4.7 ALGEBRA
ALGEBRA LESSON IDEA 1
TITLE: SNOOKER BALLS AND VARIABLES!
TOPIC: ALGEBRA
AIM:
- To introduce students to the use of variables by means of a snooker analogy.
- To give the students confidence in using variables - algebra is not just for the good classes!
RESOURCES:
A set of snooker balls or different coloured discs or coloured circles drawn on the overhead or blackboard; a large sheet showing the values of the different coloured balls in snooker; prepared worksheets.
METHOD:
- Opening discussion about snooker.
- Get the students to give the value of different snooker balls. Use the large sheet showing the values of different coloured balls in snooker as an aid.
- Write up a pretend break on the board as follows:
Pretend break.
Ask students to work out the value of the break score.
More examples can be done and the scores found. - Students can be given a prepared worksheet (number 1) to complete using similar examples which can be corrected immediately.
- The next step is to ask students to write out the pretend break in words. The teacher should write clearly what each letter stands for:
R = Red G = Green U = Blue K = Black N = Brown
So, students are asked: How many of each colour are there?
Answer: 2 red and 2 green. (answer to be given in this way) - Students are then given another worksheet (number 2) with similar examples to complete.
- When the second worksheet is corrected an identical worksheet (number 3) is given out. On this occasion students are asked to write the answer as follows:
Answer: 2R + 2G - The same worksheet (number 4) is again given to the students. Now, students are asked to substitute in the value of each snooker ball and arrive at the total break:
Answer: 1 + 3 + 1 + 3 = 8 Example: 2R + 2G = 8 - A similar worksheet (number 5) is given to students to complete. On this occasion the variable expression is given under each example and students asked to substitute in the correct value and arrive at the score break.
The teacher shows an example first:
Answer: 2R + 2G = 2(1) + 2(3) = 2 + 6 = 8
The students are now substituting values for variables.
They can then progress to using x and y or other letters.
It is possible for the teachers to combine the last two worksheets into one if so desired.
CLASSROOM MANAGEMENT IMPLICATIONS:
Students can work individually or in pairs.
NOTE:
Strictly speaking, the letters in the above analogy stand for objects and not variables per se. However, teachers have found the lesson idea effective when dealing with students who find algebra difficult.
ALGEBRA LESSON IDEA 2
TITLE: SIMPLIFYING ALGEBRAIC EXPRESSIONS WITH LIVEMATH
TOPIC: ALGEBRA
AIM:
- To give students an opportunity of using a computer algebra software package.
- To show students how LiveMath can be used to simplify algebraic expressions.
RESOURCES:
Computers with LiveMath installed. A basic working knowledge of LiveMath on the part of the teacher.
METHOD:
Follow the sequence of steps below using LiveMath
Declarations
Algebraic expressions are entered as they are written in ordinary mathematics:
3x - 7y
To represent the symbol x multiplied by the symbol y, we must separate them by a space:
x y
To divide two things, we press the "forward slash", x/y, and LiveMath arranges the fraction correctly.
Note the difference between the following:
(x)/(y+1)
(x)/(y) + 1
Note: to leave the denominator, press the "Esc" key.
The first use of LiveMath in algebra is in simplifying algebraic expressions.
We have the choice of using "Simplify" from the "Manipulate" pull-down menu, or using the brush on the Palette:
5a+3b-4a+3b-5a+6b
5a+3b-4a+3b-5a+6b=-4a+12b Simplify
To highlight the expression, click on the "square"; then choose "Simplify".
When students are comfortable with typing in the expressions correctly and using the simplify feature they can be asked to simplify the following algebraic expressions:
Simplify: 5a + 3ab - 5b + 6ab - 4a + 7b
2x2 - 3x + 5x2 - 4x - 3x2 + 7x
(x2 + 5x + 4)/(x - 4)
CLASSROOM MANAGEMENT IMPLICATIONS:
Students may initially like to work in pairs and communicate using mathematical terms such as "variables" and "expressions".
ALGEBRA LESSON IDEA 3
TITLE: GUESS MY RULE
TOPIC: ELEMENTARY ALGEBRA
AIM: To develop the concept of a variable.
RESOURCES:
No special ones.
METHOD:
- The teacher thinks of a simple (verbal) rule such as "double and add 1", and challenges the students to guess what this rule is.
The teacher asks the students to think of numbers between, say, 0 and 20.
Then:
- a student provides a number
- the teacher gives the result of applying the rule; and this cycle is repeated several times.
At any stage, a student may offer a guess as to what the rule is. If the guess is wrong, then the student cannot guess again (or put forward another number) during this game.
Eventually - hopefully - the students guess the rule.
Some time is spent checking, to see that all students understand and can apply the rule. Note: at this stage, symbols should be avoided and the rule formulated in words (unless the students themselves suggest somesymbolic representation). - The game is played again, using a different rule (perhaps formulated by, or with input from, the student who identified the rule correctly in the first game). If the students have not already seen the need to record number pairs systematically, the teacher suggests the strategy and an appropriate format is agreed.
- The students form small groups and play the game. The one who guesses the rule formulates the next rule.
Meanwhile, the teacher circulates in order to arbitrate where necessary and to check that the rules are being applied correctly. The students have to write the rules in their own words and explain them to the teacher when s/he visits the group. - For homework, students are asked to try out the game at home, and to report any interesting incidents the next day. (On the next day, the teacher asks for feedback, for instance asking if anyone suggested a "short cut" - but not yet introducing symbolic notation unless the students suggest it.)
CLASSROOM MANAGEMENT IMPLICATIONS:
It is helpful if the students are used to working in groups, and ideally to negotiating and sorting out problems themselves before appealing to the teacher as "referee".
NOTE:
- The game can be played with various types of rule - say of type ax + b, a(x + b), x2, x2 + 1, x/2, and so forth - for some time. Eventually, when the students are tired of writing the rules in English, symbolic notation can be introduced, in stages, say:
2 × number + 1
and, after a while,
2 × N + 1
(or using some symbol of the students' choice in place of N, but eventually introducing a letter to stand for "any number").
The multiplication sign should be retained for some time (as in the revised Primary School Curriculum). - This introduces the idea of a variable - which can take many values - rather than an unknown (which has a specific value, for example as in "x + 7 = 10"). Starting with a variable may help to avoid problems which can arise with the similar game, (say) "my number plus 7 is 10, what is my number?" which can lead eventually to protests of form "but yesterday we decided x was 6...".
- The game can be used also to introduce the idea of a function.
ALGEBRA LESSON IDEA 4
TITLE: I HAVE AN EXPRESSION... AND YOU HAVE AN EXPRESSION
TOPIC: ALGEBRAIC SKILLS AND CONCEPTS
AIM: To give practice in manipulating algebraic expressions.
Note: this is the same game as that described in Number systems lesson idea 6, but with algebraic expressions taking the place of numbers.
RESOURCES:
A set of cards of the form:
- I have [an algebraic expression].
- Who has [some function of it]?
Examples might include:
- I have 2x + 3; who has 5 less than this?
- I have 4y - 3; who has this expression minus 2y?
- I have x2 + 2x + 1; who has the square root of this?
(the difficulty level being geared to the class).
See Number systems lesson idea 6 for details of the structure of the set of cards.
METHOD:
See Number systems lesson idea 6.
CLASSROOM MANAGEMENT IMPLICATIONS:
See Number systems lesson idea 6.
ALGEBRA LESSON IDEA 5
TITLE: PATTERNS AND FORMULAE
TOPIC: ELEMENTARY ALGEBRA
AIM: To create a need for learning to manipulate algebraic expressions
RESOURCES:
No special ones.
METHOD:
- The teacher draws or displays a diagram consisting of a row of squares such as that shown, and challenges the class: how many matches (or lollipop sticks, or other suitable "rods") are needed to make it?
- The students are asked to draw a similar diagram, but with more or fewer squares (still arranged in a line). How many matches are needed?
- The students are then challenged to find a relationship between the number of squares and the number of matches by looking at their own diagrams and perhaps the (probably different) diagrams of their neighbours. This relationship should be expressed as a verbal rule, for example "three for each square, plus one extra for the first square" or "twice the number of squares (for the top and bottom) plus one more than the number of squares (for the uprights)."
If different versions do not emerge spontaneously, the students can be challenged to find them. - The teacher can then ask: can we be sure that the different rules will always give the same answer? To find out, the verbal rules can be written as algebraic expressions. Examples may include:
3n + 1
2n + (n + 1)
4n - (n - 1) {The teacher might provide this one if the students do not do so.}
The teacher can work with the class in order to formulate the expressions correctly on the blackboard or OHP.
In order to reconcile the various formulae, the students then have to learn how to simplify the expressions, or have to apply what they have already begun to learn about algebraic manipulation. - As a follow-up activity, or in another lesson, the "Guess my rule" game (Algebra lesson idea 3) can be used in this context. For instance:
You say: 2 7 3 1 0
I say: 12 32 16 8 4
This can be interpreted as either 4(x + 1) or 4x + 4 - The teacher can provide a set of appropriate exercises, or students can then make up their own examples.
CLASSROOM MANAGEMENT IMPLICATIONS:
None