Improving Numeracy and Algebraic Thinking Project Director: Dr. Steven R. Lay [email protected]“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
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I mproving N umeracy and A lgebraic T hinking Project Director: Dr. Steven R. Lay [email protected] “ INAT ” Partners: Lee University (mathematics.
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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.
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.
The instructional practices and assessments discussed or shown in this presentation are not intended as an endorsement by the U. S. Department of Education