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MFM 1P Foundations of Mathematics

Grade 9 Applied Mitchell District High School

Unit 2 Proportional Reasoning 9 Video Lessons

Allow no more than 14 class days for this unit!

This includes time for review and to write the test. This does NOT include time for days absent, including snow days.

You must make sure you catch up on class days missed.

Lesson # Lesson Title Practice Questions Date Completed

1 Fractions Handout

2 Working with Rationals - Multiplying and Dividing Handout

3 Working with Rationals - Adding and Subtracting

(Same Handout as Lesson 2)

4 Equivalent Ratios 12 #1-5, 7-9

5 Ratio and Proportion Page 117 #1-12

6 Ratio and Proportions Using Algebra Page 133 #1-6

7 Unit Rate Page 123 #1-9

8 Percent as a Proportion Page 137 #1-11

9 Putting it all Together - Applications of Proportion

complete the proportional problems

left on the note

Test Written on : _______________________________________

MFM 1P U2L1 Fractions

Topic : Fractions

Goal : I know what a fraction is and how to change between fractions and decimals.

A fraction compares pieces you have to the pieces that would be in a whole thing.

If we cut a pizza into 8 slices, each slice is said to be one eighth.

What is a Fraction?


Working with Fractions

Changing a Fraction to a Decimal

47

162

Pull

Pull

Rule

Divide the numerator by the denominator.

Rule

Keep the whole number for in front of the decimal point,

then divide the numerator by the denominator.

Pull

Pull

Comparing Fractions

Place the fractions in order of smallest to largest. Check you are right by turning them into decimals.

12

58

275

-2 34 - 7

3


Changing Decimals to Fractions

Change 1.002 to a fraction.

Step 1

Step 1

Take the number after the decimal place and use as

a numerator.

Step 2

Step 2

Count the number of decimal places

and find the power of 10 with that

many zeros.

Step 3

Step 3

Use the power of 10 as the denominator

and reduce to lowest terms with a

calculator.

Examples: Express in fractional form in lowest terms.

a) 3.05 b) 0.125 c) 8.75


Practice

MFM1P U2L2 Working with Rationals - Multiplying & Dividing

Working with Rational Numbers (i.e. Fractions)

Today's Topic : Fractions

Today's Goal : I can multiply and divide fractions

If you don't have a calculator with a fraction button, now is a good time to get one. For today, you can trade me for one.

Your fraction button will look like …

a b/cit may look

like this - if so we need to

talk :-)

To enter a fraction…

23 2 a b/c 3

235 2 a b/c 35 a b/c


Mixed vs. Improper FractionAn improper fraction has a bigger number on top than the bottom.A mixed number has a whole number and a fractional part.

Mixed to Improper

235

143

Improper to Mixed185123


Step 1 Change to an improper fraction

Multiplying Rationals

Step 2 Multiply the numerators

Step 3 Multiply the denominators

Step 4 Reduce to lowest terms if possible.

143 x 2

5

Dividing Rationals

143 ÷ 2

5 Step 1 Change to an improper fraction

Step 2 Flip over the divisor (2nd fraction)

Step 3 Multiply as normal.

Step 4 Reduce to lowest terms if possible.

Practice Questions - Handout Page

MFM1P U2L3 - Adding and Subtracting Fractions

Topic : Fractions

Goal : I can add and subtract Rational Numbers (fractions).

Adding and Subtracting Fractions

MFM1P U2L3 - Adding and Subtracting Fractions

Adding and Subtracting23

+ 14

a)

131 - 5

6b) 2

3 1- 15

c)

Step 1 Change to an improper fraction

Step 2 Look at both denominators and find the smallest number they both divide into. This is finding a common denominator.

Step 3 Multiply both numerator and denominator of each fraction to get the common denominator.

Step 4 Add/Subtract the numerators, keeping the denominators the same. Reduce your answer to lowest terms (if needed)

Practice Questions - Handout Page

MFM1P U2L4 - Equivalent Ratios

Topic : equivalent ratios

Goal : I can recognize and write ratios that are equal in various forms.

Equivalent RatiosA ratio is in lowest terms if there is no whole number that will divide evenly out of all parts of the ratio.

Two ratios are equivalent if they are the same when reduced to lowest terms.

Example 2. Write the ratios of Yellow:White for each of the above rectangles and reduce to lowest terms. Are the ratios equal?

Example 1. What fraction of each diagram is yellow? Reduce to lowest terms. What do you notice?

What is the difference between a fraction and ratio?

A fraction compares one thing to the WHOLE.

A ratio compares two things to each other.

This class has _____ boys and ____ girls.

a) What fraction is boys? ______ b)What fraction is girls? ______

c) What is the ratio of boys to girls? ___________

d) What is the ratio of girls to boys? ___________


Example 4. What are the two multiplication relationships in

12:36 = 24:72

Example 5. Find the missing number.

5 : 30 = 15 : ?


7 : 12 = p : 48

Example 3. For each of the following ratios, determine and equivalent ratio by either dividing or multiplying.

a) 1 : 5 b) 120 to 60 c) 3535

c)

There are three different ways to write a ratio - each are illustrated in the next example.

Ratios can have a multiplication/division relationship between them or within them - as illustrated in the following example.



12 : 30 = 18 : p

Sometimes the question becomes easier if you simplify a ratio first. This means find it's LOWEST term value. Figure out what number can be divided out of each ratio, to produce an equal ratios with smaller numbers.

Example 4. You earned $200 last week working 20 hours. How long will you have to work to earn $550?

Practice Questions - Page 112 #1-5, 7-9

MFM1P U2L5 - Ratio and Proportion

Topic : Ratio and Proportion

Goal : I can solve proportional relationships using a variety of methods.

Ratio and ProportionRatio and proportion is one of the most useful skills you'll get from your math classes. Here are some following situations where it could be useful in every day life.

A paediatric nurse (children's nurse) is working in the OR following a child's surgery. The dosage of pain medication says 5 mg / 100 kg of bodyweight. How much medication should she give a 15 kg child?

You decide to paint the living room in your house. A gallon of paint will cover about 375 ft2. You calculate your living room walls to have a surface area of 525 ft2 and you are using a dark colour so you will need to use 3 coats. How many gallons of paint do you need?

To open your swimming pool you will need to treat it with an algaecide to get rid of algae growth. The package says that you need to use 1 cup for every 1000 gallons of water. How many cups do you need for a 22 900 gallon swimming pool?

Let's set up ratios.

There are two ways to do this. Look for a multiplication relationship between ratios OR look at a multiplication relationship inside a ratio.

Example 1. A child is given pain medication based on how heavy they are. The dosage says 2mg for each 6 kg of bodyweight. How many milligrams should be given to an 18 kg child?

between inside


Example 2. What are the two multiplication relationships in


6:18 = 30:90Between ratios inside the ratio

7 : 10 = ? : 40

3 : 21 = ? : 49


18 : 30 = 21 : p


Example 5. �� !��

Example 6. A car travels 125 miles in 3 hours. How far would it travel in 5 hours?

Example 7. The scale on a diagram is 1 : 250. This means that every unit on the diagram represents 50 units in real life. If a tree measures 4 cm on the diagram, how tall is it in real life?

Practice Questions - Page 117 #1-12

MFM1P U2L6 - Ratio and Proportion Using Algebra

Using Algebra to Solve a ProportionIf we write a ratio as a fraction, we can find two different ways to solve the proportion using our algebra skills.

Method 1 : Isolate the unknown (letter)Example 1.

Example 2.

the variable would be by itself if the 15 were gone.

15 is dividing the variable, so to make it go away we do the opposite

The opposite of dividing is multiplying.

You have to do the same on the other side, to keep the equation equal.

Goal : I can solve proportional relationships using algebraic methods.

Topic : Ratio and Proportion using Algebra


Example 3. Try a few more...

a) b)

c) d)

e) f)


Method 2 : Cross Multiplying (a property of equal fractions/ratios)

This property can be useful when solving a ratio.

Example 4. Try a few more use cross multiplying...


Example 5. At the same time of day, all shadows cast by the sun will be proportional to the height of the object casting the shadow.

I'm 1.53 m tall. I measure my shadow to be 2.6 m long at the same time as I measure the shadow of a tree to be 9.8 m long. How tall is the tree.


MFM1P U2L7 Unit Rate

Unit RateA unit rate is a comparison between two things, where the second quantity being compared is ONE.

Some examples of unit rate.

* kilometres per ONE hour of driving* Cost per ONE mL of pop* goals per ONE game

A unit rate can be written in 3 different ways...

* Using words : * Using numbers, symbols and words :* Using numbers and symbols :

Topic : unit rates

Goal : I know what a unit rate is and I can solve problems with it.

90 km/h 90 km per hour

90 kilometres per hour

Since speed is one of the most common unit rates, we will take a moment to examine that.

Let's say it takes 15 minutes to drive 20 km. What is the unit rate?


Example 2. At Walmart you can buy a case of 12 cans of Coke for $6.99 or a case of 24 cans for $14.88. Calculate the unit rate for each case. Which is the better deal?


Example 1. Find the unit rate in each situation...

a) $6 for 12 cans of pop b) 650 kilometres in 7 hours


EQAO Check :

MFM1P U2L8 Percent as a Proportion

Percent as a RatioTwo students take a test on the same topic with different teachers. One student gets 30/32 the other gets 24/25. Who did better on that topic?

It's hard to compare if they aren't out of the same thing, so we change them to make them both out of 100. This is called the percent. Anytime you take a situation and change it to be out of 100, you are dealing with percent.

Example 2. Changing a percent to a decimal. Rule: divide by 100.

Example 1. Changing a fraction to a percent.

Rule: divide top by bottom then multiply by 100% (move decimal place 2 spaces left and don't forget the % sign)

14% 4% 104%

Topic : Percent as a Ratio

Goal : I know what percent means and I can use proportion to solve percentage calculations.

Method 1 - set up a proportion Method 2 - divide and X 100%

3032

2425


Example 3. Finding the percent of a number using ratios.

a) What is 10% of 45? b) What is 63% of 230?

Example 4. Finding the percent of a number by remember that 'of' in math means to multiply.

a) What is 10% of 45? b) What is 63% of 230?

Example 5. The yearbook committed conducted a survey to find out how many students in the school had a cell phone. of the 63 students they surveyed, they found that 83% said they had a cell phone. How many students were carrying a cell phone?

Method 1 - ratio method Method 2 - 'of' method


Example 7. In Ontario we pay 13% sales tax (HST). So really what we are paying is 113% of the sale price. How much money do we have to pay in total for an iPod that sells for $159.75?


Example 6. A pair of jeans you've been waiting for goes on sale for 25% off. How much will you have to pay if they were $75.99 originally?

MFM1P U2L9 Putting it all Together - Proportional Problems

Topic : Proportional Problems

Goal : I can set up ratios and proportions to solve word problems dealing with "real life" situations.

Proportional Problems

MFM1P U2L9 Putting it all Together - Proportional Problems

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