COURSE TITLE:
w MATHEMATICS FOR INTEGRATED w w
SCIENCE
COURSE CODE: C w w
REDIT UNITS: 2
w
COURSE MATERIALS DEVELOPED
w w
BY
IBRAHIM HARUNA USMAN DEPARTMEN
w w w
T OF SCIENCE EDUCATION AHMADU BE
w w w w w
LLO UNIVERSITY, ZARIA
w w
,PREAMBLE
The course, mathematics for integrated Science is meant for 200 level B.Ed and B.Sc. (Ed)
w w w w w w w w w w w w w w
wintegrated Science Students who are purposely science teachers-in-
w w w w w w w
training. As teachers of science, the course is aimed at acquainting them with some basic
w w w w w w w w w w w w w w
wknowledge of higher mathematics that will help them understand and teach very well t
w w w w w w w w w w w w w
heir teaching subjects.
w w
, MODULE 1: ALGEBRA
w w
INTRODUCTION
Inwthiswmodule,wyouwwillwbewexposedwtowalgebraicwexpressionswandwequations.wSolutionswofwsomewequations
wwillwalsowbewconsidered.wVariations,wIndices,wandwlogarithmswwillwbewdiscussed.
Thewmodulewiswdividedwintowthreewunits,wnamely:
Unitw1w Algebraicwexpressionswandwequationsw
Unitw2w Indiceswandwlogarithms
Unitw3w Variation
Unitw1w AlgebraicwExpressionswandwEquations
CONTENT
1.0 Introduction
2.0 Objectives
3.0 Meaningw andw simplificationw ofw algebraicw expressionsw andw meaningw andw solutionw ofw algebraic
wequations
4.0 Conclusion
5.0 Summary
6.0 TutoredwMarkedwAssignment
7.0 Reference/furtherw Readings
1.0 INTRODUCTION
Algebraicwexpressions,wequationswandwinequalitieswarewverywimportantwtowmathematics,wsciencewandwtech
nology.wScientificwdiscoverieswandwrelationshipwbetweenwquantitiesworwvariableswarewmostlywexpressedwinw
thewformswofwalgebraicwequationsworwinequalities.wItwiswthereforewimportantwtowunderstandwthewmeaningw
ofwsomewmethodswofwsimplifyingwandwsolvingwofwalgebraicwexpressions,wequationswandwinequalities.
2.0 OBJECTIVES
Afterwstudyingwthiswunit,wstudentswshouldwbewablewto
i. Statew thew meaningsw ofw algebraicw expressions,w equationsw andw inequalityw andw distinguishw t
hewthree
ii. Expressw wordw problemsw inw thew formsw ofw algebraicw expressions,w equationsw andw inequaliti
eswusingwalphabets.
iii. Understandwlawwofwalgebra
iv. Simplifywalgebraicwexpressions
v. Solvewsomewalgebraicwequationswandwinequalitieswusingwvariouswmethods
, 3.0wMEANINGwOFwALGEBRAICwEXPRESSION
Inweverydaywlife,wwewdealwwithwstatementswinvolvingwquantitieswsuchwasw1wbookw5wpencils,w2werasers,w
10wsharpeners,w3wdusterswandwsowon.wTakingwthewfirswletterswofwthesewitemswtowrepresentwthem,wtheyw
wouldwbewwrittenwasw1b,w5p,w2e,w10s,w3dwarithmeticalwoperationswcanwbewcarriedwoutwwithwthewabovew
quantitieswtowmakewsense.wForwexample,
1. awstudentwhavingw3wbooks,w5wpencilswandw2weraserswcanwbewalgebraicallywrepresentedwasw3bw+w5pw
+w 2e
2. iwamwtravellingwwithwawbriefwcasewcontainingw3wshirts,w2wtrousers,w4whandkerchiefs,w3wcaps,w5wbelts
.wCanwbewsaidwthatwthewbriefcasewcontainsw3sw+w2tw+w4Lw+w3cw+w5b
Representingwawstatementwusingwalphabeticalwsymbolswandwarithmeticwoperationswiswsaidwtowbewanw algeb
raicwexpression.wAnwalgebraicwexpressionwmaywcontainwoneworwmorewquantities.wEverywalgebraicwexpressi
onwiswmadewupwofwawtermwandwawcoefficient.wAwtermwrepresentswawquantityworwawnumberwinwgeneral,winwth
ewexpressionw5a;wawiswthewtermwandw5wiswthewcoefficientwofwa.
Inwthewexpressionw3yw–
w2kw+w7r;wy,wkwandwrwarewthewtermswandw3,w2wandw7warewthewcoefficientswofwy,wkwandwrwrespectively.
SELFwASSESSMENT
1. definewanwalgebraicwexpression
2. writew5wexampleswofwalgebraicwexpressions.
SIMPLIFICATIONwOFwALGEBRAICwEXPRESSIONS
LawswofwAlgebra
Algebraicwexpressionswinvolvewoneworwmorewarithmeticwexpressionswofwaddition,wsubtraction,wmultiplicati
onwandwdivision.wThesewequationswarewcarriedwoutwaccordingwtowthewfollowingwlawswofwalgebra:
1- Commutativew laww ofw additionw andw multiplication.
Twow numbersw aw andw bw canw bew addedw or
wmultipliedwinwanyworderwwithoutwaffectingwthewresult.
i) aw+wbw=wbw+wawandwii)wabw=w
bawexample
3w+w5w=w8wandw5w+w3w=w8wthus,w3w+w5w=w5w+w3w=w8
3wXw5w=w15wandw5wXw3w=w15wthus,w3wxw5w=w5wXw3w=15
Itwshouldwbewnoted,whowever,wthat
I)waw–wbw≠w 𝑏w −w𝑎
Butwaw–wbw=waw+(-b)w+w(-b)w+aw=w-b+a
1w 1w
ii) 𝑎w ÷w𝑏w ≠w 𝑎w·w =w ·w𝑎
𝑏 𝑏
example, 5w–w3w=w2w 3w–w5w=w-2,wthus,w5w−w3w ≠w 3w−w5,
1
𝑎𝑙𝑠𝑜 6w÷w2w =w 3w𝑎𝑛𝑑w2w÷w6w =w ,w 𝑡ℎ𝑢𝑠w6w÷w2w ≠w 2w÷w6
3w
2. Associativew Laww forw additionw andw Multiplication:w Thew wayw numbersw arew associatedw underw th
ewoperationwofwadditionworwmultiplicationwdoeswnotwaffectwthewresult.
aw+w(bw+wc)w=w(aw+wb)w+cw=waw+wbw+c
w MATHEMATICS FOR INTEGRATED w w
SCIENCE
COURSE CODE: C w w
REDIT UNITS: 2
w
COURSE MATERIALS DEVELOPED
w w
BY
IBRAHIM HARUNA USMAN DEPARTMEN
w w w
T OF SCIENCE EDUCATION AHMADU BE
w w w w w
LLO UNIVERSITY, ZARIA
w w
,PREAMBLE
The course, mathematics for integrated Science is meant for 200 level B.Ed and B.Sc. (Ed)
w w w w w w w w w w w w w w
wintegrated Science Students who are purposely science teachers-in-
w w w w w w w
training. As teachers of science, the course is aimed at acquainting them with some basic
w w w w w w w w w w w w w w
wknowledge of higher mathematics that will help them understand and teach very well t
w w w w w w w w w w w w w
heir teaching subjects.
w w
, MODULE 1: ALGEBRA
w w
INTRODUCTION
Inwthiswmodule,wyouwwillwbewexposedwtowalgebraicwexpressionswandwequations.wSolutionswofwsomewequations
wwillwalsowbewconsidered.wVariations,wIndices,wandwlogarithmswwillwbewdiscussed.
Thewmodulewiswdividedwintowthreewunits,wnamely:
Unitw1w Algebraicwexpressionswandwequationsw
Unitw2w Indiceswandwlogarithms
Unitw3w Variation
Unitw1w AlgebraicwExpressionswandwEquations
CONTENT
1.0 Introduction
2.0 Objectives
3.0 Meaningw andw simplificationw ofw algebraicw expressionsw andw meaningw andw solutionw ofw algebraic
wequations
4.0 Conclusion
5.0 Summary
6.0 TutoredwMarkedwAssignment
7.0 Reference/furtherw Readings
1.0 INTRODUCTION
Algebraicwexpressions,wequationswandwinequalitieswarewverywimportantwtowmathematics,wsciencewandwtech
nology.wScientificwdiscoverieswandwrelationshipwbetweenwquantitiesworwvariableswarewmostlywexpressedwinw
thewformswofwalgebraicwequationsworwinequalities.wItwiswthereforewimportantwtowunderstandwthewmeaningw
ofwsomewmethodswofwsimplifyingwandwsolvingwofwalgebraicwexpressions,wequationswandwinequalities.
2.0 OBJECTIVES
Afterwstudyingwthiswunit,wstudentswshouldwbewablewto
i. Statew thew meaningsw ofw algebraicw expressions,w equationsw andw inequalityw andw distinguishw t
hewthree
ii. Expressw wordw problemsw inw thew formsw ofw algebraicw expressions,w equationsw andw inequaliti
eswusingwalphabets.
iii. Understandwlawwofwalgebra
iv. Simplifywalgebraicwexpressions
v. Solvewsomewalgebraicwequationswandwinequalitieswusingwvariouswmethods
, 3.0wMEANINGwOFwALGEBRAICwEXPRESSION
Inweverydaywlife,wwewdealwwithwstatementswinvolvingwquantitieswsuchwasw1wbookw5wpencils,w2werasers,w
10wsharpeners,w3wdusterswandwsowon.wTakingwthewfirswletterswofwthesewitemswtowrepresentwthem,wtheyw
wouldwbewwrittenwasw1b,w5p,w2e,w10s,w3dwarithmeticalwoperationswcanwbewcarriedwoutwwithwthewabovew
quantitieswtowmakewsense.wForwexample,
1. awstudentwhavingw3wbooks,w5wpencilswandw2weraserswcanwbewalgebraicallywrepresentedwasw3bw+w5pw
+w 2e
2. iwamwtravellingwwithwawbriefwcasewcontainingw3wshirts,w2wtrousers,w4whandkerchiefs,w3wcaps,w5wbelts
.wCanwbewsaidwthatwthewbriefcasewcontainsw3sw+w2tw+w4Lw+w3cw+w5b
Representingwawstatementwusingwalphabeticalwsymbolswandwarithmeticwoperationswiswsaidwtowbewanw algeb
raicwexpression.wAnwalgebraicwexpressionwmaywcontainwoneworwmorewquantities.wEverywalgebraicwexpressi
onwiswmadewupwofwawtermwandwawcoefficient.wAwtermwrepresentswawquantityworwawnumberwinwgeneral,winwth
ewexpressionw5a;wawiswthewtermwandw5wiswthewcoefficientwofwa.
Inwthewexpressionw3yw–
w2kw+w7r;wy,wkwandwrwarewthewtermswandw3,w2wandw7warewthewcoefficientswofwy,wkwandwrwrespectively.
SELFwASSESSMENT
1. definewanwalgebraicwexpression
2. writew5wexampleswofwalgebraicwexpressions.
SIMPLIFICATIONwOFwALGEBRAICwEXPRESSIONS
LawswofwAlgebra
Algebraicwexpressionswinvolvewoneworwmorewarithmeticwexpressionswofwaddition,wsubtraction,wmultiplicati
onwandwdivision.wThesewequationswarewcarriedwoutwaccordingwtowthewfollowingwlawswofwalgebra:
1- Commutativew laww ofw additionw andw multiplication.
Twow numbersw aw andw bw canw bew addedw or
wmultipliedwinwanyworderwwithoutwaffectingwthewresult.
i) aw+wbw=wbw+wawandwii)wabw=w
bawexample
3w+w5w=w8wandw5w+w3w=w8wthus,w3w+w5w=w5w+w3w=w8
3wXw5w=w15wandw5wXw3w=w15wthus,w3wxw5w=w5wXw3w=15
Itwshouldwbewnoted,whowever,wthat
I)waw–wbw≠w 𝑏w −w𝑎
Butwaw–wbw=waw+(-b)w+w(-b)w+aw=w-b+a
1w 1w
ii) 𝑎w ÷w𝑏w ≠w 𝑎w·w =w ·w𝑎
𝑏 𝑏
example, 5w–w3w=w2w 3w–w5w=w-2,wthus,w5w−w3w ≠w 3w−w5,
1
𝑎𝑙𝑠𝑜 6w÷w2w =w 3w𝑎𝑛𝑑w2w÷w6w =w ,w 𝑡ℎ𝑢𝑠w6w÷w2w ≠w 2w÷w6
3w
2. Associativew Laww forw additionw andw Multiplication:w Thew wayw numbersw arew associatedw underw th
ewoperationwofwadditionworwmultiplicationwdoeswnotwaffectwthewresult.
aw+w(bw+wc)w=w(aw+wb)w+cw=waw+wbw+c
