Building the foundations

The­first­foundation­is­to­understand­why­a­fraction­is­written­the­way­it­is,­that­is­two­

numbers,­one­on­top­of­the­other­and­separated­by­a­line.­A­fraction­could­be­visualised­as­

incorporating­a­hidden­division­sign­(Figure­10.3).

Figure 10.3­The­hidden­division­sign

The golden rule of adding or subtracting fractions is that you can only start the addition or­subtraction­process­when­the­fractions­have­been­adjusted­to­have­the­same­name­(or­

denominator­or­bottom­number).­So,­for­example­ cannot be computated until the is renamed to be then

So­students­need­to­be­able­to­rename­fractions,­understand­what­this­means­and­why­we­

do it. This is the second foundation.

Renaming­implies­that­there­already­is­a­name.­The­name­of­ ­is­‘a­half’.­The­name­of­ is

‘a­seventh’.­The­name­of­ ­is­‘fifth’­and­there­are­two­of­them,­hence­‘two­fifths’.­So­the­name­

comes­from­the­denominator,­the­bottom­number,­the­number­at­the­bottom­of­the­fraction,­the­

number­below­the­dividing­line.

As­much­time­as­is­needed­should­be­spent­on­developing­the­concept­and­skill­of­renaming.­

Renaming­does­what­it­says­it­does,­it­takes­a­fraction,­for­example­one­fifth,­ ­­and­gives­it­a­

new­name,­for­example­two­tenths,­ ­It­does­not­give­it­a­new­value.­The­new-named­fraction­

must­be­an­equivalent,­same­value,­fraction.­It­remains­the­same­value­because­it­is­multiplied­

by­another­fraction­whose­value­is­one­(1),­for­example­ ­Examples­can­be­taken­from­every-day­experiences­such­as­half­an­hour­as­ ­half­a­pound­(£)­as­ The renaming­fraction­always­has­the­same­number­as­numerator­and­denominator­(top­number­and­

bottom­number)­because­it­has­a­value­of­1.

Tactile­materials­such­as­Cuisenaire­rods,­poker­chips­or­stacker­counters­(from­Crossbow­

Games,­see­Appendix­p.­152)­are­good­to­show­the­equivalence­of­fractions.­Folding­squares­

or circles of paper can also illustrate the concept. The written symbols should always be shown alongside­these­concrete­experiences.

Maybe­it­is­worth­using­colours,­one­colour­for­the­numerator­and­a­different­one­for­the­

denominator­just­to­add­focus­to­the­fraction­as­having­two­number­components.­Or­writing­

a­big­division­sign­÷­on­the­board­to­remind­learners­that­a­fraction­is­a­number­divided­by­

another number.

A­problem­could­occur­if­learners­have­automatic­recall­of­only­a­few­number­facts.­This­

will­handicap­the­extent­of­their­ability­to­rename­fractions,­so­this­process­will­require­a­

lot­of­carefully­structured­practice,­with­the­focus­on­the­process­rather­than­on­knowing­

all the basic facts.

A­known­or­at­least,­a­familiar­example­should­be­used­as­a­first­model,­such­as:

♦­ ­to­be­renamed­to­have­the­same­denominator­as­

♦­ ­or­to­be­renamed­to­have­the­same­denominator­as­ ­(using­the­familiar­model­of­a­

clock­again).

The­overview/­start­up­questions­which­should­be­asked­are:

♦­Do­both­fractions­have­to­be­renamed?­Not­if­one­denominator­is­a­multiple­of­the­other­

denominator. Four is a multiple of 2 so only the half has to be renamed.

♦­What­number­is­used­to­change­the­chosen­fraction,­the­half?

This­number­will­be­found­by­dividing­the­larger­denominator­(4)­by­the­smaller­denominator­

(2)­which­should­give­a­whole­number­(2)­and­thereby­avoiding­creating­a­fraction­within­a­

fraction!­In­this­example­the­renaming­factor­is­therefore­2.

The top and bottom­ numbers­ of­ the­ fraction­ which­ has­ to­ be­ renamed­ have­ to­ be­

­multiplied­by­this­factor.­In­this­example

Thus­the­fraction­remains­the­same­value,­it­is­still­a­half,­but­is­renamed­from­being­called­

one half to being called two quarters.

If­ both­ fractions­ have­ to­ be­ renamed,­ for­ example­ with­ ­ to­ have­ a­ common­

(meaning­the­same)­denominator­(meaning­bottom­number).

♦­­ Both­fractions­have­to­be­renamed.

♦­­ The­simplest,­but­not­necessarily­the­most­numerically­elegant,­is­to­take­the­two­

denominators­(bottom­numbers)­as­factors­of­the­new­denominator­and­multiply­them.­

So­the­new­denominator­becomes­3×4­and­4×3,­that­is­12.

♦­­ For­renaming,­the­numerator­(top­number)­and­denominator­both­have­to­be­

multiplied.

So­is­ multiplied by ­to­give­

and is multiplied by ­to­give­

Since­both­fractions­are­now­renamed­and­written­as­twelfths­they­can­now­be­added:

The answer ­is­less­than­1­and­a­sketch­or­estimate­will­show­this­to­be­as­expected.

A circle picture as in Figure­10.4 can be used for estimates and appraisals of fraction sums.­A­clock­is­a­good­model­(another­reason­to­use­an­analogue­watch)­for­

and­for­inter-relating­these­fractions­(Figure­10.5)

The­process­of­‘mutual’­renaming­can­be­demonstrated­with­squares­of­paper.­For­exam-ple,­the­ cannot be added to the ­because­the­fractions,­the­parts­are­not­the­same­size­

(Figure­10.6).­The­first­fraction,­ ­was­created­by­dividing­the­square­into­three­parts­and­

using­two­of­them.­The­second­fraction,­ ­was­created­by­dividing­the­square­into­four­

parts and using one of them.

Figure 10.4

Figure 10.5 Fractions and clocks To­make­both­parts­the­same:

♦­­ the­one­divided­into­3­parts­initially­is­further­divided,­but­into­4­parts­(3×4­parts=12­

parts)

♦­­ the­one­divided­into­4­parts­initially­is­further­divided,­but­into­3­parts­(4×3­parts=12­

parts).

It’s­another­example­of­the­commutative­property a×c=c×a

and once again we return to the basic principles of numbers.

Figure 10.6­Renaming­fractions­to­make­them­have­the­same­name