# Negatives,Mixed Numbers and Improper Fractions

1. Compute
algebraically. Justify each equality.

(i) Addition in Q

(ii) Addition in Z

(iii) 0 as a fraction

2. Compute
algebraically. Justify each equality.

(i) Fundamental Law

(ii) Multiplication in Z

(iii) Addition in Q

(iiii) Addition in Z

(iv) 0 as a fraction

3. Compute
algebraically. Justify each equality.

by definition of negative numbers.

4. Use fundamental law to show

5. Use fundamental law to show

6. Explain why you cannot use fundamental law to show

.

The negative sign on −4/5 is out in front of the fraction and is not attached to
either the numerator or

denominator. The Fundamental Law of Fractions will only change the numerator and
denominator, it will

never change the negative sign out in front of the fraction.

7. Show algebraically that
equals
Justify each equality.

(a) Definition of Mixed number

(b) Integer in rational form

(c) Fundamental Law

(d) Multiplication in Z

(e) Addition in Q

8. (a) You probably would not have a student convert a
mixed number to an imporoper fraction in the way

you did in problem 7. Explain to a student the standard algorithm to convert a
mixed number to an

improper fraction.

Step 1: Multiply the denominator by the whole number.

Step 2: Add the above product to the numerator.

Step 3: Put the above sum as the numerator of your new fraction, keeping the
denominator the same.

In other words, if is the mixed number then
the improper fraction is

(b) Explain why the algorithm you gave above works.

Let’s consider the mixed number Since c is
the denominator we know a whole is broken into c

pieces. Let’s break each of the A wholes into c pieces, which means the wholes
now have a total of Ac

shaded pieces. The b tells us that there are b shaded pieces in the last whole.
Therefore we have a total

of Ac + b shaded pieces. Therefore we know the numerator is Ac + b. In addition,
each of the wholes

has c pieces in it, so the denominator is still c. Thus we have the fraction

.

9. Explain to a student the standard algorithm to convert an improper fraction
to a mixed number.

Step 1: Divide the denominator into the numerator.

Step 2: The quotient is the whole number.

Step 3: The remainder is the numerator.

Step 4: The denominator remains the same.

10. Explain why the algorithm you gave above works

The denominator tells us how many pieces it takes to make a whole, so dividing
the denominator into the

numerator tells us how many wholes we have. The remainder tells us how many
pieces are left over, which

becomes the numerator. The pieces are still the same size, so the denominator
stays the same.

11. You give Max the problem
Max does the problem in the following way.

3 − 2 = 1

Therefore Max concludes that

(a) Is Max’s answer correct? Yes

(b) Is Max’s method correct? Explain.

Yes, we can rewrite subtraction as addition of negatives, then uses associative
and commutaive properties

of addition. See below.

12. You then give Max the problem
He tries to do it the same way, but gets
confused.

(a) What could be confusing Max?

Max probably got confused because when he subtracts the fractional parts, he
gets a negative.

(b) What can you say to help him?

Using his method he would get 1 for the whole number part and − 3/10 for the
fractional part. So he

need only compute to get the correct
answer.