Teaching fractions and

geometry according to

the CCSSM

Selden, NY

April 25, 2015

H. Wu

Part 1. Definition and the addition of fractions.

Part 2. Congruence and similarity (geometry of grade

8 and high school).

Recall the basic mathematical requirements:

(1) Every concept is clearly defined.

(2) Every statement is precise about what is true and

what is not true.

(3) Every statement is supported by reasoning.

(4) Mathematics is presented as a coherent story.

(5) A purpose is given to each skill and concept.

Part 1. Definition of a fraction

We first introduce the number line.

On a horizontal line, let two points be singled out.

Identify the point to the left with 0 and the point to

the right with 1. This segment, denoted by [0,1] is

called the unit segment and 1 is called the unit.

0 1

Now mark off equidistant points to the right of 1 as on

a ruler, as shown, and identify the successive points

with 2, 3, 4, . . . .

0 1 2 3 4 5

This line, with a sequence of equidistant points on the

right identified with the whole numbers, is called the

number line.

Intuitive discussion. Suppose “the whole” is taken to

be the length of the segment [0,1] on the number line

(or any of [1,2], [2,3], etc.).

Divide [0,1] into three equal parts (= three segments

of equal length). The part adjoining 0 is a third.

Denote its right endpoint by 13.

0 1 2 3 4 5

13

Fix the distance between 0 and 13. Marking off equidis-

tant points to the right of 13 as we would with whole

numbers, we obtain a sequence of points, denoted by

23, 3

3, 43, etc.

0 1 2 3 4 5

13

23

33

43

53

63 etc.

The segment [0, 13] may as well be identified with its

right endpoint, 13. Similarly, the segment [0, 2

3] may as

well be identified with its right endpoint, 23, etc.

Each of these parts-of-a-whole (in the context of thirds)

is now replaced by a point on the number line.

Two-thirds can be replaced by the 2nd point to the

right of 0, denoted by 23.

Seven-thirds can be replaced by the point that is the

7th point to the right of 0, denoted by 73.

m-thirds can be replaced by the mth point to the right

of 0, denoted by m3 .

The sequence of thirds should remind you of the se-

quence of whole numbers.

The only difference: for the whole number sequence,

we start with 0 and 1, but for the sequence of thirds,

we start with 0 and 13.

The sequence of thirds is thus entirely analogous to

the sequence of whole numbers.

This is one of several features that exhibit the paral-

lel between the study of whole numbers and that of

fractions. This puts whole numbers and fractions on

the same footing.

Contrary to common misconceptions, fractions are

not essentially different from whole numbers.

Fractions with denominator equal to 5 are similarly

placed on the number line: 85 is the 8th point to the

right of 0 in the sequence of fifths. And so on.

0 1 205

15

25

35

45

55

65

75

85

We also agree to identify 0n with 0 for any nonzero

whole number n. In this way, all fractions are unam-

biguously placed on the number line.

Intuitively, we have identified parts-of-a-whole with

points on the number line.

A fraction such as 85 is therefore not three things: 8,

and 5, and the “action” of taking 8 out of a division

of the unit segment into 5 equal parts.

Rather, it is one thing: a certain point on the number

line. Every part of the symbol 85 is needed to locate

the position of the fraction on the number line: the

8th point to the right of 0 in the sequence of 5ths.

The importance of recognizing a fraction mn as

a single object cannot be overemphasized.

Formal definition of a fraction: A fraction is one of

the points on the number line as described above.

What does this definition mean?

It means: Any time we explain something about frac-

tions, there is no need to guess what a fraction is (is it

a piece of pizza or is it part of a square?). Everything

we want to say about a fraction can be—and has to

be —realized on the number line.

Part 1. (cont.) Equivalent fractions

We are going to convince students that:

1

3=

2× 1

2× 3=

2

6,

5

3=

2× 5

2× 3=

10

6,

2

5=

3× 2

3× 5=

6

15, etc.

The general statement is:

Theorem on equivalent fractions. Given any frac-

tions k` and a nonzero whole number c, then:

k

`=

c k

c `

i.e., the two fractions k` and c k

c ` are the same point

on the number line.

Let us explain why 73 = 14

6 .

Here is the common explanation from TSM:

7

3= 1×

7

3=

2

2×

7

3=

2× 7

2× 3=

14

6

Here are two possible reactions by students:

(1) So 22 ×

73 = 2×7

2×3. Fractions are so simple! I can

now add fractions!2

2+

7

3=

2 + 7

2 + 3

Here are two possible reactions by students:

(1) So 22 ×

73 = 2×7

2×3. Fractions are so simple! I can

now add fractions!2

2+

7

3=

2 + 7

2 + 3

(2) I am supposed to know that 44 ×

23 = 4×2

4×3 ? I have

just learned that 44 and 2

3 are pieces of pizzas, and

now I am supposed to multiply two pieces of pizza? I

give up.

Let us try again: In order to show that 73 = 14

6 , we

must show that the

7th point to the right of 0 in the sequence of

thirds is also the 14th point to the right of 0 in

the sequence of sixths.

Moral: With a clear-cut definition of a fraction, there

will be no ambiguity about what must be proved.

We divide each of the thirds into 2 equal parts, getting

sixths (2× 3 = 6):

0 1 2 3 4 5

7373

Clearly the 7th point to the right of 0 in the sequence

of thirds is also the 14th point to the right of 0 in the

sequence of sixths.

Another example: prove 56 = 15

18 .

We must show that the 5th point to the right of 0

in the sequence of sixths is also the 15th point to the

right of 0 in the sequence of eighteenths.

We divide each of the sixths into 3 equal parts, getting

eighteenths (3× 6 = 18):

0

56

1

We see that the 5th point to the right of 0 in the

sequence of sixths is also the 15th point to the right

of 0 in the sequence of eighteenths.

Observe that the reasoning for each of the equalities

73 = 14

6 and 56 = 15

18

is the same. So this reasoning proves the Theorem in

general.

Application: Given 27 and 5

4, we can rewrite them

as two fractions with the same denominator:

2× 4

7× 4and

7× 5

7× 4

Generalization: Given two fractions mn and k

` , the

Theorem says we can always rewrite them as two frac-

tions with equal denominators, e.g.,

`m

`nand

kn

`n

This is the FFFP (Fundamental Fact of Fraction

pairs): Any two fractions may be regarded as two

fractions with the same denominator.

Part 1. (concluded) Addition of fractions

First, how do we add whole numbers?

The sum of 4 and 3 is the length of the concatena-

tion of a segment of length 4 and a segment of length

3. (Concatenation: joining an endpoint of one seg-

ment to an endpoint of the other and putting them

on a straight line, as shown.)

u︸ ︷︷ ︸4

︸ ︷︷ ︸3

Now, because whole numbers are also fractions, the

meaning of 45 + 3

5 should not be different from the

addition of whole numbers.

We define the sum 45 + 3

5 to be the length of the con-

catenation of one segment of length 45 and a second

segment of length 35 :

u︸ ︷︷ ︸45

︸ ︷︷ ︸35

In terms of segments of length 15, 4

5 + 35 is just the

length of the concatenation of 4 such segments and

3 such segments, and is therefore exactly 4 + 3 such

segments, i.e.,

4

5+

3

5=

4 + 3

5

The same reasoning allows us to add any two fractions

with the same denominator:

k

n+

m

n=

k + m

n

Observe: the meaning of 45 + 3

5 is not different from

that of 4 + 3. The addition of fractions with the

same denominator is not different from the addition

of whole numbers.

Next, something more complicated: 47 + 2

5. What

does it mean?

We define this sum in exactly the same way: it is the

length of the concatenation of one segment of length

47 and another segment of length 2

5:

u︸ ︷︷ ︸47

︸ ︷︷ ︸25

Therefore, by definition, 47 + 2

5 is the total length of

4 of the 17’s and 2 of the 1

5’s.

This looks forbidding, because the addition of 47 + 2

5

becomes something like adding 4 feet and 2 meters.

But FFFP tells us that there is never any need to face

two fractions with different denominators:

4

7+

2

5=

4× 5

7× 5+

7× 2

7× 5=

34

35

In general, we define the addition of k` and m

n in ex-

actly the same way: k` + m

n is the length of the

concatenation of one segment of length k` and an-

other of length mn :

u︸ ︷︷ ︸k`

︸ ︷︷ ︸mn

By FFFP,

k

`+

m

n=

kn

`n+

`m

`n=

kn + `m

`n

Using LCD:

1

6+

5

8=

(4× 1) + (3× 5)

24=

19

24

Without using LCD:

1

6+

5

8=

8× 1

48+

6× 5

48=

38

48

Same answer (of course).

By the definition of concatenation, addition of frac-

tions is commutative and associative, e.g.,

k

`+

m

n=

m

n+

k

`

This is something you cannot easily prove using LCD.

LCD is a distraction. It should be brought up only as

a specialized skill, not as part of the definition of the

addition of fractions.

The continuity from whole numbers to fractions is of

critical importance in the learning of fractions.

The continuity lightens the cognitive load: students

have less to learn.

It also enhances their incentive to learn: they see that

what they learned about whole numbers is still valid

and has an immediate payoff.

Looking back:

(1) Every concept is clearly defined.

(2) Every statement is precise about what is true and

what is not true.

(3) Every statement is supported by reasoning.

(4) Mathematics is presented as a coherent story.

(5) A purpose is given to each skill and concept.

Part 2. Congruence and similarity.

The most striking aspect of the school geometry cur-

riculum is the discontinuity from middle school to high

school.

First students are told in K–8 that congruence is same

size and same shape, and that similarity is same shape

but not necessarily the same size.

Middle school students also study rotations, reflec-

tions, and translations for artistic reasons: they learn

about the beauty of symmetries. It is fun.

We have already seen that the definition of similarity

as “same shape and not necessarily the same size” is

fraudulent as mathematics.

The definition of congruence as “same size and same

shape” is no better: Can we use “same size and same

shape” to prove theorems?

Perhaps for these reasons, all that is forgotten in high

school. High school geometry is taught using axioms.

As is well-known, axiomatic geometry is a radical de-

parture from the rest of the school mathematics cur-

riculum.

Whereas up to this point not much reasoning is given,

students are suddenly thrust into an arena where ev-

erything, no matter how trivial or obvious, has to be

proved.

Congruence and similarity are now tied down to tri-

angles and polygons, and only triangles and polygons.

They are only discussed in terms of equal angles and

equal (or proportional) sides.

Such restrictions facilitate the discussion of congruent

and similar rectilinear figures. Congruence and similar-

ity between “curvy figures” like parabolas? Nowhere

to be found.

In other words, the way we teach students about the

two cornerstones of school geometry—congruence and

similarity—is to teach them first as metaphors, and

then as abstractions unrelated to the metaphors.

This can hardly be Exhibit A of good teaching. Not

surprisingly, the rampant nonlearning in geometry classes

became a scandal.

In the 1990’s, textbooks began to appear that taught

geometry by hands-on activities alone, with no proofs.

Are these the only viable alternatives:

• teach axiomatic geometry with proofs but no un-

derstanding and no connection to the rest of the

school curriculum, or

• teach geometry with no proofs?

The CCSSM offer a third alternative.

CCSSM’s approach is roughly the following.

(i) Introduce—informally, using manipulatives such

as transparencies—translations, reflections, rotations,

and dilations in grade 8.

(ii) Use this informal knowledge to define the con-

cepts of congruence and similarity in general, and then

explore elementary facts about congruent and similar

triangles, such as why SAS, ASA are true for congru-

ent triangles and why AA is true for similar triangles.

(iii) Give precise definitions of translations, reflec-

tions, rotations, and dilations in high school and, re-

tracing the steps in grade 8, use them to define con-

gruence and similarity for all geometric figures.

(iv) Prove the basic congruence criteria for triangles

(ASA, SAS, SSS, HL) and the basic similarity crite-

ria for triangles (AA, SAS, SSS). Use these as the

foundation for developing Euclidean geometry.

This approach at least avoids making school geometry

impossible to teach from the beginning.

There is another important consideration: to rescue

the teaching of linear equations in two variables

ax + by = c

TSM never gives any reason why the graph of a linear

equation (in two variables) is a line.

This is because TSM never gives the correct definition

of the slope of a line.

Consequently, TSM forces students to learn the ge-

ometry of linear equations by rote. This is one of the

main reasons why students have trouble learning alge-

bra (V. Postelnicu and C. Greenes, NCSM Newsletter,

Winter 2011-2012).

In TSM, the definition of the slope of a (nonvertical)

line L in the coordinate plane is the following: let

P = (p1, p2) and Q = (q1, q2) be distinct points on L.

Then, the slope of L is the ratio:

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Yes, because:

If A = (a1, a2) and B = (b1, b2) are

two other points on L, then

the slope of L would be:

a2 − b2

a1 − b1.

So which of these ratios should be the slope of L:

p2 − q2

p1 − q1or

a2 − b2

a1 − b1?

This question must be answered if slope is to be

a general property of the line L and not of the two

specific chosen points on L.

It turns out thatp2 − q2

p1 − q1=

a2 − b2

a1 − b1.

The fact thatp2 − q2

p1 − q1=

a2 − b2

a1 − b1is true requires the

proof of similar triangles: 4ABC ∼ 4PQR.

This is exactly why eighth graders, who begin the

study of linear equations in two variables, need some

familiarity with similar triangles in order to learn a

correct definition of slope.

It will be difficult for these eighth graders, not to say

impossible, to solve problems related to slope without

the explicit knowledge that slope can be computed by

choosing any two points that suit one’s purpose.

CCSSM’s departure from the standard school curricu-

lum is most pronounced in the two areas of fractions

and geometry.

I hope I have given you some idea of the reasons

for and benefits of this departure from the standard

school curriculum. At least I hope you appreciate why

the departure is absolutely necessary.

References:

Understanding Numbers in Elementary School Math-

ematics, American Mathematical Society, 2011.

Also go to https://math.berkeley.edu/˜wu/ for:

Teaching Fractions According to the Common Core

Standards

Teaching Geometry According to the Common Core

Standards

Teaching Geometry in Grade 8 and High School Ac-

cording to the Common Core Standards