Category Archives: Mathematics

Inequality Problem

The above is a very instructive problem and solution.

Old Geometry Book

Reference: Course on Mathematics

For application by the student

These sections are taken from PLANE AND SOLID GEOMETRY by George Wentworth and David Eugene Smith, first published in 1888.

G00 – Contents

G01 – Introduction

G02 – BOOK I. Rectilinear Figures

G03 – BOOK II. The Circle

G04 – BOOK III. Proportion. Similar Polygon

G05 – BOOK IV. Area of Polygons

G06 – BOOK V. Regular Polygons and Circles

G07 – Appendix to Plane Geometry

G08 – BOOK VI. Lines and Planes in Space

G09 – BOOK VII. Polyhedrons, Cylinders and Cones

G10 – BOOK VIII. The Sphere

G11 – Appendix to Solid Geometry

G12 – Miscellaneous & Index


Troubleshooting Math

To troubleshoot any difficulty you first look at the broad area of that difficulty, and then you gradually narrow it down until you have defined the actual difficulty precisely.

So, to troubleshoot a difficulty in math you start with the broad area of Mathematics.

Mathema (Greek) = Learn
Mathematics = Tools for learning

Mathematics provides you with analytical tools for learning. When you are troubleshooting mathematics, you are troubleshooting the difficulties a person is having with learning analytically. You narrow down to the area of mathematics where the person cannot think analytically.

Mathematics is analytical learning and not just memorizing of materials.

If the student is having trouble with higher mathematics, such as, Trignometry, Analytical Geometry, or Calculus, then start from there. You may explain the area the student does not understand. But if the student cannot understand the explanation analytically, then the troubleshooting may lead to one of the three areas below.

When you select one of these areas, explain it per Math Overview. You do not have to explain that whole document. Keep to the trail of trouble.

Ask, “What part of this area you have most difficulty with?”

Use the answer to narrow down further to the area of difficulty. Quiz the student on the key math vocabulary in that area. From student’s answers you may narrow down the area of difficulty further.

If the student cannot answer the question, simply start with the first lesson
related to that area at Mathematics. Follow student’s attention to fish around for the actual difficulty.

As you narrow down the area of difficulty, keep asking, “What part of this area you have most difficulty with?”

Check the key math vocabulary in the narrowed down area. Soon you’ll reach the actual difficulty. Handle it using the right materials selected from the appropriate level at Mathematics, or from student’s own materials.

Once that area is handled, the student may come up with another area that he or she has attention on. Narrow down to the actual difficulty in that area as above, and handle it.

Otherwise, start all over again from the diagram above. This time you may follow a different trail to a different area of difficulty.

Ultimately, teach the student how to troubleshhoot difficulties. This is the best thing you can ever do for the student.


Middle School Review


Reference: Remedial Math

For application by the student

Lesson 1: Number to Integer

Exercise: Problems from Dubb (Word problems)


Lesson 2: Terms & Expression

Exercise: Problems from Dubbs (Word problems)


Lesson 3: Factoring

Exercise: Factoring Problems from Dubb

Prime Factors

Common Prime Factors

Greatest Common Divisor

Least Common Multiple

Word Problems


Lesson 4: Fractions

Exercise: Fraction Problems from Dubb

Integer to Fraction

Mixed Number to Improper Fraction

Improper Fraction to Mixed number

Reducing Fraction to a Denominator

Reducing Fraction to Lowest Terms

Reducing Fraction to LCM & GCD

Addition of Fractions

Subtraction of Fractions

Multiplication of Fractions

Multiplication Word Problems

Division of Fractions

Finding Part of a Number

Reducing Complex Fractions

Complex Fraction Word Problems

Word Problems for Fractions


Lesson 5: Decimals

Exercise: Decimal Problems from Dubb

Writing Decimals

Reading Decimals

Decimals to Common Fractions

Common Fractions to Decimals

Addition of Decimals

Subtraction of Decimals

Multiplication of Decimals

Division of Decimals


Lesson 6: Review of Basics

Make yourself familiar with these basics

Real Numbers

The Basics of Math

Additional Math Concepts

Equations & Transposition




To be used for students needing more details.

MS B1 Mixed Operations

MS B2 Integers

MS B3 Factors

MS B4 Frac Props

MS B5 Frac Ops

MS B6 Decimals


Mathematics & Thinking


Reference: Critical Thinking in Education


The most useful aspect of mathematics is that it provides opportunities:

  1. To think outside the box.

  2. To help learn something new.

Thinking outside the box

There’s a popular story that Gauss, a famous mathematician, had a lazy teacher in his elementary school. The so-called educator wanted to keep the kids busy so he could take a nap; he asked the class to add the numbers 1 to 100.

But Gauss found the answer in less than 10 minutes, and he interrupted the teachers nap with his answer: 5050. So soon? The teacher suspected a cheat, but when he looked at Gauss’s method, he realized that he had a genius in his class.

Here is what Gauss did. He was required to add the first 100 numbers as follows.

1 + 2 + 3 + 4 + … + 98 + 99 + 100

But he split the numbers in two groups (1 to 50 and 51 to 100), and arranged these numbers as follows

1     +   2   +   3  + … + 48 + 49 + 50

100 +  99 +  98 +  … + 53 + 52 + 51

Each row had fifty numbers. He added the corresponding numbers as follows.

1 + 100 = 101

2 + 99 = 101

3 + 98 = 101

48 + 53 = 101

49 + 52 = 101

50 + 51 = 101

Gauss found that the final sum would be

101 + 101 + 101 + … (50 times)    =    101 x 50    =    5050.

This was thinking outside the box. Mathematics provides many such opportunities.

EXERCISE: Add numbers

(a)  1 to 20

(b)  1 to 50

(c)  1 to 33


Helping learn something new

Mathematics also provides many opportunities to help learn something new. For example, the feel for numbers is very important and it helps one learn to add very quickly.

Part of the feel for numbers is to know the gap between a number and the next TEN.

This gap can be seen on a number line at the beginning of this essay, where it helps add

39 + 5  =  39 + 1 + 4  =  40 + 4  =  44.
66 + 8  =  66 + 4 + 4  =  70 + 4  =  74.

Here the gap is filled first by the second number and then the rest of the number is added easily to a TEN.

A student and his or her study partner can drill these gaps. One of them calls out a number and the other responds with the gap. Such drill is a lot of fun, when the numbers called out are random.

Number           Gap

9                              1

8                              2

7                              3

6                              4

5                              5

4                              6

3                              7

2                              8

1                              9

27                            3

49                            1

54                            6                            etc.

The fundamental aspects of mental math can be learned quite quickly with such drilling. But any such drilling must be followed by proper understanding. For example, the student must first understand that multiplication is “repeated addition” before he or she drills the multiplication tables.