Download Notes and Worksheets for Chapter 6

Chapter 6: Rational Numbers
General Notes:
(worksheets on following pages)
Section 6.1 Negative Exponents, A5 : demonstrate and explain the meaning of negative exponents for base ten
Before starting negative exponents, students need to review on Exponents and exponential notation.
(ie. p. 242 in the white box. 53 = 5x5x5)


#2 and 3 on p.242
sheet <<Powers of 10>> (which is a copy of the tables on p. 246-247 but it has been modified and students do not
have to copy it.)
textbook p.246, students should see the pattern of 101=1 and 102=100 and see the pattern going up that the number of zeros
equals the exponent.
When you do the smaller units, 10-1 = 0.1 and 10-2 = 0.01, etc, students should realize that they can say the exponent
corresponds with the number of spaces after the decimal.
1
) the fractions they place brackets around them and
10
1 2
1 1
1
1
you can show them that (
) means
x
which is
or
and is the same as 10-2.
10
10 10
100 10 2

The second table is from p.247. Make sure when writing (

Also 100 equals 1 because it has “zero zeros”

p. 250 #1,2,3,4,5,6,8,11
Section 6.2 Scientific Notation, A6 : represent any number written in scientific notation in standard form, and vice versa
 sheet « Scientific Notation »
 I do not like how the text presents the concept of changing a scientific notation by multiplying a fraction or
dividing by tens. I think students understand better by multiplying by the decimal equivalent so if the question is
2.45x10-3 then it is 2.45 x 0.001 or move the decimal 3 places to the left.
Section 6.3 Compare and Order Rational Numbers, A7: compare and order integers and positive and negative rational
numbers (in decimal and fractional forms)


sheet « Compare and Order Rational Numbers »
Students should recall that positive numbers are on the right and the negatives are on the left. Larger numbers are
closer to the right or to zero. Negative numbers are reverse to what they think. Think temperature. -10 is colder
than -4.

Rational Number: is a positive or negative decimal or fraction. The decimal must be repeating like 0. 3 or
terminating like 4.5. Decimals that are non-repeating such as pi or
have a rational number with zero for the denominator



2 are not rational. Also, it is impossible to
6
. (Impossible to divide something into groups of 0)
0
1
have students put the whole number and find the decimal so 2 and 1÷5=0.2 so 3.2.
5
1
example with negative : -1 =-1.5
2
For a mixed fraction like 3
1
is -1÷2
2
1
is 1÷-2
2
-
1
is – (1÷2)
2
So all of these are the same; it does not matter where the single negative is but if you write it with the numerator, there is
less chance of forgetting it; a good habit for next year when adding and subtracting negative fractions.
Section 6.4 Operations with Rational Numbers
B12 : add, subtract, multiply, and divide positive and negative decimal numbers with and without the calculator
B13 : solve and create problems involving addition, subtraction, multiplication, and division of positive and
negative decimal numbers
 sheet `Operations with Rational Numbers`


Grade 7 has integer outcomes: adding, subtracting, multiplying and dividing with concrete, pictures and
numerical so it should be a review. However, this year, they can use a calculator after they are
comfortable (B12).
p.274 : 1, 2,3 (do 2 and 3 by estimating not number line or tiles), 5,6,7,8,9 (with a calculator)
Section 6.5 Properties of Operations
B1 : demonstrate an understanding of the properties of operations with integers and positive and negative
rational numbers (in decimal and fractional form)
B13 : solve and create problems involving addition, subtraction, multiplication, and division of positive and
negative decimal numbers
Part 1
 sheet « Order of Operations »
 Order of operations is en essential concept that students need. It is a concept taught in grade 7 but is
worth reviewing.
 The rules are there so everyone will have the same answer no matter where they are in the world.
 Make sure to follow BEDMAS in the brackets too and that if there is a fraction, use BEDMAS in the
numerator and then in the denominator and then divide the answer.
 can use a calculator to work through but must show calculations and not do it all on the calculator. Most
calculators do not do order of operations.
 sheet “Order of Operations Exercises”
Part 2
 sheet “Order of operations with negative numbers and decimals”
 BEDMAS with negatives: Students should realize that brackets around a negative with an exponent
means to repeat the negative (-2)3 means -2x-2x-2. If there were no brackets, then the negative sign
would not be repeated.
 BEDMAS and decimals: Students estimate by rounding off to the nearest whole number before using a
calculator to find the answer. A calculator is fine to use but do not key in the wrong number…by having
an estimate, there is something to compare calculator answer.
 Exercise sheet on “BEDMAS with Negatives and Decimals”
 Once again, make sure all calculations are shown even if using a calculator.
Name:
Date:
Powers of 10 and Negative Exponents
In life, there are numbers that are really large and others that are really small. Often these numbers include many
zeros and instead of writing all the zeros, it is easier to write with powers of 10.
The table below gives suggestions of real measurements and how many meters each object would be in real life.
Object
The diameter of the sun
Unit of measure
Approximate measurement in metres
Gigameter
1 000 000 000 meters
Megameter
1 000 000 meters
Kilometer
1000 meters
Hectometer
100
Decameter
10
Meter
1
Decimeter
0.1
Centimeter
0.01
Millimeter
0.001
Micrometer
0.000 001
Nanometer
0.000 000 001
Exponential form
The diameter of an asteroid
A ten-minute walk
The length of a soccer
field
The length of a
typical classroom
The height of a
doorknob from the
floor
The diameter of a
softball
The diameter of a
marble
The width of a grain of
sugar
The length of a
bacteria
The width of an atom
10-1
Negative Exponents
Fill in the table: change decimals to fractions to exponents.
Fraction with
Exponential
Decimal
Fraction
denominator
form using
exponent
base 10
Name: _______________________________
Class: __________ Date: ________________
Scientific Notation
100
In science, there are great distances like the diameter of the
solar system at approximately 10 000 000 000 000 meters or
10 .
10
1
There are also measurements of really small objects like a
cell that measures 0.000 001 meters or 10 .
0.1
0.01
If you have to write all those zeros, there is a greater chance
of error so we use scientific notation to help write numbers
that have a lot of zeros.
0.001
0.000 1
Scientific notation uses the multiplication of a number
between 1 and 10 and an exponent with base 10.
0.000 01
0.000 001
0.000 000 1
Rules for writing Scientific Notation:
You must have a decimal number that is ___________ than or ______ to 1 AND _________ than 10 which is multiplied by an
____________________ with a base of _____.
When changing from scientific notation to standard notation and the exponent is…
…positive, you move the decimal to the ________________. Example :4.1 x 105 = ________________________________
…positive, the answer is more than _____. Example 7.1x104 = _______________
…negative, you move the decimal to the ______________. Example: 4.1x10-5 = ______________________
…negative, the answer is less than ______. Example 2.6x10-4 = ______________
…equals 1, you move the decimal ___ place to the ___________. Example 5.2 x 101 = ________________________
…0, the answer is ______________. Example 3.4 x 100 = _________________
Objects: Match on far right
A particle of sand
Scientific Notation
9.9 X 107m
A body cell
1.1 X 102m
Diameter of Earth’s orbit
Width of a finger on your hand
Diameter of the Solar System
Diameter of Earth
Object (choose from list on left)
1.2 X 10 m
The height of a child
The length of a football field
Standard Notation
0
1.3 X 10-6 cm
1.5 X 10-2m
9.75 X 1011m
2.01 X 10-4m
1.39 X 1013 km
a)
The star closest to Earth that is not the sun is called Proxima Centauri and is found at a distance of 40 200 000 000 000km. Its
scientific notation = _______________
b) The diameter of an electron is 0.000 000 000 000435 cm. Its scientific notation = ______________
c)
d)
e)
f)
g)
4.56 X 10 –3 = ____________
1.12 X 10 1 = ____________
7.21 X 10 5 = ____________
7.88 X 10 0 = ____________
4630 = 4.63 X 10 ?___
h) 0.003 45 = 3.45 X 10 ?___
i) 984 = 9.84 X 10 ?___
j) 76 900 = _______ X 104
k) 3890 = _______ X 103
l) 0.000 896 = ________ X 10 –4
Name :
Compare and Order Rational Numbers
1. Put the numbers -12 to +12 on this number line (under it).
2. What does each small vertical line represent?
3. A rational number is:
4. How do you change a fraction to a decimal?
5. Are these numbers below equivalent? ______________
1
2
1
2
-
6. Put these numbers on the number line:
5
3 3 15 1 3 1 7
, -1 , , - , , , 1 , 2
8
16 8 16 3 4 4 8
1
2
Date :
Section 6.4 Operations with Rational Numbers
B12 : add, subtract, multiply, and divide positive and negative decimal numbers with and without the calculator
B13 : solve and create problems involving addition, subtraction, multiplication, and division of positive and
negative decimal numbers
 sheet `Operations with Rational Numbers`


Grade 7 has integer outcomes: adding, subtracting, multiplying and dividing with concrete, pictures and
numerical so it should be a review. However, this year, they can use a calculator after they are
comfortable (B12).
p.274 : 1, 2,3 (do 2 and 3 by estimating not number line or tiles), 5,6,7,8,9 (with a calculator)
Section 6.5 Properties of Operations
B1 : demonstrate an understanding of the properties of operations with integers and positive and negative
rational numbers (in decimal and fractional form)
B13 : solve and create problems involving addition, subtraction, multiplication, and division of positive and
negative decimal numbers
Part 1
 sheet « Order of Operations »
 Order of operations is en essential concept that students need. It is a concept taught in grade 7 but is
worth reviewing.
 The rules are there so everyone will have the same answer no matter where they are in the world.
 Make sure to follow BEDMAS in the brackets too and that if there is a fraction, use BEDMAS in the
numerator and then in the denominator and then divide the answer.
 can use a calculator to work through but must show calculations and not do it all on the calculator. Most
calculators do not do order of operations.
 sheet “Order of Operations Exercises”
Part 2
 sheet “Order of operations with negative numbers and decimals”
 BEDMAS with negatives: Students should realize that brackets around a negative with an exponent
means to repeat the negative (-2)3 means -2x-2x-2. If there were no brackets, then the negative sign
would not be repeated.
 BEDMAS and decimals: Students estimate by rounding off to the nearest whole number before using a
calculator to find the answer. A calculator is fine to use but do not key in the wrong number…by having
an estimate, there is something to compare calculator answer.
 Exercise sheet on “BEDMAS with Negatives and Decimals”
 Once again, make sure all calculations are shown even if using a calculator.
Name :
Operations with Rational Numbers
Date :
a) The temperature is -3ºC goes up 5 ºC. Show this solution on a number line.
b) You have a debt of $6 and your father takes away $4. What is your debt now? Show this solution on a number
line.
c) You want to buy a shirt for $25 but you only have $17. How much debt will you have if you borrow the money
from your mother?
2. Show these same questions using square tiles. A white square is negative and a colored square will be
positive.
a)
b)
c)
3. Estimate each answer. Round to the nearest whole number.
a) -6.5 + 4.3
b) -10 – -2.7
c) -9.3 + -5.8
d) -5.1 – 8.2
4. Fill in the table below. What patterns do you notice?
A
B
C
D
+3  (–1) =
+3  (–1) =
+3  (+1) =
+3  (+1) =
+2  (–1) =
+2  (–1) =
+2  (+1) =
+2  (+1) =
+1  (–1) =
+1  (–1) =
+1  (+1) =
+1  (+1) =
0  (–1) =
0  (–1) =
0  (+1) =
0  (+1) =
–1  (–1) =
–1  (–1) =
–1  (+1) =
–1  (+1) =
–2  (–1) =
–2  (–1) =
–2  (+1) =
–2  (+1) =
–3  (–1) =
–3  (–1) =
–3  (+1) =
–3  (+1) =
+3 ÷ (–1) =
+3 ÷ (–1) =
+3 ÷ (+1) =
+3 ÷ (+1) =
+2 ÷ (–1) =
+2 ÷ (–2) =
+2 ÷ (+1) =
+2 ÷ (+1) =
+1 ÷ (–1) =
+1 ÷ (–1) =
+1 ÷ (+1) =
+1 ÷ (+1) =
0 ÷ (–1) =
0 ÷ (–1,5) =
0 ÷ (+1) =
0÷ (+1) =
–1÷ (–1) =
–1 ÷ (–1) =
–1 ÷ (+1) =
–1 ÷ (+1) =
–2 ÷ (–1) =
–2 ÷ (–1) =
–2 ÷ (+1) =
–2 ÷ (+1) =
–3 ÷ (–1) =
–3 ÷ (–1) =
–3 ÷ (+1) =
–3 ÷ (+1) =
Multiplying or dividing two integers with the same sign (+) X (+) or (–) X (–), gives a ____________ answer.
Multiplying or dividing two integers with different signs (–) X (+) or (+) X (–), gives a ___________answer.
Section 6.5 Properties of Operations
B1 : demonstrate an understanding of the properties of operations with integers and positive and negative
rational numbers (in decimal and fractional form)
B13 : solve and create problems involving addition, subtraction, multiplication, and division of positive and
negative decimal numbers
Part 1
 sheet « Order of Operations »
 Order of operations is en essential concept that students need. It is a concept taught in grade 7 but is
worth reviewing.
 The rules are there so everyone will have the same answer no matter where they are in the world.
 Make sure to follow BEDMAS in the brackets too and that if there is a fraction, use BEDMAS in the
numerator and then in the denominator and then divide the answer.
 can use a calculator to work through but must show calculations and not do it all on the calculator. Most
calculators do not do order of operations.
 sheet “Order of Operations Exercises”
Part 2
 sheet “Order of operations with negative numbers and decimals”
 BEDMAS with negatives: Students should realize that brackets around a negative with an exponent
means to repeat the negative (-2)3 means -2x-2x-2. If there were no brackets, then the negative sign
would not be repeated.
 BEDMAS and decimals: Students estimate by rounding off to the nearest whole number before using a
calculator to find the answer. A calculator is fine to use but do not key in the wrong number…by having
an estimate, there is something to compare calculator answer.
 Exercise sheet on “BEDMAS with Negatives and Decimals”
 Once again, make sure all calculations are shown even if using a calculator.
Name:
Date:
Order of Operations
To get your driver’s license, you must follow specific rules of the road. These rules prevent accidents.
Like the rules of the road, there are rules in math. These rules were invented in the 1500’s. You must use these
rules to find the answer to many calculations in one questions in order to make sure everyone finds the same
answer.
The rules to follow:
1.
2.
3.
4.
The key word to memorize that order is ________________.
If there are brackets, you must do BEDMAS in the brackets first. You must write all calculations vertically
below the question.
1. Practice:
a) 52 + 8 x 3
b) 52 + 8 – 3
e) (122 – 14) ÷ 26
c) 3 x 19 – (5 + 33)
f) 9 x 4 ÷ 2 + 5
d) 42 – 3 x 5 + 10
g) 18 + 3 x 4 – 24 ÷ (3 + 1)
2. If the question is in the form of a fraction, use BEDMAS on the numerator and then BEDMAS on the
denominator and then divide the answer.
a)
46  3 x 2
10  40  8
b)
3  7 x3
48
c)
13x5  2
22  11  5
d)
33  2 2  5
12  3
e)
(5  4) x3
52  2
Name:
Order of Operations Exercises
Date:
1. Find the answer. Show your calculations.
a) 12÷3 + 6 ÷ 3 b) 6 ÷ 2 x 0 + 8 c) (6+8) ÷ 2 + 5 d) 25 + 10 – 42 x 2
e) 40 ÷ 2 x 22 – 2
f) 62 – 12 ÷ 3 x 4
g) 12 ÷ (6-3) x 4
h) 50 – 10 x (3+2)
2. Find the answer. Show your calculations.
a)
22  43
6 x2  8
b)
(32  17)  3
10 2  20
c)
8  2 x(9  3)
(10  4)  2
d)
6 2  (3  1)
(1  5) x3
3. Put brackets where needed to equal the answer given.
a) 16 – 4 + 6 = 6
b) 84 – 28 ÷ 4 x 7 = 2
c) 42 + 5 ÷ 3 + 4 = 3
4. Are these questions true or false? Use BEDMAS to check.
a) 5 x 74 x 20 = 5 x 20 x 74
b) 48 ÷ (6 ÷ 2) = (48 ÷ 6) ÷ 2
c) 4781 + 2342 = 2342 + 4781 d) 64 ÷ 8 = 8 ÷ 64
e) 5 x 307 = (5x30) + (5x7)
Name:
Order of Operations with Negative Numbers and Decimals
Date:
BEDMAS and Negative Numbers
Watch to make sure you pay attention to the negative signs and the rules we reviewed the other day.
-x-=
+x+=
-x+=
+x-=
-÷-=
+÷+=
-÷+=
1. Calculate. Show your work.
a) -2 + (-3)2 – 6 b) (-5 + +10) + 20 ÷ -4 x -3
+÷-=
c) -3 + (-5)2
d) 12 – -4 + -2 x 5
BEDMAS and Decimals
When doing order of operations with decimals, it is always a good idea to estimate before calculating so you
have an idea what the answer will be before you use a calculator.
Estimate and then calculate. Show your work.
2. a) -3.8 + -7.32 + -1.2
b) 13 x -1.4 – 3 x-1.4
d) 7 x -3.6 + 3 x -3.6
e) 5 ÷ (-2.5 ÷ 2)
c) -4.1 + (-5.9 +8.7)
f) -8.2 x 5.4 + -1.8 x 5.4
3. Calculate. Show your work.
a) (-2)2 + 9
b) (-8 – 11) ÷ -1 c) 6 x (-1 – -6)
e) 62 – 6 x (-5 + -2)
f) 5 ÷ -5 – 3 x -3
d) (-2 + -3)2
g) -10 x -4 ÷ -2 – -3
h) 16 + (-3)2 x -4
4. Estimate and then calculate. Show your work.
a) -3.5  9  2
b) -5  7 + -5  0.5
c) 23  -1.6 – 5  -1.6
d) -2.7 + -3.73 + -5.3
e) -0.4 x 23 x -5
g) 3  -2.9 + 7  -2.9
h) -3.7 + 2 x (-6.3 +12.1)
f) -3.6 + -7.4 + 9.2
i) (-4.1)2 + (4.5 x -2)