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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
Checking the answer
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.
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
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.
Counting Time 2
I have mentioned that children do not understand the concept of time in Counting time 1. After months of writing that post, and now it is a new year, I want to say that adults have the responsibility to explain the concept of time to children.
Scenario 1: On 30th Dec 2013, an elder sister (7 years old) told her younger brother (5 years old) that he would be 6 years old on 1st Jan 2014. The mother, who had gone away when the conversation took place, came back and the brother asked the mum whether he would be 6 years old in two days. The mum told the boy that he would only turn 6 at his next birthday and scolded the sister, asking her not to say anything stupid like that again to her brother.
Scenario 2: On 2nd Jan 2014, my 7-year-old student said to me: “We have not met in a year.” Then, he continued that the last time we met was before Christmas 2013, and “after a year”, which is 2014, we meet for the first time (in 2014).
Children and adults think differently. It is important for the adults to explain to the children about how long is a year. From scenario 1, it is not stupid to say that you are one year older when it is New Year. When I was younger, my mum used to tell me that I am one year older on Chinese New Year. When I grew a little older, I understood that I grow one year older on my birthday. When the two concepts collide, I asked myself, so am I two years older in a year?
No one answers my question. Along the years, I learn the concept of time and understand that both concepts are correct, but you only need to take one reference point in a year, either your birthday or the New Year. Just like anniversary, it is the one reference point (a specific date) in a year that you take note and celebrate.
The mother in scenario 1 seems like she is busy juggling between family and work. But I do not agree that she said her daughter’s idea was stupid. It will only leave both the children in confusion. Though, one day, both of them will understand the concept, but the word “stupid” will leave an impact in the daughter’s memory. Since it was lunch time, I would suggest the mother can either explain the concept while they are having meal or ask the children to drop the subject for the moment and discuss later. (I am not a mother yet, I may not understand what she has gone through.)
When I heard “We have not met in a year.” in scenario 2, I was shocked. After some probing, I understand that his concept is: we last met in Dec 2013, and then we first met in Jan 2014. It does not matter which month it is, the important thing is the year has gone from 2013 to 2014, it is one year!
I explained to the young boy that from Christmas to New Year, it is only one week, so we have not met each other for two weeks at most. I hope he will understand the concept of week and year faster than any other children.
Whether you are a parent or not, if you have a chance to discuss time matters with children, please spend time to explain to them. Yes, it takes time. Nonetheless, children nowadays are learning faster. It will worth the time spent with them.
Why Do We Learn Percentage?
This is the second post in the “Why Do We Learn …?” series. The main purpose of this series is to show the real life application of different Mathematics concepts. In this post, we continue with the topic of buying a commodity.
Below is table showing the price of the commodity with different weights:
Weight |
Sell Out Price (S$) |
Buy Back Price (S$) |
Difference between SOP and BBP (S$) |
Difference in Percentage (%) |
1 g | 56.50 | 56.10 | 0.40 | 0.71 |
1 kg | 56300 | 56100 | 200 | 0.36 |
1 ounce | 1820 | 1750 | 70 | 3.85 |
10 ounce | 17900 | 17600 | 300 | 1.68 |
Concept 1: Money
The sell out price is the money that you pay to the bank when you buy the commodity.
The buy back price is the money that the bank pays you when you sell the commodity back to the bank.
From the table above, we can see that there is difference between the sell out price and buy back price and the buy back price is always lower than the sell out price.
Concept 2: Percentage
If we invest in a commodity, we would like to earn money, that’s common sense.
With the assumption that the price is the same as the above table when you buy and sell the commodity, which weight would you choose to make maximum profit?
The difference in price does not make a good indicator because the weight difference is big and thus the difference in price is big too. Thus, we should choose the difference in percentage as a better indicator.
We choose the smallest difference in percentage to make maximum profit. To calculate the difference in percentage for 1 g:
The price difference = 56.50 – 56.10
= 0.40
The percentage = (0.40/56.50) x 100%
= 0.71%
From the table, we know that the smallest difference in percentage is for 1 kg, 0.36% and the biggest difference in percentage is for 1 ounce, 3.85%.
If the price of the commodity has increased by 2%, investment in 1 kg has positive return while investment in 1 ounce has negative return.
In a nutshell, we can decide which weight will give you a good return by using the difference in percentage, i.e. 1 kg gives the best return among the four weights while 1 ounce gives the least return.
Disclaimer: This post is not encouraging children and / or adults to invest blindly. This is only a simple example where you can use the knowledge learnt in real life. Investment in real life is more complicated and involves more risks.
Counting Length
Once, my 8 year-old student said his classroom is 8 cm. I showed him how long is 8 cm with a ruler. But he still insisted that his classroom is 8 cm instead of 8 m.
For Lower Primary students at Singapore, they learn metres (m) and centimetres (cm). As they have learnt the four Mathematics operations (addition, subtraction, multiplication and division), they need to apply the four operations into length problems.
The most important concept that they need to know is — 100 cm = 1 m. At home, parents can easily show the children how long is one metre. With the help of a ruler, parents can also show how long is one centimetre.
After they understand that centimetre is much shorter than metre, parents can show them the length of different objects: how long is a pencil? How tall am I? Etc.
For real life application, parents can use the distance to different places, such as school, library, market and playground. A simple example as below figure:
(a) The distance between my house and the playground is 110 m.
(b) The distance between my house and the park is 100 m.
(c) The distance between my house and the school is (100 + 80) m = 180 m
(d) The playground is nearer to my house than the school. The playground is (180 – 110) m = 70 m nearer to my house than the school.
Length and distance are everywhere in our daily lives. Instead of doing boring assessment books, why not we learn from real life application? Enjoy your learning experience!
Counting Time 1
Children do not understand the concept of time. They have a vague idea of time. They may ask you, is 1 minute equals to 10 minutes? Is 1 second longer than 1 minute? For adults, these are silly questions. But we must teach the concept of time to children slowly as time is an important concept.
Children in Singapore learn time from hours to half-an-hour to minutes. It is a stage-by-stage learning and thus will be easily understood by the children.
At home, parents can help the children to reinforce the concept by using time whenever possible. At P1, children learn hours, as in 3 o’ clock, 5 o’clock, etc. If you are having dinner at 7pm, you can tell the child in advance and let him / her tell you when is the time for dinner.
At P2, children learn half-an-hour, as in half past two, half past seven, etc. At this stage, you have more choices to teach the child on time. Is his / her favourite cartoon showing at 7.30pm? Show him / her the time beforehand and give him / her the “authority” to switch on the TV at 7.30pm. By doing so, the child can learn independence and time at the same time.
Another way to learn time is when you are out and about. When we cross the road, we will see the green man and the time counting down in seconds. Teach your children on time and road safety at the same time.
Learn with your children and they will reinforce what they learn at school. Wishing all parents have an enjoyable learning journey with the children.
Counting Weight
In “Why Do We Learn Average?”, we learn 1 kg = 1000 g. As adults, we understand the concept of 1 kg = 1000 g, which means if I carry 1000 bags weighting 1 g each, it means I carry a total weight of 1 kg. But children are confused when they see “g” and “kg”.
Below are the pictures showing equivalence of the weight by using weighting scale, to help children to understand better:
(a) The weighting scale is balance. Thus, we know the weight on the right is equal to the weight on the left. That means if I have 10 bags weighting 1 g each, I have a total weight of 10 g.
(b) The weighting scale is balance. With the 10 g each bag from example (a), I fill 10 bags. So, I have 10 bags with 10 g each.
10 x 10 = 100
Thus, I have a total weight of 100 g.
(c) The weighting scale is balance.
With the 100 g each bag from example (b), I fill another 10 bags. So, I have 10 bags with 100 g each.
10 x 100 = 1000
Thus, I have a total weight of 1000 g, which is also equivalent to 1 kg.
On the other hand, if I just have many bags of 1 g, for 1 kg, I will have 1000 bags of 1 g each.
Why Do We Learn Average?
We learn different Mathematics concepts in Primary school. For Primary students, do you expect them to understand why they are learning average? Or percentage? Yes, they do not understand the reasons behind and when they face difficulties, they give up.
As parents or educators, when we are teaching the children, it would be better if we can explain the reasons behind why they are learning different Mathematics concepts and how they can apply the concepts in real life. Below is an example of buying a commodity, using different Mathematics concepts learnt at school:
Concept 1: Weight
1 kg of the commodity costs S$ 9780.00. As a small investor, you can only buy 10 g at a time. How much does 10 g of the commodity cost?
1 kg = 1000 g => S$ 9780.00
10 g => S$ 97.80
Concept 2: Average
Unfortunately, the price goes down and you wish to reduce the risk of losing more money. Thus, you buy another 10 g at S$ 96.90 and another 10 g at S$ 94.50.
The average price for 10 g = (S$ 97.80 + S$ 96.90 + S$ 94.50) / 3
= S$ 96.40
This concept is called average down in investment term, where it is commonly used for buying stock and/or commodity, such as gold, silver, etc.
Concept 3: Money
(a) At the end of the year, the price has gone up to S$ 9710.00 per kg. You are thinking if you can make money by selling all the 30 g of commodity that you have. The maintenance fee is S$ 20.00.
At the first glance, the price of S$ 97.10 per 10 g is not profitable because you first bought the 10 g at S$ 97.80. But after averaging (concept 2), you only spent averagely S$ 96.40 for 10 g.
Another factor to consider is the maintenance fees. You need to subtract the maintenance fees to get the total money won or lost:
For this example, the total money invested = S$ 96.40 x 3 = S$ 289.20
When you sell the 30 g of the commodity, you get = S$ 97.10 x 3 = S$291.30
Without considering the maintenance fees, you think that you have made S$ 2.10 from this investment. After subtracting the maintenance fees, you actually lose S$ 17.90 from this investment.
(b) After the calculation from (a), you know you are losing money, so you wait for the commodity price to go up further. After six months, the price has gone up to S$ 9920.00 per kg. The maintenance fee is S$ 20.00. You are thinking if you can make money by selling all the 30 g of commodity that you have.
The total money invested = S$ 96.40 x 3 = S$ 289.20
When you sell the 30 g of the commodity, you get = S$ 99.20 x 3 = S$297.60
Similarly, you need to subtract the maintenance fees to get the total money won or lost:
S$ 297.60 – S$ 20.00 = S$ 277.60
Again, after considering the maintenance fees, you are still lost of S$ 11.60, even though the price has gone up to S$ 9920.00 per kg.
As you can see, a simple example link three different concepts. Children will appreciate the concepts better if they can see the linkage and also the real life application.
Disclaimer: This post is not encouraging children and / or adults to invest blindly. This is only a simple example where you can use the knowledge learnt in real life. Investment in real life is more complicated and involves more risks.