Discuss.

To help students remember which way the inequality sign pointed, the teacher drew an alligator mouth. You know the lesson. Here’s an example from songsforteaching.com:

And the song lyrics:

And one from GradeAMathHelp.com

And this simple shortcut or trick was the cause of all of Allison’s troubles with inequalities.

She explained: “everyone knows, the bigger fish eats the small fish”:

A 42 oz mix of nuts is 6 parts peanuts to 1 part cashews. How many ounces of cashews must be added to make a make the mixture 2 parts cashews to 1 part peanuts? Does Singapore (bar) model method work here?

Sure it can! Here’s one solution.

First model the beginning mixture at 6:1.

Next, we can figure out how many units need to be added to the cashews to make the cashews to peanuts ratio 2:1.

Since there are 11 additional units and the value of each unit is 6 oz, we can find the value of the added cashews.

66 ounces of cashews must be added to the peanuts to make the mixture 2:1.
Changing Ratios is taught in level 6A in the Primary Mathematics materials (U.S. Edition), typically used in 6th grade classrooms.

From publisher Frank Schaffer and Singapore Asian Publications comes a series of books: Singapore Math: 70 Must-Know Word Problems. From the back cover:

This book is designed to help students master word problems, which are often tricky and challenging…This book is perfect for students familiar with Singapore Math and for those who just need extra practice with word problems.

The 70 Must-Know Word Problems books are marked by levels that are considered a grade above for the U.S. market. For example, the Level 6 book claims to be appropriate for students in grade 7 and the Level 4 book is listed as appropriate for students in grade 5. If your child has been learning with Primary Mathematics and you’d like to pick up one of these workbooks, you should probably pick up the level that correlates to their current grade. Meaning, if your child is currently working their way through the Level 4 Primary Mathematics textbook, then the level 4 70 Must-Know Word Problems book will work as a supplement. The problems in the books are extra practice type problems, designed to give students similar to the ones in the Primary Mathematics textbooks. If you student is look for something harder, you might want to consider the Challenging Word Problems for Primary Mathematics series.

The 70 Must-Know series does not follow the Primary Mathematics lesson progression, the questions jump between concepts. Questions #3 and #47 in the Level 6 book, for example, are both on Volume. Question #2 & 5 cover the four operations with money  and Question #4 is on percentages, so you may need to verify that a topic has been covered in class if your planning on using the book at home. The books have little instruction, however they do include a four page Introduction to Singapore Math (be forewarned, it’s written in six-point font!). Each problem has its own full page for work and there are fully worked solutions included in the answer key.

Here’s Question #70 from the Level 4 book. It’s listed in the table of contents under “Addition, Subtraction, Multiplication and Division of Whole Numbers”.

At a movie, 1/4 of the people in the theater were men, 5/8 were women and the rest were children. If there were 100 more women than children, what was the total number of people in the theater?

And Question # 48 from Level 6:

A box contained some red, blue, and green markers. For every 5 red markers, there were 2 blue markers. For every 3 blue markers, there were 5 green markers.

(a) Find the ratio of red markers to blue markers to green markers.

(b) When 6 red markers were removed from the box, 3/7 of the remaining markers were red markers. How many markers were left in the box?

Singapore Math: 70 Must-Know Word Problems was$12.99 at my local Barnes and Noble and Borders stores. The easy accessibility of these books make them great for parents looking for additional problems for their students. Teachers looking to use these materials in class need to keep on mind that these books may already be in their students’ homes!

Google Books is offering free previews of books in this series:

• Singapore Math 70 Must-Know Word Problems, Level 1
• Singapore Math 70 Must Know Word Problems , Level 2
• Singapore Math 70 Must-Know Word Problems, Level 3
• Singapore Math 70 Must-Know Word Problems, Level 5

Denise at Let’s Play Math has written up a some fourth grade level multi-step word problems (and provided bar model solutions) based on C. S. Lewis’  The Lion, the Witch and the Wardrobe. She concludes:

…as word problems become more complex, the bar diagrams offer significant help for students who struggle with the question, “What do I do?” Diagrams make visible the abstract relationships between numbers, enabling the student to decide which arithmetical operation makes sense in the context of the problem.

This is the fourth installment of her series on Pre-Algebra Problem Solving. Other posts  use Mr. Popper’s Penguins and Poor Richard to illustrate problem solving using bar models in earlier grades.

Enjoy!

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?

1. For both piles to have the same weight, which item belonging to the pile on the left should be exchanged with another item belonging to the pile on the right?

2. Replace the letters with numbers.

TEAR
+EAR
TALE

3. Fill in the boxes with appropriate numbers.

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.

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?

(Answers next week!)

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.

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
3. short answer 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” ?