Algebraic Expressions - Carl Schurz High School

1 AlgebraicExpressions468m = 5h + 1s = 4hh = no. hexagonss = no. starsm = no. matches5(x + 7)10 aChapter Contents4:01 Generalised arithmeticChallenge: Let s play with blocks4:02 SubstitutionInvestigation: The history of algebra4:03 Simplifying Algebraic expressions4:04 Algebraic fractionsAAddition and subtractionBMultiplication and divisionFun Spot: Try this maths-word puzzle4:05 Simplifying Expressions with groupingsymbolsFun Spot: What is taken off last beforeyou get into bed?4:06 Binomial products4:07 Special productsAPerfect squaresInvestigation: The square of a binomialBDifference of two squares4:08 Miscellaneous examplesChallenge: Patterns in productsInvestigation: Using special productsin arithmeticMathematical Terms, Diagnostic Test, Revision Assignment, Working MathematicallyLearning OutcomesStudents will be able to: Use the Algebraic symbols to represent word problems.

2 Simplify, expand and factorise simple Algebraic Expressions . Work with Expressions involving Algebraic fractions. Expand binomial of InteractionApproaches to Learning (Knowledge Acquisition, Problem Solving, Communication, Logical Thinking, Reflection), Human IngenuityCHAPTER 4 Algebraic EXPRESSIONS694:01 | Generalised ArithmeticIn mathematics, the method of solving a problem is sometimes hard to express in words. In cases like this, pronumerals are often used. The result could be a simple formula. Some numbers in a pattern are known. How can we find the others?For example: 9, 8, 7, 6, .. or 3, 5, 7, 9, ..Patterns like these can be written in a table of values, where n represents the position of the number in the pattern, and T the actual number (or term). Two angles of a triangle are known. How can we find the third?AConsider a numerical example.

3 = 180 (72 + 54) = 54 BShow the general result. = 180 (a + b) or 180 a b 180 (a + b) is called the general :1the sum of 7 and 52the difference between 9 and 23the number 8 less than 254the quotient of 48 and 65the product of 7 and 3612 more than 87the average of 41 and 478the total of 13 and 219the number of times 23 can be taken from 13810the number 8 less than the product of 4 and 5n12345n12345T9876T3579 Here we can see that:T = 10 nSo an Algebraic expression thatrepresents this pattern would be:10 nHere we can see that:T = 2n + 1So an Algebraic expression thatrepresents this pattern would be:2n + 1prepquiz4:01 The angle sum of a triangle is 180 .72 54 a b worked examples1 The sum of 8 and 12= 8 + 12so the sum of x and y= x + y2 The cost of 6 books at 30c each= 6 30cso the cost of x books at 30c each= x 30= 30x centscontinued 70 INTERNATIONAL MATHEMATICS 4 Match each table of values with the correct Algebraic expression from the given list tocomplete the statementT =.

4 Write down an Algebraic expression that represents each pattern of numbers, using n to represent the position of each number in the , 4, 6, 8, ..b4, 5, 6, 7, ..c7, 6, 5, 4, ..d5, 7, 9, 11, ..e, 1, 1 , 2, ..f 3, 1, 1, 3, ..For questions 3 to 8 write Expressions for sum of 5 and 7athe product of 3 and 7bthe sum of 5 and ybthe product of a and 7cthe sum of x and ycthe product of a and bathe difference between 8 and 3athe average of 8 and 12bthe difference between 8 and pbthe average of 8 and xcthe difference between q and pcthe average of w and xathe cost of 5 books at 75c eachadividing 30 cm into 5 equal lengthsbthe cost of a books at 75c eachbdividing 30 cm into t equal lengthscthe cost of a books at b cents eachcdividing Acm into t equal lengthsan1234bn1234T3456T 2 101cn1234dn1234A3nBn2Cn + 2D3n + 2E2n 1Fn 3T36916T581118en1234fn1234T14916T13573 The average of 9 and 13= so the average of a and b= 4 The change from $10 after buying 3 booksat $2 each= 10 (2 3)

5 Dollarsso the change from $10 after buying x booksat $2 each= 10 2 x= 10 2x dollars The aim of generalised arithmetic is to write an Algebraic expression that shows the steps to be taken, no matter which numbers are +2---------------ab+2------------Exercis e 4:01 Generalised arithmetic1 Write Expressions for:athe sum of 3a and 2bbthe average of m and n2 aFind the cost of x books at 75c the age of Bill, who is 25 years old, in another y Worksheet 4:0112 Use a table of valuesif you need 4 Algebraic EXPRESSIONS71aIf Steve is 15 years old, how old will he be in 6 years?bHow old will Steve be in y years?aIf a car travels at 60 km/h for 3 hours, how far does it travel?bIf the same car travels for h hours, how far does it travel?aIf three lengths of rope, each 2 m long, are cut from a piece of rope 10 m long, what length is left?

6 BIf two lengths of rope, each xm long, are cut from a piece of rope Xm long, what length is left?aA student buys x books and y pens. If each book costs 85c and each pen costs 63c, what is the total cost?bIf the books cost C cents each and the pens D cents each, what is the total cost?Mr Smith is Y years old; his son is 22 years younger. How old is his son? How old will the son be in x years time?aIf I travel x km in 2 hours, and then y km in the next 3 hours, how far have I travelled altogether?bWhat is my average speed for the whole journey?Bob and Tom have $1 between them. If Bob has x cents, how much has Tom?aWhat is the next even number after 6?bWhat is the next even number after y, if y is an even number?cWhat is the largest odd number less than y, if y is even?The expression 2n will give an even number for all values of n , where n is an integer.

7 Write down an expression that will always give an odd angles of a triangle are 25 and 79 . What is the size of the third angle?bTwo angles of a triangle are a and b . What is the size of the third angle?aHow far will a person walk at m km/h in h hours?bWhat is the average speed of a car that travels k km in h hours?cHow long will it take to travel k km at m km/h?aA TV set is bought for $P. If it is sold for $R, what is the profit?bIf a gain of $G is to be made, what should the selling price of the TV be?Write an expression for the perimeter of each figure below. All measurements are in d9101112131415161718192021ab4xysxx + 572 INTERNATIONAL MATHEMATICS 4 Write an expression for the area of each bcdIn each of the following, use grouping symbols to make the meaning product of 4 and a + 2bsubtract x + y from 8ctwice the sum of p and qdthe square of 2aethe difference between 3a and b + cfthe square of x + 2gthe product of x and y 5hthe square root of 2m + 3naWhat is the cost of 3 books at p cents and 4 pens at q cents each?

8 BWhat is the change from $5 (in cents)?Translate the following Algebraic Expressions into ac2p + qd3m 2ne5(x + y)f(a + b)2ghijTranslate the following into words to explain the difference between each pair of + b + cba bandb ac3(a + b) and 3a + bdx2 + y2and (x + y)2eandfandChallenge 4:01 | Let s play with blocksEight blocks have been stacked together here to form a cube. If the outside of the cube were painted, how many sides of each block would be painted?How many blocks make up the second cube? If this cube were painted, how many blocks would have 3 sides, 2 sides, 1 side or even no sides painted?What would be the result of painting a cube that had four blocks along each edge?22xxx 2yxhb232425xy+2------------mn a-------------uv+ab326ab+ab+ab--23---+a2 +b3+------------challenge4:01 CHAPTER 4 Algebraic EXPRESSIONS734:02 | SubstitutionAlgebra involves the use of pronumerals as well as numbers.

9 A pronumeral is usually a letter, such as x, that takes the place of a number in an expression like 3x + a number is substituted for each pronumeral, a value for the expression can then be the following Expressions if x = 3, y = 4 and z = + yb3x + 2yc2x ydx2ye(x + y)2fx2 + y2gz(x + y)hxz 10ixy xzjklz2 zmxy2 + x2ynxz yzo4(x 2y)px(y2 z2)qx 3yry2 z2sz xytuvwxyz2x4:02 Substitution4:02 Magic squaresWrite the following Algebraic Expressions in their simplest b2x y y32 x + 3 ySimplify the following:42 + 4 353 4 + 2 564 5273(6 10)8 6 5910 prepquiz4:0212---8 2 3------------53---35---worked examplesFind the value of the following Expressions , given that a = 10, b = 4, x = 5 and y = + 2b2x2 + y23ab24= 3 10 + 2 4= 52 + ( 3)2= 30 + 8= 25 + 9= 38= 3412---1x---1y---+= 10 42= 160= 8012---12---= = = 15---13 ()-----------+15---13--- 215------Exercise 4:02 Substitution1 Find the value of:a2x + 3y if x = 3, y = 52If a = 4, b = 5, c = 2, findthe value of:aa2 + bcFoundation Worksheet 4:021 Expressionshave noequals +2------------x2---y4---+xy--yx--+xy+xy- -----------xy+xy ------------12---3xy74 INTERNATIONAL MATHEMATICS 4 Find the value of the following Expressions if a = 3, b = 4 and c =.

10 Aa + bba bcabdab2e(ab)2fbcgabch10 abijc(2a + b)kl(a + b)(a b)mb2 b3nopqab + bc + acaFind the value of mx + c if:im = 3, x = 10, c = 1iim = 5, x = 4, c = 3iiim = 2, x = 7, c = 4bFind the value of if:ih = 4, a = 7, b = 5iih = 7, a = 4, b = 6iiih = 3 4, a = 9 2, b = 3 7cFind the value of r2 if:i = 3 1, r = 8ii = 3 14, r = 2 5iii = 3 142, r = 1004:03 | Simplifying Algebraic ExpressionsInvestigation 4:02 | The history of algebraFind out as much as you can about the early history of might consider: Ahmes Papyrus (Egyptian c. 1700 BC) Diophantus (Greek c. AD 250) Mohammed ibn Musa al-Khowarizmi (Arab c. AD 825) Bhaskara (Hindu c. AD 1150)Write these Expressions in their simplest + 2x29x 8x33x 2y45x x512x 4610ab 5a73a + 2b + 5a + 3b86x + 2y x y93 ( 2a) 4a103a ( 9b)212--- 112------112--- 2==1a---1b---+1a---c+1c---1b--- ab--bc-- ab--bc--cd--++3h2---ab+()investigation4: 02prepquiz4:03worked examples Like terms containidentical pronumeralparts.