Algebra

 

Activity 1: My Family.

 

The age of each person in my family is as follows.

 

I am 14.

My brother is three years older than me.

My sister is two years younger than me.

My Mum is three times my age.

My Dad is three times my age and two more years.

My dog is half my age.

 

How old is everyone in my family?

 

If I was 10 and the same rules applied, how old would everyone be?

If I was 20 and the same rules applied, how old would everyone be?

 

Complete the table:

My Age

Brother

Sister

Mum

Dad

Dog

14

 

 

 

 

 

10

 

 

 

 

 

20

 

 

 

 

 

Me

Me + 3

Me - 2

2 x Me

2 x Me + 2

Me ÷ 2

The family next door to me I know are the following ages.

 

Bart is 10 years old.

Lisa is two years younger than Bart.                      8              B - 2

Maggie is 1/10th Barts age.                                     1               B/10 (or B ÷ 10)

Marge is three times Barts age plus 4 years.        34            3B + 4 (or B x 3 + 4)

Homer is four times Barts age less 4 years.         36            4B – 4 (or B x 4 – 4)

 

How old is everyone in the Simpson family?  

If Bart was B years old, how old would everyone be?

 

The family guys across the road are the following ages.

 

Chris is 13

Meg is 4 years older than Chris.                            17            C + 4

Stewie is 12 years younger than Chris                   1               C - 12

Lois is three times Chris’ age plus one year          40            3 x C + 1 (or 3C + 1)

Peteris four times Chris’ age less on year             51           4 x C – 1 (or 4x – 1)

Brian (the dog) is half Chris’ age plus 1/2              7              C ÷ 2 + ½ (or C/2 + ½)

 

The family across the road has an average age of 24, What might be their ages?

How can I increase their average age by 1 year?

 

 


 

 

Activity 1: Apartment Algebra.

On the way to work each day as I drive through Footscray I look at all the new apartment buildings going up. There are a few new complexes that I have been keeping a close eye on. I noted how quickly each complex was being built for the first three weeks. I have drawn them below.

Apartment Algebra

Week 1

Week 2

Week 3

 

 

 

Complex A

 

 

 

 

 

 

 

 

 

Complex B

 

 

 

 

 

 

 

 

 

 

 

 

Complex C

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Complex D

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Complex E

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1. Complete a T-Chart for each apartment complex.

Week

# Apartments

1

2. Graph each complex on the same graph.

2

3

3. How long will it take each to build 50 apartments? Explain.

4

5

4. Which complex will be the first to build 50 apartments? Explain.

10

100

5. Explain how the expression for each complex (n) relates to the graph.

n

6. Make one of your own, complete a T-Chart and draw the graph.

Bruce Williams

CreatingRealMathematicians.com

 


 

 

Activity 2: Stone Tile Path.

The park near my house is having a path built with a fountain in the middle. It looks something like what’s drawn below. The builder was confused and asked me how many stone tiles she will need to complete the paths. She is unsure how many steps needed to get to the edge of the square park.

 

 

 

 

 

 


                                                                                                                                                                

                                                                                                                                                                       And so on...

 

 

 

 

 

 

 

 


What to do...

1)   Find the number of stone tiles needed and a general expression

 

Steps to edge of park

Total no. of tiles surrounding waterfall

1

0

2

4

3

 

4

 

5

 

10

 

100

 

T

 

 

Draw the Graph on graph paper (remember all labels) to represent the above table.  

 

 

 

 

 

 

Activity 3: Car Park Investigation.

The cost of parking in the Carranballac Car Park is $2 per hour. Complete the first column on how much it will cost to park here.

 

Hours

Parking Cost

Cost for students

Cost for

business

Cost at Point Cook car park

1

 

 

 

 

2

 

 

 

 

3

 

 

 

 

4

 

 

 

 

5

 

 

 

 

6

 

 

 

 

12

 

 

 

 

24

 

 

 

 

h

 

 

 

10 x

 

 

 

 

1)           Draw a graph of the cost v’s hours parked.

2)         Students were given a discount of $2 per visit. Complete a T Chart and Graph for students.

3)         Business workers were charged an extra $2 per visit. Complete the T Chart and Graph for Business workers.

4)         The cost at the oppositions Point Cook car park is 10 cents for the first hour and doubles each hour after that. Complete the T Chart and Graph for this car park.

5)         Which car park is cheaper, yours or your oppositions? Explain.

6)         Weekend rates are half price. Complete the T Chart and Graph for the three groups of people for the weekend.

7)         Another car park has $10 for the first hour, and halves in price for each subsequent hour. Complete the T Chart and Graph for the three groups of people for the weekend.

8)         Make up a charge of your own with a free parking period and different charge rates per hour. Put this information into a spreadsheet and generate the relevant graphs.

9)         Give a written explanation / interpretation of this final graph of three charge rates.


 

 

 

Activity 4: DVD Flix.

 

I want to buy all the latest DVD’s on line as they are released. At DVDFlix they have three different plans. Each plan has a membership charge per month and charge different amounts for each DVD. I am not sure which plan I should choose?

 

Plan A:     $10.00 per month, $20 per DVD.

Plan B:     $40.00 per month, $10 per DVD.

Plan C:     $100.00 per month, $5 per DVD.

 

 

 

 

 

 

Plan A                              Plan B                              Plan C

DVD’s per month

Cost

 1

 

2

 

3

 

4

 

5

 

10

 

100

 

n

 

DVD’s per month

Cost

 1

 

2

 

3

 

4

 

5

 

10

 

100

 

n

 

DVD’s per month

Cost

 1

 

2

 

3

 

4

 

5

 

10

 

100

 

n

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Graph Number of DVD’s v’s Cost

 


Which plan should I select if I purchase DVD’s each month? Explain.

 

 

 

Activity 5: Make Up Prices.

Myer has make up on sale at the moment. The sales catalogue has the following information.

Product

Price

Face Moisturiser

$24

Perfume

Twice the price of the face moisturiser

Lipstick

Half the price of the face moisturiser

Hair Dye

$10 less than the price of the face moisturiser

Eye liner

$6 more than the price of the face moisturiser

 

 

 

1)           Find the price of each product

 

Product

Price

Face Moisturiser

$24

Perfume

 

Lipstick

 

Hair Dye

 

Eye liner

 

 

2)  Find the expression for each product if the price for the face moisturiser is $F.

 

Product

Price

Face Moisturiser

$F

Perfume

 

Lipstick

 

Hair Dye

 

Eye liner

 

 

 

3) Complete the following table if the price for the face moisturisers changes.  Find the new prices of the other products.

 

Face moisturiser

Perfume

Lipstick

Hair Dye

Eye liner

$10

 

 

 

 

$20

 

 

 

 

$30

 

 

 

 

$40

 

 

 

 

$50

 

 

 

 

$60

 

 

 

 

 

4) Graph the above table on graph paper.

 

5) a)Find total of all make-ups. How much would that cost? What is the expression?

b) What if we bought two of each make-ups? Total cost and expression?

The total package for one of each item is on special for $190. How much is being charged for each item? (11F/2 - 4 = 190).

The price of Face Moisturiser varies between $19 and $28 depending on which store you buy from. How much would all the other products vary between? Write the expression for each. (Inequalities).


 

 

Activity 6: Cinema Tickets.

 

Tickets

Price

Student

$12.50

Adult

$5.00 more than student tickets

Concession

$1.00 less than student tickets

Gold class

Triple student tickets

3D

Double student tickets

Tuesdays

Half price

 

 

 

 

 

1)           Find the price of each ticket

2)         Find the expression for each ticket if the price of the student ticket is $T

Tickets

Price

Student

$T

Adult

 

Concession

 

Gold class

 

3D

 

Tuesdays

 

 

Tickets

Price

Student

$12.50

Adult

 

Concession

 

Gold class

 

3D

 

Tuesdays

 

 

 

 

 

 

 

 

3)         Complete the following table.  Where the price for the student ticket changes.  Find the new prices of the other tickets.

Student Tickets

3D

Adult

Concession

Gold Class

Tuesdays

$10.50

 

 

 

 

 

$15.50

 

 

 

 

 

$22.50

 

 

 

 

 

$25.50

 

 

 

 

 

$30.50

 

 

 

 

 

$32.50

 

 

 

 

 

4)         Graph the above table on graph paper.

5)         You decide to take your whole family to the movies. Find

a)  Total price in($)

b)  Total price in ($T)

6)         You decide to take your whole family to the movies on Tuesday

c)  Total price in ($)

d)  Total price in ($T)

 


 

 

Activity 7: The Big Party.

I am organising a party for a few friends. We will all be sitting at tables. I called the table hire place and they had Triangular tables that were a good price. I was trying to work out how many seats I would need if one person sat on each side of each table.

 

 




One seat is to be placed on each side of each table.

What to do...

1)   Find the number of seats needed for each table and find a general expression

 

Number of Tables

Total no. of seats needed

0

 

1

 

2

 

3

 

4

 

5

 

10

 

100

 

 

2) Draw the Graph on graph paper (remember all labels) to represent the above table.

 

3) Complete T-Chart, Expression and Graph for Triangular tables (seats 3), Square tables (seats 4), Trapezoidal tables (seats 5), Hexagonal tables (seats 6) and Octagonal tables (seats 8).

 

4) I want to sit 100 people at the barbeque. How many tables will I need for each size chosen?

 

5) Explain the affect the Pronumeral and the affect the Constant has on how the graph is drawn.  (Seats =m Tables +c  m is the Pronumeral, c is the Constant).

 

 


 

 

Activity 8: The Sub-Division. Which block of land?.

There is a new estate opening up near me. The dimensions of the blocks of land are given below (in metres).

 

 

             20            10        15            m            2m                m-2        m+5        2m+3

10

25

m

m + 2

M - 4

2m

3m + 3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


   

 











Which is the biggest block of land?

Students use the Area Method of multiplication as shown in prior pages and model with Algebra Tiles.

 

 

References:

Improving Mathematics in Schools (TIMES) Project. Teacher Support Modules:

http://www.amsi.org.au/teacher_modules/linear_equations.html

 

Mathematics Developmental Continuum

Conceptual Growth for Solving Equations - Progression Points

Dimension

Level

Progression Point

Structure

2.25

·         Use of ‘=’ to indicate equivalence or the result of a computation

2.5

·         Construction of number sentences

·         Calculations using notation such as ‘3 + 5 − 2 =’

3.0 Standard

… Students understand the meaning of the ‘=’ in mathematical statements and technology displays (for example, to indicate either the result of a computation or equivalence).

They construct number sentences with missing numbers and solve them.

3.25

·         Use of trial and error to find a missing number in a number sentence; for example, 4 × ? + 6 = 22

3.5

·         Consistent and correct use of conventions for order of operations

4.0 Standard

… Students solve equations by trial and error.

4.5

·         Use of inequality, equality, approximately equal and not equal, including in symbolic expressions

·         Solution of simple linear equations using tables, graphs and inverse operations (backtracking)

4.75

·         Solution of equations such as = 17 as required in measurement situations; for example, using pythagoras theorem

5.0 Standard

… Students use inverses to rearrange simple mensuration formulas, and to find equivalent algebraic expressions (for example, if P = 2L + 2W, then W = P/2 − L. If A = πr2 then r = √A/π for r > 0).

They solve simple equations (for example, 5x + 7 = 23, 1.4x − 1.6 = 8.3, and 4x2 − 3 = 13) using tables, graphs and inverse operations.

They recognise and use inequality symbols.

They solve simple inequalities such as y ≤ 2x + 4 and decide whether inequalities such as x2 > 2y are satisfied or not for specific values of x and y.

5.25

·         Solution of equations by graphical methods

5.5

·         Expansion of products of algebraic factors, for example, (2x + 1)(x − 5) = 2 − 9x − 5

·         Use of inverse operations to re-arrange formulas to change the subject of a formula

5.75

·         Factorisation of simple quadratic expressions and use of the null factor law for solution of equations

·         Formulation of pairs of simultaneous equations and their graphical solution

6.0 Standard

… Students solve equations of the form f(x) = k, where k is a real constant (for example, x(x + 5) = 100) and simultaneous linear equations in two variables (for example, {2x − 3y = −4 and 5x + 6y = 27} using algebraic, numerical (systematic guess, check and refine or bisection) and graphical methods.

 

Equivalence in Number Sentences - Progression Points

Dimension

Level

Progression Point

Structure

2.25

·         Use of ‘=’ to indicate equivalence or the result of a computation

2.5

·         Variation of order and grouping of addition (commutative and associative property) to facilitate computations; for example, 3 + 5 + 7 + 5 = 3 + 7 + 5 + 5 = 10 +10 = 20

·         Construction of number sentences.

·         Calculations using notation such as ‘3 + 5 − 2 =’

2.75

·         Use of distributive property in calculations; for example, 6 × 37 = 6 × 30 + 6 × 7

3.0 Standard

… Students understand the meaning of the ‘=’ in mathematical statements and technology displays (for example, to indicate either the result of a computation or equivalence).

They use number properties in combination to facilitate computations (for example, 7 + 10 + 13 = 10 + 7 + 13 = 10 + 20).

They multiply using the distributive property of multiplication over addition (for example, 13 × 5 = (10 + 3) × 5 = 10 × 5 + 3 × 5).

3.25

·         Use of add and subtract as inverse operations to solve simple word equations such as ‘I am thinking of a number. If I add 6 I get 18, what number did I start with?’

·         Use of trial and error to find a missing number in a number sentence; for example, 4 × ? + 6 = 22

3.5

·         Use of division and multiplication as inverses; for example, multiplication by 25 can be carried out as ‘multiplication by 100 followed by division by 4’

4.0 Standard

… Students establish equivalence relationships between mathematical expressions using properties such as the distributive property for multiplication over addition (for example, 3 × 26 = 3 × (20 + 6)).

They recognise that addition and subtraction, and multiplication and division are inverse operations.

They use words and symbols to form simple equations.

They solve equations by trial and error.

 

Structure of Algebraic Expressions - Progression Points

Dimension

Level

Progression Point

Structure

2.25

·         Use of ‘=’ to indicate equivalence or the result of a computation

2.5

·         Construction of number sentences

3.0 Standard

… Students understand the meaning of the ‘=’ in mathematical statements and technology displays (for example, to indicate either the result of a computation or equivalence).

They construct number sentences with missing numbers and solve them.

3.25

·         Use of add and subtract as inverse operations to solve simple word equations such as ‘I am thinking of a number. If I add 6 I get 18, what number did I start with?’

·         Use of trial and error to find a missing number in a number sentence; for example, 4 × ? + 6 = 22

3.5

·         Use of division and multiplication as inverses; for example, multiplication by 25 can be carried out as ‘multiplication by 100 followed by division by 4’

4.0 Standard

… Students recognise that addition and subtraction, and multiplication and division are inverse operations.

They use words and symbols to form simple equations.

They solve equations by trial and error.

4.25

·         Interpretation of a letter as a symbol for any one of a set of numbers and use in symbolic description of relationships

4.5

·         Use of inequality, equality, approximately equal and not equal, including in symbolic expressions

·         Translation from verbal description to algebraic representation, and of the structure of algebraic expressions; for example, if $500 is shared between n people, each receives 500/n

·         Solution of simple linear equations using tables, graphs and inverse operations (backtracking)

4.75

·         Graphical representation of simple inequalities such as y ≤ 2x + 4

5.0 Standard

… Students solve simple equations (for example, 5x + 7 = 23, 1.4x − 1.6 = 8.3, and 4x2 − 3 = 13) using tables, graphs and inverse operations.

They recognise and use inequality symbols.

They solve simple inequalities such as y ≤ 2x + 4 and decide whether inequalities such as x2 > 2y are satisfied or not for specific values of x and y.

5.5

·         Equivalence between algebraic forms; for example, polynomial, factorised and turning point form of quadratics

6.0 Standard

… Students solve equations of the form f(x) = k, where k is a real constant (for example, x(x + 5) = 100) and simultaneous linear equations in two variables (for example, {2x − 3y = −4 and 5x + 6y = 27} using algebraic, numerical (systematic guess, check and refine or bisection) and graphical methods.

 

 

Australian Curriculum

Year 8 Algebra

 

Generalise the distributive law to expansion and factorisation of simple algebraic expressions and use the four operations with algebraic expressions

Elaborations

understanding that the distributive law can be applied to algebraic expressions as well as numbers and also understanding the inverse relationship between expansion and factorisation

connecting the numerical use of the distributive law to its application with algebraic terms: just as 28(100 + 1) = 28 × 100 + 28 × 1, so then 4(5x + 7) = 4 × 5x + 4 × 7 and describing and generalising the results

applying the distributive law to the expansion of algebraic expressions using strategies, such as the area model

recognising that factorising is the opposite of expanding, identifying the greatest common divisor (highest common factor) of numeric and algebraic expressions and using a range of strategies to factorise algebraic expressions

understanding the process of substitution into algebraic expressions and evaluating expressions after substitution

recognising like terms, adding and subtracting like terms to simplify algebraic expressions and explaining why unlike terms cannot be added and subtracted

understanding and applying the conventions for simplifying multiplication of single term algebraic expressions, including those with positive integral indices, such as 3m3 × 2m = 6m4

understanding and applying the conventions for simplifying division of algebraic terms which can be expressed as a single fraction, including those with positive integral indices

determining whether a simplified expression is correct by substituting numbers for variables


 

 

 

Understanding the Distributive Law

28 x 150 = 28 x (100+50)

              = 28 x 100  +  28 x 50

              = 2800  +  1400

              = 4200

 

Explain to students that numbers can be partitioned, ie, 14 = 10 + 4, 36 = 30 + 6= 10 + 10 + 10 + 6, etc and that this method of partitioning can be used for multiplication, in particular the area method of multiplication.

Solve the following:

 

6 x 14

X

10

4

6

60

24

60

6  x  14  = 6 x 10   +    6 x 4

     =  60        +     24

               =  84

 

 

18 x 23 = (10 + 8) x (20 + 3)

X

20

3

10

200

30

8

160

24

 

18 x 23 = 200 + 30 + 160 + 24

              = 414

 

 

Expanding a set of brackets

Use the above as a segway to expanding an algebraic expression using the distributive law.

Earlier we saw that 6 x 14 was the same as 6 x (10 + 4), but what if we didn’t know that the number was 14, instead we just knew that it was n+4, how could we perform the same operation?

 

X

n

4

6

6n

24

 

So, 6(n+4) = 6n+24

 

 

Perform similar operations using algebra tiles

http://www.TheMathLab/toolbox/algebra%20stuff/algebra%20tiles.htm

Or google “Algebra Tiles Images”.

Also www.Math.About.com or interactive at National Library of Virtual Manipulatives (www.NLVM.usu.edu)

See Malcolm Swan Tasks PDF. A must read before starting this unit.

 

https://www.asms.sa.edu.au/wp-content/uploads/2013/04/Improving_learning_in_maths-UK.pdf

http://map.mathshell.org/download.php?fileid=1726

https://denvermathfellows.wikispaces.com/file/view/ExpressionsandEquations_TM.pdf

 

This series of activities is meant to be spread across the unit. It may be interspersed with the Malcolm Swan type problems from “Mostly Algebra. A1: Interpreting Algebraic Expressions” and “A2: Creating and Solving Equations.”

Assess students against the Progression Points for “Algebra” listed on Page 1 of this Unit Outline as well as the following Progression Points provided above.

“Conceptual Growth for Solving Equations”

“Equivalence in Number Sentences”

“Structure of Algebraic Expressions”


 

 

We want students to understand the relationship between:

the expression (2x + 1)

its equation (y = 2x + 1)

the T-Chart or table of values (when x is 3, y = 2 times 3 plus 1, y=7)

x

0

1

2

3

y

1

3

5

7

 


and the Graph. Will cross the y (vertical) axis at 1 (the constant) and will have a slope of 2                                                                y=2x+1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 






 

 



Warm Up Activities:

1

6

4

9

11

16

20

25

n

n + 5

Whats My Rule?

1

2

3

6

10

20

32

64

n

2 x n

(Or 2n)

 

Write the above tables on the board and have students work out the rule.

Use a variety so students get used to different rules. Use Addition ( E.g., n + 2) Subtraction (E.g., n – 4), Multiplication (E.g., 3 x n or 3n), Division (E.g., n ÷ 2 or n/2) and gradually introduce a mixture of two operations (n x 2 + 1 (double plus one)), (n x 4 – 1), etc.  Have students make up their own rules and challenge the class.

 

What’s the Pattern?

Put some patterns on the board and have students continue the pattern. Good patterns to use include:

1,4,7,11,... (+3)

6,11,16,21,... (+5)

100, 93, 86, 79,... (-7)

1,2,4,8,16,... (x2)

1,2,4,7,11,... (+1,+2,+3,+4...)

1,1,2,3,5,8,...(Add previous two numbers)

1,3,9,27,... (x3)

Have students make up patterns of own and challenge the class.

 

Follow Me:

Make some cards that have question one end and the solution on another card. Each card has one solution then the next question.  E.g.,

 

 

Start     2x6

12          3xb

3b        2c+3c

5c         20÷f

20/f    2(a+b)

2a + 2b   End

You can have students make these cards using the exercises in their work sheets.

Then collect and hand out to groups at the start of the lesson and have them sort into a line of cards as quick as they can.

 

Expression, Graph, T-Chart Match.

Again you can have students make these using questions from their worksheets. They have an expression (n+2) or equation (y = n + 2) on one card, the T-Chart on another

n      0      1       2      3      4      5

y      2      3      4      5      6      7

and the graph (quick sketch only) on another.

Similar is on Malcolm Swan Tasks (See Malcolm Swan Tasks PDF “Mostly Algebra” Pages A1-5 Card Set A Algebraic Expressions to A1 Card Set D Areas of Shapes).

 

 

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