REVIEWING THE MODELS FOR SOLVING EQUATIONS Robert Yen Hurlstone Agricultural High School.

REVIEWING THE REVIEWING THE MODELS FORMODELS FOR

SOLVING EQUATIONSSOLVING EQUATIONS

Robert YenRobert YenHurlstone Agricultural High SchoolHurlstone Agricultural High School

OVERVIEWOVERVIEW

To review and compare 3 models for To review and compare 3 models for teaching equationsteaching equations

Students often have trouble solving Students often have trouble solving equations because their teachers equations because their teachers teach only one method, the method teach only one method, the method they were taught themselvesthey were taught themselves

OVERVIEWOVERVIEW

To discuss and share ideas and To discuss and share ideas and classroom experiences on teaching classroom experiences on teaching equationsequations

Model 1:Model 1:GUESS, CHECK AND IMPROVEGUESS, CHECK AND IMPROVE

Alternative namesAlternative names Guess and checkGuess and check Guess, check and refineGuess, check and refine Trial and errorTrial and error ‘‘By inspection’By inspection’


DescriptionDescription The name explains the method: The name explains the method:

guess the solution, test it, make a guess the solution, test it, make a better guess, keep testingbetter guess, keep testing

It is a process that cycles, is It is a process that cycles, is repetitiverepetitive


Example 1Example 1

xx + 5 = 40 + 5 = 40

By inspection, By inspection, xx = 35 = 35

because 35 + 5 = 40.because 35 + 5 = 40.


Example 2Example 2

= 10= 10

By inspection, By inspection, dd = 30 = 30

because = 10.because = 10.

3d

330


Example 3Example 3

44xx + 5 = 57 + 5 = 57GuessGuess CheckCheck CommentComment

xx = 10 = 10


Example 3Example 344xx + 5 = 57 + 5 = 57

GuessGuess CheckCheck CommentComment

xx = 10 = 10 4 4 1010 + 5 = + 5 = 4545

Too lowToo low

xx = 14 = 14




xx = 10 = 10 4 4 1010 + 5 = + 5 = 4545

Too lowToo low

xx = 14 = 14 4 4 1414 + 5 = + 5 = 6161

Too highToo high

xx = 12 = 12




xx = 10 = 10 4 4 1010 + 5 = + 5 = 4545

Too lowToo low

xx = 14 = 14 4 4 1414 + 5 = + 5 = 6161

Too highToo high

xx = 12 = 12 4 4 1212 + 5 = + 5 = 5353

Too lowToo low

x x = 13= 13


Example 3Example 3

44xx + 5 = 57 + 5 = 57GuessGuess CheckCheck CommentComment

xx = 10 = 10 4 4 1010 + 5 = + 5 = 4545

Too lowToo low

xx = 14 = 14 4 4 1414 + 5 = + 5 = 6161

Too highToo high

xx = 12 = 12 4 4 1212 + 5 = + 5 = 5353

Too lowToo low

x x = 13= 13 4 4 1313 + 5 = + 5 = 5757

Correct!Correct!



xx = 13. = 13.


xx = 10 = 10 4 4 1010 + 5 = + 5 = 4545

Too lowToo low

xx = 14 = 14 4 4 1414 + 5 = + 5 = 6161

Too highToo high

xx = 12 = 12 4 4 1212 + 5 = + 5 = 5353

Too lowToo low

x x = 13= 13 4 4 1313 + 5 = + 5 = 5757

Correct!Correct!


Example 4Example 4

33xx + 4 = + 4 = xx – 6 – 6GuesGues

ssLHSLHS RHSRHS

x x = 10= 10 3434 44

xx = 5 = 5


Example 4Example 433xx + 4 = + 4 = xx – 6 – 6

GuesGuesss

LHSLHS RHSRHS

xx = 10 = 10 3434 44

xx = 5 = 5 1919 -1-1

xx = 1 = 1



GuesGuesss

LHSLHS RHSRHS

xx = 10 = 10 3434 44

xx = 5 = 5 1919 -1-1

xx = 1 = 1 77 -5-5



GuesGuesss

LHSLHS RHSRHS LHS-RHSLHS-RHS

xx = 10 = 10 3434 44 3030

xx = 5 = 5 1919 -1-1 2020

xx = 1 = 1 77 -5-5 1212

xx = -2 = -2



GuesGuesss


xx = 10 = 10 3434 44 3030

xx = 5 = 5 1919 -1-1 2020

xx = 1 = 1 77 -5-5 1212

xx = -2 = -2 -2-2 -8-8 66

xx = -4 = -4



GuesGuesss


xx = 10 = 10 3434 44 3030

xx = 5 = 5 1919 -1-1 2020

xx = 1 = 1 77 -5-5 1212

xx = -2 = -2 -2-2 -8-8 66

xx = -4 = -4 -8-8 -10-10 22

xx = -5 = -5



GuesGuesss


xx = 10 = 10 3434 44 3030

xx = 5 = 5 1919 -1-1 2020

xx = 1 = 1 77 -5-5 1212

xx = -2 = -2 -2-2 -8-8 66

xx = -4 = -4 -8-8 -10-10 22

xx = -5 = -5 -11-11 -11-11 00



xx = -5 = -5

GuessGuess LHSLHS RHSRHS LHS-RHSLHS-RHS

xx = 10 = 10 3434 44 3030

xx = 5 = 5 1919 -1-1 2020

x x = 1= 1 77 -5-5 1212

xx = -2 = -2 -2-2 -8-8 66

xx = -4 = -4 -8-8 -10-10 22

xx = -5 = -5 -11-11 -11-11 00


What the syllabus says What the syllabus says (p.86, PAS4.4)(p.86, PAS4.4)

‘‘Five models have been proposed to Five models have been proposed to assist students with the solving of assist students with the solving of simple equations ... simple equations ... Model 4Model 4 uses a uses a substitution approach. By trial and substitution approach. By trial and error a value is found that produces error a value is found that produces equality for the values on either side equality for the values on either side of the equation (this highlights the of the equation (this highlights the variable concept).’variable concept).’


AdvantagesAdvantages ??????


AdvantagesAdvantages Reinforces aim of solving equations and Reinforces aim of solving equations and

algebraic concepts of algebraic concepts of unknown, variableunknown, variable Reinforces checking of solutionsReinforces checking of solutions Simple to understand and applySimple to understand and apply Feedback on partial solutions, homing Feedback on partial solutions, homing

in on answer, unlike algebraic methods in on answer, unlike algebraic methods where one careless error will undermine where one careless error will undermine the solution processthe solution process


AdvantagesAdvantages Being repetitive, can be performed Being repetitive, can be performed

via technology: spreadsheet, via technology: spreadsheet, graphics calculatorgraphics calculator

With better guesses, solution can With better guesses, solution can be found quicklybe found quickly

Can improve students’ computation Can improve students’ computation skills and number senseskills and number sense


DisadvantagesDisadvantages ??????


DisadvantagesDisadvantages Guesswork is not an ‘elegant’ Guesswork is not an ‘elegant’

methodmethod Harder to apply for more complex Harder to apply for more complex

equations (such as equations (such as xx on both sides) on both sides) May be hard to test values that are May be hard to test values that are

large or negativelarge or negative More time-consuming if guesses are More time-consuming if guesses are

badbad

Model 2: BALANCINGModel 2: BALANCING

Alternative nameAlternative name ‘‘Doing the same thing to both Doing the same thing to both

sides’ (of the equation)sides’ (of the equation)


DescriptionDescription The traditional algebraic methodThe traditional algebraic method Models the equation as balance Models the equation as balance

scales, upon which the same scales, upon which the same inverse operations are performed inverse operations are performed on both sides to create equivalent on both sides to create equivalent equations until it simplifies to ‘equations until it simplifies to ‘xx = = ___’___’


DescriptionDescription Invented by Arab mathematician Invented by Arab mathematician

al-Khwarizmi al-Khwarizmi in AD 825, who wrote in AD 825, who wrote

Hisab al-jabr w’al-muqabalahHisab al-jabr w’al-muqabalah‘‘The science of restoration and cancellation’The science of restoration and cancellation’

al-jabral-jabr = restoration by balancing,= restoration by balancing,

from which we get the name ‘algebra’from which we get the name ‘algebra’ muqabalahmuqabalah = cancellation of terms= cancellation of terms


DescriptionDescription From al-Khwarizmi’s name, we get the From al-Khwarizmi’s name, we get the

name ‘algorithm’name ‘algorithm’ Can be demonstrated using concrete Can be demonstrated using concrete

objects such as cups, envelopes, objects such as cups, envelopes, counters, coins or coloured dotscounters, coins or coloured dots

Or coloured plastic bottle caps Or coloured plastic bottle caps (see (see session by Kevin Fuller tomorrow @ 2 pm)session by Kevin Fuller tomorrow @ 2 pm)


Example 1 (A concrete model)Example 1 (A concrete model)

22xx + 7 = 9 + 7 = 9


Example 1 (A concrete model)Example 1 (A concrete model)

22xx + 7 = 9 + 7 = 9

Subtract 7 coins from both sidesSubtract 7 coins from both sides


Example 1Example 1

22xx + 7 = 9 + 7 = 9 Place the remaining

coins into two equal rows


Example 1Example 1

22xx + 7 = 9 + 7 = 9

Divide both sides by 2Divide both sides by 2

Place the remaining

coins into two equal rows


Example 1Example 1

22xx + 7 = 9 + 7 = 9

xx = 1 = 1

[[Check:Check: 2(1) + 7 = 9] 2(1) + 7 = 9]


Example 1 algebraicallyExample 1 algebraically

22xx + 7 = 9 + 7 = 9

22xx + 7 + 7 – 7– 7 = 9 = 9 – 7– 7

22xx = 2 = 2 22xx//22 = = 22//22

xx = 1 = 1


Example 2 (Another concrete Example 2 (Another concrete model)model)

33xx + 2 = + 2 = x x + 10+ 10


Example 2 (Another concrete Example 2 (Another concrete model)model)

33xx + 2 = + 2 = x x + 10+ 10

SubtractSubtract x x from both sidesfrom both sides


Example 2Example 2

33xx + 2 = + 2 = x x + 10+ 10


Example 2Example 2

33xx + 2 = + 2 = x x + 10+ 10

SubtractSubtract 22 from both sidesfrom both sides


Example 2Example 2

33xx + 2 = + 2 = x x + 10+ 10


Example 2Example 2

33xx + 2 = + 2 = x x + 10+ 10

xx = 4 = 4

[[Check:Check: 3(4) + 2 = 14 3(4) + 2 = 14

4 + 10 = 14]4 + 10 = 14]



33xx + 2 = + 2 = x x + 10+ 10

33xx + 2 + 2 – – xx = = xx + 10 + 10 – – xx

22xx + 2 = 10 + 2 = 10

22xx + 2 + 2 – 2– 2 = 10 = 10 – 2– 2

22xx = 8 = 8

22xx//22 = = 88//22

xx = 4 = 4


Note that the appropriateinverse operations must be identifiedand performed in the correct order.

Aim to have x on its ownon the LHS of the equation:

‘x = ___’


What the syllabus says (p.86, PAS4.4)What the syllabus says (p.86, PAS4.4)

‘ ‘Model 1Model 1 uses a two-pan balance uses a two-pan balance and objects such as coins or and objects such as coins or centicubes. A light paper wrapping centicubes. A light paper wrapping can hide a ‘mystery number’ of can hide a ‘mystery number’ of objects without distorting the objects without distorting the balance’s message of equality.’balance’s message of equality.’

Model 2: BALANCINGModel 2: BALANCINGWhat the syllabus says (p.86, PAS4.4)What the syllabus says (p.86, PAS4.4)

‘ ‘Model 2Model 2 uses small objects (all the uses small objects (all the same) with some hidden in containers same) with some hidden in containers to produce the ‘unknowns’ or ‘mystery to produce the ‘unknowns’ or ‘mystery numbers’, eg place the same number of numbers’, eg place the same number of small objects in a number of paper cups small objects in a number of paper cups and cover them with another cup. Form and cover them with another cup. Form an equation using the cups and then an equation using the cups and then remove objects in equal amounts from remove objects in equal amounts from each side of a marked equals sign.’each side of a marked equals sign.’



‘ ‘Model 3 uses one-to-one matching of terms on each side of the equation, eg

3x + 1 = 2x + 3 x + x + x + 1 = x + x + 2 + 1

By one-to-one matching and cancelling:




3x + 1 = 2x + 3 x + x + x + 1 = x + x + 2 + 1

By one-to-one matching and cancelling: x + x + x + 1 = x + x + 2 + 1




3x + 1 = 2x + 3 x + x + x + 1 = x + x + 2 + 1

By one-to-one matching and cancelling:

x = 2.’


AdvantagesAdvantages Powerful and elegant logical methodPowerful and elegant logical method Works for all types of equations, Works for all types of equations,

including those with including those with xx on both sides on both sides If done correctly, solution emerges If done correctly, solution emerges

quicklyquickly Reinforces algebraic concepts of Reinforces algebraic concepts of

balancebalance and and equivalenceequivalence of expressions of expressions (by cancelling and simplifying)(by cancelling and simplifying)


DisdvantagesDisdvantages ??????


DisdvantagesDisdvantages Harder to model 6 – 2Harder to model 6 – 2xx = =

14, 14,

x x 22 = 10 with concrete objects: how = 10 with concrete objects: how do you represent subtraction, do you represent subtraction, division or squaring of objects?division or squaring of objects?

Difficult for some students to Difficult for some students to understand, conceptualise, reasonunderstand, conceptualise, reason

,25

3

y


DisdvantagesDisdvantages Some students have trouble knowing Some students have trouble knowing

which inverse operation to perform which inverse operation to perform firstfirst

Lines of working can appear Lines of working can appear complicated and messycomplicated and messy

Students often don’t know what they Students often don’t know what they are actually doing or why they are are actually doing or why they are doing itdoing it

3

d


Example 3 (a model for Example 3 (a model for negatives)negatives)

6 – 26 – 2xx = 14 = 14

SubtractSubtract 66 from both sidesfrom both sides

6 – 26 – 2xx – 6– 6 = 14 = 14 – 6– 6

3

d


Example 3Example 3

6 – 26 – 2xx = 14 = 14

-2-2xx = 8 = 8


3

d

28

22

x


Example 3Example 3

6 – 26 – 2xx = 14 = 14

--xx = 4 = 4

Take the negative of both sidesTake the negative of both sides

3

d


Example 3Example 3

6 – 26 – 2xx = 14 = 14

xx = -4 = -4

[[Check:Check: 6 – 2(-4) = 14] 6 – 2(-4) = 14]

Model 3: BACKTRACKINGModel 3: BACKTRACKING

Alternative namesAlternative names Un-doing or unpackingUn-doing or unpacking Reverse flowchartReverse flowchart


DescriptionDescription An alternative algebraic modelAn alternative algebraic model To undo the operations performed To undo the operations performed

on the variable on one side of an on the variable on one side of an equation, inverse operations are equation, inverse operations are performed on the performed on the otherother side of the side of the equationequation


DescriptionDescription Can be demonstrated using a Can be demonstrated using a

flowchart and reverse flowchartflowchart and reverse flowchart Models the equation as a sequence Models the equation as a sequence

of operations on a variable that are of operations on a variable that are ‘undone’ by a sequence of inverse ‘undone’ by a sequence of inverse operations that ‘backtrack’ to the operations that ‘backtrack’ to the variablevariable


Example 1Example 1

22x x + 7 = 9+ 7 = 9

Use a flowchart to go from Use a flowchart to go from xx to 2 to 2x x + + 77


Example 1Example 1

22x x + 7 = 9+ 7 = 9

But 2But 2x x + 7 = 9+ 7 = 9

==


Example 1Example 1

22x x + 7 = 9+ 7 = 9

To get back to To get back to xx, undo the operations, undo the operations

using a reverse flowchartusing a reverse flowchart

==

To undo To undo + 7, + 7,

we – 7we – 7


Example 1Example 1

22x x + 7 = 9+ 7 = 9



= == =

To undo To undo

2, 2, we we 2 2


Example 1Example 1

22x x + 7 = 9+ 7 = 9



= == =

xx = 1 = 1



22xx + 7 = 9 + 7 = 9

22xx = 9 = 9 – 7– 7

= 2= 2

x x = = 22//22

x x = 1= 1

With backtracking, With backtracking, we (inverse) we (inverse)

operate on operate on one one sideside only only


Inverse operations areInverse operations areperformed in performed in reverse reverse order.order.

For example,For example,to undo putting on our to undo putting on our socks,socks,

then our then our shoes,shoes,we take off our we take off our shoesshoes first, first,

then take off our then take off our socks.socks.


Example 2Example 2

Use a flowchart to go from Use a flowchart to go from yy to to

25

3

y

25

3

y


Example 2Example 2

Use a flowchart to go from Use a flowchart to go from yy to to

==

25

3

y

25

3

y


Example 2Example 2

Use a reverse flowchart to backtrack Use a reverse flowchart to backtrack to to yy

==

25

3

y

To undo To undo 5, 5,

we we 5 5


Example 2Example 2

Use a reverse flowchart to backtrack Use a reverse flowchart to backtrack to to yy

= == =

25

3

y

To undo To undo + 3, + 3,

we – 3we – 3


Example 2Example 2

Use a reverse flowchart to backtrack to Use a reverse flowchart to backtrack to yy

= == =

yy = 7 = 7

25

3

y



y y + 3 = 2 + 3 = 2 5 5

= 10= 10

y y = 10 = 10 – 3– 3

y y = 7= 7

25

3

y



‘ ‘Model 5 Model 5 uses backtracking or a uses backtracking or a reverse flow chart to unpack the reverse flow chart to unpack the operations and find the solution.’operations and find the solution.’


AdvantagesAdvantages Only un-doing Only un-doing one one side of equation: side of equation:

less workingless working For some students, this method is For some students, this method is

more intuitive: it’s what we do when more intuitive: it’s what we do when we solve an equation mentallywe solve an equation mentally

Conceptually easier to understand Conceptually easier to understand than balancingthan balancing


AdvantagesAdvantages Algebraic working consistent with Algebraic working consistent with

balancing methodbalancing method If done correctly, solution emerges If done correctly, solution emerges

quicklyquickly Reinforces skills in algebraic Reinforces skills in algebraic

notation and generalising formulasnotation and generalising formulas


DisdvantagesDisdvantages ??????


DisdvantagesDisdvantages Takes longer to teach as it requires Takes longer to teach as it requires

careful practice with flowchartscareful practice with flowcharts Does not work if Does not work if xx on both sides of on both sides of

equation, eg 3equation, eg 3xx + 2 = + 2 = xx + 10 + 10 Harder to model if Harder to model if xx is not in the is not in the

first term, eg 6 – 2first term, eg 6 – 2xx = 14 = 14

So which is the best method to use?So which is the best method to use?

So which is the best method to use?So which is the best method to use?

All three methods have merit and All three methods have merit and can be used together in the can be used together in the classroomclassroom

Depends on the classDepends on the class Guess, check and improveGuess, check and improve is is

good for starting the topic, and good for starting the topic, and leads to the idea of inverse leads to the idea of inverse operations and algebraic methodsoperations and algebraic methods

BacktrackingBacktracking is a useful tool for is a useful tool for students who struggle with algebra, students who struggle with algebra, again convenient for introducing again convenient for introducing inverse operationsinverse operations

Once students are confident with Once students are confident with the algebra, introduce harder the algebra, introduce harder equations that require equations that require balancingbalancing to simplify the equation first, for to simplify the equation first, for example, example,

xx on both sides on both sides x x is not in the first termis not in the first term equations with bracketsequations with brackets

x x on both sideson both sides

33xx + 2 = + 2 = xx + 10 + 10


33xx + 2 = + 2 = xx + 10 + 10

33xx + 2 + 2 – – xx = 10 = 10

Use an inverse operation to remove the x from the RHS


33xx + 2 = + 2 = xx + 10 + 10

33xx + 2 + 2 – – xx = 10 = 10

22xx + 2 = 10 + 2 = 10 Simplifies to a2-step

equation: proceed by

backtracking or balancing


33xx + 2 = + 2 = xx + 10 + 10

33xx + 2 + 2 – – xx = 10 = 10

22xx + 2 = 10 + 2 = 10

22xx = 10 = 10 – 2– 2


33xx + 2 = + 2 = xx + 10 + 10

33xx + 2 + 2 – – xx = 10 = 10

22xx + 2 = 10 + 2 = 10

22xx = 10 = 10 – 2– 2

= 8= 8


33xx + 2 = + 2 = xx + 10 + 10

33xx + 2 + 2 – – xx = 10 = 10

22xx + 2 = 10 + 2 = 10

22xx = 10 = 10 – 2– 2

= 8= 8

28

x


33xx + 2 = + 2 = xx + 10 + 10

33xx + 2 + 2 – – xx = 10 = 10

22xx + 2 = 10 + 2 = 10

22xx = 10 = 10 – 2– 2

= 8= 8

x x = 4= 4

28

x

x x is not in the first termis not in the first term

6 – 26 – 2xx = 14 = 14

3

d


6 – 26 – 2xx = 14 = 14

-2-2xx + 6 = 14 + 6 = 143

d

If backtracking, rewrite so that x

is in the first term


6 – 26 – 2xx = 14 = 14

-2-2xx + 6 = 14 + 6 = 14

-2-2xx = 14 = 14 – 6– 6

3

d

Proceed by Proceed by backtracking or backtracking or

balancingbalancing


6 – 26 – 2xx = 14 = 14

-2-2xx + 6 = 14 + 6 = 14

-2-2xx = 14 = 14 – 6– 6

= 8 = 8

3

d

Proceed by Proceed by backtracking or backtracking or

balancingbalancing


6 – 26 – 2xx = 14 = 14

-2-2xx + 6 = 14 + 6 = 14

-2-2xx = 14 = 14 – 6– 6

= 8 = 8

2

8

x


6 – 26 – 2xx = 14 = 14

-2-2xx + 6 = 14 + 6 = 14

-2-2xx = 14 = 14 – 6– 6

= 8 = 8

x x = -4= -4

28

x

REVIEWING THE MODELS FOR SOLVING EQUATIONS Robert Yen Hurlstone Agricultural High School.

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Transcript of REVIEWING THE MODELS FOR SOLVING EQUATIONS Robert Yen Hurlstone Agricultural High School.