A Problem from a Hong Kong School

A mom shared that her P1 daughter in Hong Kong was posed this mathematics problem:

AB + B = BA

A = ________B = ________

How do you teach a P1 child algebra?

Lily

In formal algebra, ab means the product of a and b. I do not think that is what the problem is about. In this problem, I believe A and B represent digits and AB is a two-digit number which when added to B gives a two-digit number which has the tens and ones digit of the original number (AB) reversed. In this case the letters are not used in the same convention as formal algebra. I would recommend that this problem is presented orally to grade one children. The teacher might say, "Each letter (or shape) stands for a digit. The same letter stands for the same digit. Different letters stand for different digits. I have a number, the tens digit is A and the ones digit is B (Teacher writes down AB). When it is added to a number B (Teacher writes down AB + B), the total is a number with B as its tens digit and A as its ones digit. (Teacher writes down AB + B = BA) Find the digits A and B."

I would illusrate with an example say 12. What is A in this illustration? What is B? A is 1 and B is 2. So, in this problem AB + B (which is 12 + 2) is supposed to be BA (21). But it is not, right? So AB is not 12. I suppose guess and check is the best strategy for grade one children to use.

Advanced students or older students may reason this way: Is A an odd or even digit? Yes, it must be an even digit. Why? Did you notice that B + B = A. Sure it could be 1A as well. But not 2A or 3A or 4A and so on, right? Since the final sum is BA, AB + B must involve renaming. Why? Otherwise the tens digit in the sum is the A, isn't it? So B must be 6, 7, 8 or 9. And A is one less than B. Think about this one! Hence, A is 5 when B is 6, A is 7 when B is 8 - both not possible. Why?

Hence, A = 6 and B = 7 or A = 8 and B = 9. Checking 67 + 7 = 74 and 89 + 9 = 98. I think the solution is A = 8 and B = 9.

Secondary students may solve it algebraically: 10A + B + B = 10B + A or 9A = 8B or the ratio of A : B = 8 : 9. For digits, A has to be 8 and B has to be 9.

I don't think P1 children are expected to do the algebraic solution or even the reasoning based on number properties. They are most likely able to solve it by guess-and-check.

Another Speed Problem

I have a word problem about speed that I cannot explain to my pupils easily.

Car A and Car B left Town X for Town Y at the same time. Car A was travelling at an average speed of 80 km/h and Car B was travelling at an average speed of 60 km/h. Car A was leading Car B by 8 km for every 1/6 of the distance from Town X to Town Y. Find the distance between Town X and Town Y. One of my friends solve it like this: 6 units x 8 km = 48 km and 80 km - 60 km = 20 km. 80 : 20 = 4. Hence, the distance from Town X to Town Y = 4 x 48 = 192 km. However, the solution is too difficult for my pupils.

Charmaine's Suggestion:

Car A travels 20km more than Car B in an hour. (80-60)

Since Car A leads Car B by 8km for 1/6 of the journey, it leads by a total of 8*6 =48km for the entire journey (6/6).

Thus, time taken for the whole journey by A: 48/20hours = 12/5 hours (leave in simplest improper for easier calculation later...)

Distance between X and Y is thus 12/5*80 = 192km.

Seow's Suggestion

http://farm4.static.flickr.com/3648/3457754184_35d7fcdc47_o.jpg

Speed Problem

I teach maths in Grade 6. There is a word problem in the maths book that I cannot teach using the model method that is usually more suitable for my pupils. The problem: A car and a lorry were travelling towards Town A. The car overtook the lorry when they were 90 km away from Town A. The car arrived at Town A 1/2 h earlier than the lorry while the lorry was 30 km away. Find the average speed of the lorry. Find the average speed of the car.

Gita

Please do not think that the model method can be used for all problems. That is not the idea. We want students to learn a variety of heuristics and they should apply it accordingly. Speed problems are rarely solved using the model method. A line diagram is more useful. Please see any Singapore Grade 6 books for such line diagrams.

In the problem posed, the lorry took 1/2 h to finish the last 30 km. So the average speed of the lorry is easily found (60 km/h). With 90 km to go, both the car and the lorry has travelled the same distance from the spot where the lorry started - that was when the car overtook the lorry. The car must be faster but started later.

I want to suspend the solution for a while. I invite readers to continue to solve the problem. Post further question if necessary.

I have a word problem: The price of a pen was $5, The price of a pencil was $2. Miss Lee bought a number of pens and pencils for $26. How many pens and pencils did she buy?

The day before yesterday I taught my pupils this: I made a list for the pens : 1 x $5 = $5 , 2 x $5 = $10 and so on. And also a list for the pencils : 1 x $2 = $2 and so on. Miss Lee bought 4 pens and 3 pencils because (4 x $5) + (3 x $2) = $26. However, I realize that this method cannot be used in big numbers.

Merry

The method you used is called make a list. It is a common problem-solving heuristic. Please continue to use it with your younger students. I wonder if Miss Lee could also buy 2 pens and some pencils. I know 5 pens is not possible because the money left is an odd number $1 and the price of a pencil is $2. Similarly, 1 pen or 3 pens are not possible. Students learn reasoning.

If students know algebra, they can set up equation 5x + 2y =26 where x is the number of pens and y is the number of pencils. As there is only one condition, you still need to use guess-and-check to solve this equation. Unless the problem says something about x + y.

If the value is not 26 but larger then the equation is 5x + 2y = k where k is the large number. A graph can be plotted and possible solutions seen on the graph. (With larger k the number of solutions increases).

For the young students, the method you use is probably the best. When they are older, they will learn to solve the same problem with larger k values.

Goggles Problem

This problem is from Fairfield Methodist Primary School, used in a seminar for their parents.

88 children took part in a swimming competition. 1/3 of the boys and 3/7 of the girls wore swimming goggles. Altogether 34 children wore swimming goggles. How many girls wore swimming goggles on that day?

I have read a word problem you wrote in the blog that asks for the number of girls who wore goggles. It is rather difficult for me. I am in Grade 3. However, I just tried to solve it and I found the answer is 21 girls wore goggles. Is it right? In solving it, I was helped by one of my cousins who is in Grade 7.

Jane, a third-grade student in Indonesia

Please go to http://math.nie.edu.sg/T3/downloads/2009%20Parents%20PSLE.pdf to view a powerpoint presentation of the model used. Your answer is correct! Keep it up.

Classroom Management

I've been using your method so far and it is amazing how kids learn, but there is a problem in the Chilean concept of discipline. Chilean kids tend to be very immature.

Carola, a teacher in Chile

It is often said that classrrom management problem will disappear when kids are engaged in the learning process. Hopefully with time, when the kids are engaged they will be better in the class, in terms of discipline. But my personal experience with teaching four research lessons in Chile was very good. Besides my observation that some kids in the public schols do not develop basic skills such as addition well enough, they are otherwise fantastic - just like kids in Singapore and everywhere.

In Singapore schools, discipline is emphasized. We still have difficult cases with a small number of students.

The photograph show a demonstration lesson done in Chile for a seminar organized by Ministry of Education Chile. The fourth graders were doing an exercise on pictorial and symbolic representations of fractions. Despite the language barrier, I did not find the kids difficult to manage. The Minister of Education opened the seminar.

A Nine-Year Old Likes Model Method

I am 9 years old and I live and go to school in Jakarta. I like solving maths word problem by the model method. I go to school that teach maths in the Indonesian language. My mother bought me some Singapore maths book two months ago. I like the model methods that are given in the books. My teacher has never taught maths using the model method. My English is not so good so I must look up a dictionary for difficult words. Can I ask you for help if I find some word ptoblems difficult?

Jane, a nine-year old in Jakarta

If Jane pose some questions that she finds difficult, I will share it here. But here is one that comes from a school in Singapore. Try it!

88 children took part in a swimming competition. 1/3 of the boys and 3/7 of the girls wore swimming goggles. Altogether 34 children wore swimming goggles. How many girls wore swimming goggles on that day?

Word Problem on Speed

I have a word problem that I cannot solve. The problem is: Carson and Daniel started driving from the same place but in opposite directions. After 5 hours, they were 60 km apart. Carson's average speed was 12 km/h less than Daniel's. What was Daniel's average speed ?

A teacher in Indonesia

Every hour, who travels a longer distance? Daniel, isn't it? How much further? That is correct - 12 km every hour. That means after 5 hours, Daniel has travelled 60 km more than Carson. Since they are 60 km apart after 5 hours - what does that mean? It seems that Carson was not moving at all and Daniel was travelling at an average speed of 12 km/h - which seems a little slow!

Is there a flaw in the reasoning? Or the problem was not well-posed. Comments?

Freudenthal Institute

I got this message from Freudenthal Institute. You may be interested if you are a school teacher in science and mathematics.

Hereby we would like to let you know that the Utrecht Summer School in Science and Mathematics Education at Utrecht University will be held from August 17 - 21, in the Netherlands. Attached you will find an invitation to participate. Target group: Science and mathematics teachers with an MSc and MA degree in one or more of the Sciences & Mathematics as well as proficiency in English.

Looking forward to meeting you in the Netherlands,

With kind regards,

Jaap den Hertog
Freudenthal Institute for Mathematics and Science Education
Utrecht University
The Netherlands

http://www.utrechtsummerschool.nl/
http://www.science.uu.nl/summerschools
http://www.fi.uu.nl/fisme/en/
http://www.science.uu.nl/summerschools/appliedsciences/

USSE 2009
c/o Ank van der Heiden
Freudenthal Institute,
PO Box 9432 3506 GK Utrecht The Netherlands
T: +31 30 - 263 55 55 F: +31 30 - 266 04 30
Email: A.vanderHeiden@fi.uu.nl

Deadline for Application: 1 May 2009