The IntegersThe IntegersThe IntegersThe IntegersOrderOrder

Think about the game!

• When working a problem on paper, think about what it means in the MathGym-1D game.

• Which direction does the answer move something? (Is answer is positive or negative?)

• How far does it move it? (What is the size of the answer? )

Order

• Order has to do with what comes first. 5 comes after 3 because it is 2 more than 3. We write either 5 > 3, read as “5 is greater than 3”

or 3 < 5, read as “3 is less than 5”

• Likewise, -3 comes after -5 since it is 2 more than -5, so

-3 > -5 and -5 < -3

Order

-2-3 0-1 1 2 3

Definition:x < y means y – x is positivex < y and y > x mean the same

thing

... ... 3 2 1 0 1 2 3

Do not get order mixed up with size.-5 < -3, but size-wise |-5| > |-3|

On the number line, if x is to the left of y, then x < y.

True or False!

T 7 < 12 12 – 7 = 527) – (-109) = 2

True or False!

T 7 < 12 12 – 7 = 5

True or False!

T 7 < 12 12 – 7 = 5

True or False!

F 3 < -6 (-6) – 3 = --27) – (-109) =

True or False!

F 3 < -6 (-6) – 3 = -9

True or False!

F 3 < -6 (-6) – 3 = -9

True or False!

T -107 > -105 (-107) - (-105) = -(-27) –

True or False!

T -107 > -105 (-107) - (-105) = -2

True or False!

F -107 > -105 (-107) - (-105) = -2

True or False!

T -27 > -109 (-27) – (-109) = 82

True or False!

T -27 > -109 (-27) – (-109) = 82

True or False!

T -27 > -109 (-27) – (-109) = 82

Sort the numbers in increasing order

4, -5, -7, 3, 2, -1, 0

All the negatives are less than the positives with 0 in between.

-5, -7, -1, 0, 4, 3, 2

Now sort the positives in increasing size and the negatives in the decreasing size

-7, -5, -1, 0, 2, 3, 4

Sort the numbers in increasing order

4, -5, -7, 3, 2, -1, 0

All the negatives are less than the positives with 0 in between.

-5, -7, -1, 0, 4, 3, 2

Now sort the positives in increasing size and the negatives in the decreasing size

-7, -5, -1, 0, 2, 3, 4

Sort the numbers in increasing order

4, -5, -7, 3, 2, -1, 0

All the negatives are less than the positives with 0 in between.

-5, -7, -1, 0, 4, 3, 2

Now sort the positives in increasing size and the negatives in the decreasing size

-7, -5, -1, 0, 2, 3, 4

Prove it!Theorem If a < b and b < c, then a < c

Proof:b – a > 0 and c – b > 0, so(c – b) + (b – a) > 0, butc – a = (c – b) + (b – a)

Prove it!Theorem If a < b and b < c, then a < c

Proof:b – a > 0 and c – b > 0.The sum of positives is positive, so(c – b) + (b – a) > 0. Butc – a = (c – b) + (b – a).So c – a > 0.

Therefor a < c.

Prove it!Theorem If a < b and b < c, then a < c

Proof:b – a > 0 and c – b > 0.The sum of positives is positive. So (c – b) + (b – a) > 0. Butc – a = (c – b) + (b – a).So c – a > 0.

Therefor a < c.

Prove it!Theorem If a < b and b < c, then a < c

Proof:b – a > 0 and c – b > 0.The sum of positives is positive. So (c – b) + (b – a) > 0. Butc – a = (c – b) + (b – a).So c – a > 0.

Therefor a < c.

Prove it!Theorem If a < b and b < c, then a < c

Proof:b – a > 0 and c – b > 0.The sum of positives is positive, so(c – b) + (b – a) > 0. Butc – a = (c – b) + (b – a).So c – a > 0.

Therefor a < c.

Prove it!Theorem If a < b and b < c, then a < c

Proof:b – a > 0 and c – b > 0.The sum of positives is positive, so(c – b) + (b – a) > 0. Butc – a = (c – b) + (b – a).So c – a > 0.

Therefor a < c.

Prove it!Theorem If a < b and b < c, then a < c

Proof:b – a > 0 and c – b > 0.The sum of positives is positive, so(c – b) + (b – a) > 0. Butc – a = (c – b) + (b – a).So c – a > 0.

Therefor a < c. TA DAH