MATHEMATICSAlgebraic Manipulation – Multiplying and Dividing Terms

Aims of the Lesson

• To review algebraic terminology and algebraic conventions.

• To look at how we can simplify expressions that involve multiplication.

• To look at how we can simplify expressions that involve division.

Algebraic Terminology

• An expression is a calculation involving letters.

• An equation is an algebraic calculation with an equal sign within it.

• The terms of an expression are separated by add or subtract signs.

• A co-efficient is the value in front of a letter or group of letters.

Algebraic Terminology - Examples

• Here is an expression: 3ab + 6p – 7

• Here is an equation:3ab + 6p – 7 = 4p + 25

• The terms in the top expression are: 3ab, +6p and – 7

• The co-efficient of ab is 3• The co-efficient of b is 3a• The co-efficient of p is 6 (or +6)

Algebraic Conventions

• In algebra we use letters to replace some or all of the values in a calculation

• If there is only 1 of a letter we don’t write the 1

E.g. 5p – 4p = p (we don’t write 1p)

• As with numbers that multiply themselves repeatedly, we use powers with letters that multiply themselves repeatedly.

E.g. 6 × 6 × 6 = 6³ and p × p × p × p = p4

• We do not write the × symbol between things we wish to multiply as it may get confused with the letter x!

• Therefore, 3a means 3 times a and pq means p times q!

• In algebra given mixed terms in a multiplication, we always write numbers in front of letters, which are written in alphabetical order

E.g. n × 2 × k is written as 2kn (not n2k)

Algebraic Multiplication• When multiplying think signs, numbers, letters…

• Remember: pairs of the same sign = +ve answer pairs of different signs = -ve answer

• Example… 5cd × –7 × 3ad =

SIGNS:5c is +ve but we have -7 which is –ve.Together these give a –ve answer which then goes together with 3cd which is +veTogether, and therefore overall, we have a –ve answer!

NUMBERS:5 × 7 = 35 then this × 3 = 105LETTERS:Dropping all the × signs we get cdadIn alphabetical order = acddIn index form we finally get acd²

–105 acd²

Algebraic Conventions (continued)• When dealing with division, we sometimes represent division as a long division or / (ie the line in a fraction) instead of this ÷ sign.

• In algebra we also use the fractional version to represent divide

• Therefore…• a/b means a divided by b • 2/n+1 means 2 divided by (n+1)!

Algebraic Division• To simplify/cancel fractions down remember to…

• Find the largest number that both the top and bottom values can be divided by.

• Write out any indexed letters in repeated form and cross out any letter that appears in the top and the bottom.

• Example… 35p²qr ÷ 60pr = 35ppqr 60pr

7 .

12=

NUMBERS:35 and 60 can BOTH be divided by 5 to become 7 (on top) and 12 (on bottom)LETTERS:There is one p on BOTH the top and bottom which can be crossed outThere is also one r on BOTH the top and bottom which can be crossed out

pq

FORMAT:Write out as a fraction, writing ‘powered’ letters out in full (i.e. as repeats)

Practice

1) Write these as simply as possible:3 × r4 × t × 2b × c × b

2) Write these as simply as possible:p ÷ m8 ÷ 2r12a² ÷ 18ax

Answers3 × r =

4 × t × 2 =

b × c × b =

p ÷ m =

8 ÷ 2r =

12a² ÷ 18ax =

Don’t write ‘×’ signNumbers before lettersMultiply the numbers togetherIgnore any other ‘×’ signNumbers before lettersDon’t write ‘×’ signWrite letters in alphabetical orderRepeated letters are written in index form

Replace the ‘÷’ sign with the line in a fraction

Replace the ‘÷’ sign with the line in a fractionIf possible, cancel down the numbers in the fraction (here: 8 ÷ 2 = 4)

Replace the ‘÷’ sign with the line in a fractionCancel down any numbers (i.e. divide by 6)Cancel down any repeated letters (i.e. a)

3r

8t

b²c

pm

4r

2a3x

bbc =

Further Practice (including brackets)

• Write these as simply as possible:4 × y × 8 7 × (t + 2) 3 × t × t × s 32y

• Write these as simply as possible:p ÷ (m – 1) (g + 3) ÷ r 6 ÷ 7 – x pm – 1

Click to find out the answers…

– x

7(t + 2) 3stt 3st²

67

g + 3 r

What next?

Make appropriate notes (including examples) on simplifying terms being multiplied or divided or printout the prepared notes and complete all the tasks within them.

Work through the MyMaths lesson (and then its online homework) called:Algebra > Algebraic Manipulation > Simplifying 2

Save and complete the worksheet: Simplify-S1.xlsx