Solving SAT problems using the model method

David Marain over at MathNotations has been periodically tweeting an SAT problem of the day via @dmarain. A recent series of tweets asked:

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.

Singapore Math bar model

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

Singapore Math Model diagram - after

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

Singapore Math units

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.


Singapore Math Must-Know Word Problems

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!


Problem Solving and Literature


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.



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!

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.


3. Fill in the boxes with appropriate numbers.