A Multiple Problem

I am a two-digit number. When I am divided by 8, there is a remainder of 3. When I am divided by 9, there is a remainder of 4. I am less than 80. What number am I?

Jane, Indonesia

One way is to guess and check. Make an intelligent guess and check if both conditions are met. I can guess 27. 27 divided by 8 gives 3 remainder 3. 27 divided by 9 gives no remainder. So is the number 27?

I think it takes time to guess this way. Let's use logical reasoning. The number is 3 more than a multiple 8 and 4 more than a multiple of 9.

Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80.

Three more than a multiple of 8: 11, 19, 27, 35, 43, 59, 67, 75 (no need to try 83 and beyond)

Multiple of 9: 9, 18, 27, 36, 45, 54, 63, 72

Four more than multiple of 9: 13, 22, 31, 40, 49, 58, 67, 76

You got the solution, right?

How about a different way? If you know some algebra, the number is 8m+3 or 9n+4 where m and n are whole numbers. 8m+3 = 9n+4 or 8m = 9n + 1 which gives a possible solution of m = 8 and n = 7. Thus, the number which is 8m+3 can be easily found.

What if the last condition that it is less than 80 is not given?

Can you make up a similar interesting problem for the others to solve?

Pre-School Numeracy & Assessment

We are writing you from Santiago, Chile. We are using Earlybirds Kindergarten Mathematics. The doubt is that in Book A, to evaluate classification, is it necessary to evaluate all the previous steps such as "different things", "things that are used together", "things that do not belong" etc.? Or can we evaluate only the final concept, that is sorting by the three attributes.

We have to tell you that this is our first year using your method in Mathematics, and, at the beginning , we had some problems, because of the language (our students don´t speak English at home, they learn it only at school). But now, we have used the book for 4 months, and we think the students, and we, the teachers, have learned a lot from it.

Anamaría, Chile

In Kindergarten, we do not want to overwhelm children with assessment and evaluation. In this unit students have learnt how to match things according to some attribute (being able to say two animals are the same despite differences in size, color, orientation; being able to say two things are the same even though they are drawn differently; matching by colors; matching by patterns). Some where in between they apply this to pick the odd item out among, say, four items. Later, they learn to classify according to more abstract attributes such as function (what they are used for). Again, the apply these skills to pick the odd one out. Finally, the use the skills to classify items into two groups according to given criteria and their own criteria. In the evaluation, we assess and evaluate if the children are able to apply these skills in picking the odd one out and in classification.

There is no need to evaluate every sub-skill. However, if any child cannot complete the main task (odd one out and classification) then teachers may want top check if they can do the sub-skills.

Test Items from Singapore National Tests

I understand that somewhere in your website I can access past primary-level mathematics tests, but I cannot find them. Do they exist, and can I get access to them? I am learning about Singapore Math.

Lee, a math coach in Utah

These are not available online. Released items from the tests are compiled into a book every year. For example, the most recent book consists of released items from 2004 to 2008 tests. About three-fourths of all items are released each year. This book is not available outside Singapore. I believe this is an agreement between the copyright holder of the test items and the publishers. In Singapore, they are easily available in any Popular Bookshops. If you are outside Singapore, get a friend to help you buy a copy.

The national test at the end of primary schooling is the Primary School Leaving Examination. Students are tested on English, Mathematics, Mother Tongue Language and Science. For mathematics. most students do the Mathematics test. A small proportion do the Foundation Mathematics test which focuses more on basics and less on problem solving.

Students complete 15 multiple-choice items (20 points), 20 short-answer items (30 points) and 13 long-answer items (50 points) which includes some challenging tasks. The first paper (50 min) includes the multiple-choice items and 15 short-answer ones. The second paper (1 h 40 min) includes the other items. Students can use a calculator in the second paper. This format will be used for the first time in 2009. Previously, these items were in one 2 h 15 min paper and students do not use calculators (and the numbers are not tedious to compute).