combining variables – developing understanding mastering mathematics © hodder and stoughton 2014...

Combining variables – Developing Understanding Mastering Mathematics © Hodder and Stoughton 2014

Collecting like terms

Multiplying numbers and letters

Finding the value of expressions

Steps



TermA term is a single number or a variable.A term can also be the product of numbers or variables.Examples:

t 5 3n ab m 7pqLike terms are terms that use exactly the same variable.5T and 3T are like terms as they use the same letter.6w and 3v are not like terms.

There were 3 ways of splitting 3r into like terms.There were 7 ways of splitting 4r into like terms.Investigate the number of ways for 5r.

What do you notice?Can you use a formula to describe your rule?Test your formula with other values.

Menu Vocabulary Opinion 1Back Forward Cont/d More Opinion 2 Answer Opinion 1 Opinion 2 AnswerQ 1 Q 2

Bruce knows that 2r + r is the same as 3r.He writes down some other sets of terms that add up to 3r.

1. How many sets of terms can you find that add to 4r?

2. Which expression is the odd one out?a) 3m + 5n + 2n + m + 2mb) 2n – 3m + 4m + 2n + 3n + 5mc) 10m + 8n – 4m – 6n + 2n

r + r + r = 3r

2 r + r = 3r

r + 2r = 3r

There are seven possible ways without using subtraction.If the like terms are collected together they are all equivalent to 4r.

3m + 5n + 2n + m + 2m = 6m + 7n2n – 3m + 4m +2n + 3n + 5m = 6m + 7n

10m + 8n – 4m – 6n + 2n = 6m + 4n

c) does not add to 6m + 7n so it could also be the odd one out.

r + r + r + r = 4r r + 2r + r = 4r2r + 2r = 4r r + 3r = 4r

There are 4 different ways.

You can also have different orders.r + r + r + r = 4rr + r + 2r = 4r r + 2r + r = 4r 2r + r + r = 4rr + 3r = 4r 3r + r = 4r2r + 2r = 4rSo there are 7 different ways.

a) is the odd one out as it has no subtracted terms.

b) is the odd one out as the expression has six terms.


2a + 3b

3a + 3b-4a + 4b

7a + 3b -2a + 3b a + 4b

9a – 2b 5a – 2b

3a – b


Like termsLike terms use exactly the same letter or combination of letters .They can be combined into a single term. This is called collecting or gathering terms.Example: 2a + 3b + 4a + 2b = 6a + 5b

Make your own algebraic magic square.Begin with a 2 by 2 square while you work out an efficient method.Now make a 3 by 3 square.

Can you make one with more than two variables?

Menu Vocabulary Opinion 1

There are 8‘a’s and ‘b’s altogether in 5a + 3b so that’s the same as 8ab.

So the answer could be: ab + 3ab + 4ab.

Back Forward Cont/d More Opinion 2 Answer AnswerQ 1 Q 2

1. Find three different expressions that can be added together to give 5a + 3b.

2. Rearrange these cards to make an algebraic magic square. Every row and every column must add to the same amount.

2a + 7b is an expression in a and b.3a – 4b is also an expression in a and b.They can be added together and the like terms can be collected.

2a + 7b + 3a – 4b = 5a + 3b

How many of each variable are there altogether? How many rows are they spread between?So what must each row total be?

Clue

Each row and column adds to 8a + 5b.Did you find a different answer?

2a + 3b

3a + 3b -4a + 4b

7a + 3b-2a + 3b

a + 4b

9a – 2b

5a – 2b

3a – b

There are lots of possible answers.The ‘a’s have to add together tomake 5a.The ‘b’s have to add together tomake 3b.

It could be: 4a + b, 3a + b and -2a + b.

How many more answers can you find?

You have to keep the ‘a’s and the ‘b’s separate.

It could be: 4a + b, 3a + b and -2a + b.



Expression

A collection of numbers and letters combined together using arithmetic signs.

The Spotted Dragon says that the cost will be £15 for the full evening and £5 for the disco.How much does the group have to pay?How many different ways can you find to work out the total cost?Which is the easiest way? Why is it easier to gather like terms first?

Menu Vocabulary Opinion 1Back Forward More Opinion 2 Answer Opinion 1 Opinion 2 AnswerQ 1 Q 2

1. The total cost in pounds of all the tickets for people in the office is 6f + d.What are the cost in pounds of tickets for each of the other departments?

2. Write an expression for the cost of tickets for the whole group.

Liz is organising an evening at the Spotted Dragon for people at work. She knows how many people are going. She doesn’t know how much the tickets will cost yet. She writes £f for the cost of afull ticket.She writes £d for the cost of a disco ticket.

It doesn’t matter how you add up the terms, but they must be kept separately. The f and d terms can’t be combined.

The correct answer is 13f + 13d.

Cutting room: 2f + 4dSewing room: 3f + 5dDespatch: 2f + 3d

The question asks for the cost of the tickets, not the number of people who are going.

Cutting room: 2f + 4dSewing room: 3f + 5dDespatch: 2f + 3d

Cutting room: 2 + 4 = 6 ticketsSewing room: 3 + 5 = 8 ticketsDespatch: 2 + 3 = 5 tickets

Total = Cutting room + Sewing room + Despatch + Office

= 2f + 4d + 3f + 5d + 2f + 3d + 6f + d= 13fd

Total = cost of full tickets + cost of disco tickets= 2f + 3f + 2f + 6f + 4d + 5d + 3d + d= 13f + 13d


2 cm

5 cm

Area = width × length= 2 × 5= 10 cm2



The width w is under one of the notes. The length l is hidden under the other note.So the Area = lw.

The same number could be under both.Call it n for the number.So A = n x n.

Back Forward Cont/d More Opinion 2 Opinion 1 Opinion 2Q 1 Q 2

The length of this rectangle hasbeen covered up. Its area is 3 × This can be written as aformula: A = 3l

1. The length and width are both unknown in this rectangle.

Write a formula for its areausing l for the length and w

for the width.2. The sides of this square are

hidden. Write a formula for its areausing s for the length of the side.

3

m

3

m

Area = s x s.This is the same as Area = s2.

Area = s x s.But there’s no need to write the sign.A = ss. Answer

The length and width are different measurements so we must use two different variables.

Call them l and w.

A = lw

Remember ‘no sign’ means multiply.

Answer

w

l

Variables are squared in the same way as numbers.

s2 means s x s.

So A = s2.

What other formulae for shapes do you know?

Can you find some formulae that use a cube like r3 instead of a square?

Variables

The letters used in a formula.

They represent numbers which can change (or vary).



Menu Opinion 1Back Forward Cont/d Opinion 2 Answer Opinion 1 Opinion 2 AnswerQ 1 Q 2

2. Each small rectanglehas width d and height e.Each small area is de.

Write an expression forthe area of the largerectangle.

Each small rectangle has an area B. Three of them make up the large rectangle. The diagram shows that B + B + B = 3B.

B

B

B

1. How many ‘C’s arethere in thisdiagram? Write anexpression thatshows how theycan be counted ingroups.

C+C=2CC

C

C

C

C

C

C+C=2C

C+C=2C

d d

e

e

e

de

There are 3 rows. Each row adds to 2C.There are 6 ‘C’s altogether.So 2C + 2C + 2C = 6C

They are set out in 2 columns. Each column adds to 3C.There are 6 ‘C’s altogether.So 3C + 3C = 6C

Each column has 3 small rectangles.The area in each column adds to 3de.The total area is 3de + 3de = 6de.

2 x 3de = 6de

Each row has 2 small rectangles.The area in each row adds to 2de.The total area is 2de + 2de + 2de = 6de.

3 x 2de = 6de

Both opinions are correct. The diagrams show that:

3 × 2C = 6C or 2 × 3C = 6C

When terms are multiplied together the numbers can be multiplied separately.

Both opinions are correct. The total area is 6de.The large rectangle has height 3e and width 2d.So its area = 3e × 2d. This shows that:

3e × 2d = 6deor 2d × 3e = 6de2d × 3e = 2 × d x 3 × e

= 2 × 3 × d × e= 6de

de

2d

3e


de

f

4e

3d2f


Product

Two numbers or variables that are multiplied together.

Menu Vocabulary Top

The area of the top is 2f x 4e.The face is made from 8 rectangles. The area of each is fe.

This shows that 2f x 4e = 8fe.

The area of the side is 2f x 3d.The face is made from 6 rectangles. The area of each is fd.

This shows that 2f x 3d = 6fd

Back Forward Side Answer Opinion 1 Opinion 2 AnswerQ 1 Q 2

1. The area of the front of the stack is3d × 4e.The face is madefrom 12 small rectangles. Thearea of each is de.This shows that 3d × 4e = 12de.Write a similar expression for the area of top and side faces.

2. a) How many small cuboids are there in the stack? What is the volume of each one? b) What is the volume of the whole stack?c) Do you get the same volume if you

multiply the length, width and height of the stack?

The small cuboid is d cm high, e cm wide and f cm long. Its volume is d × e × f cm3.This is written as def cm3.

3d

4e

de

a) There are 8 cuboids on the top layer. There are 3 layers, so that’s 3 x 8 = 24 cuboids. Each one has volume def.b) Volume of stack = 24 def.c) 3d x 4e x 2f = 3 x 4 x 2 x d x e x f = 24 def

Always multiply the numbers and the variables separately.Write the numbers before the variables in a product.

2f x 4e = 8fe2f x 3d = 6fd

a) If you slice the stack along the top there are two piles of 12 cuboids. So that’s 24 altogether. Each one has volume def.b) Volume of stack = 24 def.c) To get the volume you add up all the edges.That comes to 3d + 4e + 2f.

Opinion is correct for parts a) and b), but not c).Opinion is fully correct.The lengths can be multiplied in any order:

3d × 4e × 2f = 3 × 4 × 2 × d × e × f= 24 def

3d × 2f × 4e = 3 × 2 × 4 × d × f × e= 24 def

2f × 4e × 3d = 2 × 4 × 3 × f × e × d= 24 def


The value of8m – 2

is more than the value of 4m + 18


Menu Vocabulary Opinion 1Back Forward Cont/d More Opinion 2Q 1

The statement compares two expressions involving the variable m.

1. Is the statement always true, sometimes true or never true? If you think ‘sometimes’, when is it true and when is it not true?

Answer

ExpressionA collection of numbers and letters combined together using arithmetic signs.

VariablesThe letters used in a formula.They represent numbers which can change (or vary).

Think of any number.

Can you make up two expressions that will both give the same value when your number is substituted into them?

Example:My number is 5.I need to find two expressions that are the same when 5 is substituted into them.

They could be 3n + 2 and 22 – nThese both have the value 17 when n = 5 is substituted.

It depends on the value of m. 8m – 2 is larger than 4m + 18 when m > 5.

When m = 3 When m = 7 When m = 58m – 2 8m – 2 8m – 2 = 8 × 3 – 2 = 8 × 7 – 2 = 8 × 5 – 2 = 24 – 2 = 56 – 2 = 40 – 2= 22 = 54 = 38

4m+18 = 30 4m+18 = 46 4m+18 = 38

Smaller Larger The same

8m – 2 is always larger as there are eight ‘m’s compared with only four in

4m + 18.

When m = 34m + 18 = 30

But 8m – 2 = 22

So 8m – 2 is smaller.


k + 6k2

8 – 3k


Expressions inA quick way of saying that the expression uses this variable

An expression in terms of k would use only the variable k combined with numbers and arithmetic signs.Example: 3k + 7 4k2 – 3k + 5


When k = -14, k + 6 = x -14 + 6 = -7 + 6= -1 [k + 6 is negative]

When k = 5, 8 – 3k = 8 – 3 x 5= 8 – 15 = -7 [8 – 3k is negative]

When k = -4, k2 = -4 x -4= –16 [k2 is negative]

I agree with the first two in Opinion , but the value for k2 is wrong.

k2 can’t ever be negative.

There are many values of k for whichk + 6 and 8 – 3k are negative.

But k2 will always be positive whatever the value of k.

Remember that two negative numbers multiply to give a positive number.

Back Forward Cont/d More Opinion 2 Answer Answer 1 Answer 2 Answer 3Q 1 Q 2

1. Find values of k for which each of these expressions is negative.

2. The three expressions are to be put into order. Find a value of k for each one that gives it the middle value.

Here are three expressions in k.Remember that k2 means k × k.Also k means × k.To work this out find half of k, or k ÷ 2.

Use graphing software or graph paper to draw graphs of the three expressions above. Superimpose all three on the same graph.

Look at where the graphs cross. Can you see why each expression can have the middle value?

What are the critical values of k when the order of the expressions changes?

When k = 4, k + 6 = × 4 + 6 = 2 + 6= 8 Middle

When k = 4, 8 – 3k = 8 – 3 × 4= 8 – 12 = –4 Smallest

When k = 4, k2 = 4 × 4= 16 Largest

When k = 2, k + 6 = × 2 + 6 = 1 + 6= 7 Largest

When k = 2, 8 – 3k = 8 – 3 × 2= 8 – 6 = 2 Smallest

When k = 2, k2 = 2 × 2= 4 Middle

When k = 1, k + 6 = × 1 + 6 = + 6= 6 Largest

When k = 1, 8 – 3k = 8 – 3 × 1= 8 – 3 = 5 Middle

When k = 1, k2 = 1 × 1= 1 Smallest


House points

Achievement: 3 pointsVoluntary service: 5

pointsNo homework: –2 points


Make a spreadsheet to help the House Leader keep track of the total points.

Mike goes to Avonford Academy.He finishes the term with 34 points. He has volunteered 3 times.

How many times could he have missed his homework?

Menu Opinion 1

a is the points awarded for achievement.v is the points awarded for volunteering.h is the points taken off for no homework.

a is the number of achievement awards given.v is the number of volunteering awards given.h is the number of times homework is missing.

Back More Opinion 2 Answer Opinion 1 Opinion 2 AnswerQ 1 Q 2

1. What do the variables a, v and h represent?

2. Jo has volunteered 7 times this term. She thinks there’s been a mistake because her term total is zero points. What do you think?

Avonford Academy awards house points for achievement and voluntary service. House points are taken away if homework is not finished. The expression 3a + 5v – 2h is used to calculate the number of points gained

Jo has gained 5v points for volunteering.This is worth 5 x 7 = 35 pointsThat’s an odd number.If 2 points are taken off for every missing homework it’ll never get to zero. The lowest it could get to is 1 point.

Opinion is wrong because she could have gained some extra points for achievement.That could bring her total up to an even number before any homework points are taken off.

Some possible answers are:a = 5, v = 7, h = 15Points = 3 × 5 + 5 × 7 – 2 × 25

= 15 + 35 – 50 = 0

a = 25, v = 7, h = 55Points = 3 × 25 + 5 × 7 – 2 × 55 = 75 + 35 – 110

= 0A student is given 3 points for every achievement. So the total points for achievement is 3 × the number of awards.a is the number of achievement awards given.v is the number of volunteering awards given.h is the number of times homework is missing.


Editable Teacher Template

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Information

1. Task – fixed

2. Task – appears

Vocabulary

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Q1Opinion 2

Q1Answer

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combining variables – developing understanding mastering mathematics © hodder and stoughton 2014...

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