GCSE Maths - How to Write Algebraic Expressions for Directly Proportional Relationships #90

00:05:42
https://www.youtube.com/watch?v=eGKT1EFseJc

摘要

TLDRThis educational video discusses how to write equations to express relationships involving direct proportionality between variables. Explanation is given through examples where two variables, such as hours worked and money earned, or time taken and mass, are directly proportional. The video guides on formulating equations by explaining steps to determine the constant of proportionality and using this constant to solve for unknown values. Specific examples include calculating the time taken to boil water based on mass and determining wire diameter based on length, emphasizing understanding and application of proportional equations without graphs.

心得

  • 🔗 Direct proportionality is a key concept in relating two variables.
  • ✏️ Learn to write equations such as e = k * h for proportional relationships.
  • 🔍 Identify constants of proportionality using given values.
  • 📝 Algebraic expressions describe the relationship between variables.
  • 🧮 Practice calculating unknown values using proportionality equations.
  • 📊 Equations replace graphs in solving proportionality problems.
  • 🕒 Example calculation shows boiling time based on water mass.
  • 📏 Length and diameter example demonstrates practical use.
  • ➗ Use division for finding constants in proportional equations.
  • 🧑‍🏫 Effective learning involves applying mathematical concepts in examples.

时间轴

  • 00:00:00 - 00:05:42

    The video begins by explaining that if two variables, such as the number of hours worked and money earned, are directly proportional, then an increase in one results in a proportional increase in the other. To illustrate this, the video uses an example where money earned (e) is directly proportional to hours worked (h), described by the equation e = 12h, meaning one earns 12 pounds per hour. If one works for three hours, they earn 36 pounds.

思维导图

视频问答

  • What is direct proportionality?

    Direct proportionality means that as one variable increases, the other increases at a constant rate.

  • How is an equation for direct proportionality written?

    It is written as one variable equals a constant multiplied by another variable (e.g., e = k * h).

  • What is a constant of proportionality?

    It's a constant value that relates two variables that are directly proportional.

  • How do you find the constant of proportionality?

    By dividing the given value of one variable by the given value of the other variable.

  • How can you use the constant of proportionality to solve problems?

    Once the constant is known, you can use it in the equation to find other variable values.

  • Can you provide an example of a direct proportionality equation?

    Yes, if time is directly proportional to mass, the equation is t = k * m.

  • How is the equation t = k * m solved for specific values?

    Substitute the known values of t and m to find k, then use k to find new values for t or m.

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自动滚动:
  • 00:00:00
    [Music]
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    we saw in our last video that if two
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    variables like the number of hours
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    worked and money earned are directly
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    proportional to each other
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    then as one of them increases
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    the other one increases proportionately
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    so in this case the more hours you work
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    the more money you'll learn
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    in this video though we're going to
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    cover how to write an equation to
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    express this sort of relationship
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    so for this example if we were to say
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    that e is the money earned
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    and h is equal to the hours worked
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    then the equation that describes the
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    relationship between money earned and
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    hours worked would be e equals 12 h
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    so if you worked for three hours
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    e would equal 12 times three which is 36
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    so you'd have earned 36 pounds for those
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    three hours
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    in exams though you'll often have to
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    figure these equations out for yourself
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    without using a graph
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    so to see how we do that let's jump
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    straight in with a typical question
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    so the first thing we're told here is
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    that the time in seconds that it takes
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    to boil some water
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    is directly proportional to the mass of
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    the water measured in grams that in the
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    kettle
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    so basically the time taken which we
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    represent with the letter t is directly
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    proportional to the mass of water that
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    we have which we represent with the
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    letter m
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    or if we wanted to put it in algebra
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    form we could say that t is proportional
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    to m
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    next we're told that when t equals 600
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    m equals 200
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    or in other words it takes 600 seconds
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    to boil 200 grams of water
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    and finally the question is asking us to
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    find t when m equals 450
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    so to find the time it would take to
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    boil 450 grams of water
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    now the idea with this type of question
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    is to first of all use the values that
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    we're given to help us write an
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    expression to describe the relationship
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    in this case the relationship between
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    time and mass
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    and then once we have that we can plug
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    in something like the mass of 450 to
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    help us find out the corresponding value
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    of t
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    so if we go back to our expression so
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    far
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    which says that t is proportional to m
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    the first thing we have to do is change
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    the proportional sign to an equal sign
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    but when we do that we also have to add
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    a constant of proportionality
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    which would normally represent like this
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    using the letter k
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    this constant has an actual value though
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    like 0.2 3 or 50.
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    and to find it we need to rewrite the
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    equation using these values from the
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    question
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    which in this case would give us the
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    time of 600 equals k times the mass of
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    200
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    so then we can divide both sides by 200
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    to find that 3 equals k
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    and now that we know that k is 3 we can
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    rewrite our equation as t equals 3 m
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    so we now have an algebraic equation
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    that links time and mass
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    which we can use to convert between the
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    two of them
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    so to find the time it would take to
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    heat 450 grams of water leica asks in
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    the question all we have to do is plug
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    the 450 into our equation
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    so yet t equals three times 450
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    or t equals 1 350.
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    so it would take 1 350 seconds to heat
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    up 450 grams of water
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    let's have a go at one more
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    the length of a piece of wire is
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    directly proportional to its diameter
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    a piece of wire that's 25 centimeters
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    long has a diameter of 2 centimeters
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    write an expression for the relationship
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    between length and diameter and
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    calculate the diameter of a piece of
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    wire that's 40 centimeters long
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    so from this first part we know that
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    length is directly proportional to a
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    diameter
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    so l is proportional to d
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    then if we add our constant of
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    proportionality we get l equals kd
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    and now we can use this second sentence
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    to figure out what k is
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    so if we plug in our length of 25 and
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    our diameter of 2
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    we get 25 equals k times 2
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    and then we can divide both sides by 2
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    to find that 12.5 equals k
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    which means that our equation that
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    length length and diameter is l equals
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    12.5 d
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    then moving on to this last part of the
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    question
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    we can now use our equation to find the
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    diameter of a 40 centimeter piece of
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    wire
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    so we plug in 40 in the place of l to
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    get 40 equals 12.5 d
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    and then divide both sides by 12.5 to
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    find that 3.2 equals d
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    so the 40 centimeter wire would have a
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    diameter of 3.2 centimeters
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    anyway that's the end of this video so
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    hope you found it useful
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    and we'll see you again soon
  • 00:05:42
    you
标签
  • Direct Proportionality
  • Equation Writing
  • Mathematics
  • Constant of Proportionality
  • Examples
  • Calculation
  • Teaching
  • Learning
  • Variables
  • Problem Solving