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Maths Performance Task (Inverse/Direct Proportion)
Transcript of Maths Performance Task (Inverse/Direct Proportion)
The more ingredients you have, the more curry you would make How do you know whether two quantities are in direct proportions? Direct proportion is when a change in one quantity causes a change, or is linked to a change, in another quantity through equal factors. Determining direct proportion through tables A common misconception is that two variables are directly proportional if one increases as the other increases. Two variables are said to be directly proportional only if, their ratio is a constant for all values of each variable. Thus when one variable is divided by the other, the answer is always a constant. Determining direct proporion through graphs However, in this example, y increases as x increases. Importantly, their ratio is the same for all values so they are directly proportional. In this example, y does increase as x increases. In fact, y increases uniformly with x. However, the ratio of y to x is not the same for all values so this relationship is not of that direct proporion Direct proportion graph are straight line graphs that goes through the point of origin When A changes by some factor, B also change by the same factor, thus showing that this straight line through the origin represents a direct proportion. Determining direct proportion through equations For example, the cost of a pearl, $C is directly proportional to the square of its mass, m grams. If a pearl of mass 10g costs $240, find an equation connecting C and m. How do you apply the properties of direct proportion to solve problems? Give examples to show them. For example, suppose that you are buying pearl milk tea at Sweettalk. Let us imagine that they cost $2.00 each.
Suppose that you buy 4 cups, you would pay $8.00.
Suppose that you buy 8 cups, you would pay $16.00.
So, changing the number of cans that you buy will change the amount of money that you pay.
Notice that the number of cups changed by a factor of 2, since 4 cups times 2 is 8 cups.
Also, notice that the amount of money that you must pay also changed by a factor of 2, since $8.00 times 2 is $16.00.
Both the number of cans and the cost changed by the same factor, 2 Y varies directly as x means that y=kx where k is the constant of variation
Another way to write is k=y/x
X is the independant variable, Y is the dependant variable and K is the constant of proportionality What is inverse Proportion? When quantities are related by inverse proportion,they change by reciprocal factors. When two quantities A and B are in inverse proportion if by whatever factor A changes ,B changes by the multiplicative inverse or reciprocal of that factor. What are the difference between direct proportion and indirect proportion? Direct proportion means if you double one quantity, you double the other quantity. However, inverse quantity if you increase one quantity,you decrease the other quantity.
What are some real life application or examples of inverse proportion? Some examaples are:
When the temperature of the country rise, the sale of sweaters would go down.
The more the distance traveled, the lesser the remaining distance is.
The further you move from the Earth, the less the Gravity
How do you know if the two quantities are in inverse proportion?
The two quantities are inversely proportionate when one is multiplied by any number,the other is divided by the same number
Or when the one is divided by any number the other is multiplied by the same number
Determining inverse proportion through i)table A common example of quantities that are inversely proportional occurs with rectangles of the same area. Remember that the area of a rectangle can be found by multiplying the lengths of the sides.
Consider rectangles with areas of 36 square metres.
If you know the length of one of the sides, A, you can determine the length of the other side, B. For example, if A = 12m, B = 3m. Similarly, if A = 8m, B = 4.5m. Through this table, we can tell that, as the more the length of side A, the lesser the length there is for side B and the less the length of side B, the more the length of side B will be. Hence, the lengths of the sides of the rectangles are changing through inverse proportion. Determining inverse proportion through ii)equations. y= k/x
when y increases, x decreases and k is the non zero constant Determining inverse proportion through iii)graphs. How do you apply the properties of inverse proportion to solve problems? Give examples to show. A farmer has enough cattle feed to feed 64 cows for 2 days.
a) How long will the same food last 32 cows?
64 ÷ 32 = 2
2 x 2 =4 days
b) Complete the table below for the same amount of cattle feed: We can now see what inverse proportion is about.
From the cattle feed problem, we can tell that:
If you double the number of cows, the number of days you can feed them is halved.
If you treble the number of cows, you can feed them for a third of the number of days.
If you have ten times the number of cows, you can only feed them for a tenth of the number of days.
This graph is a typical inverse proportion graph:
It shows us that as the number of cows increases, the number of days decreases.
Obviously, the reverse is also true that if we decrease the number of cows, we will increase the number of days feed. The graph of y against x is part of a curve called a hyperbola The graph of y against 1/x is part of a straight line which passes through the origin
As the “x” value increases, the “y” value always decreases
As the “x” value decreases, the “y” value always increases