Interior Angles of a Triangle

Introduction:

This lesson demonstrates a theorem of geometry established over 2300 years ago.

Background

The original theorem, as stated by the Greek mathematician, Euclid, is the sum of the interior angles of a triangle equal two right angles. This statement requires us to know the size of a right angle. To simplify the theory in current terms, most books in geometry state the sum of the interior angles of a triangle equals 180 degrees. This statement requires us to know the meaning of 180 degrees so another rewording of the theorem is the sum of the interior angles of a triangle form a straight line.

Lesson           

Stack three pieces of paper together. Cut through the three pieces at one time to make three identical triangles – the size and shape of your choosing. Just make sure all three triangles are the same.

 

Write on the interior angles (corners) of each of the triangles a, b, c.

Next, set down one of the triangles.

Next, rotate the second triangle and place next to the first so angle b of the first triangle is next to angle a of the second triangle.

 Next, rotate the third triangle so angle c is adjacent to angle a of the middle triangle. Finally, with the edge of a ruler or the edge of a piece of paper, confirm that the sum of the three angles, a + b + c, form a straight line.

 

Historical notes:

Reviewing the original Greek proof of this theorem requires additional background in geometric terms and basic axioms of geometry – the proof results in the same addition of the three angles (A, B, C)  are shown by three different colors. See Useful Links below for additional information.

 

(By Master Uegly - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=39450205)

 

Note to parents or instructors:

It is possible that one of the triangles might flip over after cutting. This should not change the result as flipping over a triangle doesn’t change the angle of its corners. What is important is that the three different corners are put together (a, b, c); a straight line may not result if two or more of the corners placed together are the same such as (a, b, b) or (c, c, c).

 Variations:

1.      Try the experiment with three isosceles triangles (a triangle with two equal sides).

2.     Try the experiment with three equilateral triangles (a triangle with three equal sides).

 

Useful links:

https://en.wikipedia.org/wiki/Sum_of_angles_of_a_triangle

https://en.wikipedia.org/wiki/Euclidean_geometry

Even Numbers as the Sum of Two Primes (The Goldbach Conjecture)

Summary

This lesson is about an interesting property of all even numbers greater than 2. Every even number greater than 2 is the sum of two prime numbers.

Background – Prime numbers

A prime number is a number that cannot be formed by multiplying two smaller numbers. For example, 5 is a prime number because only 1 x 5 = 5. 6 is not a prime because you can multiply two smaller numbers, 2 and 3, to make 6. Prime numbers under 100 are:

2            3            5            7            11          13              17          19          23         29         31          37          41          43         47          53             59          61          67          71          73          79          83         89         97         

Numbers, other than 0 and 1, that are not prime are call composite numbers. All composite numbers can be formed by multiplying two smaller numbers.

Examples

Here are examples of how even numbers can be expressed as the sum of two prime numbers:
4 is the sum of 2 + 2

6 is the sum of 3 + 3 (the two numbers do not have to be the same as you’ll see in the next example)

8 is the sum of 3 + 5

10 is the sum of 5 + 5 (10 can also be expressed as the sum of two other primes, 3 + 7)

12 is the sum 5 + 7

 

Exercises

1.  Continue with the number 14. Using the list of prime numbers (above), find how each even number can be expressed as the sum of two prime numbers. Try to complete the exercise for all even numbers up to 40.

2. There will always be at least one way to add two primes to form each even number but in some cases, as shown above for 10, there can be two or more ways to add two primes to form the number. Find other even numbers like 10 for which there are two or more ways to add two primes to form the number.

3.  The number 100 is interesting in that there are six ways in which the sum of two primes equal 100. Can you find all six pairs of prime numbers that add up to 100?

4.  This pattern of adding two prime numbers only works for forming even numbers; however, it sometimes works for odd numbers - but not all. For example: 5 = 2 + 3; 7 = 2 + 5; but 11 cannot be expressed as the sum of two prime numbers. To check, here are all the ways two numbers add up to 11:
1 + 10 (10 is not prime)

      2 + 9 (9 is not prime)

3 + 8 (8 is not prime)

4 + 7 (4 is not prime)

5 + 6 (6 is not prime)

Continue the exercise with higher odd numbers; either find a pair of prime numbers that add up to the number or prove (as done above for 11) that the number cannot be formed by the sum of two prime numbers. 

For parents, teachers and advanced thinkers - How does it work?

This lesson is based on the Goldbach Conjecture which states that all even numbers greater than 2 can be expressed as the sum of two prime numbers.  A conjecture is a mathematical statement that is generally accepted as true but has not been proven. A formally proven statement is called a theorem. While the Goldbach Conjecture has not been proven, it has been confirmed to be true using computers for even integers beyond 1,000,000,000. Monetary prizes are available for the person who can formally prove the Goldbach statement to be true. If a single even number was found that could not be stated as the sum of two prime numbers, then the Goldbach statement would be disproven. However, given the very large numbers that have already been tested, finding such a starting number is unlikely.

For additional information:

Goldbach conjecture info: http://www.mi.sanu.ac.rs/vismath/garavaglia2008/index.html

 

Exercise key and comments:

1.      (Just one pair of prime numbers is listed for each even number; there may be additional pairings that work)
14 = 7 + 7
16= 3 + 13
18= 7 + 11
20= 7 + 13
22 = 3 + 19

24 = 5 + 19
26 = 13 + 13
28 = 5 + 23
30 = 7 + 23
32 = 3 + 29
34 = 17 + 17
36 = 7 + 29

38 = 19 + 19
40 = 3 + 37

 

 

    2. Examples of even numbers formed by multiple pairs of two primes include:
         10 = 3 + 7; 10 = 5 + 5

16 = 5 + 11; 16 = 3 + 13

20 = 3 + 17; 20 = 7 + 13

 

3.  The number 100 can be formed the sum of six different pairs of prime numbers: 

100 = 3 + 97; 100 = 11 + 89; 100 = 17 + 83; 100 = 29 + 71; 100 = 41 + 59; 100 = 47 + 53

    4.    13 = 2 + 11
            15 = 2 + 13
            19 = 2 + 17
            21 = 2 + 19
            25= 2 + 23

Additional counter examples include: 17, 23, 27, 29, 37…

Update 1/18/2022 - a recent post gives some additional insight to this problem: Math Vacation: Additional Thoughts on the Goldbach Conjecture (jamesmacmath.blogspot.com). 

The Missing Digit Magic Trick

There is an easy "magic" trick that can be performed by anyone who can add a few single digits together.

Set-up of the trick:

Give your friend a calculator or have them use the calculator function on their phone.

Without showing you, ask them to enter a private 4-digit pin or any number other than 0000.

Ask your friend to multiple their private number by 153; your friend should not tell or show you the result.

Ask your friend to pick one of the digits of their result as the “secret number” but not zero if there was a zero in the result.

Now ask for all the other digits in the result.

You then tell your friend the digit that was not revealed.

How you do it?

As your friend tells you the digits in their number (other than the secret digit), you add them up. Usually, you will have 5 or 6 single digits to add. When you get to the total, subtract that total from the next higher multiple of 9. The answer will be your friend's secret number. For example, suppose they tell you the other digits are: 1, 0, 3, 3, and 4. Adding these numbers, 1 + 0 + 3 + 3 + 4 = 11. The next multiple of 9 is 18, so 18 - 11 = 7. The secret digit is 7. Another example: suppose you are told the other digits are: 5, 3, 4, 1, and 8. The sum of these numbers is 21. The next multiple of 9 greater than 21 is 27. 27 - 21 = 6, so their secret number is 6.

How does the trick work?

The basis of the trick is an interesting property of all numbers that are multiples of 9. The sum of the digits of every number that is a multiple of 9 will equal 9 (or a multiple of 9). Examples:

2 x 9 = 18             1 + 8 = 9

3 x 9 = 27             2 + 7 = 9

21 x 9 = 189         1 + 8 + 9 = 18 (multiple of 9) or one can continue with the digits of 18 to get 1+8 =9

The trick starts with your friend choosing any number. You don’t know if the starting number is a multiple of 9 or not but 153 is a multiple of 9 (1 + 5 + 3 = 9). Therefore, any starting number multiplied by 153 will result in a multiple of 9. So now you know all the digits in the final result will add up to 9 or a multiple of 9. There is nothing extra special about 153 other than it is a multiple of 9. If you repeat the trick you might want to use a different multiple of 9 each time. Another good one to use is 117.

Some extra tips:

With practice, one can get good at quickly adding up the digits given to you. As you add up the digits, here are three tips to speed up the process.

1. Ignore any 9's. Dropping any 9 from your addition will not change the final result. For example if you are given the numbers: 3 + 1 + 0 + 9 +9, their sum is 22. The next multiple of 9 is 27; 27 - 22 = 5. Ignoring the two nines, gives you a total of 4 and, again, you need a difference of 5 to get to the next multiple of 9 so you still end up with the same result.

2. Ignore any pair of numbers that equal 9. In the second example above, the numbers given were: 5, 3, 4, 1, and 8. Ignore the 5 and 4 (their sum = 9) and ignore the 1 and 8. Now you are left with 3. 9-3 = 6 which is the same answer from the example.

3. If your final sum has more than 1 digit, add the digits of the sum together to get a single digit number. Now, the secret number will be 9 minus this new total. In the first example above, you had the numbers 1, 0, 3, 3, and 4. Their total is 11. Adding 1 + 1 = 2 and 9 - 2 = 7. Again you get the same answer.

Special notes:

1. During the set-up of the trick, remember to ask your friend if any of the digits of their multiplication result is zero. You don't want them to use zero as the secret digit. The reason why is if zero was the secret digit the sum of the remaining digits will be 9 and then you would not know if the secret digit was zero or 9.

2. Sometimes their multiplication will result in a number with repeating digits. The first example had the digits 1, 0, 3, 3, 4 and 7. If your friend picked one of the 3's as the secret digit, they still need to tell you all the other digits in the number including the other 3. You may need to reinforce this by saying "now tell me all the other digits in the result, even if one them is the same as your secret digit. You don't need to tell me if there are any repeats with your secret digit."

Google sheets to complete this trick: 

https://docs.google.com/spreadsheets/d/1WZZBPXtMPWwyX6vrd-QuQvn684i3Q1mRcAvPRSGU6c4/edit?usp=sharing

Spirals of Prime Numbers

Stanislaw Ulam ID badge photo from Los Alamos

Public Domain, https://commons.wikimedia.org/w/index.php?curid=315600

Instructions

Begin with any number (limited to 0 and positive integers: 1, 2, 3, 4…) in the center of the spiral. Continue writing down successive numbers until several layers of the spiral have been built up. You generally need at least 50-60 numbers, but more is better for observing the pattern. Use the grids below to create your spirals.

Example 1

Starting number: 0

					4	3	2
					5	0	1
					6

 

Continue adding numbers to the spiral:

		64	63	62	61	60	59	58	57	56
		65	36	35	34	33	32	31	30	55
		66	37	16	15	14	13	12	29	54
		67	38	17	4	3	2	11	28	53
		68	39	18	5	0	1	10	27	52
		69	40	19	6	7	8	9	26	51
		70	41	20	21	22	23	24	25	50
		71	42	43	44	45	46	47	48	49
		72	73	74	75	76	77	78	79	80

 

Next, highlight or circle the prime numbers. For your reference the prime numbers under 200 are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199.

		64	63	62	61	60	59	58	57	56
		65	36	35	34	33	32	31	30	55
		66	37	16	15	14	13	12	29	54
		67	38	17	4	3	2	11	28	53
		68	39	18	5	0	1	10	27	52
		69	40	19	6	7	8	9	26	51
		70	41	20	21	22	23	24	25	50
		71	42	43	44	45	46	47	48	49
		72	73	74	75	76	77	78	79	80

 

A pattern appears:

Regardless of the starting number in the middle, many of the prime numbers – but not all – will form diagonal lines in the grid. In the example above, observe the diagonal that goes from near the upper left corner down to the lower right formed by the primes: 37, 17, 5, 7, 23, 47, 79. Also, the other major diagonal that includes the primes: 71, 41, 19, 5, 3, 13, 31.

Exercises:

Create your own spirals starting with any numbers. One particularly interesting example is the spiral that begins with 17. It has a diagonal 14 primes in a row.

For parents, teachers and advanced thinkers:

The formal name of the spiral pattern is called the Spiral of Ulam, named after the Polish-American mathematician, Stanislaw Ulam (1909-1984). Ulam worked on the Manhattan Project during World War II.

Background on Prime Numbers

A prime number is any number greater than 1 that only can be divided evenly by 1 or itself. The first two prime numbers are 2 and 3. Each can only be divided by 1 or itself. The number 4 is not prime because it can be divided evenly by 2 as well as by 1 and itself. That is, 1 x 4 = 4; 2 x 2 = 4; and 4 x 1 = 4. The next prime numbers are 5 and 7. The number 9 is not prime because 3 x 3 =9. It has been proven that the list of primes goes on without end. While there is no way to predict which numbers will be prime, there are some general rules which state which numbers cannot be prime. For example, even numbers greater than 2 cannot be prime because each even number greater than two will have 2 as a divisor. Also, numbers greater than 5 that end with a 5 or a 0 (such as 10, 15, 20, 25…) cannot be prime because they are all divisible by 5.

Ulam Spiral Google Sheets Link

Image of large Ulam Spiral (red dots are prime; blue: primes of form  )

(image credits: Will Orrick)

$$
{\displaystyle 4n^{2}-2n+41}

Please see this other post for another spiral of prime numbers.