UNIT 1 LESSON 1 Comparing and Ordering Rational Numbers …2... · UNIT 1 LESSON 1 Comparing and Ordering Rational Numbers (2.1 in textbook) ... 1" ’ −2 # & 3 > - order of operations

UNIT 1 LESSON 1

Comparing and Ordering Rational Numbers (2.1 in textbook) Rational Numbers: formed when one integer is divided by another integer

where

Rational numbers can be positive, negative, or zero and include integers, fractions, mixed

numbers and many decimals.

For example, , ,

Ordering Rational Numbers

Example 1: Place -4.1, 3, -0.5 and 1/2 on the number line.

Comparing Fractions

• Fractions can be compared by finding a common denominator, then comparing the

numerators.

Example 2: Which is larger:

a) 2/3 or 3/4? b) −"# ,

%&%' )*

%&' ?

c) -0.2 or -1/4?

Comparing Decimals

• If the whole numbers are the same, compare the decimal portion of the number. To be

‘fair’, they have to have the same number of decimal places. If one number has less

decimal places, add zeros to right of the number until all numbers have the same

number of decimal places.

Example 3: Which is larger:

a) 42.15 or 32.23? b) 42.153 or 42.16? c) -3.254 or -3.23?

Comparing Decimals to Fractions

• Convert the fraction to a decimal first, and then compare your numbers OR convert the

decimals to fractions, then compare.

Example 4: Place all the numbers and their opposites on a number line:

-0.9, -7/8, 2.7, 2 2/3

Example 5: Identify a decimal and a fraction between

a) 1.4 and 1.5 b) #& +,-

'".

On your own: Write the numbers in ascending order: -1.235, -11/9, -1.23, -2 2/5

On your own: Identify a decimal and a fraction between -1.23 and -1.2

Class work/HW: p.51 #5-9 (calc), #10-13, 15-16 (no calc) Challenge: #28, 29, 30

UNIT 1 LESSON 2: Problem Solving with Rational #’s in Decimal Form (2.2 in the textbook)

Rules for operations positive and negative rational numbers are the same as for positive and

negative integers.

• To add rational numbers of with the same sign, add the number values, and keep the

sign

• To add rational numbers with different signs, subtract the smaller value from the

larger value. The answer has the sign of the bigger number.

• Subtracting a rational number is equivalent to adding its opposite • The product/quotient of 2 rational numbers having the same sign is positive. The

product/quotient of 2 rational numbers having different signs is negative.

Example 1: Estimate then calculate

a) +2.2 + 3.2 b) (-3.62) + (-7.21)

c) +7.83 + (-2.21) d) (-9.1) + (+6.3)

e) (-6.82) – (+2.51) f) (-2.3) – (-3.7)

g) (-6.2) x (-3.1) h) (-8.4) ÷ (+2.1)

ORDER OF OPERATIONS

- just like with integers, the order of operations for rational numbers follows

BE(DM)(AS):

- perform operations inside parentheses ( ) first; if there are brackets inside other

brackets, start with the inner brackets

- divide and multiply in order from left to right

- add and subtract in order from left to right

Example 2: Solve.

a) 6.2 ÷ 3.1 + 6.2 (-3.0) b) 2.5 + 5 x [3.32 x (2 - 6.22)]

PROBLEM:

A hot air balloon climbs at 0.8 m/s for 10 s. It then descends at 0.6 m/s for 6 s.

a) what is the overall change in altitude?

b) What is the average rate of change in altitude?

Class work/HW: p. 60 #4, 6, 8, 9, 11, 18

Challenge questions (optional): p.62 #26, 27, 28, 29

UNIT 1 LESSON 3: Problem Solving with Rational #s in Fractional Form (2.3 in the textbook)

• Rational numbers expressed as proper or improper fractions can be

added, subtracted, multiplied, and divided the same way as positive

fractions.

• The sign rules for integer operations also apply to rational numbers

expressed as fractions.

a) b)

c) d)

e) #/ +

&' × 2−

#/3 f) 2−2 "& +

&'3 ÷ 2−

&6 − 2

"'3

Problem: Tim had $50. When he went to the mall, he spent 1/5 of the money on

food, another 1/2 on clothes, and 1/10 on bus fare. How much money does he

have left at the end of the day?

Class work/ HW: p.68 #5-10, 21

Challenge: p.70 #19, 20, 25, 26, 27

UNIT 1 LESSON 4: Determining Square Roots of Rational Numbers (2.4 in textbook)

Activity: Calculate the area of each square.

a) b) c)

What is the side length of each square with the given area?

a) Area = 4cm2 b) Area = 100cm2 c) Area = 36cm2

side = ______ side = _______ side = _______

Summary: - To calculate the area of a square, we multiply the side length

by itself, or we square the side ( Area = side2 )

- To calculate the side length when we know the area, we do

the opposite of squaring

- The opposite of squaring is called “square root” ( Side = √+*8+ )

- A number is a perfect square if its square root is rational

Example 1: Calculate the following perfect square integers.

02= 12= 22= 32= 42= 52= 62= 72=

82= 92= 102= 112= 122= 132= 142= 152=

Example 2: Calculate the following square roots without a calculator.

a) √81 b) √225 c) <#/&=

b) √1.21 b) <"? c) √0.04

Example 3: Determine if the following are perfect squares, then evaluate the

square root of each number to the nearest hundredth.

a) 20 b) 2.25 c)#/".

Class work/HW: p.78 #4, 6, 8, 10, 11, 12, 13, 16, 17, 24

Challenge: p.80 # 25, 27, 29, 30, 31, 32, 33, 34, 35, 36

UNIT 1 LESSON 5: Categorizing Numbers

Rational Numbers:

- Can be written as a fraction

- Might look like:

o A fraction, decimal, integer

o A terminating decimal (ex: 2.3)

o A repeating decimal (ex: 0. 3$, 2. 12$$$$, 7.11111111…) o A square root of a perfect square (ex: √16 )

Irrational Numbers:

- Cannot be written as a fraction

- Might look like:

o A non-terminating, non-repeating decimal (ex: 2.347234006…)

o A square root of a number that is not a perfect square (ex: √8)

Example 1: Which categories do the numbers belong to? Place a check mark in

all the categories that apply.

Number Real Natural Whole Integer Rational Irrational

a) 8.2

b) 0

c) √0.81

d) √8.1

e) -5

f) -.

g) 0.1254084…

h) /0

i) −124

j) 3 405

k) -9.1

l) 0. 6$

m) 9.23232323…

n) √40

UNIT 1 SUMMARY – RATIONAL NUMBERS (CHAPTER 2 IN TEXTBOOK)

KEY CONCEPTS OF EACH LESSON

CONCEPTS I GET

IT (√ )

I NEED MORE

PRACTICE (√ )

Lesson 1: Comparing and Ordering Rational Numbers (2.1 in textbook) - Placing numbers on a number line < Ex1: place on a number line: 2.1, -3, 05, -1, 1 > - Comparing fractions without a calculator, by writing the fractions with a

common denominator < Ex2: Which is bigger? − 07 or − 7

. ? > - Comparing decimals with fractions with a calculator, by converting the fractions

to a decimal < Ex3: Which is bigger? 3.45 or 3.4? > - identifying a decimal or fraction that is between two given decimals or fractions,

without a calculator < Ex4: Identify a number and a fraction between 1.1 and 1.2 >

Lesson 2: Problem Solving with Rational #’s in Decimal Form (2.2 in the textbook) - operations with decimals (adding, subtracting, multiplying and dividing), with a

calculator < Ex1: Evaluate (-2.45) – (- 3.1) > - order of operations (BEDMAS) with decimals, with a calculator < Ex2: Evaluate

(1.1 + 3.2) ÷ (−3.6 − 0.5) > - Word problems with decimals < Ex3: Calculate the total cost including 12% tax,

of a sweater that costs $28.90. >

Lesson 3: Problem Solving with Rational #s in Fractional Form (2.3 in the textbook) - Operations with fractions (adding, subtracting, multiplying and dividing),

without a calculator < Ex1: 1 "' − 22#&3 >

- order of operations (BEDMAS) with fractions, without a calculator - Word problems with fractions, without a calculator

Lesson 4: Determining Square Roots of Rational Numbers (2.4 in textbook) - Calculating the square root of a perfect square that is an integer, fraction or

decimal, without a calculator - Estimating the square root of a number that is not a perfect square, without a

calculator - Calculating the square root of a number that is not a perfect square, with a

calculator - Identifying if a number is a perfect square or not, without a calculator and with

a calculator

Lesson 5: Categorizing Numbers - Categorizing numbers as Natural, Whole, Integer, Rational, Irrational and Real

To study for the test:

- Complete the review exercises - Do quiz corrections / redo quiz questions - Complete any homework questions from the unit that you did not yet do, or did not understand

(ask questions if something is not clear!) - Read the class notes - Do extra practice questions - Write your own chapter summary / cue cards

Unit 1 class work / HW:

Lesson 1: p.59 #5, 9, 12, 14; challenge: #28, 29, 30 Lesson 2: p. 60 #4, 6, 8, 9, 11, 18 Challenge questions (optional): p.62 #26, 27, 28, 29 Lesson 3: p.68 #5-10, 21; Challenge: p.70 #19, 20, 25, 26, 27 Lesson 4: p.78 #4, 6, 8, 10, 11, 12, 13, 16, 17, 24; Challenge: p.80 # 25, 27, 29, 30, 31, 32, 33, 34, 35, 36 Lesson 5: just the exercise in the lesson notes Review exercises: p.82 #1-25; Optional: p.84 #1-20