Math Antics - Number Patterns - Free Educational videos for Students in K-12

Math Antics - Number Patterns - By Mathantics

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00:03	Uh huh . Hi , I'm rob . Welcome to
00:07	Math . Antics by now , you probably know that
00:09	math involves a lot of calculations using arithmetic , but
00:13	math is about more than just calculations . In fact
00:16	. One important type of math that sometimes gets overlooked
00:18	involves number patterns . Now when you hear the word
00:21	pattern you might think of a shirt , yep .
00:24	And I'll bet you wish you had some fine threads
00:27	like these , don't you ? Actually ? Actually I'm
00:29	good with this shirt . Thanks , I get it
00:32	. Not everyone can pull off a look this red
00:35	. That's true . Of course the reason you might
00:37	think of a shirt is because the word pattern often
00:40	describes repeating images or objects . Like if I show
00:43	you this pattern . Dog , cat bird , dog
00:46	, cat blank . What animal do you think should
00:48	fill in the blank to complete the pattern ? A
00:51	bunny ? Why would you think it was a bunny
00:55	? Well , because I like bunnies . Well it's
00:58	not a bunny , it's a bird . See how
01:00	the pattern repeats . Dog , Cat bird . Dog
01:03	, cat bird . Well mr whiskers and I prefer
01:06	the pattern . Dog cat bunny . Dog , cat
01:09	bunny . Anyway . Number patterns can be formed by
01:14	repeating numbers like this . 147147 . Notice how the
01:19	order of the pattern really matters . If you switch
01:22	any of the numbers , it becomes a different pattern
01:25	In math . When you have a set of numbers
01:27	or elements where the order matters , it's called a
01:29	sequence . For example , the sequence 123 is different
01:34	than the sequence 3-1 even though they each contain the
01:38	same set of numbers . And in math the words
01:41	set refers to a group of numbers or elements where
01:44	the order doesn't matter and where any duplicates are left
01:48	out . For example , if you had the sequence
01:51	123321 the set of numbers in that sequence is just
01:56	123 Even though each number occurred twice in the sequence
02:01	, both sets and sequences used the same notation in
02:04	math . Each number or element is separated by a
02:08	comma and the whole group is put inside curly braces
02:11	like this . Some sequences of numbers repeat like the
02:14	sequence 010101 But some don't repeat like the sequence 123456
02:22	But think about both of those sequences for a second
02:25	. Right now , each of them contains a limited
02:28	or finite number of elements they each have six but
02:31	each of these sequences could be continued forever if we
02:34	wanted to we could just keep repeating 0101 forever .
02:39	Or we could just keep counting 789 10 forever .
02:42	To in other words , sequences can be finite or
02:47	they can be infinite If a sequence or set is
02:50	finite , it means that you can say there are
02:52	a specific number of elements in it like six or
02:55	20 or a million . But when something is infinite
02:58	, it means that no matter how much time you
03:00	have , you could never finish counting how many elements
03:03	are in it . You can't give it a specific
03:05	number . So you just say it goes on forever
03:08	. Of course we can't actually write numbers forever on
03:11	a piece of paper . So we need to use
03:13	a special notation for infinite sets or infinite sequences .
03:17	You just put three dots at the end of the
03:18	list to show that it keeps on going forever .
03:21	Like this , the three dots are an abbreviation .
03:24	That means the sequence continues in the same way they
03:27	can be used in the middle of a sequence to
03:29	save writing . Like this means the sequence of all
03:32	county numbers from 1 to 100 but you can also
03:35	use them at the end of a sequence to show
03:36	that it goes on forever . So this sequence is
03:40	repeating and finite because it has just six elements .
03:44	This sequence is non repeating and finite because it also
03:48	has just six elements . This sequence is repeating an
03:51	infinite and this sequence is non repeating . An infinite
03:55	. Makes sense for these last two infinite sequences .
03:59	What's the set of numbers that each contains ? Well
04:02	, the first keeps on repeating two numbers forever .
04:05	So even though the sequence is infinite , the set
04:07	it uses is finite because it only contains zero and
04:10	one . But in the second infinite sequence none of
04:13	the elements are ever repeated . So the set of
04:16	numbers is exactly the same as the sequence itself .
04:19	It's also infinite . Okay , so now you know
04:22	that some number patterns are repeating and some aren't .
04:25	You also know that some number patterns are finite and
04:28	some are infinite . We got our first non repeating
04:31	infinite sequence simply by counting . Let's see if we
04:34	can think of some others that way too . Suppose
04:37	you start counting at the number one but then skip
04:39	every other number . You'd end up with a sequence
04:42	13579 and so on . In other words you'd end
04:46	up with the infinite non repeating sequence that we call
04:49	odd numbers because none divide evenly by two . Or
04:53	suppose you start counting at the number two instead but
04:55	still skip every other number you'd end up with the
04:58	sequence 2468 10 and so on . That's the infinite
05:02	non repeating sequence of numbers . We call even numbers
05:05	because all divide evenly by two . And you can
05:08	make other sequences by skip counting by different amounts .
05:11	Like you could start with zero and skip every two
05:14	numbers to get the sequence 0369 12 and so on
05:19	. If you think about it counting and skip counting
05:22	are really just ways of making a number sequence by
05:24	following a rule . In the case of regular counting
05:27	, that rule happens to be add one to get
05:30	each new number in the sequence and when you skip
05:32	count every other number , the rule you're following is
05:35	added to each time . You can see that by
05:38	looking at the sequence we called odd numbers . You
05:40	could get from the first element to the second by
05:42	adding to one plus two equals three and you can
05:46	get from the fourth element to the fifth by adding
05:48	to seven plus two equals nine . In other words
05:51	, if you know the rule that a particular sequence
05:54	is based on , you can use it to find
05:55	any other number in the sequence . If you want
05:58	to know what number comes next in the sequence of
06:00	odd numbers , just add to to the last element
06:02	, you know , like 11 plus two equals 13
06:06	. All four arithmetic operations can be used as rules
06:09	for generating sequences . You've already seen how addition rules
06:12	produce sequences that count up or increase . But what
06:15	do you think you'd get if you base to sequence
06:18	on a subtraction rule , like subtract one , yep
06:21	, you get a sequence that counts down or decreases
06:24	five , four , 321 Lift off the rule for
06:32	this simple countdown sequence is to start with five and
06:35	then subtract one each time . Oh and some of
06:39	you who are familiar with negative numbers will realize that
06:41	this countdown sequence really doesn't have to stop at zero
06:44	. It could continue on forever in the negative direction
06:47	. But we're just going to focus on positive numbers
06:49	in this video . Another example of a subtraction sequence
06:53	is to start with 50 and use the rule subtract
06:55	five . In that case you get 50 , 45
06:59	, 40 35 , 30 and so on . Each
07:02	element in the sequence is five less than the one
07:04	before it . So it's pretty easy to see how
07:07	addition and subtraction can be the rule for a sequence
07:10	. But what about multiplication and division ? What number
07:13	sequence would you get from the rule ? Multiplied by
07:16	two ? Well , if we start with one as
07:18	the first element , the next would be one times
07:21	two which is to the next would be two times
07:24	two which is four and the next would be four
07:26	times two which is eight , Then the next would
07:28	be eight times two which is 16 and so on
07:32	. Notice that the numbers in this sequence are getting
07:34	big pretty fast . That's one of the clues that
07:37	a sequence might be based on a multiplication rule .
07:40	When you keep multiplying a previous result by the same
07:43	factor . The values can grow much faster than if
07:46	you just added a fixed amount each time . You'll
07:49	see that if we compare the sequence we just made
07:52	by multiplying by two each time with the sequence we
07:54	previously made by adding to each time , Even though
07:57	both sequences started the same number when we added to
08:00	each time we got up to 13 by the 7th
08:03	element . But when we multiplied by two each time
08:05	we got up to 64 by the seventh element .
08:08	That's quite a difference . And it works in a
08:11	similar way with division . Suppose you're asked to make
08:13	a sequence by starting with 40 and then dividing by
08:16	two . Each time . The first number is 40
08:19	. The next is 40 divided by two which is
08:22	20 . The next is 20 divided by two which
08:24	is 10 . The next is 10 divided by two
08:26	which is five . The next is five divided by
08:28	two which is 2.5 . And we could keep on
08:31	going divided by two forever to get smaller and smaller
08:34	fractions but We'll stop there so we can compare that
08:38	to the sequence . You'd get if you start with
08:39	40 but subtract two each time . In that case
08:43	you get 40 than 38 and 36 and 34 and
08:46	32 . And so on . Notice how the sequence
08:49	that's based on the division rule gets smaller much faster
08:52	than the sequence . That's based on the subtraction rule
08:55	. Just like the sequence that's based on multiplication .
08:57	Got bigger . Much faster than the one based on
09:00	addition . That's because when you keep adding or subtracting
09:03	the same amount , the sequence changes by a constant
09:06	amount . Each step just like going up or down
09:09	a normal flight of stairs . But if you multiply
09:12	or divide each time , the sequence changes by an
09:14	increasing or decreasing amount each step , that would be
09:17	a tough set of stairs to climb . In fact
09:20	, there is such a big difference in the way
09:22	these types of sequences increase or decrease , that mathematicians
09:26	to have different names for them , sequences that are
09:29	based on addition or subtraction rules are called arithmetic sequences
09:33	. While sequences that are based on multiplication or division
09:35	rules are called geometric sequences . Those maybe aren't the
09:39	most intuitive names . But since they've been used for
09:42	so long , it's important to know what people mean
09:44	when they say them . Okay by now , you've
09:47	probably realized that there are lots of different kinds of
09:50	number sequences and patterns in math . Far too many
09:53	to cover in just one video . So instead of
09:56	trying to do that , we're going to end this
09:57	video with some tips that you can use to figure
09:59	out . If a sequence is based on a simple
10:01	rule involving addition , subtraction , multiplication or division .
10:05	When you're given a sequence first , try to determine
10:08	if it's repeating or non repeating . For example ,
10:12	in this sequence , you can see that part of
10:13	the sequence keeps repeating . That means that you need
10:16	to use the pattern to fill in any missing elements
10:18	instead of a rule . But if the sequence isn't
10:21	repeating like this one , the next thing you'd want
10:23	to check is that the sequence is increasing or decreasing
10:26	? Not all sequences increase or decrease , but increasing
10:30	sequences could be based on an addition or multiplication rule
10:33	while decreasing sequences could be based on a subtraction or
10:36	division rule . This sequence is increasing since each new
10:39	element is bigger than the one before it . But
10:42	how can we tell if it's based on an addition
10:44	or multiplication rule ? To do that , we need
10:48	to look for either a common difference or a common
10:50	ratio in the sequence . Here's what that means .
10:53	Start by picking any two adjacent numbers in the sequence
10:57	and find the difference between them by subtracting . For
11:00	example , the difference between four and 8 is four
11:04	, Then pick any other two adjacent numbers and do
11:06	the same thing . I'm going to pick the last
11:08	2 , 20-16 is also for Are the differences the
11:13	same in this case ? Yes , that means that
11:15	we have found what's called a common difference for the
11:18	sequence . The common difference is a constant amount that's
11:22	either added or subtracted to each new element since the
11:26	common difference here is four and the sequence is increasing
11:29	, that means the rule for this sequence is probably
11:32	add for you can check to make sure all the
11:35	other elements are following that rule just to be sure
11:38	. But what if we don't find a common difference
11:40	for a sequence like this one ? If we take
11:43	the first two elements and subtract them , we get
11:45	four . But if we take the next two elements
11:47	and subtract them we get 12 . That means that
11:50	there's not a common difference for this sequence . So
11:52	it's not based on a simple addition or subtraction rule
11:55	but maybe we can find a common ratio instead .
11:58	Let's see what that means . To find a common
12:01	ratio . We also take two adjacent pairs of elements
12:05	. But instead of subtracting them , we divide them
12:08	. For example , if we take the first two
12:10	elements and divide them like this , six divided by
12:12	two , we get three and if we take the
12:14	next two adjacent elements and divide them like this ,
12:17	18 divided by six . We also get three .
12:20	Ha ha ! That's what we call a common ratio
12:23	. And it means that this sequence is likely based
12:26	on either a simple multiplication or division rule . Since
12:30	this is an increasing sequence , we know that the
12:32	rule is probably multiply by three . Again , you
12:36	can double check that on other pairs . So even
12:39	though not all sequences are based on simple arithmetic rules
12:42	, checking for a common difference or a common ratio
12:45	can help you identify the ones that are all right
12:48	. So now , you know a little bit about
12:49	number sequences . You know the difference between a sequence
12:52	in a set . You know that some sequences repeat
12:55	while others don't . You know that some sequences are
12:58	finite while others are infinite . And you know that
13:01	sequences can be based on arithmetic rules . If the
13:05	sequences rule involves adding or subtracting a constant amount each
13:09	time , that means you've got an arithmetic sequence and
13:12	you'll be able to figure out that constant or common
13:14	difference by subtracting pairs of adjacent numbers . But if
13:18	the sequences rule involves multiplying or dividing by the same
13:22	factor each time , that means that you've got a
13:24	geometric sequence and you'll be able to identify its common
13:27	ratio by dividing pairs of adjacent numbers . We covered
13:31	a lot in this video so be sure to re
13:33	watch it later if it didn't all sink in the
13:34	first time and remember . The best way to get
13:37	good at Math is to practice what you've learned as
13:40	always . Thanks for watching Math Antics . And I'll
13:42	see you next time . Learn more at math Antics
13:46	dot com . Yeah . Mr Whisker says to like
13:50	and subscribe

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