Isaac Newton and Fractions By Ryan and David Hello! I'm Isaac Newton. I am a British scientist and mathematician. I described gravity and the three laws of motion. I built the first reflecting telescope and developed the theory of colour. People consider me as the greatest scientist and mathematician ever, in better than Einstein! 3.1: Using Models to Multiply Fractions and Whole Numbers Today I will show you different ways to multiply fractions and whole numbers by using models. Lets start with a question

x 4

To solve this problem, we could use repeated addition There are 4 's or 4 strips One way we could solve this problem is by modeling it with fraction strips. We can also use a number line to multiply fractions Multiply 7 x 3.2 Using Models to Multiply Fractions 3.3 Multiply Fractions 3.4 Multiplying Mixed Numbers 3.5 Dividing Whole Numbers and Fractions 3.6 Dividing Fractions 3.7Dividing Mixed Numbers 3.8 Solving Problems with Fractions 3.9 Orders of Operations with Fractions Another way is to sketch a rectangle 7 There are 7 's or or 5 A fourth way is by using counters. Multiply x 21 21 counters Find thirds by splitting the counters into 3 equal groups.

Each group contains 7 counters.

of 21 is 7. So of 21 is 14. The last way is to use fraction circles There are in total or 2 + + + Multiply 4 x Now I will show you 2 ways on how to use models while multiplying fractions by other fractions The first way to model fractions is by using counters 0 1 2 3 4 Multiply x What we do is model a set of eighths with eight counters

There are 6 counters in , so we circle 6 of the eight counters.

To model thirds, arrange the counters into 3 equal groups.

Each group of 2 counters would show . So, of 6 is 4 counters.

We have 4 counters out of a total 8 counters, or or

So, x 6/8 = 1/2 Next I will be talking about Multiplying mixed numbers Next on the list is Dividing Mixed Numbers We could use reciprocal to divide mixed numbers

Recall that 8/3 ÷ 5/2 is the same as 8/3 x 2/5 Evaluate 2 2/3 ÷ 1 2/3 Use multiplication to turn it into a improper fraction 8/3 ÷ 5/2 We can use cross multiplication 8/3 x 2/5 = 8/1 x 1/5

= 16/15

= 1 3/5 We can also use Models

to check our answer. 1 1 1 3 of the possible 5 is left. Therefore it is

1 3/5 I am entitled to 1 2/3. This creates the 1 Multiply x 5 I make 5 jumps of and I landed on or 3 .

So, x 5 = 3 The other way is by uising a rectangle. Multiply x First you draw a rectangle.

Then you show of the rectangle. Then you divide the into quarters and shade 3/4. 3/4 1/2 Use broken lines to divide the whole rectangle into equal parts.

There are 8 equal parts.

3 parts are shaded. So, x = There are different ways to multipy fractions. You can simply multiply the numerators together and the denominators together. Multiply: Multiply the numerators (4 x 2 = 8) together and the denomerators (7 x 5= 35)together.

So 4/7 x 2/5 = 8/35 You can draw an area model to show We could also use a number line Divide the number line into thirds 0 1 2 3 Start at 2 2/3 and go back 1 2/3 each time

We are now left with 2/3. This creates the 2/3 in the fraction 2/3 1 2/3 Now lets test your learning on Dividing Mixed numbers! 3 1/12 ÷ 2 1/24 You can estimate to check. Lets first convert this into an improper fraction 3 1/12 ÷ 2 1/24 3 x 12 2 x 24 37/12 ÷ 49/24 Now lets use Reciprocal 37 1 __ 12 0 x 24 is closer to 0 and is closer to 1.

So 0 x 1 = 0.

is close to 0 so the product is reasonable. __ 49 1 2 = 74 49 __ We can also model it to check the answer Or you could simplify before you multiply to make it easier. 2 15 x 3 8 = 2 x 3 15 x 8 2 15 x 3 8 = 2 x 3 15 x 8 1 , 1 5 4 1 = 25 49 1 x 1 5 x 4 __ I am entitled to 2 1/24 pieces

Since I got all my pieces that gives me a whole

I am now left with 25 out of the 49 ( 2 1/24 )

This equals 1 25/49 1 1 1 20 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 = Then multiply the numerators together and the denominators together. 2 x 1 = 2 2 x 1/3 = 2 1 1 x

= x

= 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 + + + =

3 Now that you know how to use models to multiply fractions with whole numbers, try and answer these questions using models. We will now be talking about word problems with fractions. Here is an example: Mr. Anopher went to the store to buy some milk for his pancake store. He bought 20 bottles of milk. He then divided them into portions. How many portions are there? How many are left over? 1 1) I was in the apple orchard (where I dicovered gravity) and needed to pick all the apples for an apple pie. I needed to pick all the apples off of 8 trees. If I pick all the apples on 1 tree in h, how long did it take me to pick all the apples? _ 2 2 2) I went home and began making the apple pies. I made 5 pies and each took h to make. How long did it take to make all the pies? The word "divided" suggest division. 20 1 3) I invite some friends over to eat the pies. It takes h to eat one of the 5 pies between my friends. If we took the same time to eat all the pies, how long would it take to eat all the pies up? ___ ÷ 2 1 2 ___ We can now turn the mixed numbers into an improper fraction 20 1 ___ ÷ x 2 20 1 ÷ 5 ___ 2) You would do 5 x

You can use fraction strips.

Make 5 strips and divide them into thirds.

Shade two-thirds of each strip.

The answer is 3

It would take 3 to make 5 pies 3) You would do x 5

You can sketch a rectangle.

Make a rectangle with 4 rows and 5 colums

The answer is 1

It would take 1 h to eat all of the pies Now we can solve the problem

and turn it back to a mixed number. x 2 ___ Try the different methods on these questions Now, what if Mr. Anopher actually wanted to make pancakes out of all the milk he just bought. How much would he make? He would need to follow the recipe which is :

- 3 _ 4 of a full portion. The words "of a" suggest that this is division 1) When I was a professor in Maths and Physics, 2/3 of the university were in my math class and 4/7 of those students were in my Physics class. What fraction of sudents were in both of my classes? 2) I teach for 1/4 of the day. During that time, I spend 8/15 teaching Math and the rest of the time on Physics. How many hours do I spend teaching Physics? So 2 x 1 = 3 2 2 ___ 1 20 1 ___ ÷ 5 2 ___ R 20 1 ___ x 5 2 ___ 1 4 = 8 ___ 1 , 8 Since our answer is a whole number, there is no leftovers 8 _ 1 ÷ 3 _ 4 R 8 _ 1 x 4 _ 3 = 32 ___ 3 , 2 _ 3 10 Now that we have done division problems, lets try new ones such as multiplication, addition or subtraction He almost became a farmer!

After he got home from school he

worked the family farms As Newton worked the farm, he ran out of seeds. To continue the farm, he needs to buy seed packets. Each fills up of a farm plot. The farm contains 10 plots. Isaac Newton currently owns

of packet fertilizer. He only needs to use 1 packet of fertilizer for of all the seed packet. Does he have enough fertilizer? 1 _ 16 1 _ 2 10 5 _ 40 First we need to find out how much seed packets he needs. 10 __ 1 3) In my mathematics class, I gave my students a test. About 3/8 of my class failed and of those people, 4/5 of hem did not study. What fraction of my class failed and did not study? Answers: 1) You would do x

x

=

=

of the students are in both classes 2 x 4 3 x 7 8 21 2) You would do x

x 1 __ 16 ÷ R 10 __ 1 16 __ 1 x = 160 __ 1 To multiply mixed numbers, you must write them as improper fractions. Multiply 2 x 1 1 5 1 5 1 5 1 5 1 5 1 5 + + + = = x 4 4 5 Now that we know how much seed packets we need,

we can figure out how much fertilizer we need. 1 5 160 __ 1 x 1 5 5 _ 40 5 4 1 4 20 __ 1 3 = 4 Newton needs 20 packets of fertilizer 3 4 21 4 3 4 1 4 3 4 Brackets

Exponents

Division

Multiplication

Addition

Subtraction From left to right From left to right Lastly, I will be talking about BEDMAS 3 4 3 4 3 4 3 4 3 4 3 4 2 4 6 4 4 9 4 10 4 14 15 4 1 3 1 3 1 3 2 3 1 3 3 4 15 4 3 4 3 4 3 4 3 5 3 5 3 5 3 5 2 5 3 5 11 5 2 3 4 7 2 5 x 4 7 2 5 x = 4 x 2 7 x 5 = 8 35 4 7 2 5 x 4 7 2 5 4 7 2 5 4 7 2 5 8 35 2 5 2 3 2 3 If we can Multiply Fractions, we could also Divide it! There are many ways to divide fractions.

One way is to use reciprocal (Multiplication)

Other ways are by using models Newton is was a Physicist, Mathematician, Natural Philosopher, Alchemist, Theologian and an astronomer We will start with Models. 5 ÷ 2 __ 6 Think : How much 2/6 are in 5 wholes. 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 This shows that there is 15 pieces in 5 if it was divided by 2/6 We could also use a reciprocal. Its is a number that has the same product. Rule : Do nothing to the first then flip the second fraction

and turn the division sign to multiplication. 5 ÷ __ 6 2 5 x __ 2 6 = 30 __ 2 or 15 1 __ An example of this rule in place is: 3 _ 2 + 2 2 _ x 2 4 _ The multiplication will go first rather than the addition

as it has a higher "priority". = 11 __ 2 Now we will look at dividing fractions with fractions This is the order in which you must do equations Question #1 20 __ 8 4 8 _ + 4 16 __ ÷ 1 4 1 4 1 4 5 4 2 3 2 3 4 7 4 7 Since we didn't have any brackets or exponents, we would start with division. 4 _ 8 + 20 __ 8 ÷ 4 __ 16 R 4 _ 8 + 20 __ 8 x __ 16 4 1 2 5 1 6 8 2 3 6 8 1 3 2 3 4 8 1 2 2 3 1 2 3 4 1 2 1 2 1 2 Now try to incorporate these two techniques into these questions. Now use this steps to answer these questions Now lets add what we have: 4 _ 8 + 10 __ 1 x 8 x 8 4 _ 8 + 80 __ 8 = 84 __ 8 or 21 2 __ Question #2 2 5 1 8 6 7 _ _ _ - x 5 9 _ x ( ) Since we have brackets, we will start with them 7 _ 6 x 9 5 _ ( ) 2 5 1 8 _ _ - x 3 2 2 5 1 8 _ _ - x 14 __ 15 Now we do the multiplication and subtraction 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 _ 5 - 14 _ 120 ÷ ÷ 2 2 x 12 x 12 24 __ 60 - 7 _ 60 = 17 __ 60 + + + + + + + = 16 5 = 3 1 5 Answers: 1) You would do 8 x

You can display it as fraction circles.

Make 8 circles and divide them into fifths.

Then shade two-fifths of each circle.

The answer is 3

It will take 3 h to get all the apples we need. 2 5 1 5 1 5 5 strips 2 3 = 10 3 = 1 3 3 1 3 1 3 5 = 1 4 1 = 1 4 1 4 8 21 1 4 1 4 1 1 2 3 Change them to improper fractions 2 1 1) For a while, I became an alchmist (someone who makes potions) but only for the good of science. If i used 1 1/2 cups of water for a stomach medicine and 2 1/4 times more water for a pain reliever, how many cups of water did I use for the pain reliever? 2) I made 3 1/2 bottles of Medicine One and 1 2/3 times the number of bottles of Medicine One for Medicine Two. How many bottles of Medicine Two did I make? 3)I was working on an experiment when I received a letter telling me that my mother was sick. I went from Cambridge to Lincolnshire to see her. Now, in the 21st century it would take about 2 1/3 hours to get to Licolnshire, but back in the 1600's, it would take you 1 3/4 times longer to get Lincolnshire. How long does it take to get to Lincolnshire in the 1600's? There are 2 ways to divide fractions One way is to use common denominators

The other way is to use multiplication(reciprocal) We will start with common denominators. Our common denominator will be 10. After converting it, just divide the numerators! 2 _ 5 ÷ 10 __ 8 4 _ 10 ÷ 8 _ 10 = 2 We could also do this with multiplication. First, find the reciprocal of 2/5, then cross multiply! 2 _ 5 10 __ 8 R ÷ 2 _ 5 10 __ 8 x 1 4 2 1 = 4 _ 2 , 2 Now that you know the steps, you can try it on these questions! 8 15 8 15 =

=

I spend of my day teaching Physics. 1 x 8 4 x 15 2 1 2 15 2 15 4 3 8 3 Question #3 8 _ 2 - 2 _ 6 x 9 6 _ + 2 6 _ 2 ÷ 4 9 _ We will first start with multiplication and

division from left to right 8 _ 2 - 2 _ 6 x 9 6 _ + 2 6 _ ÷ 4 9 _ 1 3 3 2 8 _ 2 - 2 9 _ + 2 6 _ ÷ 9 4 _ R 3 2 1 2 = 5 2 8 2 _ Now we will do addition and subtraction

from left to right - 2 _ 9 + 2 _ 6 2 x = 18 x2 = 4 x9 = 18 x9 = 72 68 __ 18 + 2 _ 6 x3 = x3 = 18 6 18 __ 74 , 9 __ 37 3 1 1 4 3 = 4 3 5 2 x = 5 x 4 2 x 3 1 2 5 x 2 1 x 3 = 10 3 = = 3 3 1 Another way to find the product of two whole numbers is to use the cross-multiplicaton square. 6 _ 6 ÷ 3 6 24 __ 48 ÷ _ 12 __ 4 78 24 __ ÷ 64 __ 32 1) You would do 1 x 2

Use the cross multipliction square

Seperate the whole number from the fraction and place the in the correct spot.

Multiply 2 by 1, 2 by , 1 by , and by .

Add all the products together.

It equals 3

So, 1 x 2 = 3 3 x 1 = 3 3 x = 2 3 1 1 x

= x

= 1 x 2 = 2 1x = 1 2 2 x

= 1 x

= Since we alread have comon denominators, we will use that to answer this question 6 _ 6 ÷ 3 6 _ = 1 4 2 For this we will use cross multiplicaton. 1 2 1 4 1 2 1 2 24 __ 48 ÷ 12 __ 4 1 4 1 4 1 4 Since we already have common denominators, we will use that. Remember! Only multiply the numerators. 6 _ 6 ÷ 3 6 _ = 2 We will use multiplication for this one.

Remember the rule! Do nothing to the first then flip the second one. Don't forget to turn the division to multiplication. 24 __ 48 ÷ 12 __ 4 R 24 48 __ 12 4 __ x 1 1 12 = 2 12 __ We will also use multiplication for this one. Don't forget the rules! 78 24 __ ÷ 64 __ 32 R 78 24 __ x 64 __ 32 8 13 2 + 1 + 3 8 1 8 + 1 1 = = 3 13 8 ___ 1 8 1 2 1 3 2 3 2 3 1 2 2 3 1 2 1 2 1 2 2 3 4 7 1 0 and are both close to 1.

1 x 1 = 1

is close to 1 so the answer is reasonable. 4 7 2 3 8 15 1 0 8 15 1 4 is close to 0.

is close to 1.

0 x 1 = 0

is clsoe to 0 so the answer is reasonable 1 4 15 8 2 15 3) You would do x

x

=

=

=

of my class failed and did not study. is close to 0.

is close to 1.

0 x 1 = 0

is close to 0, so the answer is reasonable. 0 1 3 8 4 5 3 8 4 5 3 8 4 5 3 x 4 8 x 5 3 x 1 2 x 5 3 10 3 10 3 8 5 4 3 10 3 + 2 + + = 5 1 3 5 6 4 _ 5 8 ÷ 1) We will use multiplication for this one. 8 R x 5 4 _ = _ 1 2 1 10 __ 1 2 ÷ 3 7 _ 2) We will model this one. First, we need the 2 models. 1 1 1 2 2 2 3 4 3 3 4 4 = 4 2 _ 3 13 ÷ 2 _ 9 3) Lets try this one with multiplication. 13 R 1 __ x 9 2 _ = _ 2 117 2 1 1) I worked at the Royal Mint in London (a bank) where I supervised the making of the new coins.In the first year,

of the coins had a new design. of the new coins were gold. What fraction of the coins had a new design and gold? 2) 2 5 _ 10 ÷ 4 4 _ 25 2) The coins in england has the image of the queen or king at the back. The queen's face is on of the coins. of the coins with the queen's face on them are silver. What fraction of the coins are silver and has the queen's face on it? 25 __ 10 Evaluate. We will use multiplication. 104 25 ___ Evaluate. We will use multiplication. ÷ R 25 __ 10 x 25 ___ 104 = 5 2 50 520 ___ Evaluate. Use Multiplication. 40 1 5 _ ÷ 20 4 7 _ 201 5 ___ 144 ___ 7 R 201 ___ 5 ÷ x 144 ___ 7 = 1407 ____ 720 3) 1 2 1 2 1 2 1 2 2 3 3 5 2 3 1 2 1 5 4 10 You would do x

Use counters (or in this case coins)

Model 10 coins.

Box 4 of the 10 coins

Then circle of those 4 coins.

of 4 is 2, so circle 2 coins.

There are 2 coins out of the total coins, or or of the total coins.

So x is 3 5 2 10 4 10 4 10 You would do x

Use a rectangle model.

Make rows of 3 and columns of 5.

Shade 3 of the 5 rows.

Then highlight 2 of the 3 columns.

There are 6 boxes that are both shaded and highlighted.

There are a total of 15 boxes. So there are or .

So, x = = 3 5 3 2 6 15 2 5 6 15 2 5 2 3 3 5 3 7 4 7 2 3 1 2 1 2 1 3 1 3 1 6 1 2 1 2 3 4 1 3 1 2 1 3 1 6 2 3 1 2 1 3 1 2 1 2 1 3 1 3 2 x 1= 2 1 2 1 4 1 2 1 2 3 7 1 3 3 8 1 4 3 8 1 4 1 4 1 2 1 2 2) You would do 3 x 1

Use the cross multipliction square

Seperate the whole number from the fraction and place the in the correct spot.

Multiply3 by 1, 3 by , 1 by , and by .

Add all the products together.

It equals 5

So, 3 x 1 = 5 1 2 2 3 2 3 5 6 2 3 5 6 2 3 1 2 1 2 1 2 3) You would do 2 x 1

Use the cross multipliction square

Seperate the whole number from the fraction and place the in the correct spot.

Multiply 2 by 1, 1 by , 2 by , and by .

Add all the products together.

It equals

So, 2 x 1 = 4 1 3 3 4 1 3 5 24 3 4 5 24 3 4 3 4 1 3 1 3 1 x

= 2 x = 1 5 24 3 8 3 4 3 4 x = 2 + 1 + + = 3 = 4 3 8 29 24 3 7 1 3 4 Now that you know how to multiply mixed numbers, practise on these questions. 1 3 1 2 Multiply x Fun Facts! Divide: 1) 2) 2 3) = = 1 5 8 ___ Example: 1 1 1) 3) I spent of the years that I worked at the Royal Mint as a supervisor of something. I worked as the supervisor of the coin making process and the rest of the time as a supervisor of the whole bank. What fraction of the time was spent as a supervisor of the whole bank? 2 5 8 35 2 5 4 7 2 5 First, subtract from to how much was spent on being a supervisor of the whole bank.

x =

Use a rectangle model.

Make rows of 7 and columns of 5.

Shade 2 of the 5 rows.

Then highlight 4 of the 7 columns.

There are 8 boxes that are both shaded and highlighted.

There are a total of 35 boxes. So there are 8/35.

So x = 7 7 7 7 4 7 So you would do This is an actual coin that Isaac Newton made. This is Newton's coat of arms. Weird isn't it? It looks like a pirate flag Hoped you liked our presentation!

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