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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.
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.
Let’s Learn Mathematics (Primary Level) 3
Disclaimer: This Mathematics question is purely created for discussion purpose. Any resemblance to actual questions from books or schools is coincidental.
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We have discussed about simple algebra in Let’s Learn Mathematics (Primary Level). Though algebra is formally introduced in secondary schools, the model method is taught in primary schools as an introduction to algebra. Thus, you can see the importance of learning algebra.
The question of this post is considered advanced for primary students, nonetheless it is for primary students. Take your time to understand the question and the solution; they are helpful in learning algebra up to secondary schools.
Q: 3 shirts and a pair of trousers cost $63.50. A shirt and 2 pairs of trousers cost $27. Find the cost of a shirt.
A: First you need to draw the diagram to show the relationship of the equations, based on the question. The first sentence (3 shirts and a pair of trousers cost $63.50) forms an equation and the second equation forms another equation. [Please refer to The Almighty Algebra for notes.]
Diagram to show the relationships
In the above diagram, the blue rectangle represents the shirt and the yellow triangle represents the pair of trousers.
If the student has been solving similar questions, he / she will start by adding or subtracting the two equations.
s = a shirt, t = a pair of trousers
Equation 1, 3s + t = 63.50
Equation 2, s + 2t = 27
Adding equation 1 and equation 2, 4s + 3t = 90.50
Subtracting equation 2 from equation 1, 2s − t = 36.50
This is where students may face the problem to continue solving the problem. They have all the equations but the equations are leading them to nowhere. No matter it is adding or subtracting, you end up with more equations with no solution.
You can apply any Mathematics operation to any equation.
The Mathematics operation includes adding, subtracting, multiplying and dividing. Yes, many students have not thought of multiplying the equations. Nonetheless, we do not encourage students to divide the equations because it will lead to decimals or fractions. It is difficult to solve the question, it is more difficult to solve the question with decimals or fractions.
Back to the question, the question is only asking for the cost of a shirt. So, to simplify the question, we must “eliminate” the cost of a pair of trousers.
Equation 1, 3s + t = 63.50
Equation 2, s + 2t = 27
The fastest way to get rid of “t” is to multiply 2 to equation 1 and then subtract equation 2 from the new equation (equation 3).
Equation 1 x2, 6s + 2t = 127 (Equation 3)
Equation 3 − Equation 2, 5s = 100 [There is no more “t” because 2t − 2t = 0]
s = 100 ÷ 5
s = 20
The cost of a shirt is $20.
Let’s Learn Mathematics (Primary Level) 2
Disclaimer: This Mathematics question is purely created for discussion purpose. Any resemblance to actual questions from books or schools is coincidental.
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Q: In a seminar, the ratio of the number of men to the number of women was 2:3. On the following day, two more men attended the seminar. The ratio became 5:6. How many participants were there in the seminar at first?
Understanding the question
This is a question about ratio. The ratio has changed because the number of men has changed. Thus, steps to solve this question are:
1. Make the ratio with the same number of women on the first and second day of the seminar. It is because the number of women is unchanged.
2. Find out the change (difference) of the number of men on the first and second day of the seminar.
3. The difference of the number of men on the first and second day is 2, which is also equivalent to the answer from step 2.
4. Find the number of participants on the first day.
Step-by-step Answer
1. The men-women ratio on the first day => 2:3 = 4:6
The men-women ratio on the second day => 5:6
2. Comparing the men-women ratio on the first and second day => 4:6 and 5:6
The difference of the number of men on the first and second day = 5 – 4 = 1
3. 1 unit of difference = 2 more men attended the seminar on the second day.
4. Total units on the first day = 4 + 6 = 10
Total participants on the first day = 10 x 2
= 20 participants
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Checking the answer
On the first day, there were 20 participants with the ratio 2:3. Thus, the number of men on the first day was (20 x 2) ÷ 5 = 8.
On the second day, two more men attended the seminar. Thus, the number of men on the second day was 2 + 8 = 10.
The number of women on the first and second day was = 20 – 8 = 12.
The men-women ratio on the second day = 10:12
= 5:6
Thus, the answer, 20 participants, is correct.
Note to parents: If your child encounters difficult questions, asks him / her to check the answer. Checking the answer is a good way to reassure your child that he / she has answered the question correctly.
