§1.1 Expanding Brackets

• Wiskunde
• Middelbare school
• vwo
• Leerjaar 2

C1 Arithmetic with Letters

Mr. Fintelman (FNL)

Wednesday September 4th

2024

Date	Wednesday September 4th 2024
Paragraph	§1.1 Expanding Brackets
Pages from the handbook	Pag. 12-14
Subject	Rules: a(b + c) = ab + ac (a + b)(c + d) = ac + ad + bc + bd

Today is the day...

I can already…

1. … simplify products with letter variables.
   
2. … simplify additions and subtractions with letter variables.
   
3. … collect like terms to simplify letter variables.
   
4. … simplify letter variables by applying the order of operations.

Prior Knowledge

Examples of letter variables

After this lesson, I can…

1. … simplify an expression with the following rule: a(b + c) = ab + ac
   
2. … simplify an expression with the following rule: (a + b)(c + d) = ac + ad + bc + bd

Goals

Farmer Edward - Area of the land and the barn

Farmer Edward - Area of the land and the barn

Area I

Area II 

Total Area

=AB⋅AE

=AB⋅BE

=AB⋅AE+AB⋅BE

Farmer Edward - Area of the land and the barn

Area I

Area II

Total Area

=a⋅b=ab

=a⋅c=ac

=a⋅b+a⋅c=ab+ac

Farmer Edward - Area of the land and the barn

Area I

Area II

Total Area

=a⋅b=ab

=a⋅c=ac

=a⋅(b+c)=ab+ac

More examples

2b(5a−7)=

=10ab−14b

=2b⋅5a+(2b⋅−7)

=10ab+(−14b)

Explain

In this rule we multiply the factor outside the brackets with both factors in the brackets.

So in this example we multiply 2b with 5a and 2b with -7.

If you find it difficult, it might help to 'split' the parts with brackets, this often helps with the rule of 'two negatives makes it positive'.

More examples

−(3p−2q)=

=−3p+2q

=−1⋅3p+(−1⋅−2q)

=−3p+(2q)

Explain

Again you multiply factors, but in this case we see an 'invisible -1'.

This is another example in which it might help to 'split' the parts with brackets, because negative numbers often lead to mistakes for starters.

More examples

8−3(2a−b)=8+(−3⋅2a)+(−3⋅−b)=

=8+(−6a)+(3b)

=8−6a+3b

=8+(−3⋅2a)+(−3⋅−b)

Explain

Here we see a longer example, in which I 'split' the (-3) to make sure that I won't make mistakes later.

This is a tricky example.

According to the book

2b(5a−7)=10ab−14b

−(3p−2q)=−3p+2q

8−3(2a−b)=8−6a+3b

Explain

So if we go by the steps the book tells you, you'll see that the book doesn't show most steps.

I don't like that, this is one of the reasons why students often feel lost in this particular paragraph.

Farmer Charles - Area of the land and the barn

Farmer Charles - Area of the land and the barn

Total Area

Area I

Area II

Area III

Area IV

Total Area

=a⋅b=ab

=a⋅2=2a

=ab+2a+3b+6

=3⋅b=3b

=3⋅2=6

=(a+3)(b+2)

More examples

(a+2)(b+5)=

=(a⋅b)+(a⋅5)+(2⋅b)+(2⋅5)

=(ab)+(5a)+(2b)+(10)

=ab+5a+2b+10

Explain

This distribution is a little more, but not as different.

Now there are simply four factors, instead of three.

More examples

(c+2)(d−4)=

=(c⋅d)+(c⋅−4)+(2⋅d)+(2⋅−4)

=(cd)+(−4c)+(2d)+(−8)

=cd−4c+2d−8

Explain

But it is such longer distributions that make me want to use brackets to make sure I won't miss negative numbers.

More examples

(x+2)(x−3)=

=(x⋅x)+(x⋅−3)+(2⋅x)+(2⋅−3)

=(x​2​​)+(−3x)+(2x)+(−6)

=x​2​​−3x+2x−6

=x​2​​−x−6

Explain

If these factors share variables, you will often see that you need to simplify a bit more, by either adding or subtracting like terms.

According to the book

(x+2)(x−3)=

x​2​​−3x+2x−6

=x​2​​−x−6

(c+2)(d−4)=

cd−4c+2d−8

(a+2)(b+5)=

ab+5a+2b+10

Explain

Here it is in short.

Worktime

You work neatly by…

1. … reading the theory (again) before asking a question to your classmate.
2. … raising a hand before asking a question to the teacher.
3. … if the teacher is busy, remember your question and move on.
   

Help:

Exercises: 2, 4, 5, 8 and 9

Assignments:

Pages: 10-11

Exercises: 4, 5, 6 and 9

Assignments from the planning of WEEK 1:

Extra:

Exercises: 7

Now I can...

1. … simplify an expression with the following rule: a(b + c) = ab + ac
   
2. … simplify an expression with the following rule: (a + b)(c + d) = ac + ad + bc + bd

Reflection

Date	Monday September 9th 2024
Paragraph	§1.1 Expanding Brackets
Pages from the handbook	Pag. 15-16
Subject	Simplify expressions that contain brackets

Today is the day...

I can already…

1. … simplify an expression with the following rule: a(b + c) = ab + ac
2. … simplify an expression with the following rule: (a + b)(c + d) = ac + ad + bc + bd

Prior Knowledge

Prior Knowledge - What mistake was made?

(x+2)(x−3)=

3x​2​​+x−6

(c+2)(d−4)=

−cd−4c−2d−8

(a+2)(b+5)=

ab+10ab+10

After this lesson, I can…

1. … simplify expressions that contain brackets.

Goals

Simplify expressions

Exercise 17 (Homework)

Determine the area of the blue shape.

Exercise 17 (Homework)

Determine the area of the blue shape.

(6a+1)(4a+2)−(5a−4)(a+3)=

=24a​2​​+12a+4a+2−(5a​2​​+15a−4a−12)

=24a​2​​+12a+4a+2−5a​2​​−15a+4a+12

=19a​2​​+5a+14

Conclusion after this paragraph

Worktime

You work neatly by…

1. … reading the theory (again) before asking a question to your classmate.
2. … raising a hand before asking a question to the teacher.
3. … if the teacher is busy, remember your question and move on.
   

Help:

Exercises: 2, 4, 5, 8, 9, 11, 13, 15, 17, 18 and 19.

Assignments:

Pages: 12-16

Exercises: 4, 5, 6, 9, 11, 13, 15, 17, 18 and 19.

Assignments from the planning of WEEK 2:

Extra:

Exercises: 7 and 20.

Now I can...

1. … simplify an expression with the following rule: a(b + c) = ab + ac
   
2. … simplify an expression with the following rule: (a + b)(c + d) = ac + ad + bc + bd
3. … simplify expressions that contain brackets.

Reflection