Welcome to Math 6

This lesson is a review of

all the objectives

from Lessons

1-10.

When you finish the lesson and the assignments which follow, you will take an assessment of your progress so far. The assessment will be found in the assignments section.

OBJECTIVE:

Each student will: Apply and use each of the rules and concepts which we have covered so far.

Divisibility Rules

Divisibility Rules

2 if the last digit is 0,2,4,6 or 8

3 if the sum of the digits is divisible by 3

5 if it has a 0 or 5 in the ones place

6 if it is divisible by both 2 and 3

9 if the sum of the digits is a number that is divisible by 9

Prime Factorization

PRIME FACTORIZATION

When we write a number as the product of its prime factors, we call it the

When we find the prime factorization of a number the divisibility rules can come in very handy

Lets find the prime factorization of a number as a review.

What is the prime

factorization of 164?

Divide 164 by 2 since it is even.

Divide 82 by 2.

41 is odd. We cannot divide by 2.

What else could we divide by?

Here’s where we get to apply the

divisibility rules.

Is 41 divisible

by 3 ?

Since the sum of its digits, 4+1=5

is not divisible by three, we know that

41 is not divisible by three.

Is 41 divisible

by 5 ?

Only numbers that have a five (5) or

a zero (0) in the ones place are divisible by 5.

So we know that 41 is not divisible by five.

Is 41 divisible

by 7 ?

Since we know our 7 times tables,

we know that 42 and 49 are both divisible by 7

but 41 is not.

Is 41 divisible

by 11 ?

Since we know our 11 times tables,

we know that 44 is divisible by 11

but 41 is not.

Now we continue through the list of prime numbers

to check if 41 could be divided by any of them.

Prime and Composite Numbers

A Prime Number has exactly two factors: itself and 1.

A Composite Number is any number with more than two factors.

It is not expected that you memorize all of the prime numbers. Since the set of prime numbers is infinite number, that would be impossible anyway.

You are expected to be able to tell whether any number between 1 -100

is prime or composite.

You could do that easily if you have all of your multiplication facts memorized.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

Prime Numbers between 1 and 50 are ----------

in Gray

The prime factorization of 164 is 2 x 2 x 41 or 22 x 41

What is the prime factorization of 96?

Know the facts.

That will make this skill easy and quick to master.

I recommend that you memorize the

multiplication tables up to

12 x 12.

The prime factorization of 96 is 2x2x2x2x2x3 or 25x3.

The prime factorization of 96 is 2x2x2x2x2x3 or 25x3.

Do not circle composite numbers; only circle the prime numbers.

Write decimals as fractions or mixed numbers

First say the number. The ‘place’ that you named when you read it aloud is the denominator of the fraction.

That place will either be “tenths,” or “hundredths” or “thousandths.”

0.28 is read as “twenty-eight hundredths.” As a fraction it would be

written as 28/100.

In most cases, fractions should be simplified. To do so, divide the numerator and denominator by the greatest common factor.

5.245 is read as “five and two hundred forty-five thousandths.”As a fraction (actually a mixed number) it would be written as

5 245/1000.

20049

5510005245

51000245

5

Write fractions or mixed numbers as decimals

Writing Fractions as Decimals

To convert a fraction to a decimal, divide the numerator by the denominator.

See examples:

5

24

33÷4= 0.75

2÷5= 0.4

These two are examples of terminating decimals.

If the denominator of the fraction is a ten, one hundred or one thousand, rewrite it using place value.

3/10 = 0.3

45/100 = 0.45

In some cases it may be quicker and easier to convert a fraction to a decimal by renaming it (writing an equivalent fraction) so that it has 10, 100 or 1000 as a denominator. Then simply write as a decimal based on the place value system.

A mixed number is part whole number and part fraction. When converting to a decimal, only the fraction is converted. The whole number remains unchanged.

2 ½ = 2.5

Describe any ratio.“For every x, there are y”

There are four birds in the sky. A fraction would describe the number of vultures to the total number of birds.

41

A ratio describes the numerical relationship between one part to another part.

The ratio of vultures to pelicans is “one to three.” It can be written as 1:3. It also can be

31

So for every vulture there are three pelicans.

Identify, write and compare ratios and rates.

Ratio: is a comparison of two numbers by division.A ratio can compare “part to part” or “part to the whole.”

Recipe 2 4 cups of orange juice 2 cups of soda 4 cups of pineapple juice Yield: 10 cups punch

Recipe 1 5 cups of orange juice 3 cups of soda 2 cups of pineapple juice Yield: 10 cups punch

Comparing Ratios

1) Which recipe has the most juice in it? Use evidence from the recipes to support your answer.

2) Which recipe has the most soda in it? Use evidence from the recipes to support your answer.

Comparing Ratios

3) For Recipe #1, what is the ratio of orange juice to soda? Does this represent a part-to-part ratio or a part-to-whole ratio? Explain.

4) For Recipe #2, what is the ratio of orange juice to soda? Does this represent a part-to-part ratio or a part-to-whole ratio? Explain.

Rate- the ratio of two measurements having different units of measure. Example: 60 miles per hour

George and Juan compared the fuel economy of their cars and found these rates: • George’s car went 580

miles on 20 gallons of gas.

• Juan’s car went 450 miles on 15 gallons of gas.

a.) Compare the mileage (unit rate – miles per ONE gallon of gas).

George and Juan compared the fuel economy of their cars and found these rates: • George’s car went 580

miles on 20 gallons of gas.

• Juan’s car went 450 miles on 15 gallons of gas.

b.) How far can George travel on 2 gallons of gas? On 5 gallons of gas? On 20 gallons of gas?

4) Tonya used a RATIO table to solve another problem. This is what her table looked like to find out the mileage for George’s car. Complete the table.

Gallons of gas 1 2 3 4 5 6 7 8

Miles traveled

Simplify ratios

Write equivalent fractions,

decimals & percents.

Simplifying ratios is just like simplifying fractions. Find the greatest common factor of the two numbers and divide both numbers by it.

It looks the same as simplifying fractions because it is. In reality a fraction is a specific type of ratio.

Ratios and Percents

A percent is a ratio of a number to 100.

The symbol % is used to indicate that a

number is a percent. For example, 40%

is the ratio 40 to 100, or .

Percents can be written as fractions or

as decimals.

100

40

Write Percents as FractionsWrite 35% as a fraction in simplest form.

35% =

=

100

35

20

7

Write the percent as a fraction with a denominator of 100.

Simplify.

Writing Percents as Decimals

Write 43% as a decimal.

43% =

= 0.43

100

43 Write the percent as a fraction with a denominator of 100.

Write the fraction as a decimal.

Writing Decimals as Percents

Write each decimal as a percent.

0.07

0.07 =

= 7%

100

7 Write the decimal as a fraction.

Write the fraction as a percent.

Writing Fractions as Percents

Write each fraction as a percent.

=

5

4 Write an equivalent fraction with a denominator of 100.

Write the fraction as a percent.

205

204

x

x

100

80

¿

%80

-Using Division

83

= 3 ÷ 8

= 0.375

= 37.5%

Use division to write the fraction as a decimal.

Write the decimal as a percent.

Identify Proportions

&

Solve Proportions

An equation stating that two ratios are equal is called a proportion.

Equivalent RatiosIf two ratios are equal, they are said to be “proportional” to each other, or “in proportion.”

Use Common Denominators to Identify Proportions

• Rewrite the two ratios so that they have a common denominator.

• The ratios are equal if both their numerators and denominators are equal.

• This works just like finding equivalent fractions.

Use Cross Productsto Identify Proportions

For two ratios, the product of the numerator in one ratio and the denominator in the other ratiois called a cross product.

If the cross products of the ratios are equal, then the ratios form a proportion.

156

52

30152

3065

This shows that the two ratios are not proportional since their cross-products are not equal.

Use Cross-Productsto Solve Proportions

d

d

d

d

42

4168

41412

14412

Find the cross-products.

Solve.

The proportion checks.

144

4212

Use proportions to determine if two figures are similar.

𝒊

𝒋

𝒘

𝒛

𝒙𝒉

y𝒌

25

24

25

10

1818

15

6

Using proportions, decide if the figures are similar.

Using proportions, decide if the figures are similar.

𝑺

𝑹

𝑸𝒂𝒄

𝒃

𝟔

𝟒𝟕

𝟖𝟏𝟎𝟏𝟒

Using proportions, decide if the figures are similar.

𝟏𝟐

𝟏𝟓 𝟏𝟖 𝟏𝟐𝟏𝟎

Use proportions to find the length of corresponding sides of similar figures.

Use a proportion to find the length of the missing side (y).

𝟐

𝟓𝒚

𝟏𝟔

𝟖

𝟐𝟎

USE A PROPORTION TO FIND THE LENGTH OF THE MISSING SIDE.

𝒔

𝟔𝟏𝟐

𝟒

Find the least common multiple (LCM) of a group of numbers.

35 and 20The prime factorization of each is:

35=5x720=2x2x5

The prime factorization of the LCM of 35 and 20=2x2x5x7=140

That’s not so hard.

3, 15 and 7The prime factorization of each is:

3 is a prime number15=3x521=3x7

The prime factorization of the LCM of 3, 15 and 7=3x5x7=105

Be proud of yourself. You have accomplished a lot in this course so far. If you have already mastered these skills, its because you’ve made a commitment to learn them. If you are still working on some of these skills, don’t worry. Just reaffirm your commitment and you will succeed!

The commitment is just the first step. You might also need extra practice. You can find opportunities for that on the internet. Although there are loads of math websites out there, here are a couple you might look at:

http://www.dadsworksheets.com/

http://www.bbc.co.uk/skillswise/maths

AssignmentsTo review for the assessment of Part 1 (Lessons 1-10) go to the webpage

http://www.bbc.co.uk/skillswise/topic/ratio-and-proportion

Take some time to look at all the learning aids that are there: videos, factsheets...

AssignmentsThen go to this page:

http://www.bbc.co.uk/skillswise/worksheet/ma19rati-e3-w-mixing-a-cocktail

And answer all the problems. You can find the link to the answers right on the page.

AssignmentsWhile you’re on that website, be sure to look at the different mastery levels for that skill- and all the skills.

Try the Entry Quiz 1 & 2. After that, try Quiz Level 1.After you take the quiz, you can get your score immediately and review the correct answers.

In order to become a champion, you first must finish the race. Be sure to finish all the assignments. Then take the assessment of your progress.