1. FRACTIONS _ Voodoo Hoodoo or FFOO? Learn this little rhyme and it will
help you understand how to calculate with fractions. You will see why at the end
of ...
FRACTIONS _ Voodoo Hoodoo or FFOO? Learn this little rhyme and it will help you understand how to calculate with fractions. You will see why at the end of this section of work.
Adding and subtracting is a pain in the bot, Multiplying and dividing is definitely not! A visual way of seeing how to multiply two simple fractions…. 1 2 3 1 1. Given a number of same size rectangles, shade parts to illustrate , , , etc. 2 3 4 5 Part 1
2. Suppose you were asked to share 1/3 of a chocolate with a friend? How would you do it and what fraction of the original chocolate would each of you get? On one rectangle shade ½ in vertical direction and 1/3 in horizontal direction. This divides the original rectangle into 6 parts. When ½ is shaded yellow, 3 of the 6 parts are yellow.
When the rectangle has 1/3 shaded blue, 2 of the 6 parts are blue.
When the 1/3 blue shading is done on only the yellow parts, one part is shaded green . This suggests that 1/3 x ½ = 1/6.
1 3 x : 3 4 First divide a rectangle in 3 parts horizontally and 4 parts vertically. 4 parts this way 3 parts this way
3. To calculate
This divides the rectangle into 12 equal parts as shown. Shade ¾ yellow Now shade 1/3 of the yellow in green horizontally.
Now 3 parts of the 12 are shaded green. This suggests that 1/3 x ¾ = 3/12 Can you suggest another way of naming 3/12 from this diagram? (Hint: What fraction of the rectangle is coloured green?) 1
4. Use this method to find
i.
1 2 2 1 1 2 x ii. x iii. x 4 3 3 5 5 3
5. Using the TI-15 calculator check your answers to the calculations in Q.4 To multiply two fractions, first follow these instructions:
Press button to enter the Fraction Menu. The selection U n/d n/d is displayed. This lets you choose to display fractions results as mixed numbers (U n/d) or improper fractions (n/d). Press the left and right arrow keys to move the underline to choose the display you want. In this case, move the underline to n/d . Press the down arrow , and move the underline to MAN. This lets you choose to simplify fractions manually (using the key). Select .
1 3 x Press 3 4 (Remember the top line is the numerator and the bottom line is the denominator.) 3 1 The answer is [which can be simplified (press ) to ] 12 4 Now check your answers to the other questions.
eg
6. Using the correct answers, complete this table and look for a method that will help you get to the answer without either a diagram or a calculator (if you know your tables!) First fraction Second fraction 1 3 1 4 2 3 1 5
Answer to Simplified fraction first x second if possible
3 4 2 3 1 5 2 3
Do you see why multiplying is not “a pain in the bot”? Predict the answers to the following questions and check on the calculator i.
ii. iii.
iv.
2
Part 2 What is a FFOO ? 1. The easiest number to multiply by is 1. Why?? Do you know your “one times tables” …? 1 2. The number 1 can be written in fraction form as 1 2 2 1 2 so 1 3 3 1 3 2 5 6 We can write the number 1 in many ways as a fraction eg or or etc. 2 5 6 Write down 3 more ways to write 1 as a fraction.
These fractions can be called FFOO’s or “Funny Forms Of One”
2 2 1 2 2 3 6 2 6 and represent the same number! 1 = so 3 3 1 3 3 3 9 3 9 We call them equivalent fractions.
So
3. Using the TI-15 to multiply fractions :
Press button to enter the Fraction Menu. The selection U n/d n/d is displayed. This lets you choose to display fraction results as mixed numbers (U n/d) or improper fractions (n/d). Press the left and right arrow keys to move the underline to choose the display you want. In this case, move the underline to n/d . Press the down arrow , and move the underline to MAN. This lets you choose to simplify fractions manually (using the key). Select . 1 follow the steps below: 3 1 6 Press to see changed to . 3 6 1 Check that this is still the same as by pressing the key and then . 3 If you now press you will see a 2 to indicate that the calculator has simplified the fraction by dividing the numerator and denominator by a factor of 2.
To find a number of fraction equivalents for eg
4. Use several other FFOO’s to find 3 more fractions that are equivalent to
1 . 3
Compare your fractions with a partner’s. How many do you think are possible? Can you write down some more without using the calculator?
5. Choose another simple fraction and find at least 3 other fractions that are equivalent to it. Do them mentally if you can but use the calculator if you need to. 6. See who can write down the most fractions equivalent to
3
2 in 20 seconds. 5
Part 3. Upside downers… 1. Using the calculator fill in the blank parts in this table. If the symbol U n/d n/d appears at the top of the calculator screen, the fraction can be written in simpler form by pressing and . First fraction Second fraction
1 3 1 4 2 3 3 2 3 4 1 5
Answer to Simplified fraction first x second if possible
3 1 4 1 3 2 2 3 4 3 5 5
1
2 1 5 4
20 20
Explain how the second fraction is chosen when the first is given. Write down another two fractions that have this property and check on the calculator. 2. Dividing one fraction by another fraction. Use your calculator to complete this table giving the simplest fraction answer. First fraction Second fraction 1 3 1 4 2 3 1 5 2 5 3 2
Answer to Simplified fraction first second if possible
3 1 2 3 1 5 2 3 3 2 2 5
*** add some that need simplification! Can you see what is happening in a division question? 1 1 Guess the answer to and check on the calculator. 4 3 Try one more of your own choosing with a partner and check each other’s work. 4
3. To explain this process a VERY FUNNY FORM OF ONE is used. - ( a “VFFOO”). Check the steps shown below:
3 2 . First write down what you think the answer will be. 4 5 a Write the question like an ordinary division where a b and vice versa. b
To find
3 2 3/ 4 = 4 5 2/5
5 5 The VFFOO to choose this time is / . Can you see why? 2 2
If we multiply the two denominators first we get 1 (from Question 1 above) so we would then have a fraction being divided by 1 which of course is the same as multiplying by 1. ie. The question now becomes 3/ 4 5/ 2 x 2/5 5/ 2 3 5 2 5 = / 4 2 5 2 3 5 10 = / 4 2 10 3 5 = / 1 4 2 3 5 = 4 2 15 = 8 Did you guess correctly? Check the answer on your calculator.
4. Write down all the steps like the question above to calculate
2 4 . 3 5
Check your answer on the calculator. 5. Can you see a quicker way to do the question (without using the calculator!)? Try your 3 2 method on and check on the calculator. 5 3 6. Try these questions mentally if you can and check on the calculator. * * *
5
Part 4 Adding and subtracting fractions is a “pain in the bot”! 1 3 4 + . The answer IS NOT . 3 4 7 1 1 If you think about the numbers, is bigger than . 3 4 Would you rather share a chocolate with 3 people or 4 to get a bigger share? 1 3 Using the calculator we know that + = 1. 4 4 1 3 4 So we can assume that + is greater than 1 and we know is less than 1. 3 4 7 So something else is happening in addition (and subtraction)!
Consider
1. A visual way of seeing how to add two simple fractions…. 1 3 + . 4 4 Using the rectangles with parts shaded as in Part 1 for multiplying fractions: 1 Divide each in 4 parts and shade one part of the first red ( ) and three parts of the 4 3 other one yellow ( ) 4
Calculate
Adding the two together on one rectangle shows
4 1 3 4 so + = (or 1 - a FFOO!). 4 4 4 4 Check on the calculator, setting it up as before with Man and n/d .
That is
2. Using your calculator after you have guessed your answer, complete this table. First fraction Second fraction Guessed answer to Calculator Simplified answer first + second answer if possible 1 2 3 3 1 4 5 5 2 1 5 5 4 3 7 7 2 2 5 5 5 1 8 8 Fractions with the same denominator are easy to add together (and subtract!) 6
3. Another look at
1 3 + . 3 4
When we multiplied two fractions with different denominators like this we made rectangles with 3 x 4 parts by dividing the rectangle into 3 parts in one direction and 4 parts in the other direction to make 12 parts.
Divide a rectangle in 3 parts horizontally and 4 parts vertically. 4 parts this way
3 parts this way
Shade 1/3 red
This shows that
Shade
1 4 3 12
Then the deominators are the same! 1 3 4 9 13 SO + = 3 4 12 12 12
Check the calculator answer!
and
3 yellow 4
1 9 4 12
We can add fractions easily provided their denominators are the same! We can make their denominators the same using carefully chosen FFOO’s.
1 3 + . 3 4 What FFOO’s should we use here?
Remembering that the rectangles method used 3 x 4 parts, maybe
4. Yet another look at
4 3 and will do the 4 3
trick! Follow these steps carefully: 1 3 + 3 4 1 4 3 3 = 3 4 4 3 4 9 = 12 12 13 = YES! 12
Fractions that have the same denominator are easy to add together, so we use carefully chosen FFOO’s to set this up and then the question is EASY-PEEZY! 7
5. Predict what the denominator will be and check on the calculator to complete the table when these two fractions are added together. First fraction Second fraction 2 3 1 4 2 3 1 5
Predict the Denominator from calculator denominator
3 4 2 3 1 5 2 3
First Second FFOO Equivalent FFOO Equivalent Answer Calculator fraction fraction fraction 1 fraction 1 fraction 2 fraction 2 answer 1 3 3 4 1 2 4 3 2 1 3 5 1 2 5 3
6. Subtraction is very similar except that instead of adding the numerators you take them! Try these taking the second fraction from the first one. First Second FFOO Equivalent FFOO Equivalent Answer Calculator fraction fraction fraction 1 fraction 1 fraction 2 fraction 2 First - second answer 3 1 4 3 2 1 3 4 2 1 3 5 2 2 3 5 7.
Try these questions and check on the calculator.
Can you explain this rhyme now??
Adding and subtracting is a pain in the bot, Multiplying and dividing is definitely not! 8
