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KS3 Mathematics

KS3 Mathematics. A6 Real-life graphs. A6 Real-life graphs. Contents. A6.1 Reading graphs. A. A6.2 Plotting graphs. A. A6.4 Distance-time graphs. A6.3 Conversion graphs. A. A. A6.5 Interpreting graphs. A. Distance-time graphs. distance. 0. time.

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KS3 Mathematics

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  1. KS3 Mathematics A6 Real-life graphs

  2. A6 Real-life graphs Contents A6.1 Reading graphs • A A6.2 Plotting graphs • A A6.4 Distance-time graphs A6.3 Conversion graphs • A • A A6.5 Interpreting graphs • A

  3. Distance-time graphs distance 0 time In a distance-time graph the horizontal axis shows time and the vertical axis shows distance. The below distance-time graph shows a journey. What does the slope of the line tell us? The slope of the line tells us the average speed. The steeper the line is, the faster the speed.

  4. Label the distance-time graph

  5. Olympic swimmers

  6. A6 Real-life graphs Contents A6.1 Reading graphs • A A6.2 Plotting graphs • A A6.5 Interpreting graphs A6.3 Conversion graphs • A A6.4 Distance-time graphs • A • A

  7. Filling flasks 1

  8. Filling flasks 2

  9. Interpreting the shapes of graphs 150 Eating a bar of chocolate 100 Mass of chocolate (g) 50 0 0 10 20 30 40 50 60 70 80 90 100 Time (seconds) Jessica eats a bar of chocolate. This graph shows how the mass of the chocolate bar changes as it is eaten.

  10. Interpreting the shapes of graphs Temperature of water Time This graphs shows how the temperature of the water in a pan changes when frozen peas are added.

  11. Which graph is correct? Graph A Graph B Graph C Graph D Mass of sponge (g) Mass of sponge (g) Mass of sponge (g) Mass of sponge (g) Volume of water (cm3) Volume of water (cm3) Volume of water (cm3) Volume of water (cm3) In an experiment a group of pupils poured water onto a sponge and weighed it at regular intervals. Each time the sponge soaked up all the water. Which graph is most likely to show their results?

  12. Sketching graphs Temperature (oC) Time (minutes) A group of pupils are conducting an experiment. They fill three beakers with boiling water and record the temperature of the water over time. Beaker A has no wrapping, Beaker B is wrapped in ice and Beaker C is wrapped in insulation fibre. The temperature graph for beaker A looks as follows: How would the graphs for beakers B and C compare to this? Beaker A

  13. Sketching graphs

  14. Matching graphs to statements

  15. A8 Linear and real-life graphs Contents A8.1 Linear graphs • A A8.2 Gradients and intercepts • A A8.3 Parallel and perpendicular lines • A A8.4 Interpreting real-life graphs • A A8.5 Distance-time graphs • A A8.6 Speed-time graphs • A

  16. Real-life graphs American dollars British pounds When we use graphs to illustrate real-life situations, instead of plotting y-values against x-values, we plot one physical quantity against another physical quantity. The resulting graph shows the rate that one quantity changes with another. For example, This graph shows the exchange rate from British pounds to American dollars. It is a straight line graph through the origin and so the equation of the line would be of the form y = mx. The value of m would be equal to the number of dollars in each pound. What would the value of m represent?

  17. Real-life graphs investment value time This graph show the value of an investment as it gains interest cumulatively over time. The graph increases by increasing amounts. Each time interest is added it is calculated on an ever greater amount. This makes a small difference at first but as time goes on it makes a much greater difference. This is an example of an exponential increase.

  18. Real-life graphs mass time This graph show the mass of a newborn baby over the first month from birth. The baby’s mass decreases slightly during the first week. Its mass then increases in decreasing amounts over the rest of the month.

  19. Rates of change

  20. Filling flasks

  21. A8 Linear and real-life graphs Contents A8.1 Linear graphs • A A8.2 Gradients and intercepts • A A8.3 Parallel and perpendicular lines • A A8.5 Distance-time graphs A8.4 Interpreting real-life graphs • A • A A8.6 Speed-time graphs • A

  22. Formulae relating distance, time and speed DISTANCE distance time = SPEED TIME speed distance speed = time It is important to remember how distance, time and speed are related. Using a formula triangle can help, distance = speed × time

  23. Distance-time graphs 20 distance (miles) 15 10 5 0 0 15 30 45 60 75 90 105 120 time (mins) In a distance-time graph the horizontal axis shows time and the vertical axis shows distance. For example, John takes his car to visit a friend. There are three parts to the journey: John drives at constant speed for 30 minutes until he reaches his friend’s house 20 miles away. He stays at his friend’s house for 45 minutes. He then drives home at a constant speed and arrives home 45 minutes later.

  24. Finding speed from distance-time graphs change in distance gradient = change in time distance change in distance change in time time How do we calculate speed? Speed is calculated by dividing distance by time. In a distance-time graph this is given by the gradient of the graph. = speed The steeper the line, the faster the object is moving. A zero gradient means that the object is not moving.

  25. Interpreting distance-time graphs

  26. Distance-time graphs change in speed acceleration = time When a distance-time graph is linear, the objects involved are moving at a constant speed. Most real-life objects do not always move at constant speed, however. It is more likely that they will speed up and slow down during the journey. Increase in speed over time is called acceleration. It is measured in metres per second per second or m/s2. When speed decreases over time is often is called deceleration.

  27. Distance-time graphs This distance-time graph shows an object accelerating from rest before continuing at a constant speed. distance distance time time This distance-time graph shows an object decelerating from constant speed before coming to rest. Distance-time graphs that show acceleration or deceleration are curved. For example,

  28. A8 Linear and real-life graphs Contents A8.1 Linear graphs • A A8.2 Gradients and intercepts • A A8.3 Parallel and perpendicular lines • A A8.6 Speed-time graphs A8.4 Interpreting real-life graphs • A A8.5 Distance-time graphs • A • A

  29. Speed-time graphs 20 speed (m/s) 15 10 5 0 0 5 10 15 20 25 30 35 40 time (s) Travel graphs can also be used to show change in speed over time. For example, this graph shows a car accelerating steadily from rest to a speed of 20 m/s. It then continues at a constant speed for 15 seconds. The brakes are then applied and it decelerates steadily to a stop. The car is moving in the same direction throughout.

  30. Finding acceleration from speed-time graphs speed change in speed change in speed gradient = change in time change in time time Acceleration is calculated by dividing speed by time. In a speed-time graph this is given by the gradient of the graph. = acceleration The steeper the line, the greater the acceleration. A zero gradient means that the object is moving at a constant speed. A negative gradient means that the object is decelerating.

  31. Finding distance from speed-time graphs 20 15 speed (m/s) 10 5 0 0 15 30 45 60 75 90 105 120 time (s) The following speed-time graph shows a car driving at a constant speed of 30 m/s for 2 minutes. What is the area under the graph? The area under the graph is rectangular and so we can find its area by multiplying its length by its height. Area under graph = 20 × 120 = 240 What does this amount correspond to?

  32. The area under a speed-time graph For example, to find the distance travelled for the journey shown in this graph we find the area under it. 15 20 speed (m/s) 15 10 5 0 0 5 10 15 20 25 30 35 40 time (s) This area under a speed-time graph corresponds to the distance travelled. The shape under the graph is a trapezium so, Area = ½(15 + 40) × 20 = ½ × 55 × 20 = 550 So, distance travelled = 550 m

  33. Interpreting speed-time graphs

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