Number System grade 10
Summary
TLDRThis educational video lesson focuses on teaching interval notation, set builder notation, and number lines, which are often challenging for students. The instructor uses a step-by-step approach to explain how to represent numbers greater than three and less than seven on a number line, employing open circles to denote exclusion of the endpoints. The lesson covers interval notation with parentheses to indicate non-inclusive boundaries and set builder notation using brackets to specify conditions for variables. The instructor also introduces the concept of real numbers and infinity, using creative analogies like a hungry crocodile to help students remember the direction of inequalities. The lesson aims to demystify these mathematical concepts through practice and repetition.
Takeaways
- π The lesson introduces interval notation, set builder notation, and number lines, which are often challenging for students.
- π’ It explains how to represent numbers greater than three and less than seven, including examples like 3.1, 4.2, 5.6, and 6.54.
- π The number line is used to visualize intervals, with open circles indicating numbers not included (e.g., 3 and 7).
- π Interval notation is introduced as a concise way to express ranges, using round brackets to denote non-inclusive endpoints.
- π¦ A mnemonic is used to remember the notation: a 'hungry crocodile' facing the larger number signifies 'greater than' or 'less than'.
- π Set builder notation is explained, using specific brackets and the letter 'x' to define the set of numbers that meet certain conditions.
- π The concept of real numbers is briefly touched upon, indicating that 'x e r' signifies 'x is an element of the real numbers'.
- π The lesson reiterates the importance of practice to understand and apply these mathematical notations effectively.
- π The instructor provides another example with numbers greater than or equal to 7 and less than 12, demonstrating how to adjust the notation accordingly.
- βΎοΈ The lesson concludes with an example of numbers greater than five, introducing the concept of infinity in interval notation.
Q & A
What are the three different ways to represent numbers between 3 and 7 in mathematics?
-The three different ways to represent numbers between 3 and 7 are using a number line with open circles at 3 and 7, interval notation (x e [3,7)), and set-builder notation (x β β | 3 < x < 7).
What does the open circle on a number line represent?
-An open circle on a number line indicates that the number at that point is not included in the set.
How do you represent numbers that are not included in the set using interval notation?
-In interval notation, numbers that are not included in the set are represented with round brackets, for example, (3,7).
What does 'x e' stand for in interval notation?
-In interval notation, 'x e' is an abbreviation for 'x is an element of', indicating that x belongs to the set described within the brackets.
What is set-builder notation and how is it used to describe numbers between 3 and 7?
-Set-builder notation is a way to describe a set of numbers using a variable, a condition, and a colon. For numbers between 3 and 7, it would be written as {x | 3 < x < 7}.
What does 'x β β' mean in set-builder notation?
-'x β β' in set-builder notation means that x is an element of the set of real numbers, indicating that x can be any real number that satisfies the given condition.
How do you represent a number that is bigger than 5 but not including 5 on a number line?
-On a number line, a number that is bigger than 5 but not including 5 would be represented with an open circle at 5 and a continuous arrow extending to the right.
What is the interval notation for numbers that are bigger than 5 but not including 5?
-The interval notation for numbers that are bigger than 5 but not including 5 is (5, β).
How do you represent numbers that are bigger than 5 using set-builder notation?
-In set-builder notation, numbers that are bigger than 5 would be represented as {x β β | x > 5}.
Why is it important to practice these different notations for representing numbers?
-Practicing different notations helps to solidify understanding of how to represent sets of numbers mathematically and can be useful in various mathematical contexts and problem-solving.
What is the significance of the direction of the inequality symbols in set-builder notation?
-The direction of the inequality symbols in set-builder notation indicates whether the boundary numbers are included or excluded. For example, '>' means the number is strictly greater, excluding the boundary, while 'β₯' would include the boundary number.
Outlines
π Introduction to Interval Notation
The instructor begins by introducing the concepts of interval notation, set builder notation, and number lines, which are often challenging for students. The lesson aims to demystify these concepts through practice. A specific interval is discussed, which includes numbers greater than three and less than seven, exemplified by numbers like 3.1, 4.2, 5.6, and 6.54. The instructor emphasizes that these do not have to be whole numbers and that the lesson will cover when to use these notations. Three methods are introduced to represent this interval: using a number line with open circles to indicate non-inclusion of the endpoints, interval notation with parentheses to denote the same, and set builder notation with a mix of brackets and conditions to describe the set of numbers.
π’ Exploring Number Lines and Notations
This section delves into the practical application of the three methods introduced earlier. The instructor uses a number line to visually represent numbers greater than or equal to 7 and less than 12, including examples of numbers within this range. The number line is marked with a solid dot at 7 (indicating inclusion) and an open dot at 12 (indicating exclusion). Interval notation is simplified with 'x e' to represent 'x is an element of,' using square brackets for inclusive and round brackets for exclusive intervals. Set builder notation is further explained with its specific bracket style and the inclusion of 'x e r' to specify that 'x' represents real numbers. The instructor uses a crocodile analogy to help remember the direction of the inequality symbols. The lesson concludes with an example of numbers greater than 5, using a continuous arrow on the number line to represent an infinite range, and infinity notation in interval and set builder notations.
Mindmap
Keywords
π‘Interval Notation
π‘Set Builder Notation
π‘Number Line
π‘Open Circle
π‘Closed Circle
π‘Infinity
π‘Real Numbers
π‘Element of (β)
π‘Greater Than
π‘Less Than
Highlights
Introduction to interval notation, set builder notation, and number lines.
Students often struggle with these concepts initially due to their complexity.
Explanation of representing numbers bigger than three and smaller than seven.
Demonstration of using a number line to represent intervals.
Use of open circles on a number line to indicate non-inclusive endpoints.
Introduction to interval notation with the format (a, b) to represent ranges.
Explanation of using 'xe' in interval notation to denote the variable x.
Clarification on the use of round brackets to indicate non-inclusive endpoints in interval notation.
Introduction to set builder notation and its format.
Explanation of using set builder notation to represent numbers within a range.
Discussion on the use of 'x e r' in set builder notation to denote real numbers.
Practical example of representing numbers bigger than 3 using set builder notation.
Creative analogy using a hungry crocodile to remember the direction of inequalities.
Representation of numbers bigger than and equal to 7 on a number line.
Use of a solid dot on a number line to indicate inclusive endpoints.
Interval notation for numbers bigger than and equal to 7 and smaller than 12.
Set builder notation for the same range, including the use of 'x e r'.
Explanation of representing numbers bigger than 5 using a continuous arrow on a number line.
Interval notation for numbers bigger than 5 using infinity as an endpoint.
Set builder notation for numbers bigger than 5, including the use of infinity.
Encouragement for students to practice these concepts to overcome initial confusion.
Transcripts
hello everyone welcome to this lesson so
in this lesson we're going to start
talking about
interval notation set builder notation
and number lines now students usually
really struggle with this in the
beginning because it is weird it's super
confusing
but let's practice it and you'll see
it's actually not that bad so let's say
we have the numbers
bigger than three and smaller than
seven so those would be numbers such as
three point one four point
two five six
and six point five four for example
it's any number bigger than three
and smaller than seven it doesn't only
have to be
four five and six we don't always have
to work with
these natural numbers sometimes we will
sometimes we won't but i'll show you
when to do what so for now we're just
going to
take any number between three and seven
okay so kevin this is not an english
class class so we don't say numbers
bigger than three and smaller than seven
we need to write this in a fancy
mathematical way yes so there are three
different ways that you are going to be
doing this
the first one let's use a number line
okay so there we have our number line
what we would then do is just put the
two main numbers so you'd go
3 and seven if you want you can put some
numbers in between
that's fine and now we want to go
between three and seven so we want to
have numbers between three and seven
but they must be bigger than three so
we'll put a little open circle like this
and smaller than 7 so we'll put an open
circle like that
and then we'll just connect these two
together i'm going to do that in a
different color rather
and so the numbers that fit between 3
and 7
will be any of these on this pink line
over here
at the end i have used an open circle
what that means is that the number 3 is
not
included and the number 7 is not include
next we can use something called
interval notation
and so this one's quite easy you just
say x
e i'll explain now and then we're gonna
use round brackets
and we're gonna go from three to seven
the xe is just a way of saying that
we're using the numbers
x okay um yeah i'm actually not going to
explain too much about that you don't
really need to know you just need to
know
to you just need to remember to say xe
okay it'll become
more familiar to you as we go on
i've used a round bracket over
here and i've used a round bracket over
here what that means is that the number
seven and the number three are not
included if those numbers were included
you would use a
square bracket like that the third one
is called set
boulder notation and so the way this one
works is it typically uses these
funny brackets first step is to say what
letter we're working with you're working
with
x you then put a line like this you then
say that x is all the numbers
bigger than three and smaller than
seven if you're still confused as to the
direction that these
things should face that's okay i've got
grade twelves who still don't know
that direction some people just really
struggle with that but i will do my best
to
practice that as much as we can and then
your last step is to put a semicolon
here
and to say that we are working with all
the real numbers
so we'll say x e r so for now if this
seems like
what seems very strange it is strange i
mean why do we have to do all this weird
stuff
why can't we just say that x must be
bigger than three and smaller than seven
but trust me as we do this a few times
you'll start to get the hang of it
okay the xcr i didn't explain that part
that's just
saying that we are busy with real
numbers real numbers
are all numbers that you can see such as
three comma one one
one one three comma one four five six
eight two 0.148 it's
all the numbers that you can imagine if
we only wanted to use
numbers like 1 2
3 then we're gonna have to change that r
to something else but i'm going to
practice that as we go on
i just want to quickly practice this
part that we did here
if i want to say that a number is bigger
than 3 then i say
x which is the number is bigger than
3. so this means that the x is bigger
than the three so there's many ways that
students use to try remember this
so one way i explain to people is
imagine that this mouth here is a
crocodile and that crocodile is hungry
now if you are a
hungry crocodile are you gonna eat
something that's really big or something
that's really small
well if you're very hungry you're gonna
eat something that's big
and so the crocodile is clearly trying
to eat the x which means that the
x is bigger than the three so x is
bigger than three
this one says that it's smaller than
seven
so x must be smaller than seven that
means the 7 is bigger
and that's why the crocodile's facing
the 7. but now when you've got something
like this
and something like this you can just put
them together like this
okay so let's do another one so here
we're going to use numbers that are
bigger than
and equal to 7
and then it's also smaller than 12. so
give me some examples that would be
things like
numbers that are bigger than and equal
to 7 so that means you can use 7 as well
because it also says equal to seven
we can use numbers like seven point one
one we can use
eight we can use nine nine point one
four six eight three two one
ten eleven but we can't use 12 because
it says smaller than 12 it doesn't say
smaller than
and equal to 12. but we can get very
close to 12 we can go 11.99999
if we wanted to so what we do for the
number line is we use a
seven so we'll have a seven and a twelve
and if you want you can put a few
numbers in between you don't have to
so now for the seven it's equal to seven
so we're gonna use a
colored in dot it's not going to be an
open dot for the 12 that's going to be
an
open dot because we can't actually reach
the 12 we only get
close to it we can then connect the line
and so x which is the numbers can be any
number in between here which is these
ones that we said and we can do a whole
lot more we can do
every single number that we can see in
between those two
numbers interval notation very easy you
just say
x e just remember the x e we use a
square bracket now because we including
the 7
and a round bracket for the 12 because
we're not including that
set border notation has the funny
brackets we say we're working with
x you'd put a line we then say x can be
anything bigger than
7 but it can also include 7. so we show
it with that line like that
we then say smaller than 12 and then we
say
x e r at the end so there's three
different things with this
there's this part there's
this part and then lastly
there's that part the last one for this
lesson will be the numbers bigger than
five so that would mean 5.001
5.26 a thousand
ten thousand um
one million because it's any number
bigger than five
and you can go very close to the five as
well but you just can't touch the five
so what we'll do is we'll show one
number on our number line
and because now we don't have two
numbers we're not going to include the
five
now how do we show that it can be any
number bigger than five or you just show
a continuous arrow
like that and what that means is that
you can go on and on and on and on and
on
in interval notation you'll say xe
always remember the xe
round bracket because we're not
including the five
now what number do we call this one up
here
well a number that is so big that it
just keeps going on and on and on we're
going to use
infinity and infinity is always a round
bracket
because you can't reach infinity for the
interval notation
which always has three components you
start off by saying that we're working
with
x you can now say that x must be
bigger than 5 you could also say the
other way around
you could also do it like that it
doesn't matter x is bigger than 5 you
don't have to say smaller than infinity
you can just say bigger than 5
and then you must say x is an element of
r
so notice once again we have three
different components we've got this part
we've got this part and that part over
there
for this part over here some students
they do like to do the following
they like to say that x must be bigger
than 5
but it must be smaller than infinity you
can add that
over there if you wanted to so guys
that's all for this lesson in the next
lesson i will practice this a few more
times because i know it
it is quite weird at first
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