Solving for x, a Common Sense Approach. Purpose Solving for an unknown variable is a common theme in Algebra Curriculums. Solving for an unknown variable

Solving for x, a Solving for x, a Common Sense Common Sense ApproachApproach

PurposePurpose

Solving for an unknown variable is a Solving for an unknown variable is a common theme in Algebra common theme in Algebra Curriculums.Curriculums.

PurposePurpose

Solving for an unknown variable is a Solving for an unknown variable is a common theme in Algebra common theme in Algebra Curriculums.Curriculums.

We can build a method for solving We can build a method for solving many equations based on a common many equations based on a common sense approach.sense approach.

PurposePurpose

Solving for an unknown variable is a Solving for an unknown variable is a common theme in Algebra Curriculums.common theme in Algebra Curriculums.

We can build a method for solving We can build a method for solving many equations based on a common many equations based on a common sense approach.sense approach.

The way we will tackle this common The way we will tackle this common problem is related to the way the problem is related to the way the content is taught in traditional content is taught in traditional mathematics courses.mathematics courses.

PurposePurpose

Solving for an unknown variable is a Solving for an unknown variable is a common theme in Algebra Curriculums.common theme in Algebra Curriculums.

We can build a method for solving many We can build a method for solving many equations based on a common sense equations based on a common sense approach.approach.

The way we will tackle this common The way we will tackle this common problem is related to the way the content problem is related to the way the content is taught in traditional mathematics is taught in traditional mathematics courses.courses.

The approach we will take is different in The approach we will take is different in the sense that we will not seek to make a the sense that we will not seek to make a new algorithm for every solving situation.new algorithm for every solving situation.

What you will need to What you will need to begin.begin.

A knowledge of how to navigate A knowledge of how to navigate through a PowerPoint presentation.through a PowerPoint presentation.

What you will need to What you will need to begin.begin.

A knowledge of how to navigate A knowledge of how to navigate through a PowerPoint presentation.through a PowerPoint presentation.

A piece of paper to take notes on and A piece of paper to take notes on and complete the exercises contained in complete the exercises contained in this presentation.this presentation.

What you will need to What you will need to begin.begin. A knowledge of how to navigate A knowledge of how to navigate

through a PowerPoint presentation.through a PowerPoint presentation.

A piece of paper to take notes on and A piece of paper to take notes on and complete the exercises contained in complete the exercises contained in this presentation.this presentation.

A four function calculator (add, A four function calculator (add, subtract, multiply and divide)subtract, multiply and divide)

Lets get started!Lets get started!

We are going to think about the We are going to think about the process you would typically go through process you would typically go through to get to school.to get to school.

Getting to SchoolGetting to School

On a piece of paper write down a On a piece of paper write down a list of steps that you go through list of steps that you go through to get to school.to get to school.

Keep your list between four and Keep your list between four and eight items in length.eight items in length.

Make sure it is in the order that Make sure it is in the order that you went through.you went through.


Here is an example of a list.Here is an example of a list. Woke up.Woke up. Showered.Showered. Got dressed.Got dressed. Ate breakfast.Ate breakfast. Drove to school.Drove to school. Walked into the school.Walked into the school.


Now that you have Now that you have you list, consider you list, consider what it would look what it would look like if you watched like if you watched your morning in your morning in reverse (“un”going reverse (“un”going to school).to school).

Write down a list Write down a list what it would look what it would look like to go like to go backwards.backwards.

““Un”going to SchoolUn”going to School

Going to SchoolGoing to School1.1. Woke up and got Woke up and got

out of bed.out of bed.2.2. ShoweredShowered3.3. Got dressed.Got dressed.4.4. Ate breakfast.Ate breakfast.5.5. Drove to school.Drove to school.6.6. Walked into the Walked into the

school.school.

““Un”going to SchoolUn”going to School1.1. Walked out of Walked out of

schoolschool2.2. Drove in reverse Drove in reverse

from schoolfrom school3.3. Regurgitated Regurgitated

breakfast (nasty)breakfast (nasty)4.4. UndressedUndressed5.5. Unshowered. (also Unshowered. (also

nasty)nasty)6.6. Got back in bed Got back in bed

and went to sleep.and went to sleep.


Aside from some of the disgusting Aside from some of the disgusting ideas in your “un”going to school list ideas in your “un”going to school list there are some important things to there are some important things to notice about it.notice about it.



1.1. Every item should be the opposite Every item should be the opposite action from the original list.action from the original list.



1.1. Every item should be the opposite Every item should be the opposite action from the original list.action from the original list.

2.2. The order it is written in should start The order it is written in should start at the end of the original list and go at the end of the original list and go backwards to the start of it.backwards to the start of it.

So what’s the point???So what’s the point???

If we have a process that contains If we have a process that contains a series of steps that are in order, a series of steps that are in order, we can think about how we would we can think about how we would work backwards from the end work backwards from the end undoing each item as we go.undoing each item as we go.

If we do this we arrive at the If we do this we arrive at the beginning, where we were to start beginning, where we were to start with.with.

No really, what’s the No really, what’s the point???point???

Let’s look at some “math”Let’s look at some “math” Consider the linear function Consider the linear function

y=5x+3.y=5x+3. Let y=-3, so we have -3=5x+3. Let y=-3, so we have -3=5x+3.

What x-value (input) would make What x-value (input) would make this function give a y-value of -3? this function give a y-value of -3?

y=5x+3y=5x+3

How would we go about finding y How would we go about finding y if we knew what the x was? In if we knew what the x was? In other words what are the other words what are the calculation steps that take x and calculation steps that take x and give us a y?give us a y?

We take x and multiply it by 5. We take x and multiply it by 5. Then add 3 to this answer and we Then add 3 to this answer and we would get y.would get y.

With y=-3 we have -3=5x+3With y=-3 we have -3=5x+3

We take x and multiply it by 5. We take x and multiply it by 5. Then add 3 to this answer and we Then add 3 to this answer and we would get y.would get y.

But we know y and need to figure But we know y and need to figure out x. This is like being at school out x. This is like being at school and needing to figure out how we and needing to figure out how we got there.got there.

We need to “un”go to school.We need to “un”go to school.


ForwardForward

1.1. Start with xStart with x

2.2. Multiply by 5Multiply by 5

3.3. Add 3Add 3

4.4. We get y We get y

““un”Forwardun”Forward

1.1. Start with yStart with y

2.2. Subtract 3Subtract 3

3.3. Divide by 5Divide by 5

4.4. We would be We would be back to x.back to x.


ForwardForward

1.1. Start with xStart with x


3.3. Add 3Add 3

4.4. We get yWe get y





4.4. We would be back to x.We would be back to x.

We know y=-3 and we need to know We know y=-3 and we need to know x.x.

Which list will do this for us?Which list will do this for us?






4.4. We would be We would be back to x.back to x.

1.1. Start with -3Start with -3

2.2. -3-3=-6-3-3=-6

3.3. -6/5=-1.2-6/5=-1.2

4.4. We would have We would have x so, x=-1.2x so, x=-1.2

With y=-3 we have -3=5x+3 With y=-3 we have -3=5x+3 and we just found that x=-and we just found that x=-1.21.2 Lets check our result. We think Lets check our result. We think

that when x=-1.2 we will get a y that when x=-1.2 we will get a y value of -3 when we plug it in.value of -3 when we plug it in.

Does 5(-1.2)+3=-3?Does 5(-1.2)+3=-3? Well 5*-1.2=-6 and -6+3 is -3 so Well 5*-1.2=-6 and -6+3 is -3 so

yes it does.yes it does.

Try this example.Try this example.

What x would make What x would make -9x+8=53 ? -9x+8=53 ?

Write down the Write down the problem before you problem before you go on.go on.

We know the We know the answer and need to answer and need to know where we know where we started.started.

If we knew x how If we knew x how would we find y?would we find y?

Click on the list that Click on the list that describes this.describes this.

1.1. Take xTake x

2.2. Subtract Subtract 88

3.3. Divide by Divide by -9-9

4.4. This gets This gets us yus y

1.1. Take xTake x

2.2. Multiply Multiply by -9by -9

3.3. Add 8Add 8


1.1. Take xTake x

2.2. Add 8Add 8

3.3. Multiply Multiply by -9by -9


1.1. Take xTake x

2.2. Divide by Divide by -9 -9

3.3. Subtract Subtract 88


Click here to try againClick here to try again

No, that’s not the one.No, that’s not the one.

1.1. Take xTake x

2.2. Divide by -9Divide by -9


4.4. This gets us yThis gets us y

Remember at this point we only want Remember at this point we only want to write a list that takes us from x to to write a list that takes us from x to y. We are not trying to undo y. We are not trying to undo anything.anything.

Did you follow order of operations Did you follow order of operations correctly (PEMDAS)?correctly (PEMDAS)?



1.1. Take xTake x2.2. Add 8Add 83.3. Multiply by -9Multiply by -94.4. This gets us yThis gets us y

Remember at this point we only Remember at this point we only want to write a list that takes us want to write a list that takes us from x to y.from x to y.

Did you follow order of Did you follow order of operations correctly (PEMDAS)?operations correctly (PEMDAS)?



1.1. Take xTake x2.2. Subtract 8Subtract 83.3. Divide by -9Divide by -94.4. This gets us yThis gets us y

Remember at this point we only Remember at this point we only want to write a list that takes us want to write a list that takes us from x to y. These operations from x to y. These operations don’t do that.don’t do that.

Did you follow order of Did you follow order of operations correctly (PEMDAS)?operations correctly (PEMDAS)?

Great job!Great job!

You have You have identified the identified the steps that take steps that take us from x to y for us from x to y for the equation the equation y=-9x+8y=-9x+8

1.1. Take xTake x

2.2. Multiply by -9Multiply by -9

3.3. Add 8Add 8


Let’s keep going!Let’s keep going!

You have identified You have identified the steps that takes the steps that takes us from x to y for us from x to y for the equation y=-the equation y=-9x+89x+8

1.1. Take xTake x


3.3. Add 8Add 8


What is the process What is the process that will take us from that will take us from y to x?y to x?

Write down the steps Write down the steps before going on.before going on.

1.1. yy

2.2.

3.3.

4.4. xx

Which process did you Which process did you write down?write down?1.1. yy

2.2. Divide by -9 Divide by -9


4.4. xx

1.1. yy


3.3. Add -8 Add -8

4.4. xx

1.1. yy



4.4. xx

1.1. yy

2.2. Add -8Add -8

3.3. Multiply by 9 Multiply by 9

4.4. xx

You got it!You got it!

You have identified You have identified the steps that takes the steps that takes us from x to y for us from x to y for the equation y=-the equation y=-9x+89x+8

1.1. Take xTake x


3.3. Add 8Add 8


So if we knew a So if we knew a y-value we can y-value we can figure out what x-figure out what x-made it using this made it using this listlist

1.1. yy



4.4. xx


Oops, that’s not it.Oops, that’s not it.

Remember that Remember that the order we go in the order we go in matters.matters.

We need to undo We need to undo the operations.the operations.

Negative numbers Negative numbers stay negative and stay negative and positives stay positives stay positive.positive.

1.1. yy

2.2. Divide by -9 Divide by -9


4.4. xx






1.1. yy


3.3. Add -8 Add -8

4.4. xx






1.1. yy

2.2. Add -8Add -8

3.3. Multiply by 9 Multiply by 9

4.4. xx

-9x+8=53-9x+8=53

We know the y-We know the y-value (y=53) and value (y=53) and need to know x and need to know x and we have a list that we have a list that does just that thing!does just that thing!

1.1. yy



4.4. xx

1.1. y is 53y is 53

2.2. Subtract 8, Subtract 8, 53-8=4553-8=45

3.3. Divide by -9, Divide by -9, 45/-9=-545/-9=-5

4.4. x is -5x is -5

Let’s CheckLet’s Check

Does -9x+8=53 if x=-5?Does -9x+8=53 if x=-5? Input -5 for x and check to make Input -5 for x and check to make

sure it gives an output of 53.sure it gives an output of 53. -9(-5)+8=?-9(-5)+8=? 45+8=?45+8=? 5353

One More!One More!

Lets look at an Lets look at an example that really example that really shows why this is shows why this is such a powerful such a powerful way to “solve for x”way to “solve for x”

Write down the Write down the equation and the equation and the steps to go from x steps to go from x to y.to y.

This list will be This list will be longer since there longer since there are five operations.are five operations.

1.1. Take xTake x2.2. 3.3. 4.4. 5.5. 6.6. 7.7. To get y.To get y.

The list you made should look like The list you made should look like this.this.

1.1. Take xTake x2.2. Multiply by 5Multiply by 53.3. Subtract 2Subtract 24.4. Divide by -27Divide by -275.5. Multiply by 3Multiply by 36.6. Add 46Add 467.7. To get yTo get y

The list you made The list you made should look like should look like this.this.


But we know y and But we know y and we need x. We we need x. We need the other list need the other list again.again.

1.1. Take yTake y2.2. 3.3. 4.4. 5.5. 6.6. 7.7. To get xTo get x




1.1. Take yTake y2.2. Subtract 46Subtract 463.3. 4.4. 5.5. 6.6. 7.7. To get xTo get x




1.1. Take yTake y2.2. Subtract 46Subtract 463.3. Divide by 3Divide by 34.4. 5.5. 6.6. 7.7. To get xTo get x




1.1. Take yTake y2.2. Subtract 46Subtract 463.3. Divide by 3Divide by 34.4. Multiply by -27Multiply by -275.5. 6.6. 7.7. To get xTo get x




1.1. Take yTake y2.2. Subtract 46Subtract 463.3. Divide by 3Divide by 34.4. Multiply by -27Multiply by -275.5. Add 2Add 26.6. 7.7. To get xTo get x




1.1. Take yTake y2.2. Subtract 46Subtract 463.3. Divide by 3Divide by 34.4. Multiply by -27Multiply by -275.5. Add 2Add 26.6. Divide by 5Divide by 57.7. To get xTo get x

This list takes us This list takes us from y to x.from y to x.

1.1. Take yTake y2.2. Subtract 46Subtract 463.3. Divide by 3Divide by 34.4. Multiply by -27Multiply by -275.5. Add 2Add 26.6. Divide by 5Divide by 57.7. To get xTo get x

Lets follow the steps Lets follow the steps on our y value, 39.on our y value, 39.

1.1. 39392.2. 39-46=-739-46=-73.3. -7/3=-2.3333333-7/3=-2.33333334.4. -2.33333333*27=-63-2.33333333*27=-635.5. -63+2=-61-63+2=-616.6. -61/5=-12.2-61/5=-12.27.7. So x=-12.2So x=-12.2

We found that x=-12.2, Lets check We found that x=-12.2, Lets check to make sure it works. Plug it in for to make sure it works. Plug it in for x.x.

If you follow order of operations you If you follow order of operations you will find that it does work.will find that it does work.

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Solving for x, a Common Sense Approach. Purpose Solving for an unknown variable is a common theme in Algebra Curriculums. Solving for an unknown variable