4. Linear Inequalities

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4. Linear Inequalities
  Mathematics Grade 0                                            Teacher Guide

  Linear Inequalities
     Series overview
In this series, we build an understanding of the steps involved in solving linear inequalities, especially
when multiplying or dividing by a negative amount. Then we solve a contextual problem. We represent
the solutions graphically on number lines and using set builder notation and interval notation.
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     Curriculum links
The lessons in this series link to the following Learning Outcomes and Assessment Standards of the

National Curriculum Statement:
Learning Outcome : Functions and Algebra
10.2.5(d) Solve linear inequalities in one variable and illustrate the solution graphically.

 E Educational approach

There are three lessons in this series. The first lesson focuses on different ways to represent linear
inequalities. The second lesson shows the principles and mathematical skills needed to solve linear
inequalities. We investigate how the inequality sign changes when both sides of the inequality are
multiplied or divided by negative real numbers. In the third lesson we recap what was learnt in the first
two lessons and then extend the learners’ knowledge by solving inequalities which contain fractions.
The series begins with examples of how inequalities are used in the workplace and in real-life
situations. In the past, many teachers have taught learners to solve inequalities as a skill where the
numbers and variables are manipulated according to rules which are followed. Learners find it difficult
to see how these inequalities are used in real life. So we have included many real world examples
so that learners can realise how useful and relevant inequalities are to solve real problems. In this
rich context, we introduce the different representations of inequalities, namely on number lines, in set
builder notation and interval notation.
An important educational approach is to provide the opportunity for learners to work together in
small groups where they can discuss the concept of linear inequalities and the principles of solving
linear inequalities and by doing this they construct their own understandings and meanings. The
mathematical language and terminology used to describe set builder notation and interval notation
can become a barrier to learning. This approach also ensures that the learners have enough
opportunities to verbalise and write the terminology required.
It is important to remember that mathematics involves conceptual understanding and the development
of mathematical skills. No skill is achieved from one example! The lessons include many examples
for learners to practise the necessary skills. We suggest that you use more examples from your
classroom textbook to consolidate this work.
The tasks set for each lesson can be used as an assessment tool. The learners can complete the
tasks and these can be used to ?
                                assess whether or not they have achieved the intended outcomes.
To get the full benefit of the lessons, your learners need to engage actively with the concepts
presented. So, when you preview the videos, think about how to introduce each lesson and what
                               E 2+2=4
follow up activities will be useful. Also watch out for places in the video where you can pause to have
a class discussion, or ask learners to complete an activity or solve a problem posed in the video. We
have used this pause icon          to suggest some of these places to you.

    ?Mathematics Grade 0                                    Teacher Guide

      Linear Inequalities

E   2+2=4

            Series at a Glance

      Lesson title                     Lesson Outcomes

                                       By the end of this lesson, the learner should be able to:

      . Linear Inequalities on the    •   use a number line to represent a linear inequality
         Number Line                   •   use set builder notation and interval notation to
                                           represent a linear inequality

      . Solving a Linear Inequality   •   solve a simple linear inequality
                                       •   solve a linear inequality that involves multiplying or
                                           dividing by a negative number
                                       •   calculate the values for a linear inequality

      . More Linear Inequalities      •   solve linear inequalities that have a denominator
                                       •   represent solutions on a number line.

      Mathematics Grade 0                                           Teacher Guide

      Linear Inequalities

E   2+2=4

     G Teaching Guidelines
    Lesson : Linear Inequalities on the Number Line
    The video introduces inequalities in a real world context. Discuss further examples with the learners
    of where and how inequalities can be used to solve problems in real-life situations. The lesson guides
    learners to understand the concept of an inequality as opposed to an equation and to write an inequality
    using the appropriate inequality signs.
    You may want to stop the video after the problem on the two mountain climbers climbing Mount Everest
    is presented. Let the learners work together in small groups to discuss the problem with each other and
    then write the inequality and represent it on a number line and in set builder and interval notation. They
    can then check their answers with those on the video.
    Give the learners the opportunity to practise drawing number lines of closed and open intervals where
    x∈ ;x∈        and x ∈ . Many learners are not sure why and when they should draw a dot or a line, an
    open dot at the end of an interval or a closed dot. Learners also forget when to use a round bracket and
    a square bracket when writing in interval notation. Lots of practice will help clear up any confusion.

    Using the principles established in this lesson, learners can represent the inequality in the three
    different ways. Let them do this individually, as you have given them the opportunity to work together
    during the lesson.

    Lesson : Solving a Linear Inequality
    This lesson carefully works through the concept of keeping an inequality true when we add or
    subtract, or multiply and divide both sides of the inequality by positive and negative real numbers.
    After Busi’s problem has been presented, stop the video and let the learners discuss with each other
    and use their common sense to solve the problem. You may want to guide them to write an inequality
    to solve the problem. They can then check their thinking and their answers from the video.
    The critical learning point in this lesson is the changing of the direction of the inequality sign when
    multiplying or dividing both sides of the inequality by a negative real number. Very few learners
    understand this concept. They simply learn this ‘rule’ and apply it as best as they can remember it.
    We lead the learners through an investigation of why the sign changes direction. Give the learners the
    opportunity to discuss this concept. Let them work in pairs and demonstrate their understanding of
    this to each other.
    There are two examples of inequalities to solve. You may want to stop the video and let the learners
    solve these in small groups and then check their answers. If the learners are very unsure, let them
    watch the first inequality being solved on the video. Then they could redo this example and then try
    the second one on their own. Give them more examples to practise.
    The learners are expected to solve inequalities which include changing the direction of the sign.
    Some learners may need more practice and further explanation to achieve the outcomes of this
    lesson confidently.

  Mathematics Grade 0                                                                      Teacher Guide

  Linear Inequalities
Lesson : More Linear Inequalities
In this lesson we recap the previous two lessons and then use clear steps to solve an inequality that
contains fractions. If your learners have never been to Gold Reef City, discuss this theme park with them
and the different rides they could have there. Also discuss why there is a maximum number of people
that can go on the Big Wheel at a time. This discussion can lead to other examples of maximum and
minimum amounts, which are the limits that can be applied to linear inequalities.
After the problem on the Big Wheel has been presented, stop the video and allow the learners to work in
groups and solve the problem on their own first. The second problem, where the learners must calculate
the maximum total mass of the loaded Big Wheel, is an extension activity which the competent and
confident learners will enjoy. You can pair up confident learners with less confident ones.
The learners need to use their knowledge and understanding of all three lessons to complete this
task. The rubric below can be used to assess the learners’ ability to solve linear inequalities.

  Name of Learner:

  Criteria :                                                          4                                      6                 7
  The learner      Not Achieved      Elementary       Moderate           Adequate            Substantial         Excellent         Outstanding
  can:                               achievement      achievement        achievement         achievement         Achievement       achievement

  Use a            Number line       Number line      Number line        Number line         Number line         Number line       Number line
  number line      incorrect;        correctly        correctly          correctly           correctly           correctly         correctly drawn,
  to represent     interval          drawn;           drawn; interval    drawn; interval     drawn; interval     drawn; interval   interval marked
  a linear         marked            interval         marked             marked              marked              marked            correctly,
  inequality.      incorrectly;      marked           correctly; open/   correctly;          correctly; open/    correctly;        open/closed dot
                   open/closed       incorrectly;     closed dot not     open/closed         closed dot          open/closed       correct, sure of
                   dot not           open/closed      correct and/or     dot correct         correct; sure of    dot correct;      real or natural
                   correct;          dot not          unsure of real     but unsure of       real or natural     sure of real      numbers and
                   unsure of real    correct and/or   or natural         real or natural     numbers.            or natural        can explain
                   or natural        unsure of real   numbers.           numbers.                                numbers.          principles to
                   numbers.          or natural                                                                                    peers.

  Use set          All               Most             Some               One                 Most                All               All components
  builder          components        components       components         component           components          components        of notation
  notation         of notation       of notation      of notation        of notation         of notation         of notation       correctly written;
  and interval     incorrectly       incorrectly      incorrectly        incorrectly         correctly           correctly         uses correct
  notation.        written.          written.         written.           written.            written.            written.          terminology.

  Solve a linear   Does not          Mostly does      Sometimes          Mostly              Always              Always            Always changes
  inequality       change the        not change       does not           changes the         changes             changes the       the direction of
  that involves    direction of      the direction    change the         direction of the    the direction       direction of      the sign and
  multiplying or   the sign and      of the sign      direction of       sign but cannot     of the sign,        the sign and      clearly explains
  dividing by      cannot solve      and cannot       the sign and       explain why         but cannot          can give some     why this
  a negative       the inequality.   explain why      cannot explain     this happens.       give a clear        explanation       happens with
  numberr.                           this happens.    why this                               explanation         of why this       examples
                                                      happens.                               of why this         happens.

  Solve a linear   Cannot identify   Sometimes        Mostly             Identifies          Identifies          Multiplies        Multiplies
  inequality       common            identifies       identifies         common              common              by common         by common
  that has a       denominator;      common           common             denominator,        denominator         denominator       denominator and
  denominator.     cannot solve      denominator      denominator        sometimes           and usually         and solves        solves the linear
                   the linear        but cannot       but cannot         solves              solves the          the linear        inequality with
                   inequality.       solve the        solve the linear   the linear          linear inequality   inequality        confidence and
                                     linear           inequality.        inequality.         correctly.          correctly.        ease.
                                     inequality .