7-0708-12
Educational
INITIATIVES
Number sense, related concepts and basic number competency
Maths: Class 7
1 Why was the question asked?
Division is a concept that students are introduced to, around class 2, and
they are able to divide a number by another correctly much before class
7. What students need to do in this question is to see that P would be a
multiple of 5 + 1, and therefore the remainder would be 1.
Question
Number P is obtained on multiplying a whole number M by 5 and
then adding 6. What will be the remainder when P is divided by 5?
M
Multiply by 5
N
Add 6
P
Options
A 1
P
B
5
C
6
D
Cannot be determined unless the number is known.
A. 26.1%
P
2 What did students answer?
Only about 26% of the students answered the question correctly. About
51% of the students have chosen the wrong option D as their answer.
D. 51.1%
Possible reason for choosing B: Just about 7% of the students chose B
B. 6.7%
as their correct answer. These students probably just guessed the answer
as 5 by seeing 5 in the question.
C. 14.7%
Possible reason for choosing C: About 15% of the students chose C as their
answer. These students probably saw that N is a multiple of 5 and since 6 is added
to a multiple of 5, they might have felt that the remainder should be 6.
Only 26.1% answered correctly
Possible reason for choosing D: It is interesting to see that more than half the
students chose option D as their answer. These students probably believe that P is an
unknown number and without knowing P, the remainder can't be found out.
No. of students
7401
3
Learnings
Students have learnt divisibility rules by this stage, and should understand the patterns related to the possible remainders in a division.
They also need to start relating to expressions involving variables and understand that these might represent quantities about which something is known.
In this case, they should be able to see that P= 5M + 6 or 5(M+1) + 1, which clearly tells us that P is 1 more than a multiple of 5 since M is a whole number.
The data on this question indicate that most students are unable to arrive at this conclusion. They seem to think that a variable can represent “anything” and so nothing can be said about it or any expression involving it.
4
How do we handle this?
When helping students to shift their thinking from concrete to abstract ideas, it would help if students are first exposed to symbols representing abstract quantities,and then to the variables. Questions on framing
relationships involving one or more unknown quantities and appreciating the fact that something is known about the unknown (variable) to represent the relationship are important. For example, ask questions like:
• Ila has some (use symbol say «
) chocolates. She bought 5 more. What will be the number of chocolates with her now?
• Ila’s marks are 5 more than Tia’s marks. Express this as a relationship between Ila’s and Tia’s marks.
• A number P is multiplied by 5 to form the number M. Express this as a relationship between P and M.
Such questions, starting with symbols and moving on to variables help students understand the use of variables in generalising a relationship. As students do this, they realize that while the value of the variable
might be unknown, some attribute of it is known. Once this is understood, students will be able to work efficiently using variables.
The above can be followed by questions like ‘a number is multiplied by 2 and 7 is added to it. What form can the number take?’ – students should be able to realise that it would take a form 2k+1, for some natural
number k, though 7 is added.
Useful resources:
Websites: http://www.learner.org/channel/courses/learningmath/algebra/ (This website gives a step by step approach in introducing and teaching algebra)
Do write to us at ts@ei-india.com, using the feedback format provided to your school. Formats, Useful Resources Links, and other details are available online at http://www.ei-india.com/teacher-sheet/
M7-0708-02
Educational
INITIATIVES
Maths: Class 8
Number sense, related concepts and basic number competency
1 Why was the question asked?
A. 3.6%
Question
Ruchi, a class 3 student, is using repeated subtraction to divide
481 by 4 as shown below.
4 481
-400
81
-40
41
Students are introduced to the procedure of division, right from class 5. By
class 8, they are well acquainted with performing division involving large
numbers also. This question was asked to test whether they understand
what is actually being done in each step of this procedure, or they have
just memorized the steps mechanically.
2 What did students answer?
About 50% of the students answered it correctly and about 40% of the
students chose the wrong option D.
Remainder:
4 x 10
Possible reason for choosing A: Very few students chose this option
B. 49.3% P
and they are probably making a random guess.
B. 51%
Possible reason for choosing C: Very few students chose this option and
What is the error in her procedure?
they are probably making a random guess.
Options
Possible reason for choosing D: Students are probably finding the steps in the
question
unfamiliar, and are answering without trying to understand all the steps
Only 49.3% answered correctly
A There is no error; the procedure done is fully correct and complete.
further. They would probably subtract 80 in the second step instead of 40 in general
B The procedure done so far is correct, but she needs to subtract 40 once more.
practice. Since they are not finding 80 in this step, they are concluding that the procedure is
P
incorrect.
C The procedure is correct, but in the second step also she should have subtracted 400.
Quotient: 100
+ 10 +
D. 40.2%
No. of students
4902
C. 5.
7%
4 x 100
D
The procedure used can not work for division.
3
Learnings
By this stage, students are used to doing division sums. The student response data indicates that many students do not actually
understand the basis for the division procedure. In the question asked, children probably are used to subtracting 80, the biggest multiple of 4 in 81, but
they probably don't understand that we subtract 80 to avoid the subtraction of 40 twice or subtraction of 4 twenty times etc., mainly for our convenience and to keep the procedure short. Children are failing to
understand the meaning of division as repeated subtraction; they do not understand that whether we subtract 40 twice or 80 once, we are doing the same thing. This misunderstanding may be happening due to
too much of emphasis on 'standard procedures'. While it is important for students to be able to know standard procedures and use them efficiently, it is important for them to understand the underlying basis for
the procedure. This understanding allows them a deeper appreciation of the concept, helps them to avoid making mistakes in division. Students who don't have a conceptual understanding are unable to catch
“obvious” mistakes in the answers they get. Also, for many students, it becomes much easier to “remember” the procedure once they understand the underlying basis.
4
How do we handle this?
The basic idea is children should be explained the procedure for division only after they internalize the concept of division as repeated subtraction. Shown below are some activities to be given to children that
can help them develop the concept of division:
Starting with division of simple numbers say 10 ÷ 2, ask children to keep subtracting 2 from 10 till they get a 0. Ask how many groups of 2 they can subtract together from 10 to get 0.
Then ask them to do a similar procedure for 80 ÷ 4. First, ask them to subtract 10 groups of 4. Help them to understand that 40 is still remaining which can be made zero by subtracting 10 groups of 4 again. Help
them understand that 20 groups of 4 are required, in all, need to be subtracted from 80 to make it 0.
Relate this concept of division as repeated subtraction with the division procedure familiar to students.
Useful resources:
Websites: http://mathforum.org/library/drmath/view/58499.html (Explains the division procedure through an example of 73/4 using a bundle of sticks.)
http://curriculalessons.suite101.com/article.cfm/teaching_division_part_ii (Explains an uncommon procedure which deepens the understanding of division.)
Do write to us at ts@ei-india.com, using the feedback format provided to your school. Formats, Useful Resources Links, and other details are available online at http://www.ei-india.com/teacher-sheet/
M8-0708-02
Educational
INITIATIVES
Number sense, related concepts and basic number competency
Maths: Class 8
Question
Mr. Shah, an industrialist, is estimated to have assets
worth Rs. 5.2 crores. Which of these could be the actual
value of his assets?
Options
A
Rs. 52,07,900
B
Rs. 50,29,790
C
Rs. 5,02,56,770
D Rs. 5,23,45,990
P
A. 7.6%
B. 7.1%
1 Why was the question asked?
Students learn decimal numbers from class 5 onwards. This question was
asked to test whether students can apply their understanding of
conversion of decimal numbers in real-life situations.
2 What did students answer?
About 40% of the students answered correctly whereas about 43% of
students chose the wrong option C.
D. 39.8% No. of students
Possible reason for choosing A: Very few students chose this option
P
and are probably making a random guess.
916
Possible reason for choosing B: Very few students chose this option and
C. 43.2%
C. 15.7%
are probably making a random guess.
Possible reasons for choosing C: These students are most probably going by
the “looks” of the numbers, trying to match the numbers in the options that “look”
most like 5.2 crores. Students may be thinking that C is the answer because D has the
Only 39.8% answered correctly
digit 3 in the lakhs' place; they may be thinking that 5.2 crores shouldn't have any digits
apart from 5 and 2 in the highest 3 places. Some students probably didn't bother to look at
option D at all.
3
Learnings
There are two ways in which this question can be answered:
1. The number 5.2 crores can be expanded by multiplying it by 107 to expand it to 5,20,00,000 and then compare it with the options,
which is expected from most students at this level.
2. Students with a stronger grip on numbers, can answer just by comparing the first 3 digits in the options with the number 5.2 crores and reach the conclusion that D (5,23, something) is much closer to 5.2
crores, than C (5,02, something).
However, most students have failed to adopt either of these strategies and seem to have gone for a very rough and inaccurate method of matching options to what “looks” like 5.2 crores in their mind. Some
students probably choose C thinking that 0 can replace the decimal point between the two digits 5 and 2, as they don’t understand the significance of the decimal point.
4
How do we handle this?
It is important to train students to analyze the problem carefully and solve it systematically. The first step however would be to investigate the actual strategies children have adopted in answering this question,
whether wrongly or correctly. Do this by asking children to solve this problem in class and show their workings. You will then be able to choose the right strategies to teach the topic.
Certainly, one root of the problem lies in learning decimal numbers but not using them in real life applications. Encourage students to use decimal numbers in their day to day life, converting decimal numbers into
natural numbers and vice versa. e.g.
1. For small children, it would be a good exercise to read price tags and convert into paise, for eg. 2.75 rupees into paise etc.
2. Get students to talk in terms of decimal numbers like 2.5 lakhs, 1.5 million etc. One suggested activity is to ask students to give the population of India. Divide the class into 5 groups. Ask Group 1 to represent
the population in billions only, Group 2 in crores only, Group 3 in millions only, Group 4 in lakhs only and Group 5 in thousands only. Finally they can be asked to compare their answers.
The resources listed below contain other suggested activities to help children master decimal numbers.
Useful resources:
Websites: http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/number/N40001P.htm
Books: Real-Life Math: Decimals and Percents by Walter Sherwood – provides many suggested activities based on
(provides information on students’ misconceptions with decimal numbers)
real life applications of decimals. A link for its preview: http://www.books.google.com/books?id=BKzKGPeqU_oC&rview=1
Do write to us at ts@ei-india.com, using the feedback format provided to your school. Formats, Useful Resources Links, and other details are available online at http://www.ei-india.com/teacher-sheet/
M8-0708-12
Educational
INITIATIVES
Number sense, related concepts and basic number competency
1 Why was the question asked?
Maths: Class 9
Students by this stage are familiar with algebraic expressions involving
powers of numbers and generalizations too. The question was asked to
understand if the students are able to interpret the expression given and
generalize it to understand that the units' digit in 6n will be 6 for any n and
so the remainder, when it is divided by 5 should be 1.
A .3.7%
Question
The remainder obtained on dividing 6n by 5, where 'n' is
any natural number, is
B. 22.9%
No. of students
5040
P
C. 4.2%
D. 67.6%
Options
A
0
B 1
P
C
5
D
Cannot say as it depends on the value of 'n'.
Only 22.9% answered correctly
3
2 What did students answer?
Only about 23% of the students answered the question correctly
whereas about 68% of the students chose the wrong option D.
Possible reason for choosing A: Very few students chose this option. It is
quite likely that they made a random guess.
Possible reason for choosing C: Again, very few students chose this option.
It is quite likely that they made a random guess.
Possible reason for choosing D: These students probably thought that the value of
n should be known to find the value of 6n and is therefore required, to find the remainder.
Learnings
The main problem seems to be that students are unable to make the required generalization – that any power of 6
ends with a 6. These patterns of numbers like powers of 4 ending with either 4 or 6, powers of 5 ending with 5 etc. are
expected to be known by this stage as they do learn such patterns in class 8.
It is likely that a lack of understanding of variables caused the primary difficulty to students, rather than an understanding of divisibility. Students probably tend to think that a variable
can represent anything and so nothing can be said about it, or any expression involving it. By this stage, they also need to start relating to expressions involving variables and understand that these might represent
quantities about which something is known.
For example, in this question, students should be able to think in the following manne r: • The number 6n is divided by 5. To know the remainder, we need to know the digit in the units' place • Can we know the
units' digit in 6n, even if we don't know the number 6n? • To know that ,we need to know what can be n, which is given to be a natural number • But since the natural number n is unknown, can we generalise
something about 6n for any natural number n
4
How do we handle this?
While helping to shift students' thinking from concrete to abstract, students should first be exposed to symbols representing abstract quantities,and then to variables. Once this is done, in this context, questions
like the ones below would help students in understanding generalizations represented by variable expressions.
What is the remainder when any even number is divided by 2?
What is the remainder when any odd number is divided by 2?
After understanding is achieved with concrete numbers, we can go on to explain this using expressions- explaining that any odd number can be expressed as 2n+1 and so when it is divided by 2, 2n is divisible by
2, and the remainder would be 1.
Such practice with simple equations would help students develop understanding of variables and to work with them as required.
Useful resources:
Websites: http://www.learner.org/channel/courses/teachingmath/grades3_5/session_04/index.html (This link talks about the basics to higher algebra – quite a good one to develop understanding of variables, expression forming and working with expressions)
Do write to us at ts@ei-india.com, using the feedback format provided to your school. Formats, Useful Resources Links, and other details are available online at http://www.ei-india.com/teacher-sheet/
M9-0708-12
Educational
INITIATIVES
Number sense, related concepts and basic number competency
Maths: Class 9
Question
Two consecutive integers m and n are such that m > n.
The difference m2– n2
A
is always an odd positive integer.
B
is always an even positive integer.
C can be positive or negative but will be odd.
P
D
can be positive or negative but will be even.
Students learn about variables and also start working with algebraic
identities by class 8. This question tests mainly whether a student
interprets what an integer is correctly or just assumes it to be positive.
The student then needs to arrive at the required generalization of the
given algebraic expression.
2 What did students answer?
D. 11.5%
Only about 28% of the students answered the question correctly. And
about 41% of the students chose the wrong option A.
Possible reason for choosing A: These students figured out correctly
that the difference m2– n2 will be odd but probably considered only
positive integer values for m and n.
B. 17.9%
Possible reason for choosing B: These students also seem to have
considered only positive values for m and n and ignored the information that m
Only 28.1% answered correctly
and n are consecutive and so one of them should be odd and the other even.
Possible reason for choosing D: These students probably think that the square of a
number should be even. These students may also be making a random guess.
No. of students
C. 28.1%
5040
P
Options
1 Why was the question asked?
3
A. 40.5%
Learnings
Many students face problems in understanding that a variable can take many different values based on the conditions given. It is a
general tendency of students to consider particular values of variables instead of generalizing
by considering all the possible values. If asked 'what will be the value of x + 5 if x is a whole number', students should be able to understand that x + 5 can take
m>n and both are consecutive integers
m
n
m2 - n2
any value >= 0 + 5 i.e x + 5 can be any value greater than or equal to 5. Students might take a value 3 for x and say that x + 5 would be 8. This is true but it is
Positive
Positive
Positive
not a generalization, it is not that x + 5 can only be 8.
Negative
Negative
2
2
For example, in this question, students need to interpret the value of m – n in each of the combinations of the values of m and n as shown in the table below:
Negative
Negative
Negative Positive
Not possible since m>n
Many students seem to be completely ignoring the possibility of these variables being negative integers. The use of the identity m2– n2= (m + n) (m – n)
can also help in solving this question
4
How do we handle this?
Ask students questions like 'x is an integer, what will be 2x?' – the answer should be- it is an even integer, positive or negative
'• x, y are integers. What values can x + y take?' – the answer would be an integer; ask if the value can be -5, and check if some students believe it cannot be.
Investigate further, and check whether students are covering all combinations or completely ignoring some of them. Check if they are working with generalisations or solving by substituting values and checking.
Help them understand that while substitution may be helpful to think about the problem, it may not lead to a general rule.
With practice, students should be able to understand the different general values expression(s) can take, given a condition on the variable(s) in the expression.
Useful resources:
http://www.learner.org/channel/courses/learningmath/algebra/ (This website has a whole module to explain
Websites: http://www.emis.de/proceedings/PME28/RR/RR200_Kalyanasundaram.pdf (This research paper talks about
how algebraic thinking develops and how to teach algebra, problems children face even at higher levels)
how to teach algebra and describes the mistakes children make while understanding algebraic expressions)
Do write to us at ts@ei-india.com, using the feedback format provided to your school. Formats, Useful Resources Links, and other details are available online at http://www.ei-india.com/teacher-sheet/
M9-0708-02
Educational
INITIATIVES