Transcript (2 + 3) + 4 = 2 + (3 + 4)

Eva Math Team
th
th
6 -7
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.
Integers
A 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 …
Negative Direction
-7 -6 -5 -4 -3 -2 -1 0 1
Positive Direction
2 3
4
5
6
Graphing Integers on a number line
1) Draw a number line
-7 -6 -5 -4 -3 -2 -1 0 1
2 3
4
5
6
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
Graphing Integers on a number line
1) Graph -7, 0, and 5
-7 -6 -5 -4 -3 -2 -1 0 1
2 3
4
5
6
2) Graph -4, -1, and 1
-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.
-7 -6 -5 -4 -3 -2 -1 0 1
2 3
4
5
6
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
Adding Integers using the
Number Line
Example 1
Adding Integers Using a Number Line
Use a number line to find the sum.
a.
– 6 + 10
b.
–4 + ( – 3)
SOLUTION
a. Start at 0. Move 6 units to the left. Then move 10
units to the right.
ANSWER
The final position is 4. So,
– 6 + 10 = 4
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
Score Keeping: 5 points, 3 points, 1 point
Use a number line to find the sum.
1.
–8 + 4
ANSWER
–4
2.
–1 + ( – 6)
ANSWER
–7
3.
9 + ( – 3)
ANSWER
6
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
55
55
00
2 3
4
5 6
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) -10
2) -35
3) 12
4) 100
5) 1
6) X
Properties
Addition
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: 719
We can rearrange the numbers in our addition to make the
addition easier to compute. It’s easier to first compute
472 + 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 properties
to just go ahead and rearrange a sum in whatever way is
best.
Multiplication
Properties
Using the two pictures below, explain why
2x3=3x2
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)
··
··
··
··
··
··
··
··
··
··
··
··
··
··
··
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··
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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)
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 INTEGERS
Words
Change the
subtraction sign
to an addition
sign and change
the sign of the
second number.
Numbers
Algebra
2 – 3 = 2 + (–3)
4 – (–5) = 4 + 5
a – b = a + (–b)
a – (–b) = a + b
Additional Example 1: Subtracting Integers
Subtract.
A. –7 – 4
–7 – 4 = –7 + (–4)
= –11
Add the opposite of 4.
Same sign; use the sign of the
integers.
B. 8 – (–5)
8 – (–5) = 8 + 5
= 13
Add the opposite of –5.
Same sign; use the sign of the
integers.
C. –6 – (–3)
–6 – (–3) = –6 + 3
= –3
Add the opposite of –3.
6 > 3; use the sign of 6.
Try This: Example 1
Subtract.
A. 3 – (–6)
3 – (–6) = 3 + 6
=9
Add the opposite of –6.
Same signs; use the sign of the
integers.
B. –4 – 1
–4 – 1 = –4 + (–1)
= –5
Add the opposite of 1.
Same sign; use the sign of the
integers.
C. –7 – (–8)
–7 – (–8) = –7 + 8
=1
Add the opposite of –8.
8 > 7; use the sign of 8.
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:
−105
7
Answer:
-15
Evaluate each expression:
−4
−1
Answer:
4
Evaluate each expression:
−10
−2
Answer:
5
Evaluate each expression:
−144
12
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 pair
the 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