Improving Numeracy andAlgebraic Thinking

Project Director: Dr. Steven R. Lay

slay@leeuniversity.edu

“INAT”

Partners: Lee University (mathematics and education)

Edvantia, Inc.

Bradley County schools, Cleveland City schools

McMinn County schools, Meigs County schools

Monroe County schools, Polk County schools

Sweetwater City schools

INAT: Helping middle school students with the transition from arithmetic to algebra.

• Based on a long-term study of the kinds of errors that students make in algebra.

• Presents a new way of conceptualizing the basic arithmetic operations.

• Teaches how to think algebraically in an arithmetic setting.

This approach is called “Prelude to Algebra.”

• Uses a 2-week summer institute and 4 follow-up sessions in the fall.

What kind of mistakes do students make in

Algebra 1, Algebra 2, (and Calculus)?

signed numbers fractions exponents canceling

solving simple equations

What can be done in arithmetic and pre-algebra courses to prepare the students so that when they go on to later courses they won’t make these basic mistakes?

Prelude to Algebra

We must provide better conceptual models for the basic arithmetic operations.

These models must

• be mathematically correct

• anticipate “algebraic” thinking

• be easily understood

Prelude to Algebra

Here is the basic idea behind our approach:

Two Ways to View Addition

• Static Model • Dynamic Model

Two numbers (5 and 3) are added together to form a third number (5 + 3).

5 + 3 5 + 3

One number (5) is increased by an operator (+ 3) to become a second number (5 + 3).

5 + 3 5 + 3

addition operator

The mental shift in orientation between these two approaches is subtle, but significant.

+ : (5, 3) 5 + 3

+ 3 : 5 5 + 3

Increase operators can be written on the right or the left:

3 is increased by 5:

Operator is 5 +.

Decrease operators can only be written on the right:

7 is decreased by 4:

A study of increases and decreases prepares students for working with signed numbers, but we can do this before introducing signed numbers.

How do Operators “Operate”?

3 + 5 Operator is + 5.

35 +

7 – 4 Operator is – 4 .

Composition of Increases and Decreases

1. Like changes help each other to give a larger composite change of the same kind. Add their magnitudes.

+ 2 + 3 = + (2 + 3) = + 5

– 2 – 3 = – (2 + 3) = – 5

+ a + b = + b + a = + (a + b)

– a – b = – b – a = – (a + b)

How do increases and decreases interact with each other?

Composition of Increases and Decreases

2. Opposite changes work against each other. The larger change will dominate the smaller and determine the kind of composite change. Subtract the smaller magnitude from the larger.

Increase is larger:

+ 7 – 3 = + (7 – 3) = + 4 – 3 + 7 = + (7 – 3) = + 4

Decrease is larger:

– 7 + 3 = – (7 – 3) = – 4 + 3 – 7 = – (7 – 3) = – 4

When a b,

+ a – b = – b + a = + (a – b)

– a + b = + b – a = – (a – b)

Without using signed numbers, we develop the properties of signed numbers in a context that makes sense to students.

Composition of Increases and Decreases

It makes sense to students.

60

8090

70

50

Vector Models

The composition of a decrease of 7 and an increase of 3

– 7

+ 3

– 4

The operator point of view does not require the introduction of new symbols. It provides a new (better) way of thinking about the familiar symbols.

– 7 + 3 = – 4

Multiplication and Division as OperatorsIn the product 3 · 5 or 3(5),

3 · or 3( ) can be viewed as a multiplication operator.

In the quotient , can be viewed as a division operator. 82

( )2

Multiplying by 3 makes things larger,

so we call 3( ) an expansion operator.

Dividing by 2 makes things smaller,

so we call a contraction operator.( )2

A study of expansions and contractions prepares students for working with fractions and rational expressions, and we can do this before introducing fractions.

23 = 2 · 2 · 2

2 cubed is 2 times itself 3 times.

Do we evaluate 21 by multiplying 2 times itself 1 time?

No, this is not correct!

We only multiply twice, not three times.

What does 2n mean?

What does 2n mean?

2 cubed is a product of 2s where there are 3 factors. In general,

2n = 2 2 … 2

n factors

Is 21 a product of 2s with only one factor?

What does 20 mean?

23 = 2 · 2 · 2

The Real Meaning of an Exponent(operator definition)

The exponent counts the number of times 1 is multiplied by the base.

Exponential Number of Operator Computed Form Expansions Model Value

23 three 1 · 2 · 2 · 2 822 two 1 · 2 · 2 421 one 1 · 2 220 none 1 1

Bonus: A negative exponent counts the number of times 1 is divided by the base:

, , , .

-1 -2 -31 1 12 = 2 = 2 = etc

2 2 2 2 2 2

Applications to AlgebraInverses

Addition and subtraction are inverse operations, but the usual addition function

is not one-to-one. It has no inverse.

It is only the operator that has an inverse.

Likewise, to see multiplication and division as inverses of each other, it is necessary to view them as operators.

+ : (m, n) m + n

+ n : m m + n

Applications to AlgebraCancellation

How do you simplify to ? 55

xy

xy Do you “cancel the 5s”?

NO! NO! NO!Numbers do not cancel. Only inverse operators can cancel.

Multiplying by 5 cancels with dividing by 5.

+55

xy

xy ?

2x + 3

Is this a “sum” or a “product”? Why is it a “sum”?

The name of a form comes from the last operation to be performed.

So what??

Applications to AlgebraSolving Equations

Suppose you want to solve the equation

2x + 3 = 10

What do you do first?

You subtract 3 from both sides.

Why do some students want to divide by 2 first?

They see 2x + 3 as a product, not a sum.

The name of the form tell us what operation should be “undone” in solving the equation.

“Building” a Compound Expression

See a compound expression as a sequence of operators applied to a single number.

2(10 4)3

Starting Number: 10

Operators: – 4, 2( ),( )

,3 in that order.

For example,

125 7

4

( ),

4

Why is this important? Suppose we want to solve the equation

5 7 504x

First we divide by 5:( )5

7 104x

Then we subtract 7: – 7 34x

Then we multiply by 4: 4( )

x = 12

We performed the inverse operations in the reverse order.

12; + 7, 5( )

All of this can be done in arithmetic (with positive integers) after teaching the order of operations.

In building compound arithmetic expressions, we are doing arithmetic, not algebra, but we are thinking about the process in an algebraic way.

Even the solving of equations can be practiced in arithmetic without using negative numbers or fractions.

2 8 1 5 3 2 8 1 35

2 8 5 3 1 5 3 182

Proficiency in this kind of arithmetic manipulation greatly helps when solving an algebraic equation like

Solve the equation for 8 and do no computation.2 8 1 35

ax y bc ax yc

b

ax bc y bc yx a

Summary

• Begins in the simplest possible context: positive integers.

• Uses arithmetic to develop algebraic thought patterns.

• Provides better definitions of basic operations.

• Uses operators to build compound expressions.

• Makes sense to students.

• Reduces student errors in

“Prelude to Algebra”

signed numbers fractions exponents canceling

solving simple equations

The Results

• Long term studies have shown that when students go on to the next math class, they are successful 94% of the time.

• At our INAT summer institute, the teachers’ median score on a challenging assessment of pre-algebra topics was raised from a pre-institute score of 20% to a post-institute score of 86%.

• The Prelude approach has been used for more than 20 years in remedial classes at the college level.

Steven R. Lay, Ph.D.

Lee University

slay@leeuniversity.edu

www.preludetoalgebra.com

Prelude to Algebra“making algebra make sense”

The instructional practices and assessments discussed or shown in this presentation are not intended as an endorsement by the U. S. Department of Education