Year 8: Changing the Subject
description
Transcript of Year 8: Changing the Subject
![Page 1: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/1.jpg)
Year 8: Changing the Subject
Dr J Frost ([email protected])www.drfrostmaths.com
Last modified: 8th February 2016
![Page 2: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/2.jpg)
π₯=10
Starter
Solve the following equations:
2π₯+73 =9 ?
3 π₯2+5=152 π₯=7?
S1
S2
S3
5 π₯β2=3 π₯+3 π₯=52?
![Page 3: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/3.jpg)
Solving Linear EquationsImagine the stuck inside a prison β we gradually have to βundoβ the things around it before it can be released. Undo the last thing done to on each step by doing the opposite.
β2 π₯β5=3Click We canβt add 5 yet because
itβs βtrappedβ inside .We βsquareβ to undo the .
2 π₯β5=9Click
2 π₯=14Click
π₯=7
![Page 4: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/4.jpg)
Test Your Understanding
1 Solve Solve 2
Solve 3
? ?
?
![Page 5: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/5.jpg)
Changing the Subject of a Formula
The formula to calculate a temperature in Fahrenheit if we have the temperature in Celsius:
But what if we had say the temperature in Fahrenheit, and wanted to know it in Celsius?
πΆ=59
(πΉβ32 )The subject of the formula is the variable the appears on its own on one side of the equation (usually the left) and not on the other side.
![Page 6: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/6.jpg)
Skill #1: βUndoingβ to UnlockMake the subject of the formula. Undo the last thing done to the subject each time
by doing the opposite.
π¦=π₯β2 π₯=π¦+2
π¦=3 π₯+2 π₯=π¦β23
π¦=βπ₯+1 π₯=(π¦β1 )2
π¦=π₯2βπ4
π₯=Β±β4 π¦+π
?
?
?
?
Bro Tip: It doesnβt matter what side the subject is on, provided itβs on its own!
![Page 7: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/7.jpg)
Exercise 1In each case make the subject of the formula. (Please copy out question first)
1
2
3
4
5
6
7
8
9
10
11
12
13
N
N
??
?????
?
?
?
?
?
?
?
?
?
14
![Page 8: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/8.jpg)
RED ORANGE GREEN BLUE
Vote with your diaries
Test your understanding so farβ¦
![Page 9: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/9.jpg)
π₯=π¦+32π₯=
π¦+32 π₯=2 π¦β3π₯=
π¦β32
π¦=2 π₯β3Make the subject.
![Page 10: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/10.jpg)
π₯=π¦2π
π₯=( π¦π )2
π₯=βπ¦π π₯=(ππ¦ )2
π¦=πβπ₯Make the subject.
![Page 11: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/11.jpg)
π=Β±βπ βππ π=Β±β ππ βππ=Β±β πβππ π=Β±β ππ βππ¦=ππ₯2+1Make the subject.
![Page 12: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/12.jpg)
π=( ππ )πβππ=
(π βπ )π
π π=ππ
π βππ=ππβππ
π¦=πβπ₯+1Make the subject.
![Page 13: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/13.jpg)
Skill #2: Subject trapped in a negative term
When the subject is within the first argument of a subtraction, itβs easy to βreleaseβ.
π¦=2 πβ3 2 π=π¦ +3?
However, itβs a tiny bit harder if the subject is in the term being subtracted.
π¦=3β2π₯ π¦+2π₯=32 π₯=3β π¦
When the subject is inside a negative term, just add it to both sides.
?
?
![Page 14: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/14.jpg)
Examples
πβ π₯=π 1βππ₯=π1 2
3
ππβπ βπ₯=π¦+1
??
?
![Page 15: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/15.jpg)
Doing it in one step⦠(if you like)
How could you rearrange the numbers in to get another subtraction?
This suggests you can swap the thing youβre subtracting with the result. (i.e. Only the thing to the left of the subtraction stays put)
?
Examples:
πβ π₯=π??
?
![Page 16: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/16.jpg)
Exercise 2In each case make the subject of the formula. (Please copy out question first)
1
2
3
4
56
7
8
9
10
11
12
13
N
N
14
???????
???
?
??
?
?
?
![Page 17: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/17.jpg)
RED ORANGE GREEN BLUE
Vote with your diaries
Test your understanding so farβ¦
![Page 18: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/18.jpg)
x = y + 3 x = y β 3 x = 3 β y x = 3y
π¦=3βπ₯Make the subject.
![Page 19: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/19.jpg)
π₯=3
π¦β1 π₯=3π¦ β1π₯=
π¦3 β1π₯=
π¦2
π¦=3π₯+1
Make the subject.
![Page 20: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/20.jpg)
π₯=π¦2 β π§π₯=π¦β2π§π₯=
π¦βπ§2
π¦=2 (π₯+π§ )Make the subject.
![Page 21: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/21.jpg)
π₯=π¦β12 π₯=
1+π¦2 π₯=1β2 π¦π₯=
1β π¦2
π¦=1β2 π₯Make the subject.
![Page 22: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/22.jpg)
π₯=ππ¦ β1
Skill #3: Subject trapped in a denominatorWhen the subject is in the numerator of a fraction, itβs easy to βreleaseβ the subject from the fraction.
π¦=π₯π π₯=ππ¦?
But itβs a bit harder if the subject is in the denominatorβ¦
π¦=ππ₯+1 π¦ (π₯+1 )=π
π₯+1=ππ¦
In general, whenever you have a fraction in an equation, your instinct should be to multiply both sides by the denominator.
?
?
?
![Page 23: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/23.jpg)
Skill #3: Subject trapped in a denominator
! Isolate the fraction on one side of the equation, then multiply by denominator.
π=πβππ₯
ππ₯ +π=π
π=πβ ππ₯
1 2
3
??
?
![Page 24: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/24.jpg)
+2 first as was last thing done to
Doing it in one step⦠(if you like)
How would you rearrange the numbers in to get another division?
Another way of thinking about itβ¦ Remember your speed-distance-time triangle from physics? Once you get to a point where you have to release from the fraction, you can apply what I call the βtriangle trickβ.
?
Examples:
π=π
4β2 π₯ 4β2 π₯=ππ
π=ππ₯
πβπ2π₯+1
=π π¦=π¦ 2π₯+ π¦ β2
E1 E2E3
? ? ?
?
Thus we can swap the thing weβre dividing by and the result. The numerator is left unchanged.
![Page 25: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/25.jpg)
Skill #3b: βCross multiplyingβ
If you have just a fraction on each side of the equation, you can βcross multiplyβ.
ππ
ππΒΏ Click for
Bromanimation
Examples:Make the subject:
? ?
E1E2
![Page 26: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/26.jpg)
Exercise 3In each case make the subject of the formula. (Please copy out question first)
1
2
3
4
5
6
7
8
12
13
14
N
??
?
?
?
?
?
?
?
?
?
?
?
9
11
N
?
?
![Page 27: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/27.jpg)
Questions similar to those in Tiffin CATsMake the subject of the formula:
or ?
Make the subject of the formula:
?
Make the subject of the formula:
?
Make the subject of the formula:
?
![Page 28: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/28.jpg)
π₯=π¦
πβ1 π₯=ππ¦ β1π₯=
ππ¦β1π₯=2
π¦=ππ₯+1
Make the subject.
![Page 29: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/29.jpg)
π₯=π¦2 +1π₯=2 π¦+1π₯=π¦+2π₯=2 π¦+2
π¦=π₯2 β1
Make the subject.
![Page 30: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/30.jpg)
π₯=Β±β ππ¦ π₯=Β±β π¦π π₯=π¦Β±βππ₯=Β±βπ¦π
π¦=ππ₯2Make the subject.
![Page 31: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/31.jpg)
π₯=Β±β 2π¦π₯=2π¦ π₯=Β±β π¦2π₯=Β±β2 π¦
π¦=2π₯2
Make the subject.
![Page 32: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/32.jpg)
π₯=( π¦π+1)2
π₯=π¦2β1π π₯=
(π¦+1 )2
π π₯=π¦2+1π
π¦=πβπ₯β1Make the subject.
![Page 33: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/33.jpg)
π₯=1
(π¦β1 )2β1 π₯=
11β (1β π¦ )2π₯=π¦ 2β1
π¦=1
βπ₯+1+1
Make the subject.
![Page 34: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/34.jpg)
π₯=( πππβ1 )2π₯=ππππ₯=( πππ+1 )
2
π₯=(π (π+1 )π )
2
πβπ₯
=π+1π
Make the subject.
![Page 35: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/35.jpg)
Activity Time!
There are 3 levels, each of increasing difficulty.
Once youβve completed all the questions in Level 1, check your answers with me β after which you can advance onto the next level.
Merit on offer to anyone who can complete all Level 3 questions.
![Page 36: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/36.jpg)
Substituting
π=π ,π=ππBro Tip: Write out the values of your variables first using the information given.
π=ππ ,π=ππWhen we substitute our numbers in, we now have to to rearrange to solve!
?
?
Q
![Page 37: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/37.jpg)
Test Your UnderstandingThe maths exam mark of an 8EWS student is determined by the hours revised and the number of cats they have using the following formula:
Given that Jaimal has 2 cats and gets a mark of 80, how many hours did he revise?
Q QThe βcool coefficientβ of a boy is determined by their number of skateboards and the mass of hair gel they apply, using the formula:
a) If Max has a Cool Coefficient of 5 and has 10 skateboards, how many kilograms of hair gel does he use?
b) If he has a Cool Coefficient of 4 and uses 2kg of hair gel, how many skateboards does he have?
?
?
?
![Page 38: Year 8: Changing the Subject](https://reader036.fdocuments.us/reader036/viewer/2022062811/568161b7550346895dd183bf/html5/thumbnails/38.jpg)
Exercise 4
Given that and and , find
Given that and , find
a) Find when
b) Find when
The price of a car with initial price and age is given by the formula:
a) Find the current price of the car when it is initially Β£10 000 and is 5 years old.
b) What is its age when a car initially worth Β£10 000 has fallen to a tenth of its value?
You can calculate a temperature in Fahrenheit from Celsius using the formula:
It is currently . What is the temperature in Fahrenheit?86F
Determine when
Find when
Given that the conversion between Fahrenheit and Celsius is:
Determine the temperature which is the same in both Fahrenheit and Celsius.
Solving:
1
2
3
4
5
6
7
N
?
?
?
?
?
?
?
?
?
?