Distributing Expressions Lesson 12 Distributing Expressions

Let’s Take Notes on Two Methods for Distributing Tape Diagram – Make tapes to represent the expression. 3(4a+2c) That means I have 4a +2c three times. Make tapes to represent. a c 12a +6c

2nd Method Area Model – Make a rectangle and break apart the expression. Use length x width to fill in box. 3(4a + 2c) 4 a 2c 3 12a 6c

This is called Distributing the 2 or handing out the 2 5 3 This picture represents (5+3) If I want to represent 2(5+3) I would need to add another 5+3 5 3 Now I have 5 + 3 and 5 + 3 or 2 groups of 5+3 I can write that 2 ( 5+3) 2(5+3) = 2x5 + 2x3 This is called Distributing the 2 or handing out the 2 2(5+3) = 2 x 5 + 2 x 3 The 2 was DISTRIBUTED

If I wanted to double the expression I would need another 3x + 2y 2 groups of 3x +2y or 2(3x + 2y) 6x + 4y 2(3x+2y) = (2 · 3x ) + (2 · 2y) 6x + 4y (expanded form) This is called the DISTRIBUTIVE PROPERTY

3x + 2y 2 2 · 3x 6x 2 · 2y 4y 6x + 4y Using this model, we can now write the equivalent expression. We do this by calculating the area of each of the rectangles. Remember: a = lw

Now Let’s Try a Few Problems in your Notebooks

Both models prove that 2 (p+ z) =2p+ 2z p + z 2 2p + 2z 2 (p + z) Create two models for the expression and then write an equivalent expression (expanded form). p + z p z 2 2 · p 2p 2 · z 2z p z 2p + 2z Both models prove that 2 (p+ z) =2p+ 2z

Both models prove that 3 (2a + 3b) =6a + 9b 2a + 3b 3 6a + 9b Create two models for the expression and then write an equivalent expression (expanded form). . 2a + 3b a b 3 3 · 2a 6a 3 · 3b 9b a b a b 6a + 9b Both models prove that 3 (2a + 3b) =6a + 9b

Let’s try these using the models!! 5 (a + 7b) 2) 2(3p + r) 3) 6(4 +b) 5a + 35 b 6p + 2r 24 + 6b

Try these using the models on your OWN!! 4 (g + 7b) 2) 5(3e + t) 3) 2(7 +p) 4g + 28b g= 5 b= 2 15e + 5t e= 3 t= 6 14 + 2p p= 2 Let’s Evaluate these Expressions in our Notebooks to see if they are equivalent!

Let’s try a few without using models!!! 4 (4g + 3y) = 16g + 12y 2a (b + 10) = 2ab+ 20a

Distributive Property and Factoring Expressions Lesson 11 Distributive Property and Factoring Expressions

2x5 + 2x3 5 + 5 3 + 3 5 3 This picture represents 5+5 and 3+3 Or I could write it 2x5 + 2x3 Can I change the order of these numbers and it still represent the same amount? YES!!!! 5 3 I can write that 2 ( 5+3) Now I have 5 + 3 and 5 + 3 or 2 groups of 5+3 2x5 + 2x3 = 2(5+3) If you add up the bars in either picture the total is 16. If you follow pemdas in 2(5+3) 2(8) 16 You still get 16 2(5+3) = 2 x 5 + 2 x 3 This is called the DISTRIBUTIVE PROPERTY

Let’s try Another Example

2x9 + 2x4 9 + 9 4+ 4 9 4 This picture represents 9+9 and 4+4 Or I could write it 2x9 + 2x4 Can I change the order of these numbers and it still represent the same amount? YES!!!! 9 4 I can write that 2 ( 9+4) Now I have 9 + 4 and 9 + 4 or 2 groups of 9+4 2x9 + 2x4 = 2(9+4) If you add up the bars in either picture the total is 26. If you follow pemdas in 2(9+4) 2(13) 26 You still get 26 2(9+4) = 2 x 9 + 2 x 4 This is called the DISTRIBUTIVE PROPERTY

Now Let’s Try A Problem with Variables

YES!!!! So a+a + b+b could be written as 2a+2b or 2(a+b) This picture represents a+a and b+b Or I could write it 2a + 2b Can I change the order of these numbers and it still represent the same amount? YES!!!! a b Now I have a + b and a + b or 2 groups of a+b I can write that 2 ( a+b) So a+a + b+b could be written as 2a+2b or 2(a+b) If you look at 2a +2b you are really finding what they have in common (GCF) 2 and taking that and putting it in FRONT and then writing the rest of your expression 2a + 2b = 2(a+b) This is called the DISTRIBUTIVE PROPERTY

Finding the GCF for Two Expressions and Writing it as ONE EXPRESSION

Look at: 3f and 3g 3 What do they have in common (GCF)? Rewrite the expression taking out the 3(GCF). 3(f+g) 3f + 3g = 3(f+g) Look at: 4p and 4h What do they have in common(GCF)? Rewrite the expression taking out the (GCF) 4(p + h) 4p + 4h = 4(p+h) 4

What happens if it DOES NOT look like they have anything in common Look at: 6x + 9y I will have to break 6x and 9y apart 2 • 3 • x + 3 • 3 • y Now, you can see they have a 3 as the GCF So, let’s take out 3 the GCF 3(2x +3y) 6x + 9y = 3(2x + 3y)

3c + 11c What do you think the GCF is? C So, let’s take out the GCF C(3+11) 3c+11c = c(3+11)

Try this one on your own.. 24b + 8 GCF is 8 8(3b+1) 24b + 8 is equal to 8(3b+1)

Let’s try the Rest of the Exercises in Your Packet