vark websitecs301/sp15/fa14/vark/...vark vark response. why do some experts say that knowing your...

375:Learning Styles.

VARK website

() September 29, 2014 1 / 19

La Jolla Study.Pashler et.al.

The learning-styles view has acquired great influence within theeducation field, and is frequently encountered at levels ranging fromkindergarten to graduate school. There is a thriving industry devotedto publishing learning-styles tests and guidebooks for teachers, andmany organizations offer professional development workshops forteachers and educators built around the concept of learning styles.

Our review of the literature disclosed ample evidence that children andadults will, if asked, express preferences about how they preferinformation to be presented to them. There is also plentiful evidencearguing that people differ in the degree to which they have some fairlyspecific aptitudes for different kinds of thinking and for processingdifferent types of information.

However, we found virtually no evidence for the interaction patternmentioned above, which was judged to be a precondition for validatingthe educational applications of learning styles.

() September 29, 2014 2 / 19

Snippy, snippy

Science Samples.

Psychologist Robert Sternberg of Tufts University in Medford,Massachusetts, says the paper “does not even begin to be a seriousreview of the field. ... [In] limiting themselves to random-assignmentstudies, they ignored almost the entire literature.”

Just so, says Pashler – most of it is “weak.”

() September 29, 2014 3 / 19

Snippy, snippy

Science Samples.

() September 29, 2014 3 / 19

Snippy, snippy

Science Samples.

() September 29, 2014 3 / 19

Snippy, snippy

Science Samples.

() September 29, 2014 3 / 19

Vark response.

Why do some experts say that knowing your learning style doesnot contribute to improved learning?

The statement is true, in the same way that knowing you have adisease does not cure the disease or weighing yourself does not fixobesity. It is the next step that is important - When people makechanges to their learning, based on their VARK preferences, theirlearning will be enhanced. They do this by using strategies that alignwith their preferences. It is what you do after you learn yourpreferences that has the potential to make a difference.

() September 29, 2014 4 / 19

Vark response.

() September 29, 2014 4 / 19

Vark response.

() September 29, 2014 4 / 19

Willingham Video.

Willingham video.

People have different capabilites:visual, auditory, kinesthetic.

Information is inherent content: visual, hearing, meaning.

() September 29, 2014 5 / 19

Willingham Video.

Willingham video.

People have different capabilites:

visual, auditory, kinesthetic.

() September 29, 2014 5 / 19

Willingham Video.

Willingham video.

People have different capabilites:visual,

auditory, kinesthetic.

() September 29, 2014 5 / 19

Willingham Video.

Willingham video.

People have different capabilites:visual, auditory,

kinesthetic.

() September 29, 2014 5 / 19

Willingham Video.

Willingham video.

() September 29, 2014 5 / 19

Willingham Video.

Willingham video.

Information is inherent content:

visual, hearing, meaning.

() September 29, 2014 5 / 19

Willingham Video.

Willingham video.

Information is inherent content: visual,

hearing, meaning.

() September 29, 2014 5 / 19

Willingham Video.

Willingham video.

Information is inherent content: visual, hearing,

meaning.

() September 29, 2014 5 / 19

Willingham Video.

Willingham video.

() September 29, 2014 5 / 19

What do students mean...

“Professor should use more figures, because I am a visual leaner.”

In computer science, inherent content may lend itself to pictures, ordifferent views to understanding.

Example:Recursion:

codestacktreeinductive proof.environment diagram

Connect views even.

() September 29, 2014 6 / 19

Example:Recursion:

Connect views even.

() September 29, 2014 6 / 19

Example:Recursion:

stacktreeinductive proof.environment diagram

Connect views even.

() September 29, 2014 6 / 19

Example:Recursion:

codestack

treeinductive proof.environment diagram

Connect views even.

() September 29, 2014 6 / 19

Example:Recursion:

codestacktree

inductive proof.environment diagram

Connect views even.

() September 29, 2014 6 / 19

Example:Recursion:

codestacktreeinductive proof.

environment diagram

Connect views even.

() September 29, 2014 6 / 19

Example:Recursion:

Connect views even.

() September 29, 2014 6 / 19

Example:Recursion:

Connect views even.

() September 29, 2014 6 / 19

Example:Recursion:

Connect views even.

() September 29, 2014 6 / 19

Example: coloring.Thm: Any graph with degree at most d can be d +1 colored.

DasGupta, Papadimitriou, Vazirani, “Algorithms” -like.

“Greedily consistently color nodes with one of d +1 colors. At leastone color is available when a node is colored.”

Available = Green, Red, Yellow, BlueAvailable = Green,Yellow, BlueAvailable = Yellow, BlueAvailable = Blue

...so 4 is enough!

() September 29, 2014 7 / 19

DasGupta, Papadimitriou, Vazirani, “Algorithms”

-like.

...so 4 is enough!

() September 29, 2014 7 / 19

...so 4 is enough!

() September 29, 2014 7 / 19

...so 4 is enough!

() September 29, 2014 7 / 19

...so 4 is enough!

() September 29, 2014 7 / 19

Available = Green, Red, Yellow, Blue

Available = Green,Yellow, BlueAvailable = Yellow, BlueAvailable = Blue

...so 4 is enough!

() September 29, 2014 7 / 19

Available = Green, Red, Yellow, BlueAvailable = Green,Yellow, Blue

Available = Yellow, BlueAvailable = Blue

...so 4 is enough!

() September 29, 2014 7 / 19

Available = Green, Red, Yellow, BlueAvailable = Green,Yellow, BlueAvailable = Yellow, Blue

Available = Blue

...so 4 is enough!

() September 29, 2014 7 / 19

...so 4 is enough!

() September 29, 2014 7 / 19

...so 4 is enough!() September 29, 2014 7 / 19

The MIT-way...(CLRS)...

Thm: Any graph with degree at most d can be d +1 colored.

Graph: G = (V ,E).Colors = {1, . . .d +1}.For u ∈ V :

color[u] = FalseFor u ∈ V :

Available = Colorsfor v in N(u):

if color[v] != NULLAvailable = Colors - color[v]

color[u] = Available.first()

Proof: Invariant 1: Invariant 2.

() September 29, 2014 8 / 19

Graph: G = (V ,E).

Colors = {1, . . .d +1}.For u ∈ V :

() September 29, 2014 8 / 19

Graph: G = (V ,E).Colors = {1, . . .d +1}.

For u ∈ V :color[u] = False

For u ∈ V :Available = Colorsfor v in N(u):

() September 29, 2014 8 / 19

color[u] = False

For u ∈ V :Available = Colorsfor v in N(u):

() September 29, 2014 8 / 19

Available = Colors

for v in N(u):if color[v] != NULL

Available = Colors - color[v]color[u] = Available.first()

() September 29, 2014 8 / 19

if color[v] != NULL

Available = Colors - color[v]color[u] = Available.first()

() September 29, 2014 8 / 19

Proof:

Invariant 1: Invariant 2.

() September 29, 2014 8 / 19

Proof: Invariant 1:

Invariant 2.

() September 29, 2014 8 / 19

Todo: graph coloring.

Make lecture active!

Make into discussion activity!

Example: Use idea in problems:Graph and Complemnt can be colored with n+1 colors.

() September 29, 2014 9 / 19

Example: Use idea in problems:

Graph and Complemnt can be colored with n+1 colors.

() September 29, 2014 9 / 19

Just for fun..

Every planar graph has a node of degree 6.

6 color a planar graph!

() September 29, 2014 10 / 19

Just for fun..

Every planar graph has a node of degree 6.

6 color a planar graph!

() September 29, 2014 10 / 19

Completing the square

Version 1:

“Just the facts, Ma’am”

() September 29, 2014 11 / 19

Version 1: “Just the facts, Ma’am”

() September 29, 2014 11 / 19

Completing the square.

Solve x2 +16x−57 = 0.

Perfect square: (x +a)2 = x2 +2ax +a2

If perfect square can take square root.

Is x2 +16x−57 a perfect square?2a = 16→ a = 16

2 = 8 and a2 = 64 6=−57!Not a perfect square.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

Move 57 over.Make left hand side a perfect square!

Can take square root!2ax = 16x

= 16x2x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

= 16x2x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

= 16x2x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

Is x2 +16x−57 a perfect square?

2a = 16→ a = 162 = 8 and a2 = 64 6=−57!

Not a perfect square.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

= 16x2x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

Is x2 +16x−57 a perfect square?2a = 16

→ a = 162 = 8 and a2 = 64 6=−57!

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

= 16x2x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

and a2 = 64 6=−57!Not a perfect square.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

= 16x2x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

2 = 8 and a2 = 64

6=−57!Not a perfect square.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

= 16x2x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

2 = 8 and a2 = 64 6=−57!

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

= 16x2x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

= 16x2x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

= 16x2x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

= 16x2x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57

x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

Move 57 over.

Make left hand side a perfect square!Can take square root!

2ax = 16x

= 16x2x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

= 16x2x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

Can take square root!

2ax = 16x

= 16x2x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

= 16x2x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

Can take square root!2ax = 16x → a

= 16x2x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

Can take square root!2ax = 16x → a = 16x

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

2x = 8,

a2 =82 = 64

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64

x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121

(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121

(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

x =±11−8→ x =

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) = ±11

x =±11−8→ x =

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) = ±11x =±11−8

→ x =

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) = ±11x =±11−8→ x =

Doh!or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) = ±11x =±11−8→ x =

19 Doh!

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) = ±11x =±11−8→ x = −19

Doh!or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 12 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) = ±11x =±11−8→ x = −19

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 12 / 19

V2: Engage.

“Learn-gage”.

() September 29, 2014 13 / 19

V2: Engage. “Learn-gage”.

() September 29, 2014 13 / 19

Completing the square: engage?Solve x2 +16x−57 = 0.

Recall Perfect square: (x +a)2 = ?

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

A and C. A is simplification.

If perfect square can take square root.Is x2 +16x−57 a perfect square?

(A) Yes

(B) No

B. No.2a = 16→ a = 16

Later in semester. Yes? No? Uses less space! Students are trained.

() September 29, 2014 14 / 19

Completing the square: engage?Solve x2 +16x−57 = 0.Recall Perfect square: (x +a)2 =

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

Completing the square: engage?Solve x2 +16x−57 = 0.Recall Perfect square: (x +a)2 = ?

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a

= (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

and C. A is simplification.

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

A and C.

A is simplification.

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

Is x2 +16x−57 a perfect square?

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.

2a = 16→ a = 162 = 8 and a2 = 64 6=−57!

Not a perfect square.Later in semester. Yes? No? Uses less space! Students are trained.

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16

→ a = 162 = 8 and a2 = 64 6=−57!

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

and a2 = 64 6=−57!Not a perfect square.

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

2 = 8 and a2 = 64

6=−57!Not a perfect square.

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

2 = 8 and a2 = 64 6=−57!

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

Later in semester.

Yes? No? Uses less space! Students are trained.

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

Later in semester. Yes?

No? Uses less space! Students are trained.

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

Later in semester. Yes? No?

Uses less space! Students are trained.

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

Later in semester. Yes? No? Uses less space!

Students are trained.

() September 29, 2014 14 / 19

(A) x2 +2ax +a2

(B) x2 +a2

(C) (x +a)x +(x +a)a = (x2 +ax)+(xa+a2)

(A) Yes

(B) No

B. No.2a = 16→ a = 16

() September 29, 2014 14 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

Can take square root!What should a be? 16? 8? 2ax = 16x

= 16x2x

a2 =82 = 64

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

= 16x2x

a2 =82 = 64

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

= 16x2x

a2 =82 = 64

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57

x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

Move 57 over.

Make left hand side a perfect square!Can take square root!

What should a be? 16? 8? 2ax = 16x

= 16x2x

a2 =82 = 64

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

= 16x2x

a2 =82 = 64

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

Can take square root!

What should a be? 16? 8? 2ax = 16x

= 16x2x

a2 =82 = 64

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

Can take square root!What should a be?

16? 8? 2ax = 16x

= 16x2x

a2 =82 = 64

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

Can take square root!What should a be? 16?

8? 2ax = 16x

= 16x2x

a2 =82 = 64

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

Can take square root!What should a be? 16? 8?

2ax = 16x

= 16x2x

a2 =82 = 64

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

= 16x2x

a2 =82 = 64

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

Can take square root!What should a be? 16? 8? 2ax = 16x→ a

= 16x2x

a2 =82 = 64

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

Can take square root!What should a be? 16? 8? 2ax = 16x→ a = 16x

a2 =82 = 64

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

2x = 8,

a2 =82 = 64

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64

x2 +16x +64 = 121(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121

(x +8)2 = 121(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121

(x +8) =

11x =±11−8→ x =

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) =

x =±11−8→ x =

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) = ±11

x =±11−8→ x =

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) = ±11x =±11−8

→ x =

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) = ±11x =±11−8→ x =

Doh!or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) = ±11x =±11−8→ x =

19 Doh!

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) = ±11x =±11−8→ x = −19

Doh!or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 15 / 19

Solve x2 +16x−57 = 0.

x2 +16x = 57x2 +2ax +a2 = 57+a2

x2 +16x +64 = 57+64x2 +16x +64 = 121(x +8)2 = 121(x +8) = ±11x =±11−8→ x = −19

or x = 11−8 = 3.

2x = 8, a2 =82 = 64.

() September 29, 2014 15 / 19

Discover.

() September 29, 2014 16 / 19

Completing the square: discover.Find x when 16x = 4?

Here it is!How? Divide both sides by 16. Generally: inverse of multiply by 16.

x = 4/16 = 1/4.

Find x when x2 = 9? x =

3.Find x when x2 = 10? x =

√10 or x =−

√10.

Inverse operation is square root!

Find x when x2−8x +16 = 0?

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

If left hand side is perfect square...can solve!Can we always make it so?

() September 29, 2014 17 / 19

Completing the square: discover.Find x when 16x = 4? Here it is!

How? Divide both sides by 16. Generally: inverse of multiply by 16.

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

Completing the square: discover.Find x when 16x = 4? Here it is!How?

Divide both sides by 16. Generally: inverse of multiply by 16.

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

Completing the square: discover.Find x when 16x = 4? Here it is!How? Divide both sides by 16.

Generally: inverse of multiply by 16.

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

Completing the square: discover.Find x when 16x = 4? Here it is!How? Divide both sides by 16. Generally: inverse of multiply by 16.

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

= 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16

= 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

Find x when x2 = 9?

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

Find x when x2 = 10? x =√

10 or x =−√

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

Find x when x2 = 9? x = ±3.Find x when x2 = 10?

x =√

10 or x =−√

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

Find x when x2 = 9? x = ±3.Find x when x2 = 10? x =

or x =−√

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16

, x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16

Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do?

x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16

= (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2

= 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0

So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x =

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

=±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

= 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10

→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

If left hand side is perfect square...can solve!

Can we always make it so?

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

If left hand side is perfect square...can solve!Can we always

make it so?

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

If left hand side is perfect square...can solve!Can we always make

it so?

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

If left hand side is perfect square...can solve!Can we always make it

() September 29, 2014 17 / 19

x = 4/16 = 1/4.

√10 or x =−

√10.

x2 = 8x−16 , x =√

8x−16 Uh oh.

What to do? x2−8x +16 = (x−4)2 = 0 So x = 4 !!

Does x =√

10 ? =±√

10? = 4±√

(x−4)2 = 10→ (x−4) =±√

10→ x = 4±√

() September 29, 2014 17 / 19

Completing the square:discover.

..complete this.

() September 29, 2014 18 / 19

Wrap up.

Learning styles.

Maybe?

Still...use hardware of students to get at understanding.

() September 29, 2014 19 / 19

Wrap up.

Learning styles.

Maybe?

() September 29, 2014 19 / 19

Wrap up.

Learning styles.

Maybe?

() September 29, 2014 19 / 19

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