- Lew W.S.
This story board is used to guide the design and development of the Basic Inequalities package.
We start by discussing about comparison of quantities and values. Two examples (volume and number of items) Leading on to " less than' or "more than" phrases
Next, using some illustrated real life examples, we talk about less than, less than or equal to, more than and more than or equal to, and show the symbols used to represent the comparison of values and quantities Less than < Less than or equal to More than > More than or equal to
Quiz 1 Short interactive assessment on use of correct symbols for each type of basic inequality
Next, we discuss about situations where we want to describe more than one possible answer to a given problem. We start with simple cases. For example, one has a given amount of money to buy some item. The item can cost any value less than or equal to amount one has. But it must not cost more money than what one has. There are many possible values of the cost of the item which one may be willing to pay for and buy the item, but it must not exceed what one has. If the price is low, a person can possibly buy one or more units. Second, consider an overhead gantry with a height limit of 2.1 m. This means that the height of a vehicle, in metres, represented by the letter h must be less than 2.1 m in order to pass through the gantry. If the height is more than 2.1 m, the top of the vehicle will hit the gantry and be damaged. Even if h is exactly 2.1 m, we can expect the gantry to scratch the surface of the top of the vehicle. So h must be less than 2.1 m ! Lastly consider a given size of a container (length L) and how many spherical balls can be placed in. How many balls (n) depends on the diameter (d) of each ball. ie n x d <= L, but n has to be integer values.
We then illustrate how to represent the basic inequalities on the number line Less than < Less than or equal to More than > More than or equal to
Quiz 2 Short interactive assessment on representing basic inequalities on a number line
Now that the learner has grasped how to represent the basic inequalities, we illustrate with numerical examples, the effects of multiplying or dividing by a positive or negative number on both sides of an inequality as in the following cases, For n < k, (where, k and n are rational numbers) Multiplying by positive a (a > 0) we get an < ka Dividing by positive a (a > 0) we get n/a < k/a Multiplying by negative a (a < 0) we get an > ka (inequality sign changed) Dividing by negative a (a < 0) we get n/a > k/a (inequality sign changed) For n > k, (where, k and n are rational numbers) we show only the examples Multiplying by negative a (a < 0) we get an < ka (inequality sign changed) Dividing by negative a (a < 0) we get n/a < k/a (inequality sign changed), while the positive a (a>0) cases for multiplication and division can be reasoned by considering that n>k => k< n and there would be no change in sign and hence after multiplication or division k< n is still n > k. We also briefly mention that we can extend this "change direction of inequality sign when multiplying both sides by negative numbers" rule to the less than or equal to and more than or equal to
We move on from basic inequalities using integers, and the effect of multiplying or dividing by positive or negative numbers on the inequality sign (in the previous section), to making sense of , where a is positive (a>0) and how it can be reduced to We introduce the term solving an inequality as the reduction of to Borrowing the analogy of solving equations by balancing through performing arithmetic operations on both sides equally, we develop the idea of solving inequalities (another term, inequation) by keeping the unbalanced status while performing arithmetic operations on both sides equally. Then the representation of the solution on a number line will be shown.
We continue in the same approach as in previous segment to solve inequalities for ; and where a is positive (a>0) and show how we can reduce it to , and respectively.
Quiz 3 Short interactive assessment on solving ; and where a is positive (a>0) , including representing solution on a number line
We repeat the same explanations for the previous segment to solve inequalities for , ; and where now a is negative (a<0) Again we will show the solutions, ie the basic inequalities on the number line.
* Short interactive assessment on solving ; and where a is negative (a<0) including representing solution on a number line
We conclude the package with a summary, and mini assessment covering cases where a is positive or negative for all the four types of inequality signs
Link to Draft Package : https://tube.geogebra.org/material/show/id/uMTgeOlc