TMN3704
ASSIGNMENT 02
STUDENT NUMBER: 69999481
ZEENAT BHYAT
2.1 Briefly discuss what learners' mathematical thinking involves. How can you, as the
teacher, support learners to express and clarify their own thinking?
Learners’ mathematical thinking involves how the learner makes sense of mathematics by
means of the strategies that the learner applies in problem situations, the mathematical
representations that the learner creates the arguments that the learners make and the
conceptual understandings that the learner demonstrates (Empson & Jacobs 2008).
As a teacher your knowledge of how the learner thinks provides you with a framework that
informs and guides your attempt to understand, and to explain, how the learner is making
sense of mathematics. As a Mathematics teacher’s knowledge of how a learner thinks and
reasons about Mathematics is a key component of PCK (Karp 2011). Knowledge of content
and learners informs teaching as well as learning practices and guides teaching and learning
decision-making by providing an important lens through which to view and interpret how
learners respond to lesson activities generally and to assigned mathematics tasks.
2.2 Learners may begin to lose track of some numbers when they use the break-up strategy
to do calculations. Using brackets is helpful to show the grouping of numbers and so help
learners to keep track of what they are doing. Use the distributive property to multiply (96 ×
85) and check the reasonableness of the answer. (The example on page 76 of the CAPS
document provides guidance on how to simplify this problem.)
96 x 85 = 96 x (80 + 5) -------------(breaking up one number)
=96 x 80 + (96 x 5) --------------(using the distributive property)
1.1
=7 680 + 480
=8 160
2.3 In Grade 5, learners are expected to demonstrate knowledge and skills in multiplying a 3-
digit number with a 2-digit whole number. The skill assessed in Figure A was the ability to
use the vertical-column method in multiplication. Closely look at how Phina solved the
problem and answer the questions that follow.
, 2.3.1 What is the common error that Phina made? Explain fully and use the correct
mathematical language when interpreting your facts.
The common errors that Phina made are both factual and procedural errors. The procedural
error is that in the first place is that Phina should have carried down the 4 as a zero but
instead took the 6 in 64 as units instead of tens. The factual error is that when Phina
multiplies they do not carry the values over to add after multiplication. It shows that they
have a slight idea of how it is done but have not fully grasped the concept.
2.3.2 Where do you think the main problem is (root cause)? Explain
The root cause is that the Phina does not understand the number system in the first place like
for example, 64 refers to 6 tens and 4units. This also affects addition that they are supposed
to do for example when they multiply 6x4 which gives 24 they are supposed to carry 2 and
write down 4.
2.3.3 What is the possible remediation? Explain and illustrate.
₃₂ ₃₂ 6
• The first thing you do is place the large number (the 3-digit number) above the smaller
4 5 •
number (the 2-digit number).
Make sure you line them up, so the ones places are in a column, and the tens places
x 6 4 are in a column.
• Now that we have the numbers lined up, it’s time to start multiplying. We start with
2 4 2 4 multiplying the ones place column of the lower number: 4 into 456.
• We have a two-digit number, so we need to pay special attention to where the
2 7 3 6 0 numbers go in the columns. The ten number in our sum: the 2 will need to be carried.
Some people put that number on the side, but a better place for the carried number is
2 9 7 8 4 on top of the column. That way, you won’t forget it. We also usually put a line through
the carried number once used, so as not to confuse ourselves when we work out the
sum
• In our next equation, we won’t forget that the 2 needs to be added to the sum.
• We do the same for the tens. But first, we must add a zero down the ones column
before we move onto the ten’s column. This will help us not get confused over where
the numbers for the ten’s multiplications start on this new line.
• From here on, we add the two lines of sums
2.4 "The teaching and learning of Mathematics aims to develop a critical awareness of how
mathematical relationships are used in social, environmental, cultural and economic
relations" (DBE 2011, p.8).
2.4.1 Design an instructional activity that could help learners recognise that decimal fractions
can be applied across environmental situations (relating to activities in real life).
An application of decimals is a cooldrink in the first shop the price is R12,50 and in the
second one it is R12,55 so the question to the students would be where would they buy the
cooldrink and why. TO guide the students, we need to understand that the second value can
be rounded to one decimal place by looking at the second 5 we round up the amount to
R12,60 which shows that the cooldrink in the second store is expensive and therefore it is
cheaper to buy from the first shop.
2.4.2 Learners must demonstrate knowledge and skills in calculating quantities such as mass,
length, perimeter, temperature, and the volume of objects. When teaching conversions,
emphasis must be placed on multiplication by a thousand (since "kilo" means thousand) and
one thousandth (since "milli “means one thousandth). Design an instructional activity to
ASSIGNMENT 02
STUDENT NUMBER: 69999481
ZEENAT BHYAT
2.1 Briefly discuss what learners' mathematical thinking involves. How can you, as the
teacher, support learners to express and clarify their own thinking?
Learners’ mathematical thinking involves how the learner makes sense of mathematics by
means of the strategies that the learner applies in problem situations, the mathematical
representations that the learner creates the arguments that the learners make and the
conceptual understandings that the learner demonstrates (Empson & Jacobs 2008).
As a teacher your knowledge of how the learner thinks provides you with a framework that
informs and guides your attempt to understand, and to explain, how the learner is making
sense of mathematics. As a Mathematics teacher’s knowledge of how a learner thinks and
reasons about Mathematics is a key component of PCK (Karp 2011). Knowledge of content
and learners informs teaching as well as learning practices and guides teaching and learning
decision-making by providing an important lens through which to view and interpret how
learners respond to lesson activities generally and to assigned mathematics tasks.
2.2 Learners may begin to lose track of some numbers when they use the break-up strategy
to do calculations. Using brackets is helpful to show the grouping of numbers and so help
learners to keep track of what they are doing. Use the distributive property to multiply (96 ×
85) and check the reasonableness of the answer. (The example on page 76 of the CAPS
document provides guidance on how to simplify this problem.)
96 x 85 = 96 x (80 + 5) -------------(breaking up one number)
=96 x 80 + (96 x 5) --------------(using the distributive property)
1.1
=7 680 + 480
=8 160
2.3 In Grade 5, learners are expected to demonstrate knowledge and skills in multiplying a 3-
digit number with a 2-digit whole number. The skill assessed in Figure A was the ability to
use the vertical-column method in multiplication. Closely look at how Phina solved the
problem and answer the questions that follow.
, 2.3.1 What is the common error that Phina made? Explain fully and use the correct
mathematical language when interpreting your facts.
The common errors that Phina made are both factual and procedural errors. The procedural
error is that in the first place is that Phina should have carried down the 4 as a zero but
instead took the 6 in 64 as units instead of tens. The factual error is that when Phina
multiplies they do not carry the values over to add after multiplication. It shows that they
have a slight idea of how it is done but have not fully grasped the concept.
2.3.2 Where do you think the main problem is (root cause)? Explain
The root cause is that the Phina does not understand the number system in the first place like
for example, 64 refers to 6 tens and 4units. This also affects addition that they are supposed
to do for example when they multiply 6x4 which gives 24 they are supposed to carry 2 and
write down 4.
2.3.3 What is the possible remediation? Explain and illustrate.
₃₂ ₃₂ 6
• The first thing you do is place the large number (the 3-digit number) above the smaller
4 5 •
number (the 2-digit number).
Make sure you line them up, so the ones places are in a column, and the tens places
x 6 4 are in a column.
• Now that we have the numbers lined up, it’s time to start multiplying. We start with
2 4 2 4 multiplying the ones place column of the lower number: 4 into 456.
• We have a two-digit number, so we need to pay special attention to where the
2 7 3 6 0 numbers go in the columns. The ten number in our sum: the 2 will need to be carried.
Some people put that number on the side, but a better place for the carried number is
2 9 7 8 4 on top of the column. That way, you won’t forget it. We also usually put a line through
the carried number once used, so as not to confuse ourselves when we work out the
sum
• In our next equation, we won’t forget that the 2 needs to be added to the sum.
• We do the same for the tens. But first, we must add a zero down the ones column
before we move onto the ten’s column. This will help us not get confused over where
the numbers for the ten’s multiplications start on this new line.
• From here on, we add the two lines of sums
2.4 "The teaching and learning of Mathematics aims to develop a critical awareness of how
mathematical relationships are used in social, environmental, cultural and economic
relations" (DBE 2011, p.8).
2.4.1 Design an instructional activity that could help learners recognise that decimal fractions
can be applied across environmental situations (relating to activities in real life).
An application of decimals is a cooldrink in the first shop the price is R12,50 and in the
second one it is R12,55 so the question to the students would be where would they buy the
cooldrink and why. TO guide the students, we need to understand that the second value can
be rounded to one decimal place by looking at the second 5 we round up the amount to
R12,60 which shows that the cooldrink in the second store is expensive and therefore it is
cheaper to buy from the first shop.
2.4.2 Learners must demonstrate knowledge and skills in calculating quantities such as mass,
length, perimeter, temperature, and the volume of objects. When teaching conversions,
emphasis must be placed on multiplication by a thousand (since "kilo" means thousand) and
one thousandth (since "milli “means one thousandth). Design an instructional activity to
