Eva Math Team6th-7th Grade Math

Pre-Algebra

1. Relate and apply concepts associated with integers (real number line, additive inverse, absolute value, compare and order integers).

2. Perform calculations involving addition and subtraction of integers.

3. Perform calculations involving multiplication and division with integers.

Number Line

The number line is shown above. It goes on forever in both directions. Every number that we will consider is somewhere on the number line. The tick marks on the number line above indicate the integers. An integer is a number without a fractional part: …-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7…

The symbol … at either end of the above list means that the list goes on forever in that direction.The … symbol is called an ellipsis. However, as you know, there are many numbers on the number line other than integers. For instance, most fractions such as 1/2 are not integers.

IntegersA number is called negative if it is to the left of 0 on the number line. That is,

a number is negative if it is less than 0. A number is called positive if it is to the right of 0 on the number line. In other words, a number is positive if it is greater than 0. For example, 2 is positive, while -2 is negative. Note that 0 itself is neither positive nor negative, and that every number is either positive, negative, or 0.

... -3, -2, -1, 0, 1, 2, 3 …

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Negative Direction Positive Direction

Graphing Integers on a number line

1) Draw a number line

2) Graph an Integer by drawing a dot at the point that represents the integer.

Example: -6, -2, and 3

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Graphing Integers on a number line

1) Graph -7, 0, and 5

2) Graph -4, -1, and 1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Order Integers from Least to Greatest

• You need to know which numbers are bigger or smaller than others, so we need to order them from least to greatest.

Example: Order the integers -4, 0, 5, -2, 3, -3 from least to

greatest.

The order is -4, -3, -2, 0, 3, 5.

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Order Integers from least to greatest

1) Order the integers 4,-2,-5,0,2,-1 from least to greatest.

2) Order the integers 3,4,-2,-5,1,-7 from least to greatest.

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Adding Integers using the Number Line

Example 1 Adding Integers Using a Number Line

Use a number line to find the sum.

a. 10+6– b. +4– ( )3–

ANSWER The final position is 4. So, 10+6– = 4

SOLUTION

a. Start at 0. Move 6 units to the left. Then move 10 units to the right.

Example 1 Adding Integers Using a Number Line

b. Start at 0. Move 4 units to the left. Then move 3 units to the left.

ANSWER The final position is -7. So, =+4– ( )3– 7–

Competition Problems

Use a number line to find the sum.

Score Keeping: 5 points, 3 points, 1 point

1. +8– 4 ANSWER 4–

ANSWER 7–2. +1– ( )6–

ANSWER 63. +9 ( )3–

Integers and Absolute Value

Objectives: Compare integers.Find the absolute value of a number

Absolute Value

• Absolute value of a number is the DISTANCE to ZERO.

• Distance cannot be negative, so the absolute value cannot be negative.

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

5 5

5 5

0 0

Using Absolute Value in Real Life

• The graph shows the position of a diver relative to sea level. Use absolute value to find the diver’s distance from the surface.

Competition Problems!

Find the absolute value of a number…

Absolute Value

Evaluate the absolute value:Ask yourself, how far is the number from zero?1) | -4 | =2) | 3 | =3) | -9 | =4) | 8 - 3 | =

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Absolute Value

Evaluate the absolute value:Ask yourself, how far is the number from zero?1) | 12 ÷ 4 | =2) | 3 ● 15 | =3) | -9 + 1 | - │1 + 2│ =4) | 8 - 3 | + │20 - 20│=

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Opposites

• Two numbers that have the same ABSOLUTE VALUE, but different signs are called opposites.

Example -6 and 6 are opposites because both are 6 units away from zero.

| -6 | = 6 and | 6 | = 6

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Competition Problems!

Find the opposite of a number…

Opposites

What is the opposite?1) -102) -353) 124) 1005) 16) X

PropertiesAddition

Multiplication

Addition Properties

Using the two pictures below, explain why

2 + 3 = 3 + 2

First let’s look at the picture on the left. The first row has 2 squares; the second row has 3 squares. So in total, there are 2 + 3 squares. Now let’s look at the picture on the right. The

first row has 3 squares; the second row has 2 squares. In total, there are 3 + 2 squares. The picture on the right, however, is just an upside-down version of the picture on the left.

Flipping a picture upside down doesn’t change the number of squares. So we conclude that 2 + 3 = 3 + 2.

Whenever we add two numbers, the order of the numbers does not matter. Therefore,

Addition is commutative: Let a and b be numbers. Then

a + b = b + a

Using the two pictures below, explain why(2 + 3) + 4 = 2 + (3 + 4)

Let’s start with the picture on the left. It has (2 + 3) light squares and 4 dark squares. So altogether it has (2 + 3) + 4 squares. Next, let’s look at the picture on the right. It has 2 light

squares and (3 + 4) dark squares. So altogether it has 2 + (3 + 4) squares. What’s the difference between the two pictures? The only difference is the color of the middle row.

Changing the color doesn’t change the number of squares. So we conclude that(2 + 3) + 4 = 2 + (3 + 4).

We get a similar equation for any three numbers: (a + b) + c = a + (b + c). In other words, first adding a and b and then adding c is the same as adding a to b + c. This property is called the

associative property of addition.Important: Addition is associative: Let a, b, and c be numbers. Then

(a + b) + c = a + (b + c)

Find the sum:

472 + (219 + 28)

Answer: 719We can rearrange the numbers in our addition to make the

addition easier to compute. It’s easier to first compute472 + 28, and then compute 500 + 219, than it would have been to start with 219 + 28 and then add that sum to 472.In a similar way, any addition problem can be rearranged without changing the sum. Usually we won’t bother to

write all the individual steps of the rearrangement, like we did in the previous problem. Instead, we’ll use our

knowledge of the commutative and associative propertiesto just go ahead and rearrange a sum in whatever way is

best.

MultiplicationProperties

Using the two pictures below, explain why

2 x 3 = 3 x 2

First let’s look at the picture on the left. The first row has 3 squares; the second row has 3 squares. So in total, there are 2 x 3 squares. Now let’s look at the picture on the right. The first row has 2 squares; the second row has 2 squares, the third row has 2 squares. In total,

there are 3 x 2 squares.

Flipping a picture upside down doesn’t change the number of squares. So we conclude that 2 x 3 = 3 x 2.

Whenever we add two numbers, the order of the numbers does not matter. Therefore,

Multiplication is commutative: Let a and b be numbers. Then

a · b = b · a

Using the picture below, explain why(2 · 3) · 4 = 2 · (3 · 4)

····

····

····

····

····

····

The picture has 2 rows of 3 squares each and 4 dots in each square. So altogether it has (2 · 3) · 4 dots. Next, let’s look at 2 · (3 · 4). It has 4 dots in each square and 3 squares in each row. There is 2 rows. So altogether it has 2 · (3 · 4) dots. Changing the order doesn’t change the

number of dots. So we conclude that (2 · 3) · 4 = 2 · (3 · 4)

We get a similar equation for any three numbers: (a · b) · c = a · (b · c). In other words, first multiplying a and b and then multiplying c is the same as multiplying a to b · c. This property

is called the associative property of multiplication.Important: Multiplication is associative: Let a, b, and c be numbers. Then

(a · b) · c = a · (b · c)

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Let’s look at more properties…

Additive Inverse(or negation property)

-1 + 1 = 0 -2 + 2 = 0-3 + 3 = 0 -4 + 4 = 0

-0.5 + 0.5 = 0

The sum of any number (a) and its inverse (-a) is zero (0)

-a + a = 0

Properties of Arithmetic…

Addition, Subtraction, Negation

Learn to subtract integers.

Subtracting a smaller number from a larger number is the same as finding how far apart the two numbers are on a number line. Subtracting an integer is the same as adding its opposite.

SUBTRACTING INTEGERSWords Numbers Algebra

Change the subtraction sign to an addition

sign and change the sign of the

second number.

2 – 3 = 2 + (–3)4 – (–5) = 4 + 5

a – b = a + (–b)a – (–b) = a + b

Additional Example 1: Subtracting Integers

A. –7 – 4

–7 – 4 = –7 + (–4)

B. 8 – (–5)

8 – (–5) = 8 + 5

C. –6 – (–3)

–6 – (–3) = –6 + 3

= –11

= 13

= –3

Add the opposite of 4.

Add the opposite of –5.

Add the opposite of –3.

Same sign; use the sign of the integers.

Same sign; use the sign of the integers.

6 > 3; use the sign of 6.

Subtract.

Try This: Example 1

A. 3 – (–6)

3 – (–6) = 3 + 6

B. –4 – 1

–4 – 1 = –4 + (–1)

C. –7 – (–8)

–7 – (–8) = –7 + 8

= 9

= –5

= 1

Add the opposite of –6.

Add the opposite of 1.

Add the opposite of –8.

Same signs; use the sign of the integers.

Same sign; use the sign of the integers.

8 > 7; use the sign of 8.

Subtract.

Lets work some math problems!

3, 2, 1…Go!

Find the sum:

(-6) + 12

Answer:

6

Find the sum:

(-12) + 7

Answer: -5

(-12) + 7 (-5 +(-7)) +7 associative property

-5 +((-7)+7) negation property

-5 + 0 adding 0

-5

Find the sum:

(-10) + (-7)

Answer:

-17

Find the difference:

(-1) -10

Answer:

(-1) -10 (-1) + (-10)

-11

Find the difference:

8 - 7

Answer:

1

Find the difference:

(-38) - 30

Answer:

-68

Find the difference:

18 - 41

Answer:

b-a = -(a-b)18-41 = -(41-18)

= -(23)

-23

Evaluate each expression:

(-10) - 47

Answer:

-57

Evaluate each expression:

(-29) - 29

Answer:

-58

Evaluate each expression:

13 + (-29)

Answer:

-16

Evaluate each expression:

38 + 22

Answer:

60

Evaluate each expression:

(-32) - 44

Answer:

-76

Evaluate each expression:

(-12) + (-11)

Answer:

-23

Evaluate each expression:

16 +(−13) +5

Answer:

8

Simplify:

−12 + 8 + (−6)

Answer: −12 + 8 + (−6)

-4 + (-6)-10

Simplify:

12 − 37 + 19

Answer: 12 − 37 + 19 12 + 19 - 37

31 – 37-6

Simplify:

–53 + (32 – 47)

Answer: –53 + (32 – 47)

-53 + (-15)-68

Simplify:

|15 - 18| + |-7 + 5|

Answer:

|15 - 18| + |-7 + 5|

|-3| + |-2| 3 + 2

5

Which statement is true?

A. |–9| = –9

B. –9 > |–9|

C. –5 > |–9|

D. |–5| < –9

E. NOTA

Which statement is true?

A. |–9| = –9

B. –9 > |–9|

C. –5 > |–9|

D. |–5| < –9

E. NOTA

CHALLENGE PROBLEM:

What is the difference of their elevations?(Write an equivalent expression that represent the situation.)

“An airplane flies at an altitude of 26,000 feet. A submarine dives to a depth of 700 feet below sea

level.”

Answer:

26,000 – (-700)=26,700

Multiplying and Dividing integers

Lets work some math problems!

3, 2, 1…Go!

Evaluate each expression:

6 × −4

Answer:

−24

Evaluate each expression:

4 × 2

Answer:

8

Evaluate each expression:

5 × −4

Answer:

-20

Evaluate each expression:

−2 × −1

Answer:

2

Evaluate each expression:

−8 × −2

Answer:

16

Evaluate each expression:

11 × 12

Answer:

132

Evaluate each expression:

−12 × 7

Answer:

-84

Evaluate each expression:

8 × −12

Answer:

-96

Evaluate each expression:

9 × 10 × 6

Answer:

540

Evaluate each expression:

−6 × −10 × −8

Answer:

-480

Evaluate each expression:

7 × 9 × 7

Answer:

441

Evaluate each expression:

9 × 9 × −5

Answer:

-405

Evaluate each expression:

7 × 5 × −5

Answer:

-175

Evaluate each expression:

−5 × −4 × −10

Answer:

-200

Evaluate each expression:

8 ÷ 4

Answer:

2

Evaluate each expression:

12 ÷ 4

Answer:

3

Evaluate each expression:

35 ÷ −5

Answer:

-7

Evaluate each expression:

−24 ÷ 4

Answer:

-6

Evaluate each expression:

−24 ÷ 8

Answer:

-3

Evaluate each expression:

−21 ÷ 7

Answer:

-3

Evaluate each expression:

−8 ÷ −2

Answer:

4

Evaluate each expression:

−132 ÷ −11

Answer:

12

Evaluate each expression:

−60 ÷ −15

Answer:

4

Evaluate each expression:

−52 ÷ −4

Answer:

13

Evaluate each expression:

75 ÷ 15

Answer:

5

Evaluate each expression:

65 ÷ −13

Answer:

-5

Evaluate each expression:

−168 ÷ −12

Answer:

14

Evaluate each expression:

−1057

Answer:

-15

Evaluate each expression:

−4 −1

Answer:

4

Evaluate each expression:

−10−2

Answer:

5

Evaluate each expression:

−14412

Answer:

-12

Evaluate each expression:

24−12

Answer:

-2

Evaluate each expression:

60 (−15)

Answer:

-4

Using the commutative and associative

properties

Find the sum:

(2 + 12 + 22 + 32) + (8 + 18 + 28 + 38)

We could start with 2, then add 12, then add 22, and so on, but that’s too much work. Instead, let’s try to rearrange the sum in a useful way. Let’s pair up the numbers so that each pair has the same sum. Specifically, let’s pair each number

in the first group with a number in the second group:

(2 + 38) + (12 + 28) + (22 + 18) + (32 + 8):

The first pair of numbers adds up to 40; so does the second pair, the third pair, and the fourth pair. So our sum becomes

40 + 40 + 40 + 40:

The answer is 160.

Find the sum:

1 + 2 + 3 + … + 18 + 19 + 20

We definitely don’t want to add the 20 numbers one at a time. Instead, let’s try again to rearrange the numbers into pairs, so that

each pair has the same sum. We pairthe smallest number with the largest, the second-smallest with the

second-largest, and so on:

(1 + 20) + (2 + 19) + (3 + 18) + … + (10 + 11)

We have grouped the 20 numbers into 10 pairs. Each pair adds up to 21. So our sum becomes

21 + 21 + 21 + 21 + 21 + 21 + 21 + 21 + 21 + 21

Adding 10 copies of 21 is the same as multiplying 10 and 21. So the answer is 210.

Find the product:

25 · 125 · 4 · 6 · 8

We can rearrange the numbers to make the multiplication easier to compute. Let’s try to rearrange the numbers in a

useful way:

(25·4) · (125·8) · 6

The first pair of numbers product is 100; the second product is 1000; and then you multiply by 6. So our product

becomes 100 · 1000 · 6

The answer is 600,000.

Find the sum of the numbers 1 to 100

1+2+3+4+5+6+7+…+94+95+96+97+98+99+100

Answer:(use the commutative and associative properties)

(1+100)+(2+99)+(3+98)+(4+97)+…

101+101+101+101+101+…The numbers are paired together, therefore there is

50 pairs. Or, (Last#-1st#) +1 total amount of numbers. This is 100 numbers, then divide by 2

(pairs) = 50 pairs of the number 101.

101(50)=5050