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# Tag Archives: Mathematics

## Let’s Learn Mathematics (Primary Level) — Model Method 4

Continuation from Let’s Learn Mathematics (Primary Level) — Model Method 3

Here is another example on comparison model to compare two or more items.

Question: 8 similar dictionaries and 12 similar assessment books cost S\$196. Each assessment book costs S\$2 less than each dictionary. How much does each dictionary cost?

First, draw the model based on the question.

Comparison model based on the question

Each assessment book costs S\$2 less than each dictionary, thus 8 similar dictionaries cost S\$16 more than 12 similar assessment books.

S\$2 x 8 = S\$16

From the model, we can see that the total cost is 8 units of dictionaries, 12 units of assessment books and S\$16. To find the cost of a single unit, we subtract S\$16 from the total cost.

S\$196 − S\$ 16 = S\$180

Total units = 8 + 12

= 20 units

S\$180 ÷ 20 = S\$9

The cost of a single unit is S\$9. The question asks for the cost of each dictionary, thus we need to add S\$2 to get the final answer.

S\$9 + S\$2 = S\$11

Each dictionary costs S\$11.

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Disclaimer: The Mathematics questions are purely created for discussion purpose. Any resemblance to actual questions from books or schools is coincidental.

## When Good Memory Doesn’t Help

I have been emphasizing on the understanding of Mathematics questions umpteenth time. Most of the time, a student is unable to solve the problem because he or she does not understand the Mathematics question.

There is this student who has very good memory but the good memory does not help him in solving Mathematics questions.

Background of the student: When he was in Primary One (7 years old), his Mathematics teacher at school has scolded him a lot of times and almost given up on him. He has developed a phobia for Mathematics since then. Thus far, in Primary Two, he is surviving in Mathematics by “memorizing” Mathematics questions.

Let’s look at the example 1,

Jane had S\$ 5. Her father gave her S\$ 10 as pocket-money. How much money does she have now?

5 + 10 = 15

Now she has S\$ 15.

Example 1 is a common question for lower Primary Mathematics. The keyword is “Her father gave her”, so the operation used is addition. The question is straightforward and can be solved easily.

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Let’s look at example 2,

Jenny had S\$ 5. Her mother gave her some amount of money. Now she has S\$ 15. How much money did her mother give her?

Both examples look similar, but they are different. After reading the question to the aforementioned student, he insists to use the operation addition to solve the problem because of the phrase “her mother gave her”. He told me, his school Mathematics teacher taught him, gave = addition.

Therefore, I spent time explaining the question to him and finally we solved the problem using the operation subtraction.

I can understand the frustration of the school Mathematics teacher when he or she needs to spend time to explain one question to only one student who does not seem to understand. He or she has the whole class to take care of; giving attention to only one student is not an option.

Nonetheless, I hope that school Mathematics teachers do not teach the students, especially younger students, to memorize questions. THIS DOES NOT HELP!

I have upper primary students who face the same problem because of the understanding issue. With the phobia and the memorization, the same problem is repeated in a cycle until they meet a teacher who can patiently explain the questions to them. By then, whether they are willing to accept the “understanding of the question” method is another problem to solve.

For parents, if you find out that your children face problem at school, please communicate with the school teachers in a tactful way. Extra lessons or remedial lessons for your children are meant to help your children. Also, listen to the advice with an open heart. It may be difficult to accept that your children are slow in learning (Who would want his or her child to be labelled as stupid?), but cooperation with the school teachers will help the children more.

Furthermore, parents can also help the children at home by doing revision together with the children. No matter how busy is your schedule, I believe there is nothing more important than the children. When the children grow old, it is more difficult to mend the problem.

Even if there is no feedback from the school teachers, parents may take the initiative to ask the school teachers on the progress of the children and how they are coping at school. Some children might be too shy to ask questions even though they do not understand. This will create bigger problem in future.

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Disclaimer:

1. The Mathematics questions are purely created for discussion purpose. Any resemblance to actual questions from books or schools is coincidental.

2. The student is a case study for parents, teachers and tutors alike to understand why your children or students have phobia in Mathematics. Hopefully we can create a joyful environment for children to learn Mathematics.

## Counting Speed 2

Q: Jeff drives at the speed of 50 km/h from destination A to B. Jackson, who departs 30 minutes after Jeff, drives at the speed of 70 km/h from destination A to B. Both Jeff and Jackson reach destination B at the same time. What is the distance between destination A and destination B?

A: At first glance, the students may question whether there is an answer. Although the speed and the time are given, the time is not the time taken to travel from destination A to destination B.

No worry. The distance travelled is the same for Jeff and Jackson and we can use this information to solve the question. Please note that the unit for time is minute, converting to hour, it is 0.5 hour.

Let’s take t = time take for Jeff to reach destination B.

The distance travelled for Jeff = speed x time

= 50 x t

The distance travelled for Jackson = speed x time

= 70 x (t – 0.5)

The distance travelled is the same,

50t = 70 (t – 0.5)

Solving the above equation,

50t = 70t – 35

t = 35 ÷ 20

= 1.75 h

From the speed triangle, distance (travelled by Jeff) = speed x time

Distance = 50 x 1.75

= 87.5 km

The speed triangle

The distance from destination A to destination B is the same. Thus, the distance travelled by Jeff and Jackson should be the same.

Distance (travelled by Jackson) = speed x time,

Distance = 70 x 1.25

= 87.5 km

Note:

1. Look for all the necessary information in the question and use all of them.

2. Speed is a simple concept. Thus, questions about speed are normally combined with other concepts, for example, circles. Other than the speed formula and the unit conversion, the students must be prepared to use other concepts to solve speed questions.

## Counting Speed 1

After the students learn time and length, the next concept they learn in Higher Primary school is speed. The length is equivalent to the distance travelled.

The formula is

Speed = Distance ÷ Time

To help the students to remember the formula, a speed triangle is taught, as below.

The speed triangle

Taking the example from Counting Length, if I cycle to school at the speed of 15 km/h, how much time do I need to reach school?

From the speed triangle, time = distance ÷ speed

From the figure below, the distance between my house and the school is 180 m.

Please take note that the unit for speed is km/h and the unit for distance is m. Thus, the first step is to convert the two different units to the same unit.

1 km = 1000 m

15 km = 15 000 m

With speed = 15000 m/h and distance = 180 m,

Time = (180 ÷ 15000) h

= 0.012 h

= 0.72 min

= 43.2 sec

Distance

Speed is a simple concept, once the students remember the speed triangle (and the speed formula) and familiar with unit conversion, the answer is somewhere near 🙂

In our daily lives, we travel by cars and public transport. Use the speed concept in daily life and maybe the student can manage time better too. Enjoy your learning experience!

Note:

1. In Secondary school, the students will learn the term “velocity” in Physics, which is speed with a direction.

2. (a) Take note of the unit used. For distance, common units are km and m. For time, common units are hour, minute and second. The units used for distance and time determine the unit for speed.

(b) Be fluent with unit conversion for distance and time.

## Let’s Learn Mathematics (Primary Level) — Model Method 3

Continuation from Let’s Learn Mathematics (Primary Level) — Model Method 2

2. Comparison model

Comparison model is used to compare two or more items.

Example (a): Jenny has 6 dresses and 2 pairs of pants. How many more dresses than pants does she have?

6 – 2 = 4

She has 4 more dresses than pants.

Model for example 2(a)

Example (b): A pair of pants costs S\$35. A dress costs S\$ 10 more than a pair of pants. If Jenny buys 3 dresses and 2 pair of pants, how much does she need to pay?

The cost of a dress = 35 + 10

= 45

Total cost = (3 x 45) + (2 x 35)

= 135 + 70

= 205

She needs to pay S\$ 205.

Model for example 2(b)

To be continued in Let’s Learn Mathematics (Primary Level) — Model Method 4.

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Notes:

1. If you cannot draw nicely, cannot draw to scale, etc, no need to worry, this is Mathematics, not arts. Remember, it is the understanding of the questions that is the most important.

2. There are different types of Mathematics questions but there is only one model method. Do more Mathematics questions to practise on the model method.

3. Be flexible about the models, you draw the model according to the question, not to what you remember from previous question(s).

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Disclaimer: The Mathematics questions are purely created for discussion purpose. Any resemblance to actual questions from books or schools is coincidental.

## Let’s Learn Mathematics (Primary Level) — Model Method 2

Now we know that understanding of Mathematics questions is more important than the tools or methods, let’s look at different models that we can draw to help us in solving Mathematics questions.

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1. Taking the example from Let’s Learn Mathematics (Primary Level) — Model Method 1, the model is simple yet powerful. It can be used to find the total (addition) or the difference (subtraction) of the same item.

Example (a): Jane had S\$ 5. Her father gave her S\$ 10 as pocket-money. How much money does she have now?

5 + 10 = 15

Now she has S\$ 15.

Model for example 1(a)

Example (b): Jenny has 10 stamps, Kenny has 8 stamps and Lenny has 15 stamps. How many stamps do they have altogether?

10 + 8 + 15 = 33

They have 33 stamps altogether.

Model for example 1(b)

Example (c): Jenny had 12 erasers. She gave 2 erasers to Lenny. How many erasers does she have now?

12 – 2 = 10

She has 10 erasers now.

Model for example 1(c)

To be continued in Let’s Learn Mathematics (Primary Level) — Model Method 3.

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Disclaimer: The Mathematics questions are purely created for discussion purpose. Any resemblance to actual questions from books or schools is coincidental.

## Let’s Learn Mathematics (Primary Level) — Model Method 1

Do you have a model ruler?

This is the question that my students ask me when they are solving Mathematics questions that need to draw model.

When the model method was first introduced, many students and parents face problems on how to use the method to answer the questions. Along the years, students and parents are more acceptive to the method. Somebody even came out with a model ruler and gave names to different models drawn for different types of questions. For example, you draw a comparison model to solve questions that ask you to compare.

For me, this is bizarre. As long as you understand the question, you do not need a model ruler or a name for your model to solve the question. There are two similar questions in this post. I would like to show how the students may confuse themselves if they do not understand the questions.

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Q1: There are a total of 110 participants in a seminar and 60 of them are women. How many of the participants are men?

The model for Q1 is as below:

Bar model for Q1

To find the number of male participants, the operation used is subtraction.

110 – 60 = 50

There are 50 men.

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Q2: There are 50 men and 60 women in a seminar. How many participants are there altogether?

The model of Q2 is as below:

Bar model for Q2

To find the number of total participants, the operation used is addition.

60 + 50 = 110

There are 110 participants altogether.

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From the above two questions, the same model is used, but with a slight difference. We need to understand the question in order to put the numbers correctly and subsequently solve the question. As you can see, the operation used in Q1 and Q2 is different too.

If we do not understand the question, we do not even know which operation to use. Teachers must always remind the students to understand the question rather than memorize a certain type of model. If you can draw a model, but do not understand the question, it is pointless. The main aim in Mathematics is to find the answer, if you do not understand the question, how are you going to solve the question correctly?

I hope students, teachers and parents put the focus at the right place. The focus is the understanding of the question, not the tools or method used.

To be continued in Let’s Learn Mathematics (Primary Level) — Model Method 2.

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Disclaimer: The Mathematics questions are purely created for discussion purpose. Any resemblance to actual questions from books or schools is coincidental.

## Creativity for Sales

Singapore is well-known for its education system, especially for Mathematics and Science. Nonetheless, you may have read about critics on lack of creativity at the same time.

During schooling, I memorized English words, Mathematics formula, etc. I thought that was normal. As a student, you do whatever the teacher tells you to do. The teachers nowadays have more tools to help them in teaching, but there is a troubling issue behind. Let’s look at three different aspects below:

1. Composition （作文）

In English language, we have a term called “creative writing”. But for Chinese language, my students have feedback to me that their teachers (from different schools) have a favour towards “happy ending”. If you have a sad ending for your composition, too bad, no matter how well you write, you will not score high.

I always encourage my students to let their imagination runs wild. The students nowadays are exposed to worldwide news and information, why limit their imagination to only “good things”? Writing a composition with a happy ending may help to cultivate an optimistic environment. Nonetheless, when the students grow up, they will find out that this world is realistic, not everything will end happily.

2. Oral （口试）

For Primary school oral, the student is given a picture and he / she needs to say what is in the picture, where is the place and describe the people inside the picture. There is a template, it is good to help the weaker students to memorize and excel in oral. But the whole Singapore is using the same template! Okay, saying the whole Singapore is exaggerating, but different students from different schools have shown me the same template.

If the limitation in composition writing is to cultivate an optimistic environment, what about oral? What type of adults are you looking for? A batch of like-minded students?

3. Mathematics （数学）

I love Mathematics because it is fun. But many students face a problem: The school Mathematics teacher teaches you method A and all other methods are not allowed. This may confuse the students when other methods are introduced to them. Because of limitation on the use of certain method, the “stubborn” student may face problem when they see a different type of question.

To solve a Mathematics problem, the most important part is to understand the question, not to draw a beautiful model. Maybe the emphasis on the model method has been misunderstood by students. Two of my students from different schools, again, can draw beautiful model according to what the teachers have taught. The problem is, they do not understand the question and ask me how to solve. What is the point of drawing a model when you cannot solve the problem?

Furthermore, a different method is helpful in checking the answer. I believe the method introduced at the school is suitable for most students from proven record of the best Mathematics teaching. The question to ask is, how about the minor group who cannot grasp the idea? Shouldn’t the teacher allow some flexibility?

Sales!

The post is written based on my experience with my students, not from a single student, but from different students from different schools. I am not sure how many people out there realize the underlying problem. I will follow the school template, while at the same time, encourage my students to have some creativity whenever it is possible.