Connections Across Strands Provides a sampling of connections that can be made across strands, using the theme (fractions) as an organizer

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3 Fractions Context Fractions have a variety of meanings. The fraction 5 can be interpreted as parts of a whole that has been divided into 5 equal parts (part of a whole). This fraction also expresses parts of a group of 5 (part of a set) where the elements of the set are not necessarily identical, e.g., out of 5 books on a shelf. As ratios or rates, fractions are used for comparisons. A fraction can represent a division or a measurement. Fractions are used daily in construction, cooking, sewing, investments, time, sports, etc. Since many occupations require workers to think about and use fractions in many different ways, it is important to develop a good understanding of fractions. Using manipulatives and posing higher-level thinking questions helps build understanding of what fractions represent. Understanding builds when students are challenged to use a variety of representations for the same fraction (or operation) and when students connect fractions to ratios, rates of change, percents, or decimals. Context Connections Cooking Music Sharing Retail/Shopping Interest Rates Measurement Slope Time A = bh Circle Graphs Construction Formulas Other Connections Manipulatives cubes colour tiles pattern blocks geoboards tangrams coloured rods grid paper base ten blocks Technology spreadsheet software The Geometer s Sketchpad 4 calculators/graphing calculators word processing software Other Resources t_.html TIPS4RM: Continuum and Connections Fractions

5 Summary of Prior Learning In earlier years, students: relate fractions to decimals and percents; compare and order fractional amounts, with like and unlike denominators (including proper and improper fractions and mixed numbers); explain the concept of equivalent fractions, with and without concrete materials; represent fractions using words, concrete materials, and notations. In Grade 7, students: continue to develop proficiency by using fractions in mental strategies and in selecting and justifying use; develop proficiency in adding and subtracting simple fractions; use fractions to solve problems involving the area of a trapezoid; describe the reduction of two-dimensional shapes to create similar shapes and to describe reductions in dilatations; write fractions to determine the theoretical probability of an event or two independent events. In Grade 8, students: develop proficiency in comparing, ordering, and representing fractions; develop proficiency in operations with fractions with and without concrete materials; solve problems involving addition, subtraction, multiplication, and division of simple fractions; substitute fractions for the variables in algebraic expressions up to three terms; continue to write fractions to determine the theoretical probability of an event, e.g., complementary event. In Grade 9 Applied, students: continue to solve problems involving percents, fractions, and decimals; use rationals as bases when evaluating expressions involving natural number exponents; use fractions in solving problems involving area of composite figures with triangles and/or trapezoids, and in solving problems involving volumes of pyramids or cones. In Grade 0 Applied, students: solve first degree equations and rearrange formulas involving variable in the first degree, involving fractional coefficients; use ratios and proportions in problem solving when working with similar triangles and trigonometry; rearrange measurement formulas involving surface area and volume to solve problems; graph equations of lines, with fractional coefficients. In later years Students choice of courses will determine the degree to which they apply their understanding of concepts related to fractions. TIPS4RM: Continuum and Connections Fractions 4

7 Suggested Instructional Strategies Helping to Develop Understanding Grade 9 Applied Extend students working knowledge of positive fractions to negative fractions, e.g., negative rates of change. Introduce the new terminology rational number, i.e., any number that can be expressed as a positive or negative fraction and written as a terminating or repeating decimal number. Encourage estimation in problem solving to verify accuracy of solutions. Encourage the conversions between fractions, decimals, and percents wherever appropriate to simplify a problem. Consolidate understanding of fractions within the context of the course. Grade 0 Applied Consolidate operations with rational numbers within the context of the course. Encourage students whenever appropriate to leave numbers presented in the question in fraction form and to express answers in fraction form. TIPS4RM: Continuum and Connections Fractions 6

8 Connections Across Strands Note Summary or synthesis of curriculum expectations is in plain font. Verbatim curriculum expectations are in italics. Grade 7 Number Sense and Numeration generate multiples and factors of given numbers solve problems involving whole number percents demonstrate an understanding of rate solve problems involving unit rates see Connections Across the Grades, p. Measurement use fraction skills in solving problems involving measurement, e.g., the area of a trapezoid Geometry and Spatial Sense use fractions to describe reductions in dilatation and in reducing twodimensional shapes to create similar figures use fractions to describe related lines, e.g., perpendicular lines meet at 90 which is of 80 plot points on the Cartesian plane with simple fractional coordinates Patterning and Algebra model everyday relationships involving rates translate phrases into algebraic expressions Data Management and Probability use fractions to express the experimental and theoretical probability of an event research and report on realworld applications of probabilities expressed in fraction, decimal, and percent form determine the theoretical probability of a specific outcome involving two independent events Grade 8 Number Sense and Numeration determine common factors and multiples solve problems involving proportions solve problems involving percent see Connections Across the Grades, p. Measurement use fraction skills in solving problems involving measurement, e.g., the area of a circle Geometry and Spatial Sense graph the image of a point on the Cartesian coordinate plane with simple fractional coordinates determine relationships: area, perimeter, and side length of similar shapes, e.g., if triangles are similar and the perimeter of one is the perimeter of the other, compare their areas Patterning and Algebra evaluate algebraic expressions with up to three terms, by substituting fractions, decimals, and integers for the variables translate statements into algebraic expressions and equations Data Management and Probability use fractions to express the experimental and theoretical probability of an event identify the complementary event for a given event, and calculate the theoretical probability that a given event will not occur TIPS4RM: Continuum and Connections Fractions 7

9 Grade 9 Applied Number Sense and Algebra Measurement and Geometry Linear Relationships see Connections Across the Grades, p. solve problems involving area of composite figures, involving triangles and/or trapezoids develop, through investigation, the formulas for the volume of a pyramid or cone solve problems involving the volume of pyramids or cones interpret points on a scatterplot collect data; describe trends construct tables of values and graphs for data determine and describe rates of change use initial value and rate of change to express a linear relation determine graphically a point of intersection Grade 0 Applied Measurement and Trigonometry use proportional reasoning (by setting two ratios written as fractions equal) to solve similar triangle problems determine, through investigation, the trigonometric ratios of sine, cosine, and tangent as ratios represented by fractions solve problems involving the measures in right-angled triangles rearrange measurement formulas when solving problems involving surface area and volume, including combination of these figures in the metric or imperial system perform conversions between imperial and metric systems solve problems involving surface area and volume of pyramids and cylinders Modelling Linear Relations graph lines (with fractional coefficients) by hand, using a variety of techniques solve equations involving fractional coefficients connect rate of change and slope determine the equations of a line determine graphically the point of intersection of lines Quadratic Relations of the Form y = ax + bx + c collect data that can be represented as a quadratic relation solve problems involving quadratic relations TIPS4RM: Continuum and Connections Fractions 8

10 Developing Proficiency Grade 7 Name: Date: Expectation Number Sense and Numeration, 7m4: Add and subtract fractions with simple like and unlike denominators using a variety of tools and algorithms. Knowledge and Understanding (Facts and Procedures) 5 a) b) Knowledge and Understanding (Conceptual Understanding) Use a model to represent the following addition: 5 + = 6 Record a drawing of your model. c) Problem Solving (Reasoning and Proving) Three friends, Ahmed, Anja, and Eric, have a lemonade stand. They decide to share any profits in the following way. Ahmed will get Anja will get 4 of the total profits. of the total profits. Problem Solving (Connecting, Representing) Vesna used 8 eggs from a new carton of one dozen eggs. Vesna says she used of the carton and her friend, Sue, says she has 4 carton remaining. of the Explain why Eric will get of the total profits. Explain why they are both right, using a diagram or concrete materials. Give reasons for your answer. TIPS4RM: Continuum and Connections Fractions 9

11 Developing Proficiency Grade 8 Name: Date: Expectation Number Sense and Numeration, 8m9: Represent the multiplication and division of fractions, using a variety of tools and strategies. Knowledge and Understanding (Facts and Procedures) a) 4 8 Knowledge and Understanding (Conceptual Understanding) Complete the diagram to illustrate: = 4 b) Problem Solving (Reflecting, Reasoning and Proving) Jason incorrectly thinks that 6 divided by onehalf is. Show that 6 =. Illustrate your answer with a diagram. Problem Solving (Connecting) Sanjay s father went on a short trip in his truck. The 60-litre gas tank was full when he started his trip. He used about 4 of the gasoline during the 5 trip. How many litres of gas does he have left in his tank? Show your work. TIPS4RM: Continuum and Connections Fractions 0

12 Developing Proficiency Grade 9 Applied Name: Date: Expectation Number Sense and Algebra, NA.06: Solve problems requiring the expression of percents, fractions, and decimals in their equivalent forms. Knowledge and Understanding (Facts and Procedures) Bradley s homework question was to put 55%, 0.45, and in order from smallest to largest. 5 He drew this chart to help his thinking. Knowledge and Understanding (Conceptual Understanding) Explain why of 4 is 5 5. Illustrate with a diagram or concrete materials. Complete the following table for Bradley. Percent Decimal Fraction 55% Problem Solving (Reasoning and Proving, Connecting) Ahmed, Anja, and Eric invested in a business. They each invested a different amount of money. Ahmed invested \$ Anja invested \$ Eric invested \$ In the first year of the business, the profit is \$ What should each person s share of the profits be? Problem Solving (Connecting, Selecting Tools and Computational Strategies) In shop class, you are asked to cut a round tabletop with a diameter of 7 cm. 4 What is the area of the tabletop? Show your work. Give reasons for your answer. TIPS4RM: Continuum and Connections Fractions

13 Developing Proficiency Grade 0 Applied Name: Date: Expectation Modelling Linear Relations, ML.0: Solve first-degree equations involving one variable, including equations with fractional coefficients. Knowledge and Understanding (Facts and Procedures) Solve for the variable. 4 0 a) n = Knowledge and Understanding (Conceptual Understanding) Explain why the solution to x + = 5 is equivalent to the solution of 4x + 0 = 5. b) t + = 5 4 Problem Solving (Reflecting, Reasoning and Proving) Philip knows that a pyramid has a height of 6 m and volume 900 m. He determines the area of base as: V = ( area of base)( height ) 900 = ( area of base)( 6) 900 = ( area of base)( 6) 00 6 = area of base Problem Solving (Connecting, Selecting Tools and Computational Strategies) A small order of popcorn at the local fair is sold in a paper cone-shaped container. If the volume of the cone is 500 cm and the radius is 6 cm, determine the height of the paper cone. Area of base of pyramid is 94 m. Prove that this is incorrect and state Philip s error. TIPS4RM: Continuum and Connections Fractions

14 Problem Solving Across the Grades Sample Name: Date: Betty cut 4 m from a piece of rope m long. Is there enough rope left for two pieces 5 6 m long each? Show your answer in more than one way TIPS4RM: Continuum and Connections Fractions

15 Grade 7 Although the teacher may expect a student to apply specific mathematical knowledge in a problemsolving context, the student may find some unexpected way to solve the problem. Have available a variety of tools from which students can choose to assist them with their thinking and communication. a. A student operating at the concrete stage may use coloured strips of paper. A visual learner may draw a picture of the lengths of rope. The following diagrams illustrate what the first student may cut out and the second student may draw. Use manipulative representation Draw a diagram Divide into sixths Therefore, there is enough rope left for two pieces 5 6 m long. There will be m left over. 4 TIPS4RM: Continuum and Connections Fractions 4

16 Grade 7 b. A student operating at the concrete stage may use coloured strips of paper. A visual learner may draw a picture of the lengths of rope. The following diagrams illustrate what the first student may cut out and the second student may draw. The total rope cut off is 9 with or 4 left over. There is enough rope for the two 5 m 6 lengths. TIPS4RM: Continuum and Connections Fractions 5

17 Grade 7 c. Use concrete materials such as pattern blocks. Subtract 4 Remaining Subtract pieces of 5 6 Remaining Note: a double hexagon represents.. Some students may already be comfortable using symbols and operations with fractions Use symbolic representation = = = = 4 There is enough rope for all three pieces. TIPS4RM: Continuum and Connections Fractions 6

18 Grade 7. Some students using symbols may approach the solution differently = = = = 0 = > Use symbolic representation There is enough rope for all three pieces. 4. Some students may convert fractions to decimals = = = Use symbolic representation There is enough rope for all three pieces. Grade 8 Students solutions could include any of the Grade 7 answers. 4 Use symbolic representation 8 = 4 9 = = 0 0 = or > 5 ( ) = 0 = Therefore, there is enough rope. Note: Students in Grade 8 multiply whole numbers and fractions. TIPS4RM: Continuum and Connections Fractions 7

19 Grades 9 and 0 Applied Students solutions could include any of the Grades 7 and 8 answers.. Some students may choose to use a scientific calculator with a fraction key. Use a scientific calculator + = a b a b a b 4 ( 5 a b 6 5 a b 6 ) c c c c c Since the display shows all three pieces. 4, meaning 4, there is some rope left over, indicating there is enough for Note: Different calculators may show fractions differently.. Some students may represent the problem numerically using order of operations as follows: Use symbolic representation = = = 9 = = = (which is greater than zero, indicating there is more than enough rope for all three pieces) 4 TIPS4RM: Continuum and Connections Fractions 8

20 Problem Solving Across the Grades Sample Name: Date: So-Jung has a collection of 550 building blocks, whose sides measure store them in boxes with dimensions of 9 0 cm by 9 0 cm by 9 0 cm. 5 cm. She wants to How many of these boxes will So-Jung need to store her collection? Show your answer in two ways... TIPS4RM: Continuum and Connections Fractions 9

21 Grade 7 Although the teacher may expect a student to apply specific mathematical knowledge in a problemsolving context, the student may find some unexpected way to solve the problem. Have available a variety of tools from which students can choose to assist them with their thinking and communication.. Students may build a ratio table using doubling. Number of Blocks Measure 5 Comments Use a ratio table Use logic 4 6 = double double again = = 8 add block 9 8 = more block = > There isn t enough room for 6 blocks. Therefore, the bottom layer has 5 5 = 5 blocks. How many layers in 550 blocks? = Since you can only have 5 layers in a box, there would be 4 boxes with layers left. So-Jung will need 5 boxes.. It is possible that coloured rods could be used with the 0 rods as the unit. However, this might be a bit unwieldy given that the length of the box would have to be represented using rods. Some students may still need the comfort of concrete materials though. (See solution.) The rods could be used to determine the number of building blocks that would line up in one dimension only. That number, 5, would then have to be cubed. The rods would not need to be used for this operation = 5 So, 5 building blocks would fit into one box. Then, the students will determine the number of boxes needed for 550 blocks. They could do this in a number of ways: a) = 4.4 Students should interpret this as 5 boxes. b) = 550 Students should interpret this as 5 boxes. Use manipulative representation Draw a diagram Use logic Note: It is assumed in all answers that the building blocks will be placed in the box face to face in layers and that there will be some air space in each of the three dimensions. TIPS4RM: Continuum and Connections Fractions 0

22 Grade 7. A student might use a scale model in one dimension. Cut out a strip of paper 9 cm long. Cut out several strips cm 0 5 Use a scale model long. 9 cm 0 cm 5 Five is the maximum number of 5 cm strips that can be placed in 9 0 cm or less. So, 5 building blocks will fit along each inside edge of the box. Since the box has three dimensions, the number building blocks are = 5. Note: A student may employ the logic in this answer but use diagrams of rectangles instead of cut-outs. Then, the students will determine the number of boxes needed for 550 blocks. They could do this in a number of ways: a) = 4.4 Students should interpret this as 5 boxes. b) = 550 Students should interpret this as 5 boxes. Grade 8 Students solutions could include any of the Grade 7 answers.. A student might use symbols, beginning by guessing and checking. Will 4 blocks fit in one dimension? 8 4 = 4 = = 6 < building blocks will fit and leave enough room for more building block Guess and check Use representation Draw a scaled diagram Try 5 blocks: 8 5 = 5 = 8 < Try 6 blocks: = 6 = = 9 > building blocks will fit and the 9 8= cm left is not 0 0 enough to fit in a 6 th building block The maximum number of blocks for each dimension is 5. Then, the students will determine the number of boxes needed for 550 blocks. They could do this in a number of ways: a) = 4.4 They should interpret this as 5 boxes. b) = 550 They should interpret this as 5 boxes. TIPS4RM: Continuum and Connections Fractions

23 Grade 8. A student might use a scale drawing in two dimensions. Use a scale drawing 5 5 = 5 5 building blocks per layer There are five layers like this. 5 5 = 5 The maximum number of building blocks that will fit in the box is 5. Then, the students will determine the number of boxes needed for 550 blocks. They could do this in a number of ways: a) = 4.4 They should interpret this as 5 boxes. b) = 550 They should interpret this as 5 boxes.. A student might use a scale drawing in three dimensions. The student could determine that there are 5 building blocks per dimension as was done in previous answers. Use a scale drawing Then, the students will determine the number of boxes needed for 550 blocks. They could do this in a number of ways: a) = 4.4 They should interpret this as 5 boxes. b) = 550 They should interpret this as 5 boxes. TIPS4RM: Continuum and Connections Fractions

24 Grades 9 and 0 Applied Students solutions could include any of the Grades 7 and 8 answers.. Some students might choose to use exponents and fractions to solve the problem Use symbolic representation 9 8 = = OR Note: a calculator would be used from this point on = = = = 6 = = = = = Note: a calculator would be used from this point on. Therefore, she can fit 8 building blocks into one box. Note: Students would need to think about the practical context to realize that this method gives an answer that is too big since it assumes use of space in the box that cannot accommodate the building blocks. If this solution is used, guide the students to determine the maximum number of building blocks that would fit along one side of the box. To determine the maximum number 7557 of building blocks, students could evaluate and see that it equals So, five building blocks will fit along each inside edge of the box. Since the box has three dimensions, the number of building blocks is 5 or 5. Then, the students will determine the number of boxes needed for 550 blocks. They could do this in a number of ways: a) = 4.4 They should interpret this as 5 boxes. b) = 550 They should interpret this as 5 boxes. TIPS4RM: Continuum and Connections Fractions

25 Grades 9 and 0 Applied. Some students might choose to use decimals =. 6 5 =. Use decimals 9..6 = So, 5 building blocks can fit in one dimension. Since the box has three dimensions, the number of cubes is 5 or 5. Then, the students will determine the number of boxes needed for 550 blocks. They could do this in a number of ways: a) = 4.4 They should interpret this as 5 boxes. b) = 550 They should interpret this as 5 boxes.. A student could choose to use a scientific calculator with a fraction key. 9 a b a b 0 a b a b 5 = c c c c Use a scientific calculator Since the display shows 5 6, meaning the box has three dimensions, the number of cubes is 5 or , So-Jung can fit 5 along one of the edges. Since Then, the students will determine the number of boxes needed for 550 blocks. They could do this in a number of ways: a) = 4.4 They should interpret this as 5 boxes. b) = 550 They should interpret this as 5 boxes. TIPS4RM: Continuum and Connections Fractions 4

26 Is This Always True? Grades 7 0 Name: Date:. Jagdeep claims that if you increase both the numerator and denominator of a fraction by the same amount, the result will always be greater than the original fraction. For example, start with. Now add, let s say 5, to the numerator and denominator = which is greater than. Is Jagdeep s claim true for all fractions?. Jeff declares that if you subtract a smaller fraction from a larger fraction, the result will always be between the two original fractions. For example, 9 = Is Jeff s statement true for all fractions?. Graham states that if you add to any fraction, the common denominator will always be even. Is Graham s statement true for all fractions?. No. Counter-example: start with, add to the numerator and denominator: + 5 = (which is ) is smaller than (which is ) 4 Find a counter-example Follow-up question: Under what conditions will this statement be true?. No. Counter-example: 9 8 = (not between the two original fractions) Find a counter-example. Yes. If the denominator of the first fraction is odd, you will need to multiply it by to get the lowest common denominator, so the common denominator will be a multiple of two and therefore, even. If the denominator of the first fraction is even, then it will be the common denominator since two will divide evenly into every even number. Use logical reasoning TIPS4RM: Continuum and Connections Fractions 5

27 Is This Always True? Grades 8 0 Name: Date:. Is it always true that if you multiply two fractions the product will be smaller than the original two fractions? For example, = 4 6. Anuroop asserts that the square of any rational number always results in a smaller rational number. Is Anuroop s statement always true?. Ana argues that if you multiply the numerator and denominator of a fraction by the same number, the original fraction is unchanged. Is Ana s statement always true? 4. Susan notices that the formula for the area of a trapezoid is stated as: a+ b h A= h, A= a+ b, A= h a+ b Are the formulas always equivalent to each other? ( ) ( ) No. Counter-example: =, which is larger than the original fractions. Find a counter-example 4. No. Counter-example: = = = 4, (4 is larger than ) or 9 9 = = =, ( = is smaller than = ) Find a counter-example. Yes. If you multiply the numerator and denominator by the same number, it is the same as multiplying by the number. Multiplication by never changes a number. Use logical reasoning 4. Yes. multiplied by h is the same as h. a+ b h A= h a b h a b = + = + So, ( ) ( ). Use equivalent statements TIPS4RM: Continuum and Connections Fractions 6

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