Pearson's Ramp-Up Mathematics

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Maximilian Hart
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Introducing Slope L E S S O N CONCEPT BOOK See pages 7 8 in the Concept Book. PURPOSE To introduce slope as a graphical form of the constant of proportionality, k.

1 Introducing Slope L E S S O N CONCEPT BOOK See pages 7 8 in the Concept Book. PURPOSE To introduce slope as a graphical form of the constant of proportionality, k. The lesson identifies k as the ratio of the side lengths of right triangles drawn along a line. The side lengths of the right triangles are the rise and run, and are measured using the coordinates of points on the line. Mathematics of the Lesson The concept of the slope of a line. Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio ( rise over run ) is called the slope of a graph. The geometric context for ratio and proportion; similar right triangles on a graph. Using right triangles and ramp diagrams to illustrate slope as a measure of steepness Constant of proportionality and constant ratio Preparation, Organization, and Materials Read Lesson in the Student Edition, and review Concept Book pages 7 8. Work all of the problems for the lesson. Prepare an Opening to set the stage for the lesson. Opening Begin the Opening by graphing a line with a positive slope other than y = x. Sketch a right triangle with a hypotenuse along the line and with one vertex at the origin. Have students identify the coordinates of STUDENT EDITION, page the other vertex that is on the line. Use the triangle you have sketched to measure the rise and run of the line between the origin and the second vertex of the triangle. Have students find the ratio of rise to run between these two points. Emphasize the central concepts of the lesson: the constant ratio of rise to run is called slope, and the slope is equal to the constant of proportionality between variables in proportional relationships. To do so, you may want to make a ratio table that includes several pairs of x- and y-coordinates for points on the line to show that the constant ratio derived from the table is equal the ratio of rise to run. Ramp-Up to Algebra

2 Lesson Scaffolding for Success Since the concept of slope is a new concept, color-code the rise and run in the Opening example. Color the word rise and the three y -values,, and in blue. Then color the word run and the three x -values,.5, and in red. Color-coding makes the abstract concept of slope more concrete by visually showing the relationship between the graph and the slope of the line. Take time to go over the second diagram in the student material showing the rise and run of different right triangles. The key point to emphasize is that rise and run are both measurable lengths, not just coordinates or numerical values. This is conceptually very different from how students have been working with ratios so far. Slope is a ratio of lengths, not a ratio of values. In this lesson, each rise and run extends between a point on the line and the origin, which makes the lengths easy to measure: the rise is equal to the y-coordinate of the chosen point; the run is equal to the x-coordinate of the chosen point. In later lessons, however, students will need to find the rise and run between two points on the line other than the origin. To do so, they will need to find the difference between the y-coordinates of the two points (rise) and the difference between the x-coordinates of the two points (run). STUDENT EDITION, pages 5 Showing Relationships with Graphs

3 Introducing Slope To introduce the concept, measuring the rise and run from the origin keeps things relatively simple, but is only possible for graphs of proportional relationships because they intersect the origin. Students should recognize that the ratio of rise to run in the diagram is constant, no matter which right triangle is used. Again, remind students that when finding the slope, the constant ratio they are looking for is a ratio of lengths, not a ratio of values or coordinates. If students are unsure about k being the constant ratio between y and x, make the following ratio table for y = x: x 0.5 y 0 Confirm that students recognize that measurements of the legs of each triangle correspond to the coordinates of points on the line. Then have students identify the points (, ), (.5, ) (, ) on the line y = x in one of the diagrams given in the student material. Have students practice identifying the rise and run of different triangles in the diagrams. Although students will not be working with negative slopes in this lesson, it may come up in the discussion. Be prepared to address negative slope in connection with rise and run. If a line slopes up to the right, the rise and run will each be positive, giving a positive constant ratio so the slope will be positive. If a line slopes down to the right, the rise will have a negative value to indicate a drop, giving a negative constant ratio, so the slope will be negative. Work Time. The slope would be rise run = feet 0feet = 5.. The run would have to be feet. Though computationally simple, this problem may be difficult for students who have not understood the concepts of rise and run. Explain that the rise is, as in the previous problem, but the run is unknown. To achieve a slope of, the computation involves finding an equivalent ratio with as a numerator. STUDENT EDITION, page Ramp-Up to Algebra

$The problem is like the others, involving equivalent ratios, but in this case, students must find the numerator of a fraction equivalent to with a denominator of. Preparing for the Closing 5.$

4 Lesson. The run would have to be 8 feet. Again, this problem involves finding an equivalent ratio to, this time with a numerator of. STUDENT EDITION, page. The ramp will reach half a foot. The problem is like the others, involving equivalent ratios, but in this case, students must find the numerator of a fraction equivalent to with a denominator of. Preparing for the Closing 5. You may want to have students work in small groups, or instead do this problem together as a class. Discussion is important in this problem, and sketching diagrams and graphs as a group should take less time than having each student work individually. Be sure that students understand that while the rise and run of each ramp is different, the slope of each STUDENT EDITION, pages 7 ramp is the same:. The diagrams of the ramps do not need to be any more elaborate than right triangles with the correct rise and run measurements. Have students think carefully about what scale should be used for the axes in part b. The rise (y-values in the coordinate plane) has values that are not very large, but the run (x-values) has larger values. a. 8.5 b. y 5 5 (8,) (,) (, ) x Showing Relationships with Graphs

5 Introducing Slope c. The formula y = x can be used to represent all three ramps. If students have trouble generating this formula, make a ratio table with a y-value for each rise and an x-value for each run. Include each corresponding rise : run pair in the table to see that the constant ratio, k, is. By now students should recognize that the formula should have the form y = kx because y is proportional to x. d. This question should promote the most discussion. A single formula can be used to represent all three ramps. Assuming the graph representing each ramp begins at (0, 0), all three graphs will fall along the same line because they have the same slope (they climb at the same rate). The longest ramp (reaching the greatest height of feet), will extend the farthest, but the relationship between rise and run is the same.. The run also must increase as the rise increases to maintain the same slope. 7. A ramp with a rise of and run of is steeper than a ramp with a slope of because >. Closing Begin the Closing by going over problem 5 in detail. Then check that students understand that the rise and run are variables, but that they vary in constant proportion to each other. The emphasis should be on the constant ratio between rise and run, and not on the constant ratio between values in the tables. This is because in later lessons, students will learn that many linear relationships (with a constant slope) do not have a constant English Language Learners Remember to pronounce words clearly and write them on a board or overhead. ratio between values. The constant slope derives from the constant ratio of change in y-values relative to change in x-values, not a constant ratio between each pair of single y- and x-values. While at this point we want students to see the connection between constant ratio (as shown by a ratio table) and slope (as shown by a graph), they should be encouraged to get in the habit of finding the rise and run by measuring the lengths of each. By measuring the rise and run, students are better prepared to find the slope between any two points on the line. You may want to end the Closing with a demonstration: Choose any two points on the line from part b of problem 5 other than the origin. Sketch a right triangle with a hypotenuse that extends between these points. Find the rise and run of the line between these two points by measuring the legs of the right triangle. To measure the legs, you will need to find the difference between the y-values of the two points (rise) and the difference between the x-values of the two points (run). The ratio of rise to run using this triangle will be :, because the ratio of rise to run all along the line is constant, which makes the line straight. Scaffolding for Success Have students color-code the rise and run values when demonstrating how to find the slope of the line. Ramp-Up to Algebra 5

6 Lesson Skills Students write each of these ratios as a percent. a. 0% b. % c. 5% d..5% STUDENT EDITION, page 7 Review and Consolidation. Students answers will vary depending on the points that they choose. Their answers could contain: x y. Diagrams will vary depending on the values students included in their tables for problem. 8 STUDENT EDITION, pages 7 8. Students answers will vary depending on the points they choose. Their answers could contain: x y Showing Relationships with Graphs

7 Introducing Slope. Diagrams will vary depending on the values students included in their tables for problem. 5. Students answers will vary depending on the points they choose. Their answers could contain: x y. Diagrams will vary depending on the values students included in their tables for problem. 7. Students answers will vary depending on the points they choose. Their answers could contain: x y 8. Diagrams will vary depending on the values students included in their tables for problem..5 Ramp-Up to Algebra 7

Lesson 9. y 5 x 5 5 7 8 9 0. Homework y 5. The rise is 0 and the run is 5.. The rise is 7 and the run is.5.. The rise is and the run is 0.

8 Lesson 9. y 5 x Homework y 5. The rise is 0 and the run is 5.. The rise is 7 and the run is.5.. The rise is and the run is The ratio of rise to run is =. 5. y = x STUDENT EDITION, page 8 x 8 Showing Relationships with Graphs

Section 1.3. Slope of a Line

Section 1.3. Slope of a Line Slope of a Line Introduction Comparing the Steepness of Two Objects Two ladders leaning against a building. Which is steeper? We compare the vertical distance from the base of the building to the ladder