Teaching Students to Discover Math Anthony Chan. What product is sold by math teachers that most of their customers don’ t like or don’t want but must

Teaching Students to Discover Math

Anthony Chan

What product is sold by math teachers that most of their customers don’ t like or don’t want but must have?

RIDDLE

With permission from Alpha Educational Products and Services

What product is sold by math teachers that most of their customers don’ t like or don’t want but must have?

MATH

RIDDLE


A TEACHER is a salesman.


Selling Math

“Appropriate examples are the key to reaching Math students .”

Selling Math


Don’t burn them out!

Selling Math


“I hear and I forget. I see and I remember. I do and I understand.”

Chinese proverb

“If you tell me, I may forget. If I discover, I will remember--for it is

mine. Anthony Chan

Mathematics was discovered and not invented.


Mathematics was discovered and not invented.

When someone discovers something, he owns it.


Today we are going to view through a variety of tools and strategies how to sell basic algebra and geometry to students . Also, we will view how students can discover math principles on their own.

Would you want your students to eat the whole hamburger in one bite?

Give them a bite at a time—only enough that they can handle.

Information overload = limited learning

A major difference between arithmetic and algebra

In arithmetic we say: •There are 10 pigs•There are 10 animals

In algebra we say: •There are 8 little pigs and 2 big pigs• There are 8 brown pigs and 2 black pigs

What’s the difference between the two?

Misinformed concepts our students bring to class

6 – 9 =What were we taught in first grade about this?



Which is better to tell your students?:“This can’t be done,”

or“You will learn how to do this later”.



3 2+ =





3 2+ =



5



3 2+ =



5

Bad habits create future problems for everyone.

4

Use errors to benefit your students—Have THEM correct you.

81

84

+ =85

52

52

+ =5

41

42

+ =43

73

72

+ =84

Remember, whatever your students discover, they own.

Also, let your students observe some things on their own so they can discover principles for themselves.7

5

Try to figure out the pattern

x1

x4

+ =x5


81

84

+ =85



4

81

84

+ =85

52

52

+ =5



4

81

84

+ =85

52

52

+ =5

41

42

+ =43



4

81

84

+ =85

52

52

+ =5

41

42

+ =43


x1

x4

+ =x5


4

81

84

+ =85

52

52

+ =5

41

42

+ =43

73

72

+ =84


x1

x4

+ =x5


4

81

84

+ =85

52

52

+ =5

41

42

+ =43

73

72

+ =84


x1

x4

+ =x5


4

81

84

+ =85

52

52

+ =5

41

42

+ =43

73

72

+ =84

75


x1

x4

+ =x5


4


81

84

+ =85

52

52

+ =5

41

42

+ =43

73

72

+ =84

75


x1

x4

+ =x5


4


81

84

+ =85

52

52

+ =5

41

42

+ =43

73

72

+ =84


75


x1

x4

+ =x5


4


81

84

+ =85

52

52

+ =5

41

42

+ =43

73

72

+ =84


Also, let your students observe some things on their own so they can discover principles for themselves.7

5


x1

x4

+ =x5


Now let’s take a look at some basic geometry

point

dot

dotA dot is not a point.

dot

A dot is a shape that has height, length, or width.

A dot is not a point.

line


line

Here is a pencil tracing on the line.


line

It is here.


line

NOW let’s look at the piece of line under a magnifying glass.


line

A line is also an undefined term in Geometry.We know what it is but we can’t put it in words

NOW let’s look at this piece of line under a magnifying glass.


line

A line is also an undefined term in Geometry.Which means that we know what it is but we can’t put it in words

NOW let’s look at this piece line under a magnifying glass.

Click

Click

Here we have a line.


A B

Here I have drawn 2 points on the line.


A B

A Line segment is a piece of a line

between two points.

Any piece of line between points A and B is a called a Line Segment


Here we have a line.


P R

Here I have drawn 2 points on the line.


P

Notice that we have a piece of a line starting from a point and moving toward an endless direction--the right.

R


P REnd point

This is point P---- The endpoint of the piece of line that is moving toward an endless direction.


Here we also have a piece of a line that starts from point R and also goes toward an endless direction—the left .

P R


Notice that the end point here is at point R.

PR End point


In math, a piece of a line that starts from the end point and goes toward an endless direction is called a

P R End point


ray

In math, a piece of a line that starts from the end point and goes toward an endless direction is called a .

P R End point


ray

In math, a piece of a line that starts from the end point and goes toward an endless direction is called a .

P R End point


Look at the pattern above. How do we name a ray?

P REnd point

PRRP

RP

This is read-- ray PR.

This is read-- ray RP.

End point

ab

cd

ef

Help me find the line segments and the rays.

g

h

i

c

ab

cd

ef

Help me find the line segments and the rays.

g

h

i

Line segment

Line segment

Line segment

Line segment

ray

ray

ray

ray

c

We are angles. What two thing are we made up of?

RayRayRay

Line segment

Line segment

Line segment

Line segment

Line

seg

men

tpy a Line segment

Line segment

What do you notice about us?

vertex

vertex

vertex

side

side

sideside

side

side

py a

vertex

side side

side

side

vertex

What do you notice?

angle cag or angle gac

a

c

g

fd

j

angle fdj or angle jdf

y

w c

cf

g

angle ywc or angle cwy

angle gcf or angle fcg

ᴧ cag ᴧ cagor

ᴧ ywc ᴧ cwy or

ᴧ fdj ᴧ jdfor

What do you notice?

angle aa f

angle 11

3

angle f

angle 3

What do you notice?

angle a =

a f

angle 11

3

angle f

angle 3

ᴧ a

angle 1 = ᴧ 1 angle 3 = ᴧ 3

Angle f = ᴧ f

These angles have one thing in common. What is it?

Hint: Try naming them.

Compare these two angles?

fa

Compare these two angles?

fa

Here we must let the students tell you what they think. Remember whatever they discover they own.

fa

Compare these two angles.


Discuss with your students the following: How are angles measured?

Which of these two angles is larger?

a f

1st Simple Rule

1

23

4

h

r

f

5

py

f

g

m t

2

Angles that look like these are

called right angles.

Find the right angles

y

p cd

r

m

What does perpendicular mean?

Define perpendicular.

py

gt

t

1p y

90

90

90

g αr m

90

a 45

All right angles measure . 90

py

3

t

t

p y

gα

r ma

acute angles

right angles

py

fr

f

yCompare these two sets of angles?

py

3

tt

p y

g αr m

aobtuse angles

right angles

py

fr

f

yCompare these two sets of angles?

py

tt

t

p

y

g αr m

aobtuse angles

straight angles py

fr

f

y

Compare these two sets of angles?

py

a

r

r1

23

m

Angle in the red and angle in the yellow are adjacent angles. Angle in the yellow and angle in the blue are adjacent angles. Angle in the blue and angle in the green are adjacent angles. Angle in the green and angle in the purple are adjacent angles. Angle in the purple and angle p are adjacent angles. Angle in the p and angle red are adjacent angles.

t

y m

Some mathematicians call 1 and 2 opposite angles. Other call them vertical angles.Why do you think they are called opposite angles?

r 1

f

2

ᴧ

ᴧ

tt

py

m

On the figure to the left, angles 1 and 2 are opposite angles. On the figure to the right angles 3 and 4 are not opposite angles. Why not?

r r1

3

f

2 4

y

mt r c

m

On the figure to the left, we have line m and line y cut by a transversal t. On the figure to the right, we have line c and line m cut by a transversal r.

y

mtr c

m

gd

p

1 234

5 678

1 234

5 678

1 234

5 687

In Fig 1 angles 3, 4, 5, and 6 are exterior angles

Fig 1Fig 2

Fig 3In Fig 2 angles 1, 2, 7, and 8 are exterior angles

In Fig 3 angles 2, 3, 5, and 8 are interior angles

Thank you for joining us. I hope this module has helped you.

Documents

Teaching Students to Discover Math Anthony Chan. What product is sold by math teachers that most of their customers don’ t like or don’t want but must