Category Archives: Mathematics

Mathematics & Thinking

Gap

Reference: Critical Thinking in Education

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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

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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 = 44, and 66 + 8 = 74. Here the gap is filled first by the second number and then the rest of the number is added easily to 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.

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Math Tutorial: Opening Sessions

maths-cafe-4400x182

Reference: Critical Thinking in Education

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The following outline may help you get started as a tutor with a new student. Simply use the introductory materials form Remedial Math Level 1.

There are many ways to start with a new student; but the following is the best way to start with students from middle and high schools. This outline is good for couple of beginning sessions.

NOTE: For youngers students, start from step 4, and go over each lesson very closely. Explain each paragraph to them in simple language so they can understand the materials. 

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(1) Ask the student the meaning of the word MATHEMATICS.

Use SECTION 1 from Math Overview to explain it. It is also okay if you just show him the two points of SECTION 1, while explaining it to him. Have him give an example per the exercise in that section.

(2) Ask the student, “Why should somebody study mathematics?”

Use SECTION 2 from Math Overview to explain it. Ask him the three puzzles. Give him some hints if he is having trouble solving them. If he can’t solve a puzzle show him the answer.

(3) Ask the student, “What are the three main parts of mathematics?”

Use SECTION 3 from Math Overview to explain it. Explain each part and its meaning as given. Demonstrate the use of each part per the sample problem provided.

(4) Ask the student about the difference between a NUMBER and a DIGIT.

Use SECTION 2 from Lesson 1: Arithmetic & Numbers to explain it.

(5) Ask the student about the place values on an abacus.

Use the understanding from the reference MS A1 Numbers to explain what an abacus is, and to demonstrate the place values on it.

(6) Go over Lesson 1: Arithmetic & Numbers with the student.

Have the student read each numbered section silently one at a time. You read silently along with him. Have him do the exercises after each section. Explain anything that he does not understand.

(7) Lightly supervise the student as he studies subsequent lessons.

Keep an eye on the student as he is starts to study the subsequent lessons on his own. Make sure he does the exercise of each section till he can apply the materials. Check his answers for correctness. Explain anything that he does not understand.

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Remedial Math Level Pre-0

bookwagon

Reference: Remedial Math

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For application by parents

Lesson 1: ORIENTATION & SPATIAL SENSE

Lesson 2: QUANTITY & NUMBER SENSE

Lesson 3: PATTERNS & RELATIONAL SENSE

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Remedial Math Level 0

Abacus

Reference: Remedial Math

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For application by parents

Lesson 1: Orientation & Spatial Sense

Lesson 2: Numbers & Place Values

Lesson 3: Units & Fractions

Lesson 4: Counting & Measurements

Lesson 5: Numbers & Operations

Lesson 6: Patterns & Relational Sense

Lesson 7: Data Analysis & Probability

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Some Old Books

bk05ray1866

ARITHMETIC TEXTBOOKS

1885 Ray’s Practical Arithmetic

1893 Dubb’s Arithmetical Problems

ALGEBRA TEXTBOOK

1896 Elementary Algebra

GEOMETRY TEXTBOOK

1913 Wentworth Smith

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