Multiplying and Dividing, Decimals and Fractions Student Signature: Parent/Carer signature:

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1 Preparin yourself for mathematics at Guilsborouh Multiplyin and Dividin, Decimals and Fractions Student Sinature: Parent/Carer sinature:

2 Dear student, This booklet contains work which you need to complete over the summer. It is desined to help you develop key skills in mathematics which you need at secondary school. The first part of the booklet needs to be completed at home. The second part will be completed in lessons at school in September. You will need to brin the completed booklet to school on your first day. We look forward to seein you in September Guilsborouh Academy Mathematics Department

3 How does it work? Whole Numbers Lon multiplication As you did when addin and subtractin, keep your place value columns lined up neatly. You need to be aware of the place value of the number you are multiplyin by. Calculate # # # Line up matchin place values Multiply the by Carry over any tens values after multiplyin For # tens, put a 0 in the ones column and multiply by Carry over any tens values after multiplyin # + Add the two new numbers toether ` # 78 Here is another example. Be careful to line up the columns correctly. Calculate # # Line up matchin place values Multiply the by Carry over any tens values after multiplyin # For # 0 tens, put a 0 in the ones column first and multiply by # For # hundreds, put a 0 in the ones and tens columns and multiply by Carry over the tens value after multiplyin # Add the two new numbers toether ` # 0 08 Whole Numbers Mathletics Passport P Learnin H SERIES TOPIC

4 How does it work? Your Turn Whole Numbers Lon multiplication Calculate each of these multiplication questions showin all workin. Check your answers on the calculator. a b 8 0 c d 8 e 7 f 8 Whole Numbers Mathletics Passport P Learnin H SERIES TOPIC

5 How does it work? Whole Numbers Short and lon division Short and lon division are only different due to the size of the number you are dividin by. Here is a short division question. Calculate 7 08 ' Step : Step : Step : Step : Step : Divide 7 by Put the whole number answer () above the 7 Make the remainder () the tens diit for the next number Divide by Put the whole number answer () above the Make the remainder () the tens diit for the next number Divide by Put the whole number answer () above the Make the remainder () the tens diit for the next number Divide 0 by Put the whole number answer () above the 0 Make the remainder () the tens diit for the next number Divide 8 by Put the answer (8) above the 8 ` 708 ' 8 If there is a remainder at the end, always write it as a fraction. remainder fraction the amount left over the divisor Calculate 8 ' Step : 8 Divide by Put the whole number answer () above the Make the remainder () the tens diit for the next number Step : 7 8 Divide by Put the whole number answer (7) above the There is no remainder this time Step : divisor 7 8 amount left over Divide 8 by Put the whole number answer () above the 8 Write the remainder as a fraction ( ) to the riht ` 8 ' 7 H Whole Numbers SERIES TOPIC Mathletics Passport P Learnin

6 How does it work? Your Turn Whole Numbers Short and lon division WHOLE NUMBERS * WHOLE NUMBERS * Calculate each of these short division questions showin all workin. Check your answers on the calculator. a 77 ' b 80 ' Short & Lon Division.../.../0... c ' d 0 ' Calculate each of these short division questions showin all workin. (psst: remember to write any remainders as a simplified fraction) Check your answers on the calculator. a 8 ' 7 b 87 ' c ' 8 d 80' H Whole Numbers SERIES TOPIC Mathletics Passport P Learnin

Where does it work? Decimals Multiplyin with decimals Just write the terms as whole numbers and multiply. Put the decimal point back in when finished.

7 Where does it work? Decimals Multiplyin with decimals Just write the terms as whole numbers and multiply. Put the decimal point back in when finished. The number of decimal places in the answer the number of decimal places in the question! Calculate #. 8 # Multiply both terms as whole numbers 8 decimal place in question decimal place in answer ` #. 8. Calculate 0.0 #. 0 # Multiply both terms as whole numbers 0 ` 0.0 # decimal places in question decimal places in answer How does this work when multiplyin with decimals? Excellent question! Very lad you asked! Let s do the second one aain but this time chane the decimals to equivalent fractions first 00. #. # # 00# Chanin the decimals to fractions Multiply numerators and denominators toether Number of zeros in denominator total of decimal places in question 0' Dividin by moves decimal point four places to the left 000. ` decimal places in question decimal places in answer Try this method for yourself on the first example above, rememberin that as a fraction. H Decimals SERIES TOPIC Mathletics Passport P Learnin

Where does it work? Your Turn Decimals Multiplyin with decimals Calculate these whole number and decimal multiplications, showin all you workin: a # b. c 0.

8 Where does it work? Your Turn Decimals Multiplyin with decimals Calculate these whole number and decimal multiplications, showin all you workin: a # b. c 0.8 # 0. # 0. # # # d e f Calculate these decimal multiplications, showin all your workin:.8 # # #. a b c ULTIPLYING WITH DECIMALS.../.../0... MMULTIPLYING WITH DECIMALS 7. #.. #. 7. #. d e f Decimals Mathletics Passport P Learnin H SERIES TOPIC

9 Where does it work? Decimals Dividin with decimals Opposite to multiplyin, we move the decimal point before dividin if needed. To find the quotient involvin decimals, the question must be chaned so the divisor is a whole number. dividend ' divisor quotient Calculate 8. ' Divisor already a whole number so no chane needed `.8 '.07 Calculate 00. ' ' ' 000. Move both decimal points riht until divisor is a whole number. ' Quotient Dividend if divisor ` 0.0 ' Drop off any 0s at the front of the answer Here s another example showin how to treat remainders Calculate. ' 08.. ' 0.8. ' 0.8 Move both decimal points riht until divisor is a whole number. ' Add 0s on the end of the dividend for each new remainder `. ' Drop off any 0s at the front H Decimals SERIES TOPIC Mathletics Passport P Learnin

Where does it work? Your Turn Decimals Dividin with decimals Calculate these decimal and whole number divisions:.' 7. '. ' a b c.../.../0... IVIDING WITH DECIMALS DDIVIDING WITH DECIMALS `.' ` 7. ' `.

10 Where does it work? Your Turn Decimals Dividin with decimals Calculate these decimal and whole number divisions:.' 7. '. ' a b c.../.../0... IVIDING WITH DECIMALS DDIVIDING WITH DECIMALS `.' ` 7. ' `. ' 0. ' 08. ' 0. 7' 7 d e f ` 0. ' ` 0.8' ` 0.7 ' 7 Calculate these decimal divisions, showin all your workin:. ' 0.. ' ' 08. a b c `.' 0. `.' 0. ` 0. ' ' ' ' d e f `.8 ' 0. ` 0.8 ' 0.0 `.8 ' 0.00 Decimals Mathletics Passport P Learnin H SERIES TOPIC 7

$Fractions Improper fractions and mixed numerals An improper fraction has a bier numerator (top) than denominator (bottom) Improper fractions numerator denominator Mixed numerals have a whole$ $number and a proper fraction. Mixed numerals A mix of whole numbers and proper fractions. means bier than Mixed numerals are simplified improper fractions.$

11 How does it work? Fractions Improper fractions and mixed numerals An improper fraction has a bier numerator (top) than denominator (bottom) Improper fractions numerator denominator Mixed numerals have a whole number and a proper fraction. Mixed numerals A mix of whole numbers and proper fractions. means bier than Mixed numerals are simplified improper fractions. Simplify these Improper fractions to mixed numerals (i) ' numerator numerator 'denominator denominator r Whole number answer remainder same denominator (ii) 7 7' Simplify if possible picture form r remainder Whole number answer same simplified denominator Mixed numerals to improper fractions + (i) # # + same denominator (ii) + # # + same denominator 8 H Fractions SERIES TOPIC Mathletics Passport P Learnin

$Your Turn Fractions Improper fractions and mixed numerals Write the mixed numerals represented by these shaded diarams: a b c d Make sure you write the fraction in$ $simplest form where possible. e f Simplify these improper fractions by writin them as mixed numerals. a b c.../.../0.$

12 How does it work? Your Turn Fractions Improper fractions and mixed numerals Write the mixed numerals represented by these shaded diarams: a b c d Make sure you write the fraction in simplest form where possible. e f Simplify these improper fractions by writin them as mixed numerals. a b c.../.../0... IMPROPER FRACTIONS AND MIXED NUMERALS Write these fractions in simplest form first, then chane to the mixed numerals. a b c 8 Write the equivalent improper fraction for these mixed numerals. a b c Write the equivalent improper fraction for these mixed numerals after first simplifyin the fraction parts. a b c 7 Fractions Mathletics Passport P Learnin H SERIES TOPIC

13 Fractions equivalent fractions To find equivalent fractions without drawin diarams we use the numerators and denominators to uide us. Imaine your share of a cake is half. It is too bi to pick up so you cut your half into halves. You now have quarters of the cake. You have doubled the number of parts (the denominator) and by doin this you have doubled the number of parts (the numerator). This method can be used to find all equivalent fractions. Use the clues to help you make the equivalent fractions: a b c d 8 0 e f 8 h We can also reduce the number of parts in a whole. We divide to do this: a b c 8 Whatever we do to the top, we do to the bottom. Whatever we do to the bottom, we do to the top d 8 e f 0 8 Answer the followin: a Cassie s table won a pizza for havin the most table points at the end of term. There are pupils at the table. What fraction of the pizza will they each receive? b The pizza has been cut into pieces. How many slices does each pupil et? What is this as a fraction? c Stavros reckons that because they ot slices they ot more than they would have if the pizza had been cut into pieces. Is he riht? Explain your answer with words or diarams. G SERIES TOPIC Fractions, Decimals and Percentaes Copyriht P Learnin

14 Fractions simplifyin fractions These fractions are all equivalent to one half: Which is the simplest? A fraction is in its simplest form when is the only number that both numbers can be divided by. We simplify fractions to make readin and workin with fractions easier. Circle the simplest fraction in each roup: a 0 00 b c 00 0 d To find the simplest fraction, we divide both the numerator and the denominator by the same number. It makes sense for this to be the biest number we can find so we don t have to keep dividin. This number is called the Hihest Common Factor (HCF). Look at: 8?? What is the biest number that oes into both and 8? is the biest number that oes into 8 and. 8 Find the hihest common factor and then simplify: a 0 HCF is 0 b 0 HCF is 0 c HCF is d HCF is Wally says he has simplified these fractions as far as he can. Is he riht? If not, find the simplest fraction: a b c d 0 G SERIES TOPIC Fractions, Decimals and Percentaes Copyriht P Learnin

15 Fractions renamin and orderin fractions Sometimes we have to order and compare fractions with unrelated denominators such as, and. To do this, we have to find one common denominator we can convert all the fractions to. You have cakes for a class party. One has been cut into halves and one into thirds. The problem is that you want each slice to be a fair fraction of the cakes. a Continue cuttin the cakes so that each cake has the same number of fair slices: b If you had one of these new slices, what fraction of the cake would you receive? That is an example of how we rename fractions. We find a way to re-divide the wholes so that they have the same number of parts. To do this efficiently we find the smallest shared multiple. This is then called the Lowest Common Denominator (LCD): The multiples of are,,, 8, The multiples of are,,,,, is the LCD so we convert both fractions to sixths: Rename these fractions by first findin the shared LCD and then convertin the fractions. Use the multiplication table on the riht to help you find the LCD: a b c G SERIES TOPIC Fractions, Decimals and Percentaes Copyriht P Learnin

16 Fractions renamin and orderin fractions Look at each roup of fractions. Predict which you think is the larest and circle your prediction. Now, rename the fractions in the work space below so that each fraction in the roup has the same denominator. Use a different colour to circle the larest fraction. Are there any surprises? a b c 8 d 8 This time, rename the fractions and circle the larest. Underline the smallest. a 8 b 7 c 8 d For each fraction write a larer fraction below. The new fraction must have a different denominator. It can have a different numerator. If you can do this, you are a whiz! This is real extension Maths. Fractions, Decimals and Percentaes Copyriht P Learnin G SERIES TOPIC

17 Work to be completed at school in September You will complete the followin set of work at school in September. Please do not complete it at home.

18 Date: MULTIPLICATION AND DIVISION - RED Student Assessment

19 Date: MULTIPLICATION AND DIVISION- AMBER Student Assessment

20 Date: MULTIPLICATION AND DIVISION- GREEN Student Assessment

21 DECIMALS - RED Date: ) x. ) x. ). x. ). x. ). x.78 ) 7. x. 7) 0. x. 8) 0. x. ). x ).8 x 0.00 Student Assessment

22 Date: DECIMALS - AMBER ). ) 8. ). ). ) ) 0. 7) 7.0 8).80 ). 0) 7. Student Assessment

23 DECIMALS - GREEN Date: ) ).8 0. ) ).. ).0. )..8 7).0 8 8).8. ).0. 0).8.8 Student Assessment

24 Date: FRACTIONS - RED Convert the followin improper fractions to mixed numbers: Convert the followin mixed numbers to improper fractions: Student Assessment

25 Date: FRACTIONS - AMBER Fill in the missin numbers to make these fractions equivalent Write the followin fractions in their simplest form. Student Assessment

26 FRACTIONS - GREEN Date: Write the correct symbol in the box to show whether the fraction on the left is more than (>) or less than (<) the fraction on the riht. ) 8 ) 0 7 ) 8 ) ) ) 7 7) 7 8 8) 8 Write the followin fractions in ascendin order of size. Student Assessment

27 Developin Mathematical Reasonin Task Student Assessment

28 Developin Mathematical Reasonin Task Student Assessment

What is a Fraction? Fractions. One Way To Remember Numerator = North / 16. Example. What Fraction is Shaded? 9/16/16. Fraction = Part of a Whole

What is a Fraction? Fractions. One Way To Remember Numerator = North / 16. Example. What Fraction is Shaded? 9/16/16. Fraction = Part of a Whole // Fractions Pages What is a Fraction? Fraction Part of a Whole Top Number? Bottom Number? Page Numerator tells how many parts you have Denominator tells how many parts are in the whole Note: the fraction