## Word Problem Progression

These word problems come from  Practice A in the 5B Standards Edition Textbook (p. 68). This particular practice follows six lessons on “Percentage” and “Writing Fractions as Percentages”.  Notice how the challenge  progresses through the seven problems.

Let’s look at the underlying concepts of each problem:

5.  No computation is needed. “15 out of 100”

6. Variation in 2 levels

1. Find complement: 37 marbles are green, how many aren’t?
2. Convert to percentage.

7. Connects percentage to a fraction.

8. Find complement of percentage.

9. Convert fraction to percentage.

10. Corresponds to problem #5. Can you visualize: “if 14/50 how many out of 100?”

11. Multi-step. “Can you imagine how many of 100?” Demonstrates thinking proportionally.

12. First application to measures. Measurement is a continuous quantity, making this problem more abstract.

How does this progression lead to a deep understanding of the concepts?

## Creative Thinking Problems

From Accelerated Learning Primary Mathematics 3 by Ching Kheng Huat published in 1999 in Singapore. Have fun!

TEAR
+EAR
TALE

### The National Center on Teacher Quality released a report entitled: Race to the top: Colorado may be used to high altitudes but can it compete in Race to the Top?

Commissioned by the Piton Foundation, the Donnell-Kay Foundation, the Colorado Children’s Campaign and the Public Education & Business Coalition, the report suggests 7 strategies the state might take while applying for the Race To the Top (RTT) funds.

• Strategy 1: Performance management (Teacher Evaluation, Tenure & Dismissal) – Given the tremendous impact teachers have on learning, no strategy a state will take on is likely to have a greater impact on student achievement than one that seeks to maximize teacher and principal performance.
• Strategy 2: Equitable Distribution of Teachers and Principals – Schools serving children living in poverty are more apt to employ teachers with lower qualifications than schools serving more affluent children.
• Strategy 3: Induction – CO should develop a statewide system of induction support for new teachers, particularly in its high needs and remote rural schools.
• Strategy 4: Compensation Reform – CO needs to move away from lockstep salary schedules towards a system that differentiates salary on a number of factors, including teacher effectiveness, the relative difficulty of a school setting and the demand for teachers with particular skills or knowledge.
• Strategy 5: Teaching in STEM fields: CO should develop a coherent state strategy to address the difficulty school districts face in attracting and retaining sufficient numbers of qualified STEM teachers.
• Strategy6: Statewide Adoption of an Effective Curriculum: Students achieve when 4 elements are in place: Standards, Curriculum, Teachers & Assessment.
• Strategy 7: Educator Preparation (Including Alternate Certification) – In spite of countless studies looking at the value of teacher education, we have only been able to learn (apparently) that no single method of teacher preparation yields more effective teachers than another.

I’ll be honest, I haven’t read through the entire report as yet, however I managed to get through Strategy 6, in which the authors recommend statewide adoption of Singapore Math at the elementary level. The report notes that:

…curriculum has been troublingly absent in conversations about education reform as well as ignored in the indifferent approach some educators take to curricula adoptions.

… the current emphasis on human capital and effective teachers has been at the expense of an equally urgent emphasis on the importance of good curricula.

And when discussing common standards, the report flat-out states:

We would go so far as to say that if the standards were in conflict with the Singapore curriculum, a state ought to consider opting out of the new standards.

Well, you don’t hear that everyday!
.

(Cross-posted at KTM-2)

## Speed and Rate Problems

Rate problems provide some of the biggest challenges to students and adults. These come from Problem-Solving Processes in Mathematics -6 B by Fabian Ng.

1. At 10:15 am, a car left Town X for Town Y at an average speed of 86 km/h, while a truck left Town Y for Town X at an average speed of 74 km/h. At 3:15 pm, the two vehicles were 12 km apart. How far apart were the two towns?
2. At 10:30 am, a cyclist started traveling on a road at an average speed of 60 km/h. At 2:30 pm, a motorist started from the same place, traveling on the same road. If the motorist took 4 hours to catch up with the cyclist, find his average speed.
3. The distance from Town P to Town Q was 312 km. Winston started from Town P at an average speed of 76 km/h. He maintained this speed for 2 hours before increasing it by 4 km/ for the rest of the journey to Town Q.
4. a. how long did he take to complete the whole journey?
b. What was his average speed from Town P to Town Q?

Can you draw a model or diagram to illustrate each of these problems?

## Singapore Math – Old School

Old School claims to be Singapore’s #1 Primary School site. From the site:

Old School aims to be the premier resource for FREE educational material from Singapore, especially Singapore Math and Singapore Science. We have a large database of test questions and assessments and are always adding new content.

Currently, they offer test papers as well as the ability to take tests online in English, Mathematics and Science. Registering on the site allows you to track your progress on the online tests (usually the “A” part of each). Best of all, you can search the questions by topic and select questions with either a multiple choice or free-response format.

As an example, the Primary Four 2009 Mathematics page offers:

• 3  Continual Assessment 1 tests
• 4  Mid-Year Examination tests
• 3  Continual Assessment 2 tests
• 4 End Year Examination tests

That include the following topics (with the percentage of questions):

• Angles (7%)
• Area and Perimeter (8%)
• Decimals (4%)
• Factors and Multiples (6%)
• Four Operations (22%)
• Fractions (18%)
• Geometry (2%)
• Graphs (2%)
• Measurement (7%)
• Non-Standard Questions (6%)
• Perpendicular and Parallel Lines (4%)

The majority of mathematics papers are a 2 out of 4 on the site’s scale of difficulty. There were some assessments with a difficulty rating of 1.

Here’s a problem from the 2009 Primary Four End Year Assessment for your enjoyment:

Fill in the missing number.

108 x 99 = 110 x 99 + 10 x 99 – ( ? ) x 99
1. 8
2. 2
3. 12
4. 228

## Intermediate Word Problem

Published in Singapore, Challenging Problems in Mathematics for Primary Schools: Intermediate by Dr. Y.H. Leong is a series specially written to provide enrichment activities for students.  The intermediate edition is designed for Primary 4 & 5 students.

Enjoy!

A farmer planted 22 rambutan trees in a straight row. The trees were spaced out equally. If the distance between the 3rd tree and the 10th tree was 42m, find the distance between the 2nd tree and the 22nd tree.

## Comparing Singapore Math Materials: Workbooks

In Part 1 and 2,  I shared examples from the Teacher’s Guides and Textbooks from four sets of materials used in Singapore and the United States. The materials are all from the third grade level:

1. Primary Mathematics U.S. Edition (2003)  from SingaporeMath.com
2. Primary Mathematics Standards Edition (2008)  from SingaporeMath.com
3. My Pals Are Here Maths (2007) obtained in Singapore from Marshall Cavendish
4. Shaping Maths (2007) obtained in Singapore from Marshall Cavendish

Following are the practice pages from each workbook that correspond with the lesson on addition within 10,000 that introduces regrouping in the hundreds. As before, each thumbnail links to a full-sized file. Once again, there are minimal differences between the U.S. and Standards editions of Primary Mathematics.  Problem #1 changes pictures from towels hanging on a clothesline to boats. Problem #2 has one small change. The equation for  letter B changes from  4107 + 5 to 4105 + 5. Finally, on problem #4, “Weihua” becomes “Will” U.S. Edition Workbook 3A: Standards Edition Workbook 3A: The My Pals Are Here Workbook is perforated and 3 hole punched. Perforated pages would be a great change to make to the Primary Mathematics workbooks! Neither My Pals Are Here nor Shaping Maths have any word problems tied to this practice lesson, in fact, there are very few word problems in the books at all. My Pals Are Here Workbook 3A Part 1: Shaping Maths Activity Book 3A part 1: Were you expecting less practice in the materials from Singapore?

## Creative Thinking Problems

I had coffee with another fan of Singapore Math this week and he loaned me some supplemental materials from his travels to Popular Books in Singapore. These two problems come from Accelerated Learning Primary Mathematics 3 by Ching Kheng Huat published in 1999.

The front page states:

In this book, “Accelerated Learning” fosters in pupils the ability to learn faster, to remember more and to think creatively.

The Ministry of Education has initiated IT in education for pupils. To make time for IT, pupils need to learn faster to cover the syllabus. And this requires ACCELERATED LEARNING – a new dimension in pedagogical skills. MOE has also emphasized the need to develop CREATIVE THINKING, which we have incorporated in this book.

The book has seven pages of “Infographic  Images” at the beginning (visual dictionary of terms), then practice pages for 12 units. Each unit of six to seven pages has:

1. two worked examples
2. multiple choice practice
4. problem sums practice
5. creative thinking problem (one or two)

Here are the Creative Thinking problems from the unit on fractions:

And a Creative Thinking problem from the unit on time:

It takes 4 h 15 min to repair 3 computer. Repairing a radio takes 47 minutes less that the time needed to repair a computer. If a worker works 9 hours a day, he needs to complete repairing 10 items that can include both computers and radios. How many computers can he repair if he needs to repair as many computers as possible?

Do you feel that these problems will help students “develop creative thinking” ?

## Comparing Singapore Math Materials: Textbooks

In Part 1, I shared some examples from the Teacher’s Guide from four sets of materials used in Singapore and the United States.

The materials are all from the third grade level:

1. Primary Mathematics U.S. Edition (2003)  from SingaporeMath.com
2. Primary Mathematics Standards Edition (2008)  from SingaporeMath.com
3. My Pals Are Here Maths (2007) obtained in Singapore from Marshall Cavendish
4. Shaping Maths (2007) obtained in Singapore from Marshall Cavendish

Following are the pages from each textbook unit on addition within 10,000  that introduces regrouping in the hundreds. As before, each thumbnail links to a full-sized file.

There are minimal differences between the U.S. and Standards editions. The Standards edition is in color and there are two additional prompts asking students to estimate their answer first, then check for reasonableness. The first example shows regrouping in the hundreds. Problems 1-5 ask students to recall addition with regrouping the ones or tens or hundreds.

U.S. Edition Textbook 3A:

Standards Edition Textbook 3A:

My Pals Are Here includes two pages of instruction, another page with directions to a game and a final page exploring regrouping in the hundreds. Problem 5b on page 29 is the only problem that demonstrates  regrouping in both the ones and hundreds places,  although students are only asked to find the missing ones value in one addend.

Note the example #2 on page 29 that spells out the concept in words  (5 hundreds + 8 hundreds). This is a great reminder of how teachers can model this concept in a classroom and is included in the Teacher’s Guide for Primary Math both U.S. and Standards editions ( 7 ones + 5 ones = 1 ten 2 ones).

My Pals Are Here Textbook 3A:

The Shaping Maths lesson is two pages of slightly more abstract description than My Pals Are Here. Place value disks are used instead of images of base-10 blocks.

Shaping Maths Coursebook 3A:

## Comparing Singapore Math Materials: Teacher’s Guides

### First in a three-part series

I thought it might be interesting to provide examples of how a lesson is presented in four different sets of Singapore Math materials. Part 1 compares the materials where a lesson begins – the Teacher’s Guide. The rest of the series will include textbooks and workbooks.

All four sets of materials are listed below. The two editions of Primary Mathematics are currently in use throughout the United States. My Pals Are Here and Shaping Maths are currently in use in Singapore.

This overview doesn’t include the Math in Focus series by Marshall Cavendish and exclusive United States distributor Great Source (A division of Houghton Mifflin Harcourt)  which will be available soon in the United States.  Representatives at the NCTM Conference in Washington, D.C. stated that the Math in Focus content is based on the  Singaporean edition of  My Pals Are Here, with U.S. money and measurement the main additions. A listing of key topics can be found on the Great Source site.

The materials are all from the third grade level:

1. Primary Mathematics U.S. Edition (2003)  from SingaporeMath.com
2. Primary Mathematics Standards Edition (2008)  from SingaporeMath.com
3. My Pals Are Here Maths (2007) obtained in Singapore from Marshall Cavendish
4. Shaping Maths (2007) obtained in Singapore from Marshall Cavendish

Most people will display the cover of a book. You get the title, authors and little else. The back of the book,  however, contains more interesting information. (Exception? The Primary Mathematics-U.S. Edition) These are from the 3A Teacher Guides and provide a brief overview of each series. For your reading ease, each thumbnail links to a full-sized file.

For comparison, I will be using the first lesson in the 3A materials from the unit on addition within 10,000  that introduces regrouping in the hundreds. All materials use the term “renaming” except My Pals Are Here, which uses “regrouping”.

Below are the names, pages and the stated objectives for the lesson from the corresponding Teacher’s Guide along with one page as an example:

Primary Mathematics-U.S.: Adding Ones, Tens, Hundreds and Thousands (p. 22)

Primary Mathematics-Standards: Adding Ones, Tens, Hundreds and Thousands (p. 74)

• Review of addition of numbers up to 3 digits.
• Adding thousands with another number up to 4 digits with renaming once.

My Pals Are Here: Addition With Regrouping in Hundreds (p. 40)

• Add two 4-digit numbers with regrouping in hundreds using concrete representation.
• Show regrouping of hundreds to thousands and hundreds.
• Carry out vertical column addition by adding the hundreds first then the thousands with regrouping in the hundreds place
• Add without place value charts.

• To add 4-digit numbers with renaming once.
• To add 4 digit numbers with renaming more than once.

Both My Pals Are Here and Shaping Maths have a larger sized Teacher’s Guide (A4 size). This allows the publisher to include each page from the textbook, surrounded by:

My Pals Are Here: instructional objectives, instructional procedures, key concepts, materials, additional activities, individual work, heuristic for problem solving and thinking skills

Shaping Maths:  objectives, lesson, materials, classroom organisation, vocabulary, general learning difficulties, IT, notes and textbook practice

The Teacher’s Guide for Primary Mathematics – Standards provides much more guidance.  The layout of the material is more familiar to most teachers and looks a bit more like a typical American teacher manual.  Although the Standards Edition provides California Standards, it would be easy to correlate these to another state’s standards. As a design, I like the spiral binding that allows the book to lay flat.

As we will see in the rest of the series, you should not choose a set of materials based on the Teacher’s Guide alone.