Can you see this. For the same run A rises most. Can you see the vertical change over the horizontal change?

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1 Concept of slope transcript: Slide 1:Concept of slope This presentation is about the concept of slope. Slide 2: Connections Slope is a fundamental mathematical concept that you meet in all five strands of mathematics. You will meet several representations of slope and its important that you can recognise the concept however it is represented. Slide 3: A Physical Representation You can have a physical representation of slope. The slope or gradient can be measured as a measure of steepness. Consider these three lines, which do you think has the largest slope? In other words which is the steepest? Slide 4: A Geometric Representation The slope can also be regarded as a geometric ratio of the rise, how far something goes up vertically to the run, how far something goes out horizontally. Or the vertical change over the horizontal change. Slide 5: Rise over run Can you see this. For the same run A rises most. Can you see the vertical change over the horizontal change? Slide 6: Visualising slope Combining these two ideas leads to an algebraic representation. Slope m equals y 2 minus y 1 over x 2 minus x 1. Slide 7: A trigonometric ratio Slope m is equal to the rise over the run, which is equal to the change in the y value from A to B, which is equal to the change in the x value from A to B which is y 2 minus y 1 over x 2 minus x 1. Here in the diagram you can see that equals 4 take away 2 over 4 take away 2 which is equal to 2 over 2 which is one. Slide 8: A Parametric Representation The slope or gradient can be represented as the tan of the angle of inclination. You can examine this trigonometric representation in the geometry and trigonometry presentation by clicking on the web link.

2 Slide 9: A Calculus Concept In general a linear equation is written in the form y = mx + c. Where m is the slope and y is where the line cuts the x axis. Slope may also be regarded as a calculus concept. Slide 10: Sample Question (1) Calculus begins with the study of derivatives and rates of change. Slopes of lines help develop these concepts. Slide 11: Sample Question (2) Look at the examination question and think about which representation is being used. What are you given, what are you required to do? Slide 12: Write down the equation of the line Here you need to be able to compare slopes with l 2 having the same slope as l 1 and l 3 having a smaller slope and different intercept. Slide 13: Higher Level task What do you need to be able to do to write down the equation of the line shown? What information can you get from the diagram? Draw the line with equation y = 2x 4. Slide 14: Lines sloping downwards What do you know about these three lines? Could you sketch a diagram? You know their slopes, what about their y intercepts? What about the x intercepts of these lines? Can you see these on the diagram? Does this help you to write the equation of these three lines? Have a go. Slide 15: Lines sloping downwards Now what happens as the line slopes downwards as you go across to the right? How does the downward direction effect the value of the slope? Slide 16: Recognising slopes Try to calculate the slope. Use your equation, can you see the rise? Can you see the run? What can you say about the sign of the slope? You can find out more about positive and negative slopes by clicking on the weblinks. Slide 17: Predict.. This question requires you to recognise the slopes of the lines from their sketches. Have any of the lines got negative slopes? How do you know? Have any of the lines got positive slopes? How do you know? Check out the climbing analogy by clicking on the web link.

3 Slide 18: Put these in order Plot each of the pairs of points, join them to form a line and predict whether the slope of each line will be positive or negative. Check your predictions. Slide 19: Identify slopes Examine the lines, decide whether their slopes are positive or negative and then list the lines in order of increasing slope. Slide 20: Lines with the same steepness Of the four lines pictured above, one has a slope of zero, one has a slope of one another has a slope of negative one and another has an undefined slope. Decide which is which. Slide 21: Perpendicular lines Now think about lines which have exactly the same steepness, sketch two lines with the same steepness. What can you say about these lines? Think about it will they ever meet? How will their slopes compare now check your prediction. Write a statement about the slopes of parallel lines. Slide 22: Some websites Look at the two lines L 1 and L 2 they meet at an angle of 90, this means they are perpendicular. When two perpendicular lines meet there is a relationship between the two slopes. If you multiply the two slopes, they will result in the answer of negative one. Think about why this would be true. Slide 23: Question 1 By clicking on the weblinks you will get lots of practice in working with lines that are parallel or perpendicular to each other. Slide 24: Questions 2 This examination question requires to you to use these facts about parallel and perpendicular lines. Slide 25: Questions 3 Look at the diagram, say whether the following statements is true or false and in each case give a reason for the answer. Slide 26: Question 3 continued Look at the lines J, K, L, M, N what do you notice about them? Are there any lines parallel to each other? What will that tell you about their slope? Are there any lines perpendicular to each other? What will that tell you about their slope? Do any lines share y intercepts? Once you have had a chance to look at the lines, look at the next slide and answer the questions. Slide 27: Right angled triangles Where else would you see lines that are perpendicular to each other? Right angled triangles.

4 Slide 28: Rectangles Rectangles Slide 29: Tangents to circles Tangents to circles Slide 30: Triangles inscribed in semi circles Triangles inscribed in semi circles Slide 31: Using slope to prove that lines meet at 90 Now you can prove that angles are 90 by using the concept of slope. If an angle is 90 then the two lines that join to form them are perpendicular and the product of their slopes is negative 1. Slide 32: Tangents, point of contact Consider the following question. P is the point (0,7) and Q is the point (8, 11). Find the equation of the circle with diameter PQ. Find the equation of the tangent at Q. Make a sketch, what do you know about the tangent to a circle and the diameter or radius at the point of contact. Will this help you solve the problem? Slide 33:Question 4 Or this question. Two points A ( -3, 2) and B( 4, -1) are shown on the diagram below. Plot two suitable points C and D so that ABCD is a parallelogram. Label their coordinates, by performing suitable calculations, show that the figure you have drawn is a parallelogram. Slide 34: Parallelogram What are the properties of a parallelogram? Visit the website maths is fun to read about parallelograms and their properties, how will these properties help you show that the parallelogram ABCD you have drawn is a parallelogram. Will the slope help? Can you see any parallel lines? Can you see any perpendicular lines? Have a go Slide 35: Question 5 Now look at this question from a higher level sample paper. A circle has equation x 2 + y 2 2x + 4y 15 =0 Find the value m for which the line m + 2y 7 = 0 is a tangent to the circle. This is very similar to a question we spoke of earlier. What do you know about the tangent and the diameter or radius at the point of contact? Make a sketch. Slide 36: Task 1 Try this plot the points (2, 3) and ( 2, 5) join these points to form a line L from the point (5, 4) construct a line perpendicular to L. Show all the construction marks. Show using slopes that these two lines are perpendicular to each other. What do you know about perpendicular lines?what do you know about the slopes of these lines?have a go.

5 Slide 37: Other representations of slope In the real world the graphical axes are not x and y but represent real life quantities such as velocity, time, flow of electricity, productivity of farmlands, efficiency of assembly lines, cost of electricity number of units used and many other quantities. Relationships between these quantities may appear as stories or situations, tables graphs or formulae. Slope has meaning in each of these representations. Most fields of study do graphical analysis of data. One of the most important way to analyse a graph is to investigate it s slope or rate of change. Slide 38: Slope in real life Consider the following real life situation. The cost of transporting document scan be represented by the following straight line. Two points are given on the line A and B. What does A represent (0, 5) what does the zero represent in this context. What does the 5 represent in this context? Similiarly B is (6, 23) what does the 6 represent in this context? What does the 23 represent? Slide 39: Calculations You should be thinking that the point A represents zero on the distance axis and 5 on the cost axis. So, when the documents are transported zero kilometres, the cost is 5 euro. And the point B represents 6 on the distance axis and 23 on the cost axis. So when the documents are transported 6km the cost is 23 euro. Now you might be wondering why would anyone pay 5 euro to transport documents zero kms? Well this is usually called a standing charge. So when you employ the courier you re bill is usually calculated with a standing charge and a charge for every kilometre. So how do we calculate this amount for every km travelled? Well if you calculate the slope or gradient, that is the change in the y axis over the change in the x axis and in this instance that s the change in cost over the change in distance, 23 5 over 6 0 which equals 18 euro over 6 km or three euro per kilometre. So the rate charged by the courier is 3 euro per kilometre plus a standing charge of 5 euro. Can you see this on the graph? Can you see the standing charge? Where does it appear on the graph? Can you see the 3 euro per kilometre how is this represented on the graph? What woud the graph look like if it was 2 euro per km or 5 euro per km? Now take a look at this question. Slide 40: Slope in real life A cyclist cycles for 20 minutes at a constant speed and covers a distance of 15km as shown in the diagram. Find the gradient of the line and describe it s meaning in this context. What does the y intercept mean in this instance? Look at the two graphs that show two varying quantities: cost and number of units. Slide 41: Example Calculate the gradient of each line and attach a meaning to it. What is the meaning of the y intercept in this context. Write an equation for both lines. Attach meaning to each equation.

6 Slide 42: Task 2 Have a look at the following real world situation. You and your friends plan to visit the carnival next weekend. You don t know how much it costs to enter the carnival but you have the following incomplete information. If you buy eight tickets the cost is 9 euro. If you buy 12 tickets, the cost is 11. It costs to get into the carnival but you don t know how many tickets you get. So what are you asked to do? You re asked to write a linear equation in which y represents the total cost and x represents the number of rides selected. You re asked to identify the slope and y intercept in the equation and explain what each of them means in the context of the problem. Then you have 20 to spend at the carnival. If you need three tickets for each ride, how many rides will you be able to go on? Use maths to explain your answer. Use words, symbols or both. Let s try to be mathematical detectives: this is a linear relationship. What does that mean? If you plot your data points and join them, what will the graph look like? What does the y-intercept mean? What does slope mean in this situation? Have a go. Slide 43: Line of best fit In other real world situations varying quantities are not related in such an exact way. When you plot points that represent experimental data obtained in science or statistics, you draw the straight line that best represents those points. You draw by eye a straight line such that it represents the best fit. With the distance of the plotted points from this line as small as possible. This may mean that the line does not go through any of the plotted points and that some points will be above the line and some below it. Slide 44: Task 3 Try this real world example. Slide 45: Question 6 Consider activity 5.1 on page 121 and the questions from pages in the NCCA student materials. You can download these at Primary_Education/Project_Maths/Resources/Revised_LC_Student_Resources_Strand_1.pdf These deal with real life situations where you have to decide the extent to which varying quantities are related. Now assess your understanding by trying these examination style questions. Slide 46: What s slope got to do with it? Lets summarise what we ve been talking about in this presentation. Slope has meaning in formulae, tables, physical story situations and is closely related to the concept of derivative. Awareness and understanding of this will help you solve problems. The next five slides contain problem relating to the concept of slope. Use the many representations of slope discussed in this presentation to help solve problems. Slide 47 Question Slide 48 Question

7 Slide 49 Question Slide 50 Question Slide 51: Non-constant slope So far we have looked at situations where the slope is constant. That is where it doesn t change. Are there any situations where the slope is not constant? What would these look like in a table, graph, equation. Look at these graphs what would you say about the slope in each case? Is it constant or changing? How do you know? Slide 52: Slope of a tangent to a curve The Geogebra file should help you see what is happening to the slope in this situation. Download geogebra at the link shown it is a very powerful investigational tool for mathematics. Slide 53: Geogebra Can you see from the Geogebra investigation that the slope is different at every point along the curve? The slope of the curve at any point is the same as that of the tangent line at that point. Does this give you a clue as to how to find the slope of a curve? If someone said to you find the slope of that curve you would have to say at what point would you like me to find that slope because the slope is always changing. Slide 54: Real life curves But what type of situations produce graphs like this? Look at the staircase towers problem on the slide. Would a graph of this table be linear how can you tell? What is different about this situation and one that produces a linear graph? Draw the graph. What is happening to the slope of this graph? Can you see the slope in the table? Can you explain why the graph looks the way it does? Slide 55: Generalising Listen to Pete as he explains how he generalised the relationship between the tower number and the number of tiles. Pete: I knew the graph was going to be a curve because the number of tiles went up by a different amount each time. To make a second tower I added two more tiles and to make the third tower I added three more tiles. So the slope is not constant it went up by two and then by three and then by four. When I looked at the change of the change it was constant, it went up by one each time and I knew there was a quadratic relationship between the tower numbers and the number of tiles. This meant that the relationship is y = ax 2 + bx + c, all I had to do was find the a, b, c values. Three values so three pieces of information. I chose the first three values in the table (1, 1), (2, 3), (3,6) and then I used simultaneous equations to find the values of a, b, c, so I could generalise the relationship to say the number of tiles equals half the tower number by the tower number plus one. So if the tower number is ten there will be 55 tiles and that s easy to check.

8 Slide 56: Generalising 2 Listen to Anne who is following the Ordinary level syllabus as she explains her strategy. Well first of all I looked at the towers and I could see that it was like growing rectangles, but only half of it. So the first one is half of 1 times 2, the second is half of 2 x 3, the third is half of a 3 x 4 so the n th one is half of a n(n+1) so the n th term for this one is pattern. Slide 57 Verifying Slide 58: The Deal. Verify Anne s strategy by building the The Deal- what do you think of this deal? Make a table to show how much money I will get for each day for the first ten days of the month. Make observations about the values in the table. What would the graph look like would it be linear? How do you know? Make a graph to check your prediction. What is happening to the slope of this graph? Can you see the slope in the table? What is the difference between this graph and the one you drew for staircase towers? Which one goes up a faster rate.how do you know? Slide 59: Generalising 3 Listen to Peter again as he explains how he generalised the relationship between the number of days and the amount of money. I used the table to put the values into. It was a good way to see what type of relationship you have. I knew it wasn t linear because the amount of money it went up by each day was different and the amount it went up by had a pattern in itself 2, 4, 8 it was getting bigger very fast and when I looked at the differences between these differences they weren t constant. It wasn t a quadratic relationship either. It must be exponential. Knowing this helped me with the generalisation. I put another column to the table and wrote the number out in the way I got it like 2, 2 x 2 because to get this number I doubled the two and then 2 x 2 x2 because I doubled the two by two to get this number and so on. It made it really easy for me to generalise the relationship. The amount of money my Dad would give me on the thirtieth day of the month would be 2 30 which is a huge amount of money 1073,741,821 cent. Slide 60: Classifying Relationships Think about the relationships you have seen there are three main types. What does a linear relationship look like in a context, in a table, in a graph, in a generalised expression? What does a quadratic relationship look like? In a context, in a table, in a graph, in a generalised expression. What does an exponential relationship look like in a context, table, graph, expression? Slide 61: Think What is an arithmetic sequence? Look it up in your textbook or ask your teacher or Google it.

9 Side 62: Searching for connections The extract from Wikipedia defines an arithmetic sequence or arithmetic progression. Try to graph this data. How would you represent it in a table? Now graph it, does it look like a linear relationship? Can you see the slope in the table? In the sequence, generalise the linear relationship. Is this the same as the nth term in the Arithmetic sequence. Slide 63: Generalising Generalise the linear relationship and use the arithmetic sequence formula to find the nth term of the sequence in two different ways. Slide 64: Examining student work Examine this student s work, can you see the slope of the linear relationship? What does this represent in the arithmetic sequence? Look at the formula for the n th term of the arithmetic sequence, compare this with how you generalise the linear relationship. Slide 65: Think Now think of Geometric sequences and think about how they compare with exponential relationships. What is a geometrical sequence? Look at this extract from Wikipedia that defines a geometric progression or geometric sequence. Slide 66: And more connections.. Slide 67: Generalising Find the nth term. Now generalise the exponential relationship. Slide 68: Examining student work Examine this student s work. Compare each method, why did the student include the third column in the table. What does the three represent? Slide 69: Classifying relationships You have now seen several ways to represent relationships between variables. What characterises a linear relationship? What makes it different to an exponential or a quadratic relationship, how could you determine a relationship between variables from a graph tables. Slide 70: Providing evidence Have a look at the understanding Financial Maths file, You will see how this work is applied in financial situations. When you are done with that check out the making sense of the derivative file to see how the concept of slope relates to differentiation and the derivative.

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11 Slide 20 Of the lines pictured above, one has a slope of zero. One has a slope of one