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GCSE Maths AQA A Higher Lesson Plan Chapter - Collins Education

By Cheryl Hayes,2014-07-09 11:18
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GCSE Maths AQA A Higher Lesson Plan Chapter - Collins Education

    Linear graphs and equations 16

Overview

     16.1 Linear graphs This chapter covers all the methods (grades D,

    16.2 Finding the equation of a line from its C, and B) for drawing linear graphs, finding

    graph their equations, and a few uses for them.

    16.3 Uses of graphs

    16.4 Parallel and perpendicular lines

    Context

    Much of our knowledge and use of science is, or can be, displayed in graph form everything from braking distances of cars, to break-even points of a company‟s economic future.

    AQA A references

    AO2 Number and algebra: Equations, formulae and identities 16.3 2.5j “... interpret [linear simultaneous] equations as lines and their common solution as the point of

    intersection”

    AO2 Number and algebra: Sequences, functions and graphs 16.1, 16.2, 16.3, 16.4 2.6b “use conventions for coordinates in the plane; plot points in all four quadrants;

    recognise ... that equations of the form y = mx + c correspond to straight-line graphs in the

    coordinate plane; plot graphs of functions in which y is given explicitly in terms of x, or

    implicitly; ...”

    16.2, 16.4 2.6c “find the gradient of lines given by equations of the form y = mx + c ...; understand that

    the form y = mx + c represents a straight line and that m is the gradient of the line and c is

    the value of the y-intercept; explore the gradients of parallel lines and lines perpendicular to

    each other”

    16.3 2.6d “construct linear functions ... arising from real-life problems;...”

    Route mapping

    Exercise D C B A A*

    A all

    B all

    C all

    D all

    E all

    F all

    G all

    H 14 510

     ? HarperCollinsPublishers Ltd 2006

    Answers to diagnostic Check-in test 1 A = (1, 1) B = (1, 3) C = (3, 1) D = (1, 2) 2 y = 4 3 y = 5 4

5

     ? HarperCollinsPublishers Ltd 2006

Chapter 16 • Linear graphs and equations • Check-in test

    1 What are the coordinates of the points shown on the grid?

     A =

     B =

     C =

     D =

2 Find the value of y when x = 3, using the rule y = 2x 2.

    3 Find the value of y when x = 3, using the rule y = 2x + 1.

    4 Plot the points in the table on the grid below. Join the points with a straight line.

    x 2 0 2 4

    y 2 1 0 1

5 Plot the points (4, 6), (1, 3), (0, 2), (2, 0) on the grid below. Join the points with a straight line.

     ? HarperCollinsPublishers Ltd 2006

    16.11.1 Linear graphsSolving real problems

    Incorporating exercises: 16A, 16B, 16C, Key words

    16D axis: (pl: axes) linear graphs

    coefficient

    Homework: 16.1 gradient-intercept

     Example: 16.1

    Learning objective(s)

     draw linear graphs without using flow diagrams Prior knowledge

    Pupils must be able to read and plot coordinates (even if given in table form). They must also be able to substitute

    into simple algebraic functions (Chapter 5, Section 5.1a). Starter

    Write a rule on the board, for example ;2 + 1, and ask for the results when this rule is applied to

    0, 1, 2, 3 and 4.

    Give other rules and find the results, for example ;4 5, ?2 3.

    Give other input numbers for these rules, for example 3, 2, 1.

    Main teaching points

    Pupils need to be able to input values given as both „–3 to +3‟ and „–3 ? x ? 3‟. After drawing a graph, always write

    its equation on one end of it. When an equation is arranged as y = something, for example, y = 3x + 2, the 3 is the steepness (gradient) of the line (1 unit across, 3 up), the +2 means the line will cut across the y-axis at +2. Common mistakes

    Working out two points and drawing the line not checking with a third point that the line is correct. Algebra skills

    not being adequate.

    Plenary

    Draw a set of axes on the board.

    Get a volunteer to draw the line with equation y = 2x + 1 (for example).

    If it looks correct, ask for the (0, ?) and (?, 0) coordinates (and other points if suitable). 1Repeat with other volunteers and other lines, for example y = 3x 2, y = 6x, y = x + 4. 2

    Go over the different methods of drawing linear graphs plotting points, gradient-intercept, and „cover-up‟.

     ? HarperCollinsPublishers Ltd 2006

    16.21.1 Finding the equation of a line from its graphSolving real problems

    Incorporating exercise: 16E Key words

    coefficient intercept

    Homework: 16.2 gradient

     Example: 16.2

    Learning objective(s)

     find the equation of a line using its gradient and intercept Prior knowledge

    Pupils should have understood and succeeded in Exercise 16C. Starter

    Draw a set of axes numbered from 5 to +5 on both axes.

     Draw the line y = 2x + 1 by plotting three points.

     Draw the line y = 2x 1 by using the gradient-intercept method. What would you do if asked to draw 4x + 5y = 20? (Use cover-up method and draw.) Main teaching points

    Tell pupils that for y = mx + c, the „m‟ is the gradient. For example, in y = 3x + 2, m = 3, so the gradient = 3; so for every 1 unit the line goes to the right, it goes 3 units up. The „c‟ is +2, so the intercept is 2; so the graph cuts the y-axis at +2.

    Common mistakes

    Missing the negative on the gradient, when the line is a negative gradient line.

    Differentiation

    Lower ability pupils find negative and fractional gradients much more of a challenge.

    Plenary

    Draw three or four sets of axes on the board and sketch a graph on each one. For example: 1y = 2x + 2, y = 2x 4, y = x + 2, y = x 2

    DO NOT tell pupils the equations of the lines.

    Ask the pupils to explain how to work out the equation of each line.

     ? HarperCollinsPublishers Ltd 2006

    16.31.1 Uses of graphsSolving real probl ems

    Incorporating exercises: 16F, 16G Key words

    formula (pl: formulae)

    Homework: 16.3 rule

     Examples: 16.3

    Learning objective(s)

     use straight-line graphs to find formulae

     solve simultaneous linear equations using graphs

    Prior knowledge

    Pupils need to be able to give the equation of a line using gradient and intercept or cover-up method, and be able to draw a line given its equation.

    Starter

    Draw a line (for example y = 2x 3) on the board, and ask the pupils to find the equation.

    Ask the pupils to draw a set of axes (numbered 0 to 10 on both), and get them to draw the lines y = 2x + 1 and x + y = 10 on the same axes.

    Ask the pupils to give the coordinates of the point of intersection of the two lines (answer should be (3, 7)). Use this to explain about solving simultaneous equations.

    Main teaching points

    Whenever possible, on conversion graph questions, draw the horizontal and vertical „tracking‟ lines on the graph for maximum accuracy. For solving simultaneous equations, drawing graphs accurately is the main concern. Common mistakes

    When answering conversion graph questions, not drawing the horizontal and vertical lines often means accuracy is lost using a finger is not good enough. When solving simultaneous equations questions, not taking time to ensure that the individual lines are drawn accurately.

    Plenary

    Draw this diagram on the board and show that 40x = 12y:

Ask, “What is y when x = 100?” (30)

    “What is x when y = 1200?” (4000)

     ? HarperCollinsPublishers Ltd 2006

    16.41.1 Parallel and perpendicular linesSolving real problems

    Incorporating exercise: 16H Key words

    negative reciprocal perpendicular

    Homework: 16.4 parallel

     Examples: 16.4

    Learning objective(s)

     draw linear graphs parallel or perpendicular to other lines and passing through a specific point

    Prior knowledge

    Pupils need to have been taught what a negative reciprocal is, and have had some practice at finding them. They

    must be familiar with gradient and intercept in y = mx + c, and they need to know what parallel and perpendicular lines are.

    Starter

    Ask the following questions:

     What does parallel mean?

     What does perpendicular mean?

     What do the m and c mean in y = mx + c?

    1 What is a reciprocal? What is the reciprocal of 2? () 2

    1111 What is a negative reciprocal? What is the negative reciprocal of 3, 2, , ? (,, 4, 2) 4232

    Main teaching points

    Parallel lines have the same gradient (m) but different intercepts (c). Perpendicular lines have gradients which are the negative reciprocals of each other.

    Common mistakes

    Forgetting the negative when finding the gradient of a perpendicular line, that is, only finding the reciprocal.

    Differentiation

    The difference between A and A* questions is usually that A grade lines go through a given intercept.

    Plenary

    Write this equation on the board:

    y = 2x + 3

    Ask the pupils:

    a What is the gradient of this line?

    b What is the gradient of a parallel line?

    c What is the gradient of a perpendicular line?

    11Ask the same questions for other lines such as: y = 2x + 3, y = x + 3, y = x 1 22

     ? HarperCollinsPublishers Ltd 2006

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