Introduction to Slope
Summary
TLDRIn this 'Anywhere Math' video, Jeff Jacobson explains the concept of slope, which measures the steepness of a line. He uses skiing as an analogy, comparing different ski slopes to illustrate the idea. The video demonstrates how to calculate slope as the ratio of the change in the y-axis (rise) to the change in the x-axis (run) using any two points on a line. Jacobson walks viewers through examples, including finding the slope of a line given two points and graphing a line from points to determine its slope. He emphasizes that slope calculation is consistent regardless of the order of points used, and simplifies fractions when necessary, concluding with additional practice problems for viewers.
Takeaways
- đ The video is an educational tutorial on finding the slope of a line using two points.
- đ Slope is a measure of the steepness of a line, similar to how ski slopes have varying degrees of steepness.
- đ Slope is a ratio that compares the change in the Y-axis (rise) to the change in the X-axis (run).
- đ The formula for slope is expressed as the change in Y over the change in X, or rise over run.
- đ The slope can be determined using any two points on a line, and it's important to choose points that are easy to work with.
- đ The direction of the points used to calculate slope does not affect the result; the slope remains the same regardless of the order.
- đ In the first example, the slope of a line passing through points (0,0) and (3,4) is calculated to be 4/3.
- đą The second example involves simplifying the slope when it's not in its simplest form, such as reducing 3/6 to 1/2.
- đ The video includes an exercise where viewers are guided to graph a line through points (-3,5) and (4,-6) and find its slope.
- ⏠The slope of the line in the exercise is negative, indicating the line goes down as it moves from left to right.
- đŻ The final slope calculation in the exercise results in -1/7, confirming the negative slope expectation.
Q & A
What is the main topic of the video?
-The main topic of the video is how to find the slope of a line using any two points on that line.
What is the definition of slope according to the video?
-Slope is a way to measure the steepness of a line, defined as the ratio comparing the change in the Y (rise) to the change in the X (run) between any two points on a line.
What is the formula for calculating the slope of a line?
-The formula for calculating the slope of a line is the change in Y over the change in X, or rise over run.
How does the video illustrate the concept of slope using skiing?
-The video uses skiing as an analogy to explain slope, stating that different ski slopes have different steepnesses, similar to how lines can have different slopes.
Why are whole numbers chosen for points when calculating slope in the video?
-Whole numbers are chosen for points when calculating slope because they are easier to work with and simplify calculations.
What happens to the slope if you reverse the order of the points used for calculation?
-The slope remains the same regardless of the order of the points used for calculation, as long as the changes in Y and X are correctly identified.
How does the video demonstrate the process of finding the slope of a line?
-The video demonstrates the process by showing a line with two labeled points, calculating the change in Y and X, and then forming the slope as a fraction of these changes.
What is the slope of the line that passes through the points (0,0) and (3,4) according to the video?
-The slope of the line that passes through the points (0,0) and (3,4) is 4/3.
What is the significance of simplifying the fraction when finding the slope?
-Simplifying the fraction when finding the slope is important to express the slope in its simplest form, which makes it easier to interpret and compare with other slopes.
How does the video explain the direction of the slope in relation to the line's direction?
-The video explains that if a line goes down from left to right, the slope is negative, and if it goes up from left to right, the slope is positive. A horizontal line has a slope of zero, and a vertical line has an undefined slope.
What is the slope of the line that passes through the points (-3,5) and (4,-6) according to the video?
-The slope of the line that passes through the points (-3,5) and (4,-6) is -1/7 after simplification.
Outlines
đ Introduction to Finding Line Slope
In this educational video, Jeff Jacobson from 'Anywhere Math' introduces the concept of slope, which is a measure of the steepness of a line. He explains that slope is a ratio comparing the change in the y-coordinate (rise) to the change in the x-coordinate (run) between any two points on a line. Jeff emphasizes the importance of selecting points that are easy to work with, such as whole numbers, to simplify calculations. He demonstrates the process of finding the slope using two points on a line, showing that the slope remains consistent regardless of the direction in which the points are considered. The example given illustrates calculating the slope as 4/3 for a line passing through the points (0,0) and (3,4), highlighting the concept of 'rise over run'.
đ Graphing and Calculating Line Slope
The second part of the video focuses on a practical application of the slope concept. Jeff provides a step-by-step guide on how to graph a line given two points and then calculate its slope. He uses the points (-3,5) and (4,-6) as an example, plotting them on a graph and drawing the line that passes through both. After graphing, he discusses the expected direction of the slope based on the line's orientation, correctly anticipating a negative slope due to the line's downward trend from left to right. Jeff then calculates the slope by determining the change in y (-11) and the change in x (7), resulting in a slope of -11/7. The summary concludes with an invitation for viewers to try similar exercises on their own and an encouragement to subscribe for more educational content.
Mindmap
Keywords
đĄSlope
đĄSteepness
đĄRatio
đĄRise
đĄRun
đĄCoordinates
đĄGraph
đĄSimplification
đĄPositive Slope
đĄNegative Slope
đĄUndefined Slope
Highlights
Introduction to the concept of slope as a measure of the steepness of a line.
Slope is defined as the ratio comparing the change in Y (rise) to the change in X (run) between any two points on a line.
The importance of choosing points on a line that are easy to work with for calculating slope.
Example calculation of slope using the points (0,0) and (3,4), resulting in a slope of 4/3.
Explanation that the direction of point selection does not affect the calculated slope value.
Demonstration of calculating slope with points (-2,-2) and (4,1), simplifying the fraction to find a slope of 1/2.
The necessity of simplifying the fraction when calculating slope to ensure accuracy.
Guidance on graphing a line given two points and the subsequent calculation of the line's slope.
The method for determining whether to expect a positive or negative slope based on the line's direction.
Example of calculating the slope of a line passing through points (-3,5) and (4,-6), resulting in a negative slope.
The concept that the slope remains the same regardless of the direction of calculation between two points.
Illustration of the process for finding the slope by counting changes in Y and X coordinates.
The significance of understanding the vertical and horizontal changes in coordinates when calculating slope.
Final calculation of the slope as 1/7 for the line passing through the given points.
Encouragement for viewers to try calculating slopes on their own with provided examples.
Conclusion and appreciation for watching the video on finding the slope of a line.
Transcripts
welcome to anywhere math I'm Jeff
Jacobson and today we're going to talk
about how to find the slope of a line
using any two points on that line let's
get
[Music]
started oh before we do an example what
exactly is slope well slope is just a
way to measure the steepness of a line
and if you uh have ever skied or
snowboard snowboarded you would know uh
ski slopes and different ones are have
different steepnesses some are really
Steep and and very difficult and some
are a lot flatter and a lot easier so
that's that's what slope is it's just
how steep a line is uh it's a ratio and
it's a ratio uh comparing the change in
the Y
which we call the rise to the change in
the X which we call the Run uh between
any two points on a line we can find the
slope with any two points on a line um
so here we go it's a ratio so we're
going to write it as a fraction so slope
again is the change in the Y over the
change in the x or you can think of it
as the rise over the run so that's what
slope is now let's do an example to
figure out how to actually find the
slope example number one find the slope
of each line so for a uh with this graph
here is our line in green we've got two
points labeled 0 0 and 34 uh and again
like I said before if we have two points
on a line we can find the slope all we
need is two points and typically uh
you're going to want to pick points that
are easy to work with so if you look at
the line
carefully I could have picked points
anywhere along here but some of these
like if I looked here well that looks
like it would be two and then maybe
2.8 right that decimal is not going to
be very nice to work with so there's a
reason we chose uh 34 and 0 0 because
they're both uh the X and the Y
coordinates are both whole numbers so
that's nice and easy to work with so
slope change in the y or the rise over
change in X which we call the Run well
uh from this point to this point what
was the change in the Y values well we
went up four okay so that's going to go
in our numerator and from 0 0 to 34 what
was our change in X well we went over to
the right three uh and that's a positive
three when we move to the right just
like when we go up that's positive4 so
in this situation our slope then is 4/3
4
over3 okay uh now you might be asking
well Mr Jacobson what happens if I go
the other way what happens if I want to
go from 3 4 down to 0 0 would the slope
be the same and yes it would and the
reason is because if I go from uh 34 to
0 0 my change in y is actually ne4
because I would be going down
four and then I would have to go over to
the left through that would be my change
in X so that would be -3 for my change
in X so that would look
like I'll get rid of that you would have
negative four because we're going down
four and you would have -3 uh for your
changeing X because we're going to the
left three well if you look I got
negative divided by a negative which
would simplify to 4/3 which was a same
as what we had before so the nice thing
with slope it doesn't matter if you go
from this point to this point or the
other way around this point to this
point uh you should get the same answer
if you're doing it correctly let's look
at B so again we're going uh we have our
our line here uh that passes through -2
-2 and passes through
41 so slope change in the Y the vertical
change over the change in the X the
horizontal change well it's very simple
we've got the arrow here the vertical
change is three over the horizontal
change which is six so that's going to
be 3 over 6 now notice that's not in
simplif simplest form so what we need to
do is simplify that first and that would
give us 1 12 so our slope here would
be2 I'll get rid of that and again our
slope over here would be a positive 4/3
here's some to try on your
own all right here's example number two
it says graph the line that passes
through the following points then find
the slope of the line so our two points
are -35 and 4 -6 so first let's graph
those points so first thing I'm going to
do is graph the line I've got my snazzy
little um line graph maker thing here uh
so first my first point is -35 so I'm
going to take this point and I'm going
to put it
AT3 and then up five so -35 is right
there my next point 4 -6 so again I go
over four first and then down
-6 so 4 -6 is going to be right there so
those are my two points now I am ready
to draw my line I'm going to choose
those two points and there we go there
is my line now let's try to find the
slope now that we have graphed our line
uh now it's time to find the slope so uh
again like I said before it does not
matter if I go from this point to this
point or vice versa this point to this
point uh I just need to be consistent um
with going with my change in y and my
change in X now before I even do that
let's use a little bit of logic looking
at this line would I expect my slope to
be positive or negative well if you look
this line is going down as we go from
left to right um which means it should
be a negative slope if the line is going
this way as we go from left to right
it's going up that would be a positive
slope a horizontal line zero slope and a
vertical line is a slope that's
undefined um but let's get back to this
problem at hand so we're expecting a
negative slope so let's keep that in
mind uh so first well what is my change
in y well I'm going to go from this
point here uh
-35 down to 46 so my change in y is
right here
okay so let's count that well it goes
from five down to zero here so that's
five and then we go from zero down six
more so neg5 and6 that gives
me1 okay and if you want maybe kind of
the slow way would be just to count all
the squares you could do that but but
really think uh you're going from five
all the way down to six okay and also
you can also look at that over here with
our original points I'm going from five
in my y-coordinate down to -6 uh in this
y-coordinate here and that is a change
of1 uh so next let's do my change in in
X my horizontal change well that's going
to be uh right
there and again you could count it uh
but I can also think well I was at -3
on my x coordinate here now I am all the
way up to four so from3 to zero that's a
change of positive3 and then from zero
four more that's going to be positive s
so from -3 up to four that's a change of
seven and because we're going uh towards
the right that's positive 7 so now we're
ready to write the slope slope again is
the vertical change the Rise um the
change in y values over the change in
the X so my slope is
now
117 okay here's some more to try on your
own thank you so much for watching and
as always if you like this video please
subscribe
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