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SMSG Della Penna, Tricia Glatz, Ryan House, Dan

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S CHOOL OF M ATHEMATICS S TUDY G ROUP 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

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O BJECTIVES OF THE P ROJECT 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

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I MPROVING 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.

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C URRICULUM D EVELOPMENT 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.

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T HE A UDIENCE 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.

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A DDITION OF R EAL N UMBERS 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

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L OOKING AT THE S ALES D ATA The chart shows the profit and loss over two day periods.

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F INDING 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.

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T HE P ROCESS 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

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C ONTINUED.

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F LUENCY 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)

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SMSG S UBTRACTION OF I NTEGERS - 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 $100 cashier gives back $17 by using additive inverse 83 + x = 100 83 + (-83) + x = 100 + (-83) x = 17

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SMSG D EFINITION OF SUBTRACTION

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SMSG P ROPERTIES OF S UBTRACTION 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.

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SMSG S UBTRACTION AS D ISTANCE 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

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SMSG S UBTRACTION AS D ISTANCE While subtraction is used for distance, usually sign is not important so we only use positive values. |a – b|

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Multiplication Let’s consider a chart… 3 x 3 = 9 3 x 2 = 6 3 x 1 = 3 3 x 0 = 0 Because of our studies with the number line and integers, we know that we can multiply 3 by (-1) and subtract 3 from zero

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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.

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Let’s look at another one, Consider the following chart… -3 x 2 = -6 -3 x 1 = -3 -3 x 0 = 0 Again, 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) = 9 Here we see the result of a negative times a negative is a positive integer.

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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

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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.

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Division We 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

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