## Category Archives: Mathematics

### Inequality Problem

The above is a very instructive problem and solution.

## 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.

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### 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) = LearnMathematics = 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.

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## Lesson 6: Review of Basics

### Equations & Transposition

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## MS B6 Decimals

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### Mathematics & Thinking ##### Reference: Critical Thinking in Education

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

## 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

### 100 +  99 +  98 +  … + 53 + 52 + 51

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

### 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.

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## 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

### 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.

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