SMSGDella Penna, TriciaGlatz, RyanHouse, Dan

SCHOOL OF MATHEMATICS STUDY GROUP Formed by the cooperation of various

mathematical organizations in the US Included college and university

mathematicians, teachers of all levels, experts in education, and representatives of science and technology

Project initiation: March 1958 Yale University

OBJECTIVES OF THE PROJECT Bring classroom teachers and research

mathematicians together to improve pre-college mathematics curriculum

Foster research and development in the teaching of school mathematics

Developing courses, teaching materials, and teaching methods

IMPROVING THE TEACHING OF MATHEMATICS A prerequisite to this is an improved

curriculum New curriculum should take account of the

increasing use of math in science and technology

The text developed is of most value to “all well-educated citizens in our society to know and that it is important for the precollege student to learn in preparation for advanced with in the field” although the presentation is such that any student can readily grasp the information.

CURRICULUM DEVELOPMENT A mix of the old and the new: Some material was meant to be familiar

where as other material was new to the traditional curriculum

This fused the “old and the new” in hopes to lead students to better understanding in basic concepts, mathematical structure, and firmer foundations.

THE AUDIENCE Material was hoped to awaken an interest in

mathematics in a large group of students. Aimed particularly at students who had

mathematical ability and had yet to realize it.

ADDITION OF REAL NUMBERS To imply the concept of negative numbers,

the section is begun by introducing a situation where we discuss the business ventures of an ice cream salesman over the course of twelve days.

On days where he makes money, we recognize as profit

On days where he loses money, we call it a loss

LOOKING AT THE SALES DATA The chart shows the profit and loss over two

day periods.

FINDING LOSS OR PROFIT To find the loss or profit over the two days we

“put together” the profits and/or losses. To better understand how to add the

positives and negatives, addition with a number line was used.

THE PROCESS Start at zero Move |a| units to the right if a is positive, left

if negative. From new location, move |b| units to the right

if positive, left if negative. This location represents the final position

which is the sum of the two numbers

CONTINUED.

FLUENCY Students were encouraged to think of

addition of integers as profits and losses as they completed work requiring these skills until it becomes second nature.

This process also gave a visual representation of the additive identity (days where the sales man rested) and also the additive inverse (days where he lost what he previously profited)

SMSG SUBTRACTION OF INTEGERS- Introduces subtraction as “adding of the opposite”

or “adding of the additive inverse- Gives an example of a cashier counting back

change for an purchase:

$83 purchase, customer gives $100cashier gives back $17 by using additive inverse

83 + x = 10083 + (-83) + x = 100 + (-83)

x = 17

SMSG DEFINITION OF SUBTRACTION

SMSG PROPERTIES OF SUBTRACTION Shows through examples that if subtraction is the

opposite of adding, whether the properties of adding hold for subtraction.

Not associative, nor commutative Distribution over subtraction holds

When showing examples of subtraction and its properties, algebraic expressions were used.

SMSG SUBTRACTION AS DISTANCE

Shows example of how subtraction is used to find the distance between two integers (or real numbers) using the number line.

a – b and b – a Moving left on the number line means

negative; moving right means positive

SMSG SUBTRACTION AS DISTANCE While subtraction is used for distance, usually sign

is not important so we only use positive values. |a – b|

Multiplication

Let’s consider a chart…3 x 3 = 9

3 x 2 = 6

3 x 1 = 3

3 x 0 = 0Because of our studies with the number line and integers, we know that we can multiply 3 by (-1) and subtract 3 from zero

3 x 3 = 9

3 x 2 = 6

3 x 1 = 3

3 x 0 = 0

So, 3 x (-1) = -3

3 x (-2) = -6

Our observation seems to lead to a conjecture that a positive integer times a negative integer equals a negative integer.

Let’s look at another one,Consider the following chart…

-3 x 2 = -6

-3 x 1 = -3

-3 x 0 = 0Again, we know the pattern continues past zero. This time we add three to our previous product.-3 x (-1) = 3

-3 x (-2) = 6

-3 x (-3) = 9Here we see the result of a negative times a negative is a positive integer.

If the charts weren’t enough to convince you then consider the following…0 = 3 x 0 Based on our studies of “opposites” we can

rewrite zero as follows

0 = 3 x (2 + (-2))By way of the distributive property we obtain

0 = (3 x 2) + (3 x (-2))

Suppose we never looked at the previous charts and didn’t know the product of 3 and (-2).But we do know 3 x 2

0 = 6 + (3 x (-2))

We know the opposite of 6 is -6.Thus, the product of 3 and -2 must be -6.

Using the above method let’s find the product of two negatives

0 = -3 x 0Based on our studies of “opposites” we can rewrite zero as follows

0 = -3 x (2 + (-2))By way of the distributive property we obtain

0 = (-3 x 2) + (-3 x (-2))

Suppose we never looked at the previous charts and didn’t know the product of -3 and (-2).But we do know -3 x 2

0 = -6 + (-3 x (-2)) We know the opposite of -6 is 6.Thus, the product of -3 and -2 must be 6.

DivisionWe know how to divide whole numbers. Dividing integers is the same.

If you know how to multiply integers, then you know how to divide integers

Turn the division problem into a multiplication problem and use the properties we just discovered for multiplication of integers