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

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


Math Antics - Number Patterns - By Mathantics



Transcript
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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