Mindful Manipulation


Definition:

Mindful Manipulation is a method we broadly use in algebra. It can be thought of as a way of thinking rather than a strictly defined method. It is how we approach mathematical problems, roughly figure out what we have to do, look at what we know, and then “play around” with our understanding of mathematical techniques.

Question 1

We have two expressions 3x + 1 and 9x + 3

Calculate the value of each expression when x = 2

i) What is the ratio of the bigger value to the smaller value?

ii) Look at the expressions and determine why, algebraically, the ratio is this way.

Extra Tip: Start to get in the habit of when you are first reading a question, see if there is anything familiar or any sort of similarities or patterns in what the question is telling us (three properties, common factors etc.) even if the question has not directly asked you to.

Calculate the value of each expression when x = 2

3x + 1

3(2) + 1 =

6 + 1 = 7

9x + 3

9(2) + 3 = 21

There we have our two values for x = 2, see any patterns / common factors / similarities? Good, if you can, let us move onto the next part.

i) What is the ratio of the bigger value to the smaller value?

So from the first part we may have already spotted this similarity, but do not worry if not!

The bigger value is 21 and the smaller value is 7,

Looking into these two values, we can see 21 is 3 times bigger than 7, or,

21 = 3(7)

For this purpose, it may help to write it like,

1(21) = 3(7),

Here we should recognize that the ratio of the bigger to smaller value is 3 to 1.

ii) Look at the expressions and determine why, algebraically, the ratio is this way.

So, the question lets us know how to begin, “Look at the expressions…”,

3x + 1 and 9x + 3, this is when we start to do mindful manipulation, applying some of the techniques we have learned,

What similarities can we see between the two expressions? Try using the distributive property,

You may have noticed 9x + 3 has a common factor of 3,

3(3x) + 3(1) =

3(3x + 1),

Aha! Now we see a similarity,

Our two expressions can be looked at like 3x + 1 and 3(3x + 1), or in other terms,

The second expression is 3 times bigger than the first, or,

There is a ratio of 3 to 1, which explains the ratio of our two values for x = 2

Question 2

An expression 2x + 3 is equal to 5 for a certain value of x. Determine the values of the following expressions for the same value of x:

4x + 6

2x + 5

Firstly, even if you can, do not try to solve for the value of x, the question is not asking that. Sometimes, questions will be laid out in a way that looks familiar to a normal question you get asked, but will be asking something different. Start to build the skill of reading and understanding what the question is asking you and what information the question is giving you.

It can help to simply rewrite some parts of the question again in your working,

An expression 2x + 3 is equal to 5 for a certain value of x,

2x + 3 is equal to 5,

2x + 3 = 5,

So, what we know is the value of 2x + 3, now we want to use that to solve these questions.

Now look at 4x + 6, one good habit to get into is recognizing common factors in expressions,

4x + 6 =

2(2x + 3)

Another aha! moment,

We see 2x + 3 in the expression, and what we know from what the question has given us is that 2x + 3 = 5 (for the value of x), so all we do is sub that in,

2(5) = 10


Now for 2x + 5

Again, we know the value of 2x + 3, and we want to know the value of 2x + 5, we will use mindful manipulation to determine this,

So, we know what we know, we see what we want to know, and we “play around” with our algebraic methods to get there,

How can we rewrite 2x + 5 in a way that relates to what we know (2x + 3)

With mindful manipulation,

Re write 2x + 5 as, (5 = 3 + 2)

2x + 3 + 2 =

Associative property tells that equals,

(2x + 3) + 2 =

We know the value of 2x + 3,

(5) + 2 = 7

Question 3

The expression 3x + 5 has a value of 6 for some value of x, find the value of the following expressions for the same value of x (again, not solving for x!)

  • 6x + 10
  • 3x + 15
  • 3x + 2
  • -3x - 5
  • 12x + 20
  • 3x - 5
  • x + 7/3

So, we know the value of 3x + 5, lets manipulate to find the value of these expressions,

Starting to get in the habit of immediately seeing common factors in expressions, (even if sometimes we wont need to)

For 6x + 10:

Distributive Property,

2(3x + 5) =

We know 3x + 5 = 6

2(6) = 12


For 3x + 15:

Seeing the similarity of,

3x in (3x +15)

and

3x in what we know (3x + 5 = 6),

Hints that it is the 15 in 3x + 15 that we will have to mindfully manipulate to get be able to work with 3x + 5,

3x + 15 =

(15 = 5 + 10),

3x + 5 + 10 =

(3x + 5) + 10 =

Subbing in 3x + 5 = 6,

(6) + 10 = 16


For 3x + 2:

Again, seeing the similarity of 3x in (3x +2)

and

3x in what we know (3x + 5 = 6),

Hints that it’s the 2 in 3x + 2 that we will have to manipulate to relate it the two expressions we are working with,

3x + 2 =

With (2 = 5 - 3),

3x + 5 - 3 =

(6) - 3 = 3


For -3x - 5:

First, as always, get in the habit of spotting common factors, the negative sign in front of both expressions is the hint,

-3x - 5 =

(-1)3x + (-1)5 =

Distributive Property,

(-1)(3x + 5) =

We know 3x + 5 = 6,

(-1)(6) = -6


For 12x + 20:

Again, spot the common factor,

There is 2 which would give us 2(6x + 10) which would not help,

There is also 4, which gives us,

4(3x) + 4 (5) =

4(3x + 5) =

4(6) = 24


For 3x - 5:

We see the 3x, which hints we will work with the - 5,

To avoid tripping over the negative write,

3x + (-5)

Even though this looks similar to our 3x + 5, it is not, dont be tricked into subbing in 3x + 5 = 6 just yet,

Let’s manipulate (-5),

-5 = 5 - 10, so

3x + (-5) =

3x + (5 - 10) =

Associative property,

(3x + 5) - 10=

(6) - 10 = -4


For x + 7/3:

A slightly daunting one to look at, this one might involve more than one step,

We know 3x + 5 = 6,

We want to make x + 7/3 similar to 3x + 5,

Let’s try and make the x in x + 7/3 the same as the 3x in 3x + 5,

To get x to 3x, we multiply by 3, so

Multiply, x + 7/3 by 3,

3(x + 7/3) =

3(x) + 3(7/3) =

3x + 7,

Starting to look like 3x + 5,

One last step,

3x + 7 =

3x + 5 + 2 =

(3x + 5) + 2 =

(6) + 2 = 8

[ we multiplied by 3, then added 2 ]

Question 4

(4x - 3) is equal to -5 for some value of x, what is the value of (4x - 3)2 + 8x - 6 for the same value of x?

Obviously, a bigger expression this time, we may have to break it into parts to deal with it,

Write what we know, for a certain value of x,

4x - 3 = -5

For the same value of x, whats the value of the expression,

(4x - 3)2 + 8x - 6,

Lets deal with this in two parts:

(4x - 3)2 and 8x - 6

For (4x - 3)2 we can see we know 4x - 3 value, so although this is to the power of 2, dont be thrown off and stick to our process,

Subbing in 4x - 3 = - 5 into,

(4x - 3)2 =

(-5) ^ 2 =

25

Not forgetting the second part, 8x - 6,

Looking at it we see a common factor, 2,

8x - 6 =

2(4x - 3) =

We know 4x - 3 = - 5,

2(-5) =

-10

So subbing (4x - 3)2 = 25 and 8x - 6 = -10 back into

[ (4x - 3)2 ] + [ 8x - 6 ]

[25] + [-10] =

25 - 10 = 15

Question 5 - Advanced

Suppose a + b = 4a, what would a - b equal?

So, let's use what the question gives us plus midful manipulation to rearrange to give us a - b,

How can we get from a + b = 4a to give us a - b = something?

Let's try subtracting b from both sides on a + b = 4a,

[remember you can add or subtract or multiply or divide an equation as long as its done to both sides]

a + b - b = 4a - b,

Gives us,

a = 4a - b,

Now we see we have a on one side, which is getting closer to a - b, let’s try subtracting b again from both sides.

a - b = 4a - b - b,

Which gives us,

a - b = 4a - 2b,

There we have our expression for what a - b is!

4a - 2b or 2(2a - b) if you want to show off your factoring skills!