This includes:- 1. To find a common denominator and use it to write equivalent fractions 2. To add and subtract fractions with unlike denominators 3. To add and subtract mixed numbers 4. To solve fraction problems
1. Add and Subtract Fractions Fractions have been a part of our lives for a long time. In ancient times, Sumerians, Babylonians, Romans, Egyptians, Greeks, Chinese, and Arabians all used fractions. The Eye of Horus was used by ancient Egyptians to represent fractions. It was a powerful symbol that they believed protected against evil. According to an old myth, Horus’ eye was torn out, broken into pieces, and then put back together. What You Will Learn Key Words φ to find a common denominator and use it multiple to write equivalent fractions common denominator φ to add and subtract fractions with unlike improper fraction denominators mixed number φ to add and subtract mixed numbers φ to solve fraction problems MATH LINK Each part of the Eye of Horus was a –1 8 fraction. When all the parts were put together, it was believed that the eye –1 4 1 was whole again. Ancient Egyptians –1 –– 2 16 thought that the parts had a combined value of 1. Were they correct? In this chapter, you will explore fractions 1 1 –– –– through history and around the world. 64 32 Web Link To learn more about the ancient Egyptian myth of the Eye of Horus, go to www.mathlinks7.ca and follow the links. 228 NEL • Chapter 7
2. ;DA967A:H / Make the following Foldable to organize what you learn in Chapter 7. Step 1 Staple seven sheets of notebook paper together along the top edge. Step 2 Make a line 9 cm up from the bottom of the top page. Cut across the entire page at this mark. Step 3 Make a line 7.5 cm up from the bottom of the second page. Cut across the entire page at this mark. Step 4 Cut across a line 6 cm up from the bottom of the third page. Step 5 Cut across a line 4.5 cm up from the bottom of the fourth page. Step 6 Cut across a line 3 cm up from the bottom of the fifth page. Step 7 Cut across a line 1.5 cm up from the bottom of the sixth page. Step 8 Label the tabs formed as shown. Chapter 7 Add and Subtract Fractions Key Words . 71 . 72 . 73 . 74 What I Need to Work On Literacy Link As you work through Chapter 7, take notes under the appropriate tab. Include information about the key words, examples, and key ideas. Chapter 7 • NEL 229
3. Common Denominators Focus on… After this lesson, you will be able to... φ find a common denominator for a set of fractions φ compare and order positive fractions Jasmin and Tyler collect trading cards. Jasmin has collected __31 of a set. Tyler has collected 1 __ of a set. They want to know who has more cards. 4 Jasmin and Tyler need to compare the fractions. It is easier to compare fractions when the denominators are the same. So, Jasmin and Tyler need to find a common denominator. How can you determine a common denominator? • coloured pencils 1. Fold a piece of paper into 3 equal parts. Shade 1__ of the paper red. 3 2. Fold the same piece of paper into 4 equal parts the other way. 230 NEL • Chapter 7
4. 3. a) How many equal parts is the paper divided into? b) Count how many parts you shaded red. Name an equivalent 1 using your answer to part a) as the denominator. fraction for __ 3 1 of the 4. Fold a different piece of paper into 4 equal parts. Shade __ 4 paper blue. 5. Fold the piece of paper into 3 equal parts the other way. 6. Count how many parts you shaded blue. Name an equivalent 1. fraction for __ 4 Reflect on Your Findings 7. a) What is the relationship between the denominators 3 and 4, and the denominator 12? common b) What is one method for determining a common denominator ? denominator • a common multiple of the denominators of a Example: Determine a Common Denominator set of fractions • a common a) Determine a common denominator for _2_ and 1 __. denominator for _1_ 3 2 4 2 _ _ 1 __ and 1__ is 12 because a b) Determine equivalent fractions for and using the common 6 3 2 common multiple of denominator from a). 4 and 6 is 12 Method 1: Use Paper Folding or Diagrams a) Divide a rectangle into 3 equal parts. Either fold a piece of paper, or draw a rectangle. Fold the paper or divide the rectangle into 2 equal parts the other way. There are 6 parts in the rectangle. A common denominator for __ 2 and 1 __ is 6. 3 2 b) Shade 2__ of the rectangle red. 3 4 of the 6 parts are red. 2=4 __ __ 3 6 Turn the paper over, or draw another rectangle and divide it as in step a). Shade 1__ of this rectangle blue. 2 3 of the 6 parts are blue. 1=3 __ __ 2 6 7.1 Common Denominators • NEL 231
5. Method 2: Use Multiples 1 is 2. a) The denominator of __ You can use divisibility 2 rules to find multiples. multiple Multiples of 2 are 2, 4, 6, 8, 10, 12, … 6 is divisible by both 2 and 3. • the product of a given The denominator of 2 __ is 3. So, multiples of both 2 and 3 number and a natural 3 will be multiples of 6: number like 1, 2, 3, 6, 12, 18, … Multiples of 3 are 3, 6, 9, 12, 15, … and so on • for example, some The first multiple divisible by both 2 and 3 is 6. multiples of 3 are 3, 6, 9, 12, and 15 A common denominator is 6. You could use any multiple of 6 as the common denominator, but the first multiple is often better to use. The denominator will be a smaller number, which is easier to work with. b) Write equivalent fractions using 6 as the denominator. ×3 ×2 To determine equivalent fractions, __ = 3 1 __ 2=4 __ __ multiply the numerator and denominator 2 6 3 6 by the same number. This process does ×3 ×2 not change the value of the fraction. Check: 2 __ = 4 __ Strategies 3 6 Use pattern blocks. Model It Refer to page xvi. = –1 –3 2 6 = –2 –4 3 6 Determine a common denominator for each pair of fractions. Then use the common denominator to write equivalent fractions. Show two different methods. 1 a) __ and _3_ 5 b) __ and _1_ 3 4 8 6 232 NEL • Chapter 7
6. • You can use paper folding, diagrams, or multiples to determine a common denominator. Paper Folding or Diagrams 5 of the 10 parts are blue. 6 of the 10 parts are red. 5 1 = ___ __ 3 6 __ = ___ 2 10 5 10 Multiples 1 is 2. Multiples of 2 are 2, 4, 6, 8, 10, … The denominator of __ 2 3 is 5. Multiples of 5 are 5, 10, 15, 20, … The denominator of __ 5 A common denominator is 10. • To write fractions with a common ×5 ×2 denominator, determine equivalent fractions. 5 1 = ___ __ 3 = ___ __ 6 2 10 5 10 ×5 ×2 1. Tina wanted to find a common denominator and equivalent 3 and 2 fractions for __ __. This is what she did: 5 3 a) Was she correct? If not, what was her error? b) Draw diagrams to show what she should have done. c) Discuss your diagrams with a classmate. 2. Ian says, “A common denominator for __ 3 and 5__ is 12.” Meko 4 6 says, “I think it is 10.” Do you agree with Ian or Meko? Why? 3. How can you use multiples to find a common denominator for 1, __ the fractions __ 3? 2, and __ 2 5 4 7.1 Common Denominators • NEL 233
7. 7. Use a diagram to determine a common denominator for each pair of fractions. For help with #4 to #9, refer to the Example on Then write equivalent fractions using the pages 231–232. common denominator. 4. Use the folded papers shown to determine a) _3_ and 1 __ b) _5_ and 3 __ c) _1_ and _1_ 8 3 6 4 5 2 a common denominator and equivalent fractions for each pair of fractions. 8. Use multiples to determine a common a) denominator for each set of fractions. Then write equivalent fractions using the common denominator. a) _1_ and _2_ b) _1_ and _1_ 5, __ c) __ 5 1, and ___ 2 5 3 4 8 6 12 1 __ 2 __ 4 3 9. Using multiples, determine a common b) denominator for each set of fractions. Then use the common denominator to write equivalent fractions. a) _3_ and _1_ b) _1_ and _1_ 1, __ c) __ 7 2, and ___ 8 4 6 4 5 3 10 1 __ 3 __ 2 4 5. Look at the diagrams to determine a common denominator and equivalent fractions for each pair of fractions. 10. Determine a common denominator for a) each pair of fractions. Which is the larger fraction in each pair? 3 13 a) __, ___ 5 36 b) __, ___ 4 16 7 49 1 __ 3 __ 11 3 c) ___, ___ 12 d) ___, _4_ 3 5 30 10 27 9 b) 11. Draw a Venn diagram like the one shown to list common denominators that are less 1 and __ than 50 for __ 1. 5 __ 1 __ 4 6 6 4 Multiples Multiples 6. Draw a diagram to determine a common of 4 of 6 denominator for each pair of fractions. Then use the common denominator to write equivalent fractions. a) _1_ and _1_ b) _2_ and _1_ c) _1_ and _2_ 2 3 3 5 6 5 234 NEL • Chapter 7
8. 12. Fill in the blanks to make equivalent 16. 5 of a schoolyard is taken up by grass. ___ 12 fractions. 7 is the track. The rest is pavement. ___ 1 � � � � � � a) __ = __ = ___ = ___ = ___ = ___ = ___ 18 4 8 12 16 20 24 28 a) What common denominator could be b) 1 __ = = 3 = __ 2 __ __ 5 = __ 4 = __ 7 = ___ 11 used to compare these fractions? 5 � � � � � � 24 12 = __ 6 = __3 = ___ 48 = __9 b) Does the grass or the track take up c) ___ = ___ 56 � � � � � more space? d) 30 ___ = ___ 10 = __ 15 = ___ � = ___ 5 = ___ � 48 � � � 96 32 13. Fill in each blank with a numerator to 17. a) Copy the shapes. For each shape, make the statement true. Provide as many 3. answers as possible. Use diagrams to show colour in __ 8 how you determined your answers. 1 � a) __ < __ < _3_ 4 2 4 1 � b) __ < __ < _5_ 3 6 6 2 � c) __ < ___ < _4_ 5 10 5 14. Determine a common denominator for b) Which shapes were more difficult to the set of fractions. Use the common colour in? Which were easier? Explain. denominator to write an equivalent c) Imagine you are using paper folding to fraction for each fraction. Then list the determine a common denominator for fractions in order from least to greatest. 3 2. Which of the shapes would it __ and __ 8 5 1 5, __ 1, __ __, __ 3, _1_ 2, __ be possible for you to use? Show the 3 4 6 3 4 2 work by drawing the fold lines on the 15. The ancient Greeks thought of numbers shapes. as being represented by rectangles. They d) Compare your drawings with a would have made a rectangle like this to classmate’s. represent 6: 18. Write as many different proper fractions in lowest terms as you can that have denominators from 2 to 9 and numerators that are positive numbers. a) How could this rectangle be used to find a common denominator for __ 1 and 1 __? 19. Which of the following fractions is closest 2 3 3? Explain. to ___ 10 b) Use a rectangle to find a common 1 B ____ 21 9 A __ C ___ D 2__ denominator for __3 and __ 1. 4 100 40 5 4 7 7.1 Common Denominators • NEL 235
9. 20. You have three beakers that are the same 21. The table shows the fraction of the total size. _2_ of beaker 1 contains oil. _1_ of number of students at Maple Leaf 3 4 Elementary School that are in each grade. beaker 2 contains water. Beaker 3 is empty. When you pour the liquids into Kindergarten 7 ___ beaker 3, the level of the combined liquids 40 corresponds exactly to one of the Grade 1 3 ___ markings on the side of beaker 3. Which 20 of the following beakers is beaker 3? Grade 2 11 ___ 72 A B 5 Grade 3 ___ 36 Grade 4 26 ____ 180 Grade 5 17 ____ 180 Grade 6 13 ___ C D 90 a) Which grade has the greatest number of students? b) Which grade has the least number of students? c) Which two grades have the same number of students? d) If there are 54 students in grade 1, what is the total number of students in the school? MATH LINK –1 8 a) Determine a common denominator for the fractions in the Eye of Horus. Show your work. –1 4 1 –1 –– b) Use this common denominator to determine an 2 16 equivalent fraction for each part in the eye. 1 1 –– –– 64 32 236 NEL • Chapter 7
10. Add and Subtract Fractions With Unlike Denominators Focus on… 1 1 1 1 – – – After this lesson, you 6 3 2 will be able to... φ add and subtract fractions How could you use pattern blocks to model addition and subtraction? with unlike denominators φ solve problems involving the addition and How can you add and subtract fractions with unlike subtraction of denominators? fractions φ check that your 1. a) What two pattern blocks would you use to represent _1_ and __ 1? 2 3 answers are 1 + __ b) Can you tell what the answer to __ 1 is using these two pattern reasonable using 2 3 blocks? Explain. estimation 2. a) Use the green triangles to represent _1_ and 1 __. What fraction does 2 3 each green triangle represent? b) Can you tell what the answer to __1+1 __ is now? Explain. 2 3 3. a) What pattern blocks would you use to represent 1 1? __ and __ 2 6 • pattern blocks 1−1 b) Can you tell what the answer to __ __ is using these two • coloured pencils 2 6 pattern blocks? Explain. 4. a) 1. Use the green triangles to represent __ 2 b) Can you tell what the answer to __1−1 __ is now? Explain. 2 6 c) How many green triangles are left? Reflect on Your Findings 5. How can you use pattern blocks to help you add and subtract fractions with unlike denominators? 7.2 Add and Subtract Fractions With Unlike Denominators • NEL 237
11. Example 1: Add Fractions With Unlike Denominators Add. Write the answer in lowest terms. 1 1 __ + __ 3 6 Solution Is the sum closest to Method 1: Use Fraction Strips 0 , 1 + __ __ 1 + 1 , 3 6 – 2 or To add, you need to use parts that are the same size. + 1 ? 1. Count the parts. Each part represents __ 6 1 __ = 3 __ + 1 __ 3 6 6 Write the answer in lowest terms. 3 = __ __ 1 6 2 Method 2: Draw a Diagram 1 + __ __ 1 3 6 + To add, you need to use parts that are the same size. 1+1 __ __ = 3 __ 3 6 6 Strategies Solve a Simpler Problem Refer to page xvii. + = Write the answer in lowest terms. 3 __ = 1 __ 6 2 238 NEL • Chapter 7
12. Method 3: Use a Common Denominator The denominator of __1 is 3. 3 Multiples of 3 are 3, 6, 9, 12, … The denominator of 1 __ is 6. 6 Multiples of 6 are 6, 12, 18, 24, … The first multiple divisible by both 3 and 6 is 6. A common denominator is 6. Write equivalent fractions with 6 as the denominator. 1 + __ __ 1 = __ 2+1 __ 3 6 6 6 =2 + 1 Add the numerators. _____ 6 3 = __ 6 Write the answer in lowest terms. ÷3 3 = __ __ 1 6 2 ÷3 Use pattern blocks. Web Link To learn more about Is closer to or ? adding fractions, go to www.mathlinks7.ca –1 + –1 –1 1 and follow the links. 3 6 2 It equals 1 __. The answer is reasonable. 2 Literacy Link When the numerator equals the Add. Write each answer in lowest terms. denominator, the 1 __ fraction is equal to 1. a) 1 __ + 2 __ 1 +4 b) ___ __ c) __ + 4 3=1 __ 16 ___ = 1 6 3 10 5 2 8 3 16 7.2 Add and Subtract Fractions With Unlike Denominators • NEL 239
13. Example 2: Subtract Fractions With Unlike Denominators Subtract. 1 − __ __ 2 2 5 Solution Is the Method 1: Use Fraction Strips difference 1−2 __ __ – closer to 2 5 0 To subtract 2__, you need or 5 – ? parts that are the same size. 1 – 2 Subtract. 1 − __ __ 2 = ___ 1 2 5 10 Check: 1? 1 closer to 0 or __ Is ___ 10 2 0 = ___0 1 5 __ = ___ Compare this to the 10 2 10 estimate you made ___ 0 , or 0. 1 is a little more than ___ before you 10 10 subtracted. Method 2: Draw a Diagram 1 2 __ − __ 2 5 – To subtract 2__, you need 5 parts that are the same size. Subtract. 1 − __ __ 2 = ___ 1 2 5 10 Method 3: Use a Common Denominator The denominator of __1 is 2. 2 Multiples of 2 are 2, 4, 6, 8, 10, … The denominator of 2 __ is 5. 5 Multiples of 5 are 5, 10, 15, 20, … 240 NEL • Chapter 7
14. The first multiple divisible by both 2 and 5 is 10. A common denominator is 10. Write equivalent fractions with 10 as the denominator. 1 5 − ___ 2 = ___ __ − __ 4 2 5 10 10 5−4 = _____ Subtract the numerators. 10 1 = ___ 10 Subtract. Write each answer in lowest terms. a) 3 1 __ − __ 2 __ b) __ − 1 3 __ c) __ − 1 4 2 3 4 4 8 • When adding and subtracting fractions using models or diagrams, show each fraction using parts of the whole that are of equal size. Pattern Blocks Diagram = – = 1 + __ __ 3 + __ 1 = __ 2 2 − __ __ 1=4 1 __ − __ 2 3 6 6 3 6 6 6 • To add or subtract fractions with unlike denominators, use a common denominator. • You can estimate when adding or subtracting fractions by comparing fractions to 0, 1 __, or 1. 2 1. How are _1_ and 2 __ alike? How are they different? 3 6 2. a) 1+1 How would you use diagrams to calculate __ __? 4 2 b) Compare your answer with a classmate’s. 3. Why is it difficult to calculate __ 1 without changing 1 1 − __ 4? __ to __ 2 8 2 8 7.2 Add and Subtract Fractions With Unlike Denominators • NEL 241
15. 8. Write each addition statement shown by the pattern blocks. Then add. For help with #4 to #7, refer to Example 1 on a) pages 238–239. + 4. Write each addition statement shown by the fraction strips. b) Estimate and then add. a) + + b) + c) + For help with #9 to #12, refer to Example 2 on pages 240–241. 5. For each diagram, write an addition statement. Then add. 9. Write each subtraction statement a) shown by the fraction strips. + Estimate and then subtract. a) – b) – b) + c) – 10. For each diagram, write a subtraction c) statement. Then subtract. + a) – b) 6. Add. Write your answers in lowest terms. 5+1 – a) _2_ + ___ 1 b) __ __ 5 10 8 4 5 1 + ___ c) __ 1 __ d) __ + 3 3 12 4 5 c) e) 1 _ _ 1 + __ 3 __ f) __ + 1 2 5 8 6 – 7. Determine the sum. Write your answers in lowest terms. 11. Subtract. Write your answers in lowest 3 1 + __ 1 +5 terms. a) __ b) ___ __ 2 8 12 6 3 − ___ a) __ 3 5−1 b) __ __ 2 + __ c) ___ 4 d) 1 __ + 2 __ 5 10 6 2 10 5 3 9 1 − ___ c) __ 1 d) 7 __ − 1 __ 2 + __ e) __ 1 __ + 3 f) 1 __ 2 10 8 2 5 2 6 4 e) 2 __ − 2 __ 5 − ___ f) __ 5 3 5 8 12 242 NEL • Chapter 7
16. 12. Determine the difference. Write your 15. Zach was leading in a swimming race by _5_ 8 answers in lowest terms. 1 __ of a length. He won the race by a length. 3−1 a) __ __ 11 − 5 b) ___ __ 2 4 8 12 6 By how much did the second-place 2−1 c) __ __ d) 1__ − 1 __ swimmer catch up by the end of the race? 3 2 6 9 2−1 e) __ __ 5 − 11 f) __ ___ 5 4 6 15 16. A friend shows you the following work for an addition problem. 13. Write each subtraction statement shown 1 1 = __ __ + __ 2 by the pattern blocks. Then subtract. 4 3 7 a) Explain the error in your friend’s work. a) b) Use a diagram to show the correct – answer. b) 17. An airplane was loading in Pond Inlet for its flight to Iqaluit, Nunavut. The plane – was _1_ full of passengers and _1_ full of 6 3 cargo. How much space was left? 18. You can use a number line to show 2 9. 1 = ___ __ + ___ 14. The students made 3 12 12 muffins in cooking 2 8 1 – = –– –– class. They get to 3 12 12 take some muffins home. There are 0 8 9 –– –– 1 12 muffins in a 12 12 muffin tray. Draw number lines to add the fractions. 1 of a tray.” 1 __ 3 +3 a) John says, “I’m taking __ a) __ + 1 b) _1_ + 1 __ c) ___ __ 4 4 4 2 8 10 5 Katie says, “I’m taking 1 __ of a tray.” 3 19. You can use a number line to show What fraction of a tray are John and 7. 2 1 = ___ __ − ___ Katie taking altogether? 3 12 12 b) Marjoe says, “I’m taking _1_ of a tray.” 2 8 1 –– 6 – = –– 12 Sandeep says, “I’m taking ___1 of a tray.” 3 12 12 What fraction of a tray are Marjoe and 0 7 8 1 –– –– Sandeep taking altogether? 12 12 Draw number lines to subtract the fractions. 1−1 a) __ __ 1 − ___ b) __ 1 c) _5_ − 1 __ 2 8 4 12 6 4 7.2 Add and Subtract Fractions With Unlike Denominators • NEL 243
17. 22. The sum of each row, column, and diagonal in this magic square must equal 1. Copy the 20. The ancient Egyptians thought the fractions square and fill in the blanks. in the Eye of Horus added up to 1. Were 5 they correct? Show your work. � � ___ 12 7 1 –1 ___ 12 __ 3 � 8 1 __ 4 � � –1 4 –1 1 –– 2 16 23. A tangram is a square G C puzzle that is divided D 1 into seven shapes. E 1 B –– 64 –– 32 1. a) Suppose piece A is __ 4 F A What are the values 21. Water is pumped into a pool. After of pieces B, C, D, E, one hour, 1 __ of the pool is filled. F, and G? 5 b) What is the sum of A and B? a) After 3 h, how full is the pool? c) Subtract the value of D from the whole. b) How long does it take in total to fill the pool? d) Which two tangram pieces add up to the value of C? e) Make a problem of your own using tangram pieces. Have a classmate solve it. MATH LINK The Egyptians of 3000 B.C.E. used only unit fractions. These are fractions with a numerator of 1, such as __ 1, and __ 1, __ 1. They wrote all other fractions as sums of unit 23 4 fractions. For example, = + = + 3 1+1 __ = __ __ 5 = __ __ 1+1__ 4 2 4 6 2 3 These sums are called Egyptian fractions. a) Add the unit fractions. Use diagrams to show your work. __ 1 1 + __ 1+1 __ __ 4 8 3 9 b) Which one of the two sums in a) is greater? By how much? 5 c) How would ancient Egyptians have written ___ as the sum of two unit fractions? 12 244 NEL • Chapter 7
18. Add Mixed Numbers Focus on… After this lesson, you will be able to... φ add mixed numbers with like and unlike denominators After the class pizza party, there φ solve problems involving the were 1 _5_ cheese pizzas and 1 __ 5 vegetarian pizzas 6 6 addition of left over. How many pizzas were left over in total? mixed numbers To find out, you need to add mixed numbers . φ check that your answer is reasonable using estimation How do you add mixed numbers? mixed numbers Example 1: Add Mixed Numbers With Like Denominators • a number made up of Add. Write the answer in lowest terms. a whole number and a 5 + 15 1 __ __ 1 fraction, such as 2 __ 6 6 3 Solution Method 1: Use Pattern Blocks Strategies Model It + Refer to page xvi. Add the whole yellow hexagons. + Add the green triangles. + 5 + 15 1 __ 5+5 __ = 1 + 1 + __ __ 6 6 6 6 7.3 Add Mixed Numbers • NEL 245
19. Move 1 green triangle to the hexagon with 5 green triangles to make 1 whole hexagon. There are now 3 whole hexagons and 4 green triangles. 1+1+1+4 __ = 3 _4_ 6 6 Write the answer in lowest terms. 34 __ = 3 2 __ = 6 3 Method 2: Use an Addition Statement Strategies 15 5=1+1+5 __ + 1 __ 5 Add the whole numbers. __ + __ Solve a Simpler 6 6 6 6 Problem 5+5 = 2 + _____ Add the fractions. Refer to page xvii. 6 10 = 2 + ___ 6 improper fraction =2+6 __ + 4 __ Write the improper fraction as a mixed number. 6 6 • a fraction that has a numerator greater = 2 + 1 + _4_ 6 You can use diagrams. than the denominator, 4 _ _ 9 =3 such as __ 6 + + + 8 Write the answer in lowest terms. ÷2 5+5 1 + 1 + __ __ = 3 _4_ 6 6 6 34 __ = 3 _2_ 6 3 ÷2 Check: 15 5≈2+2 __ + 1 __ 6 6 2+2=4 32 __ is a little less than the estimate of 4. The answer is reasonable. 3 Add. Write each answer in lowest terms. 1 + 21 a) 1 __ 1 + 25 __ b) 3 __ __ 2+2 c) 3 __ __ 3 3 6 6 3 3 246 NEL • Chapter 7
20. Example 2: Add Mixed Numbers With Unlike Denominators How many pies are there in total? 1 apple pies. 1 __ 2 1 apple pies 2 __ 3 Method 1: Use Pattern Blocks Strategies + Model It Refer to page xvi. __ + 2 _1_ 2 3 Add the whole yellow hexagons. + Add the red trapezoid and blue rhombus. + 1+2+1 __ + 1 1+1 __ = 3 + __ __ 2 3 2 3 __ and 1 To add 1 __, you need to use pattern blocks that are the same size. 2 3 + There are 5 green triangles altogether. 3 + __ 1=3+5 1 + __ __ 2 3 6 =3 5 _ _ 6 5 __ There are 3 pies. 6 7.3 Add Mixed Numbers • NEL 247
21. Method 2: Use an Addition Statement Use multiples to determine Strategies Solve a Simpler 1 + 2 __ 1 __ 1=1+2+1 1 __ + __ a common denominator. Problem 2 3 2 3 Multiples of 2 are 2, 4, 6, 8, … Refer to page xvii. =1+2+3 2 Add the whole numbers. __ + __ Multiples of 3 are 3, 6, 9, 12, … 6 6 Use 6 as a common =3+3 __ + 2 __ denominator. 6 6 3+2 = 3 + _____ Add the numerators. 6 =3+5 __ 6 You can use diagrams. =3 5 _ _ 6 + There are 3 5 __ pies altogether. 1+2=3 The Arabs were the 6 first people to use a fraction bar: 1 __ Check: fraction bar 5 11 1≈2+2 __ + 2 __ + 2 3 2+2=4 3+2 __ __ = _5_ 6 6 6 35 __ is a little less than the estimate of 4. 6 The answer is reasonable. Add. Write each answer in lowest terms. 1 + 41 a) 2 __ __ 2 + 13 b) 1 __ __ 3+1 c) 3 __ __ 2 6 3 4 5 2 • When adding mixed numbers with like denominators, you can – add the whole numbers – add the fractions • When adding mixed numbers with unlike denominators, you can – determine a common denominator for the fractions – add the whole numbers – add the fractions • To check your answer, compare the answer to an estimate. 248 NEL • Chapter 7
22. 1. After dinner, 1 1 __ ham sandwiches and 2 __3 egg salad sandwiches are 2 4 left. Jeremy and his sister want to use 4 of the leftover sandwiches for their lunches tomorrow. a) Are there enough sandwiches, not enough sandwiches, or more than enough sandwiches left over? b) Describe the method you used to find your answer. c) Solve the problem using another method. 2. Which method that you used in #1 did you prefer? Explain. 3. a) How would you use estimation to check your answer in #1? b) Compare your estimate with a partner’s. 6. Add. Write your answers in lowest terms. Check your answers using estimation. For help with #4 to #7, refer to Example 1 on a) 1 _1_ + 1 1 __ b) 3_1_ + 5 5 __ c) _3_ + 1 _1_ pages 245–246. 3 3 8 8 4 4 d) 2 1 ___ +3 7 ___ _2_ e) 3 + 1 4 _ _ f) 4 + 1 7 8 __ __ 4. Write each addition statement shown. 10 10 5 5 9 9 a) + 7. Determine the sum. Write your answers in lowest terms. b) 2 + 21 a) 1 __ 1 + 15 __ b) 3 __ __ c) 5 4 __ + 2 __ + 5 5 8 8 9 9 1 + 23 d) 2 __ 3+4 __ e) 2 __ __ 7 + 6 11 f) 4 ___ ___ c) 4 4 5 5 12 12 + For help with #8 to #11, refer to Example 2 on pages 247–248. 5. For each of the following, write the 8. Write each addition statement shown. addition statement. a) + a) + b) + b) + c) + c) + 7.3 Add Mixed Numbers • NEL 249
23. 9. For each of the following, write an 14. The camp cook uses 1 _1_ dozen eggs to 2 addition statement. make pancakes. She uses another 3 _1_ dozen a) 3 + for scrambled eggs. How many dozen eggs does she use altogether? Check your b) answer using estimation. + 15. Chef Dimitri finished cutting 1 _1_ trays 4 c) of spinach pie before his break. After his + break he cut another 2 2 __ trays. How many 3 trays of pie in total did he cut? Include diagrams with your answer. 10. Add. Write your answers in lowest terms. 3 + 1 ___ a) 2 __ 1 1 + 21 b) 3 __ __ 16. Jenny studied 1 __1 h for her math 5 10 2 6 3 c) 3 __ +2 5 _ _ 1 __ d) 5 + 3 ___ test and 3 __ h for her science test. For 4 6 2 10 4 1 __ e) 4 + 3 5 ___ 2 + 75 f) 2 __ __ how long did she study in total? Check 4 12 3 7 your answer using estimation. 11. Determine the sum. Write your answers in 17. Jonas and Amy collect comic books. Jonas lowest terms. has 21___ boxes of Granite Guy comics and 1 + 21 a) 4 __ __ 1 + 32 b) 3 __ __ 10 3 6 2 5 2 __ 2 boxes of Quest of Koko comics. Amy 1 __ c) 1 + 5 1 _ _ d) 4 4 ___ 1 + 5 ___ 3 5 4 15 10 has 2 5 __ boxes of Alpha Woman comics and 3 + __ e) 1__ 5 f) 6 __ 9 1 + ___ 6 4 6 2 10 3 __ 1 boxes of Quest of Koko comics. 5 a) Who has the larger collection? b) How many boxes of Quest of Koko comics do Jonas and Amy own 12. 3 h and then walked for Susie ran for 1 __ 4 altogether? 7 21 __ h. For how long did she travel? c) Jonas trades ___ 4 10 of a box of 13. 3 pages of homework Kathleen did 1 __ comics to Amy 4 for her Granite before dinner. After dinner, she did Guy DVD. How another 7 __ of a page. In total, how many boxes of 8 many pages of homework did comics does she Kathleen do? have now? 250 NEL • Chapter 7
24. 19. At the school’s spring fair they sold 3 pepperoni pizzas, 1 vegetarian pizzas, 6 __ 5 __ 18. Melissa is in training for a rowing 3 4 competition. She keeps track of the hours and 4 5__ cheese pizzas. 6 she practises. At the end of the week, she a) Draw diagrams to show how much of totals her hours. each pizza was sold. Hours Practised b) Estimate, then calculate how much Sun Mon Tues Wed Thurs Fri Sat pizza was sold altogether. 2 3 __ 21 __ 1 _3_ 11 __ 11 __ 1 _1_ 4 4 4 2 4 2 20. The movie started 2 h 12 min ago. a) This week she had a goal to practise 5 h. The movie will end in 1 __ for a total of at least 10 h. Estimate 6 whether she met her goal. a) What is the total length of the movie in b) How many hours did she practise? hours written as a fraction? c) Was your estimate reasonable? Explain. b) If the movie started at 2:15 p.m., when did the movie end? Write the time as a fraction. MATH LINK Egyptian fractions can be useful today. Suppose you have 13 sacks of rice to divide among 8 people. That means each person would 5 sacks. get 1 __ 8 How can you give each person 1 5 __ sacks of rice if you do not have a 8 calculator or scale? First, give each person 1 whole sack. Then, use Egyptian fractions to determine how to give each person __ 5 sack: 8 Strategies 5 = __ __ 1 + _1_ 8 2 8 Solve a Simpler The Egyptian fraction shows that you should give each person 1 __ sack, plus 1 __ sack. Problem 2 8 1 sack to each person, there will be 1 sack left. Refer to page xvii. After you give __ 2 You then divide this sack into 8 and give each person __ 1. 8 The diagram shows that each person gets 1 sack and 1 1 whole sack, plus __ __ sack. 2 8 How would you divide the following? a) 7 sacks of potatoes among 4 people b) 7 bags of flour among 5 people c) 9 loaves of bread among 5 people 7.3 Add Mixed Numbers • NEL 251
25. Subtract Mixed Numbers Focus on… After this lesson, you will be able to... φ subtract mixed numbers with like and unlike denominators φ solve problems involving the subtraction of mixed numbers φ check that your answers are reasonable using After Lucy worked on her art project, she had 2 __43 jars of paint left. estimation Later, she used 1 1 __ jars of paint to finish her painting. How much paint 4 is left now? How do you subtract mixed numbers? Example 1: Subtract Mixed Numbers With Like Denominators Subtract. Write the answer in lowest terms. 3 − 11 2 __ __ 4 4 Solution Method 1: Use Fraction Strips 3 − 11 2 __ __ 4 4 Subtract. – 252 NEL • Chapter 7
26. 2 fraction strips. There are now 1 __ 4 Write the answer in lowest terms. __ = 1 _1_ 4 2 Method 2: Use a Subtraction Statement Strategies Subtract the whole numbers. Solve a Simpler Problem Refer to page xvii. Subtract the fractions. You can use diagrams. 3 1=2 __ − __ __ 4 4 4 23 1 = 1 _2_ __ − 1 __ 4 4 4 2–1=1 Write the answer in lowest terms. ÷2 3 __ 1 __ 2 __ 4–4=4 __ = 1 _1_ 4 2 ÷2 3 − 1 __ 2 __ 1≈3−1 4 4 11__ is close to the estimate of 2. 2 The answer is reasonable. Subtract. Write each answer in lowest terms. 2 1 a) 2 __ − 1 __ 7 3 b) 3 __ − 1 __ 3 __ c) 4 __ − 1 3 3 8 8 4 4 7.4 Subtract Mixed Numbers • NEL 253
27. Example 2: Subtract Mixed Numbers With Unlike Denominators Subtract. 3 − 11 3 __ __ 8 2 Solution Method 1: Use Fraction Strips – 3 - 11 3 __ __ 8 2 To subtract 3 __ and 1 __, you need to use parts that are the same size. 8 2 You cannot subtract _4_ from 3__. 8 8 8. Take 1 whole strip from 3 _3_ and make it the equivalent fraction __ 8 8 Subtract. – 7 strips left. There are 1 __ 8 33 1 = 1 _7_ __ − 1 __ 8 2 8 Method 2: Use a Subtraction Statement and Regroup Use multiples to determine a common denominator. Multiples of 2 are 2, 4, 6, 8, … Multiples of 8 are 8, 16, … Use 8 as a common denominator. 33 3 − 1 _4_ 1 = 3 __ __ − 1 __ 8 2 8 8 You cannot subtract 4 3. You need to regroup. __ from __ 8 8 254 NEL • Chapter 7
28. Regroup 1 whole from 3 __ 3. 8 3=2+8 3 __ __ + 3 __ You can use diagrams. 8 8 8 = 2 11 ___ 8 3 __ 3 −1 =24 __ 11 ___ − 14__ Subtract the whole 8 8 8 8 numbers and subtract = 1 _7_ the fractions. 8 11 – 1__ 2___ 4 = 17 __ 8 8 8 Method 3: Use a Subtraction Statement and Improper Fractions Determine a common denominator. 3 − 1 __ 3 __ 3 − 1 _4_ 1 = 3 __ 8 2 8 8 You cannot subtract 4 __ from 3__. 8 8 You can use diagrams. You can change to improper 3 − 1 __ 3 __ 4 = 27 ___ − 12 ___ Subtract. 8 8 8 8 15 = ___ 27 15 12 = ___ ___ – ___ 8 8 8 8 = 1 _7_ = 17 __ 8 8 33 1 ≈ 3 __ __ − 1 __ 1 − 1 _1_ 8 2 2 2 31 1=2 __ − 1 __ 2 2 __ is a little less than the estimate of 2. The answer is reasonable. 8 Subtract. Write each answer in lowest terms. 3 a) 3 __ − 1 1 1 − 32 __ b) 4 __ __ 1 __ c) 4 __ − 7 8 2 4 5 4 8 Chinese fractions do not have a fraction bar. A symbol is used that represents the 1 is written or spoken as “1 part of 2.” words “part of” or “parts of.” __ 2 7.4 Subtract Mixed Numbers • NEL 255
29. • When subtracting mixed numbers with like denominators, you can – subtract the whole numbers – subtract the fractions • When subtracting mixed numbers with unlike denominators, you can – determine a common denominator for the fractions – subtract the whole numbers – subtract the fractions • Sometimes, mixed numbers need to be regrouped or changed to improper fractions before subtracting. Regroup Change to Improper Fractions 43 5 = 3 ___ __ − 1 __ 11 − 1 5 __ 3 − 1 __ 4 __ 5 = 35 ___ − 13 ___ 8 8 8 8 8 8 8 8 =2 6 _ _ = 22 ___ 8 8 = 23 __ = 2 _6_ 4 8 = 2 _3_ 4 • To check your answer, compare to an estimate. 1. After Jack’s party, 2 3 __ bottles of pop were left. The next day, Jack’s 4 1 __ family drank 2 bottles. How much pop is left now? Discuss with a 4 partner how you would solve this problem. 2. a) What do you need to do before you can calculate 2 __ 5? 1 − 1 ___ 6 12 5. 1 − 1 ___ b) Determine the answer to 2 __ 6 12 c) Use estimation to check your answer. What method did you use? d) With a partner, compare how you calculated the answer to 21 5 . Then compare the method you used to check your answer. __ − 1 ___ 6 12 256 NEL • Chapter 7
30. For help with #7 to #10, refer to Example 2 on pages 254–255. For help with #3 to #6, refer to Example 1 on 7. Write a subtraction statement for each pages 252–253. set of fraction strips. 3. For each set of fraction strips, write the a) subtraction statement. a) – – b) b) – – c) – c) – 4. Write a subtraction statement to represent each diagram. a) – 8. For each diagram, write a subtraction b) statement. – a) – c) b) – – c) 5. Subtract. Write your answers in – lowest terms. Check your answers using estimation. 2 − 1 _1_ a) 1 __ b) 6 7 __ − 5 _5_ 5 5 8 8 1 __ c) 3 − 1 1 _ _ d) 3 1 ___ − 1 ___7 9. Subtract. Write your answers in 4 4 12 12 lowest terms. Check your answers e) 2 __ 5 1 − __ f) 4 − 1 _1_ using estimation. 6 6 7 7 a) 6 ___ − 3 2 __ 1 1 b) 4 __ − __ 6. Determine the difference. Write your 10 5 2 5 7 − 31 c) 7 ___ __ 5 d) 5 __ − 2 2__ answers in lowest terms. 9 15 6 3 5 a) 4 __ − 3 1 __ b) 2 1 __ − 2 _1_ 4 __ 2 _ _ 3 ___ 6 __ 9 9 3 3 e) 1 − 1 f) 2 − 5 3 14 7 2 __ c) 5 − 1 __ 4 d) 4 3 ___ − 2 ___9 5 5 10 10 7 e) 5 − 4 ___ f) 3 5 __ − 2 _7_ 12 8 8 7.4 Subtract Mixed Numbers • NEL 257
31. 10. Determine the difference. Write your 15. Mark and Lin race to see who can collect answers in lowest terms. the most hockey cards. Mark has collected 3 2 − 1 ___ 1 − 11 3 sets. Who has 5 _1_ sets. Lin has collected 4 __ a) 3 __ b) 1 __ __ 3 4 5 10 3 4 5 1 1 5 collected more sets? How much more? c) 7 __ − 5 __ d) 4 __ − 2 ___ 9 6 4 12 e) 3 __ 3 1 − __ 3 f) 2 __ − 1 4 __ 16. Alex has just completed 2 _3_ h of a 6 4 4 5 4 babysitting course. He must complete 13 1 __ h to get his certificate. 2 a) How many more hours does he need? Karen goes to swimming practice for 1 __1h 11. b) Check your answer using estimation. 3 each day. In the morning she has 2 __ h of 3 17. 5 laps. Mei For gym class Ben ran 1 ___ practice. How many hours of practice 12 does she have in the afternoon? ran 18 ___ laps. Who ran farther and by 12 how much? 12. A large Thermos™ has enough water to fill 9 3 __ water bottles for a team of soccer 18. You can subtract a mixed number and 4 an improper fraction. Determine each players. Halfway through practice, difference. the players drink 4 __1 bottles of water. 2 a) 3 3 __ − 3 __ 7 −6 b) 2___ 1−7 __ c) 5__ __ How much water is left for the rest of the 4 2 10 5 3 4 practice? 3 trays of dinner rolls are for sale in the 19. 1 __ 4 bakery window. A customer comes and buys 5__ of a tray. How much is left? 6 Include diagrams with your answer. 20. Daniel spends 9 __1 h sleeping. He spends 3 h to complete the 13. It takes Ria 3__ 2 4 61__ h at school. 1 h ago. marathon. The race started 1 __ 4 2 a) How much more time does he spend a) How much longer will Ria be running? sleeping than at school? b) Check your answer using estimation. b) How much time does he spend at school and sleeping altogether? 14. 1 packages of A pie recipe calls for 3 __ 2 c) How much time is left in the day to do Saskatoon berries. Julia has 1 1 __ packages. other things? 3 How much more does she need? Include diagrams with your answer. 258 NEL • Chapter 7
32. 21. Diana is allowed to use the computer for a) How much more paper does Bella use? 3 h each weekend. She used it for _1_ h on b) How much paper do Bella and Shelly 2 use in total? Saturday morning, 1 1__ h on Saturday 4 3 __ night, and h on Sunday morning. 23. There are 12 golf balls in a package. The 4 2 packages. Cindy a) For how much time can Diana use the Takeda family has 2 __ 3 computer on Sunday night? takes 1 __ package, her dad takes 1 package, 2 b) Show how you would check your and her brother takes 4 golf balls. answer using estimation. a) What fraction of a package is left? 22. Bella uses 4.1 pieces of construction paper b) How many golf balls is this? to make an art project. Shelly uses 3 _1_ 4 pieces. For each of the following questions, calculate your answer using only fractions. Then calculate using only decimals. Compare the answers. MATH LINK The Babylonian system of numbers was based on 60, not 10. 3 , ___ 2 , ___ Babylonian fractions were expressed as numbers out of 60, e.g., ___ 5 , 12 ___ . 60 60 60 60 Many things we use today come from the Babylonian times. Our clock is based on the number 60. 10. The time can be written as a fraction out of 60 min. For example, 9:10 a.m. = 9 ___ 60 For a) to e) write your answers as fractions. a) Write each time as a fraction out of 60. 8:10 p.m. 9:20 a.m. 7:48 a.m. 12:12 p.m. b) The time now is 2:15 p.m. What was the time 1 h and 12 min ago? c) The time now is 4:30 p.m. What will be the time 2 h and 36 min from now? 1 d) Amanda studied for __ of an hour. She started studying at 9:15 a.m. At what time 3 did she finish studying? e) How much time passed between 1:07 p.m. and 3:42 p.m.? between 5:45 p.m. and 9:20 p.m.? 7 f) Sam started reading the newspaper at 9:45 a.m. and finished reading it in ___ h. 12 1 h more to read the paper than Sam did. She started at 10:30 a.m. At Mila took __ 4 what time did she finish reading the paper? 7.4 Subtract Mixed Numbers • NEL 259
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