Unit 8: Proportional Reasoning
Lessons with Approximate Time: Days
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U8L1: Comparing and Interpreting Rates
Main Concepts in Lesson: | Achievement Indicator: | AFL's Assigned to Assess |
You can compare rates by writing them as – rates with the same units, where the second are the same. – unit rates When comparing rates, it is helpful to round the values. This enables you to do mental math and express each rate as an approximate unit rate. In a graph that shows the relationship between two quantities, the slope of a line segment represents the average rate of change for these quantities. The slope of a line segment, when reduced to lowest terms, representing rate of change is a unit rate. | 1.1 Interpret rates in a given context, such as the arts, commerce, the environment, medicine or recreation. 1.3 Determine and compare rates and unit rates. 1.4 Make and justify a decision, using rates. 1.5 Represent a given rate pictorially. 1.6 Draw a graph to represent a rate. 1.7 Explain, using examples, the relationship between the slope of a graph and a rate. 1.8 Describe a context for a given rate or unit rate. 1.9 Identify and explain factors that influence a rate in a given context. 1.10 Solve a contextual problem that involves rates or unit rates. | Pages 451 - 452 Questions #4, 7b, 8 (Key Question), 10, 13 and 15 |
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U8L2: Solving Problems That Involve Rates
Main Concepts in Lesson: | Achievement Indicator: | AFL's Assigned to Assess |
• Often, a problem that involves rates can be solved by writing an equation that involves a pair of equivalent ratios. To be equivalent ratios, the units in the numerators of the two ratios must be the same, and the units in the denominators must be the same. Paying attention to the units in each term of the ratios will help you write the equation correctly. • A multiplication strategy can be used to solve many rate problems, such as problems that require conversions between units. Including the units with each term in the product and using unit elimination helps you verify that your product is correct. • When a rate of change is constant, writing a linear function to represent the situation may be useful when solving problems. | 1.1 Interpret rates in a given context, such as the arts, commerce, the environment, medicine or recreation. 1.2 Solve a rate problem that requires the isolation of a variable. 1.3 Determine and compare rates and unit rates. 1.4 Make and justify a decision, using rates. 1.8 Describe a context for a given rate or unit rate. 1.9 Identify and explain factors that influence a rate in a given context. 1.10 Solve a contextual problem that involves rates or unit rates. | Pages 459 - 461 Questions #3, 6a, 6b, 15(Key Question) and 18 |
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U8L3: Scale Diagrams
Main Concepts in Lesson: | Achievement Indicator: | AFL's Assigned to Assess |
• You can multiply any linear dimension of a shape by the scale factor to calculate the corresponding dimension of a similar shape. • When determining the scale factor, k, used for a scale diagram, the measurement from the original shape is placed in the denominator: • When a scale factor is between 0 and 1, the new shape is a reduction of the original shape. • When a scale factor is greater than 1, the new shape is an enlargement of the original shape. | 2.1 Explain, using examples, how scale diagrams are used to model a 2-D shape or a 3-D object. 2.2 Determine, using proportional reasoning, the scale factor, given one dimension of a 2-D shape or a 3-D object and its representation. 2.3 Determine, using proportional reasoning, an unknown dimension of a 2-D shape or a 3-D object, given a scale diagram or a model. 2.4 Draw, with or without technology, a scale diagram of a given 2-D shape, according to a specified scale factor (enlargement or reduction). 2.5 Solve a contextual problem that involves a scale diagram. | Pages 473 - 474 Questions #6, 9 (Key Question), 11, 12 and 19 |
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U8L4: Scale Factors and Areas of 2-Dimensional Shapes
Main Concepts in Lesson: | Achievement Indicator: | AFL's Assigned to Assess |
If the area of a similar 2-D shape and the area of the original shape are known, then the scale factor, k, can be determined using the formula | 2.1 Explain, using examples, how scale diagrams are used to model a 2-D shape or a 3-D object. 2.2 Determine, using proportional reasoning, the scale factor, given one dimension of a 2-D shape or a 3-D object and its representation. 2.3 Determine, using proportional reasoning, an unknown dimension of a 2-D shape or a 3-D object, given a scale diagram or a model. 2.4 Draw, with or without technology, a scale diagram of a given 2-D shape, according to a specified scale factor (enlargement or reduction). 2.5 Solve a contextual problem that involves a scale diagram. 3.1 Determine the area of a 2-D shape, given the scale diagram, and justify the reasonableness of the result. 3.3 Explain, using examples, the effect of a change in the scale factor on the area of a 2-D shape. 3.6 Explain, using examples, the relationships among scale factor, area of a 2-D shape, surface area of a 3-D object and volume of a 3-D object. 3.7 Solve a spatial problem that requires the manipulation of formulas. 3.8 Solve a contextual problem that involves the relationships among scale factors, areas and volumes. | Pages 479- 482 Questions #4b, 5a, 6(Key Question), 7, 13b, and 17 |
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U8L5: Similar Objects: Scale Models and Scale Diagrams
Main Concepts in Lesson: | Achievement Indicator: | AFL's Assigned to Assess |
You can multiply any linear measurement of an object by the scale factor to calculate the corresponding measurement of the similar object. • You can determine the scale factor k, used to create a scale model of an object by using any corresponding linear measurements of the object and the scale model: • When a scale factor is between 0 and 1, the new object is a reduction of the original object. • When a scale factor is greater than 1, the new object is an enlargement of the original object. | 2.1 Explain, using examples, how scale diagrams are used to model a 2-D shape or a 3-D object. 2.2 Determine, using proportional reasoning, the scale factor, given one dimension of a 2-D shape or a 3-D object and its representation. 2.3 Determine, using proportional reasoning, an unknown dimension of a 2-D shape or a 3-D object, given a scale diagram or a model. 2.4 Draw, with or without technology, a scale diagram of a given 2-D shape, according to a specified scale factor (enlargement or reduction). 2.5 Solve a contextual problem that involves a scale diagram. | Pages 491 - 493 Questions #5, 8, 11(Key Question), and 14 |
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U8L6: Scale Factors and 3-Dimensional Objects
Main Concepts in Lesson: | Achievement Indicator: | AFL's Assigned to Assess |
• If you know the dimensions of a scale diagram or model of a 3-D object, as well as the scale factor used to enlarge/reduce from the diagram or model, you can determine the surface area and volume of the enlarged/ reduced object, without determining its dimensions. | 2.2 Determine, using proportional reasoning, the scale factor, given one dimension of a 2-D shape or a 3-D object and its representation. 2.5 Solve a contextual problem that involves a scale diagram. 3.2 Determine the surface area and volume of a 3-D object, given the scale diagram, and justify the reasonableness of the result. 3.4 Explain, using examples, the effect of a change in the scale factor on the surface area of a 3-D object. 3.5 Explain, using examples, the effect of a change in the scale factor on the volume of a 3-D object. 3.6 Explain, using examples, the relationships among scale factor, area of a 2-D shape, surface area of a 3-D object and volume of a 3-D object. 3.7 Solve a spatial problem that requires the manipulation of formulas. 3.8 Solve a contextual problem that involves the relationships among scale factors, areas and volumes. | Pages 500 - 503 Questions #1d, 4, 7, 11(Key Question), 15 and 17 |
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