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Which is correct? Which is correct?

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Alex and Morgan were asked to solve

Alex and Morgan were asked to solve

Alex’s “combine like terms” way

Morgan’s “combine like terms” way

Alex’s “combine like terms” way

Morgan’s “combine like terms” way

I first combined like terms on the left side of the equation. Then I subtracted both sides by 60y.

First I subtracted 45y on either side; 60y − 45y is 15y. Then I divided both sides by 15 to get the answer.

I first combined like terms on the left side of the equation. Then I subtracted both sides by 60y.

Hey Morgan, what did we learn from comparing these right and wrong ways? 

First I subtracted 45y on either side; 60y − 45y is 15y. Then I divided both sides by 15 to get the answer.

Then I divided both sides by 75 to get the answer.

Like terms contain the same variable or group of variables Then I divided both sides by 75raised to the same power. In to get the answer. order for two or more terms to be “like terms,” their coefficients can be different, but the terms need to have the same variables raised to the same powers. Unlike terms cannot be combined by addition or subtraction. 

* How did Alex solve the equation? * How did Morgan solve the equation? * Why did Alex combine the terms on the left as a first step? * Why did Morgan subtract 45y as a first step? * Which way is correct, Alex's or Morgan's way? How do you know? * Can you state a general rule about combining like terms that describes what you have learned from comparing Alex's and Morgan's ways of solving this type of problem?

* How did Alex solve the equation? * How did Morgan solve the equation? * Why did Alex combine the terms on the left as a first step? * Why did Morgan subtract 45y as a first step? * Which way is correct, Alex's or Morgan's way? How do you know? * Can you state a general rule about combining like terms that describes what you have learned from comparing Alex's and Morgan's ways of solving this type of problem?

3.2.2

3.2.2

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Which is better?

Take-away page
Which is better?

Alex and Morgan were asked to simplify
Alex’s “rewrite the exponent” way 

16

3 4

Alex and Morgan were asked to simplify
Alex’s “rewrite the exponent” way 

16

3 4

Morgan’s “use perfect squares to rewrite the exponent” way

Morgan’s “use perfect squares to rewrite the exponent” way

16

3 4

16

3 4

3

3

16 4

16 4

I rewrote the fractional exponent as 3 times 1/4.

(16 )

1 3 4

(16 )

1 4 3

I rewrote the fractional exponent as 1/4 times 3.

I rewrote the fractional exponent as 3 times 1/4.

Hey Morgan, what did 1 we learn from1 comparing 3 4 (16 these two (16 4 )3 ) ways?

I rewrote the fractional exponent as 1/4 times 3.

I expanded the expression in the parentheses.

(16 ⋅16 ⋅16)

1 4

(2)3

I simplified the expression in the parentheses. Since 24 is 16, I know that 161/4 is 2.

I expanded the expression in the parentheses.

I got 4096. Then I applied the exponent.

(4096)

1 4

8

This is my answer.

I got 4096. Then I applied the exponent.

When deciding which factors to use to rewrite the fractional exponent, be 1 on the lookout for perfect (4096) 4 squares. 

(16 ⋅16 ⋅16) 4

1

(2)3

I simplified the expression in the parentheses.

8

This is my answer.

This is my answer.

8

This is my answer.

8

* How did Alex simplify the expression? * How did Morgan simplify the expression? * What are some similarities and differences between Alex’s and Morgan’s ways? * Which strategy do you think is more efficient for this problem? Why?

* How did Alex simplify the expression? * How did Morgan simplify the expression? * What are some similarities and differences between Alex’s and Morgan’s ways? * Which strategy do you think is more efficient for this problem? Why?

9.3.2

9.3.2

Why does it work?

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3x(5x + 2) + 4(5x + 2)

Why does it work?

Take-away page
3x(5x + 2) + 4(5x + 2)

Alex and Morgan were asked to simplify the expression
Alex’s way 
 Morgan’s way

Alex and Morgan were asked to simplify the expression
Alex’s way 
 Morgan’s way

3x(5x + 2) + 4(5x + 2) 3x(5x + 2) + 4(5x + 2)

3x(5x + 2) + 4(5x + 2) 3x(5x + 2) + 4(5x + 2)

First I expanded the expression using the distributive property.

15x + 6x + 20x + 8
2

(3x + 4)(5x + 2)

First I factored the expression.

First I expanded the expression using the distributive property.

25x + 10x + 10x + 4 Alex, what did+we Hey (5x + 2)(5x 2) learn from comparing these two ways?
2

First I factored the expression.

Then I simplified the expression.

15x + 26x + 8
2

15x + 6x + 20x + 8
2

Then I expanded the expression.

Then I simplified the expression.

25x + 20x + 4
2

25x + 20x + 4
2

Then I expanded the expression.

15x + 26x + 8
2

Then I simplified the expression.

Like expressions enclosed by grouping symbols, such as parentheses, can be combined as like terms are combined.

* How did Alex simplify the expression? How did Morgan simplify the expression? * What are some similarities and differences between Alex's and Morgan's ways? * Is Morgan’s way OK to do? Why or why not?

* How did Alex simplify the expression? How did Morgan simplify the expression? * What are some similarities and differences between Alex's and Morgan's ways? * Is Morgan’s way OK to do? Why or why not?

7.4.3

7.4.3

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How do they differ? Alex was asked to graph the equation y = 2x, and Morgan was asked to graph the equation y = −2x. Alex’s “graph y = 2x” way ! Morgan’s “graph y = -2x” way How do they differ?

Take-away page
Alex was asked to graph the equation y = 2x, and Morgan was asked to graph the equation y = −2x. Morgan’s “graph y = -2x” way

Alex’s “graph y = 2x” way 

y = 2x
I rewrote the equation in y = mx + b form.

y = −2x
y = mx + b y = −2x + 0
y

y = 2x
I rewrote the equation in y = mx + b form. I rewrote the equation in y = mx + b form.

y = −2x
I rewrote the equation in y = mx + b form.

y = mx + b y = 2x + 0
y

y = mx + b y = mx + b In the slope-intercept y = 2x + 0 y = −2x + 0
y

I graphed the yintercept, (0,0) and counted up 2, right 1 and down 2, left 1 to plot other points on the line. I connected the points to draw the graph of the line.

x

x

I graphed the yintercept, (0,0) and counted down 2, right 1 and up 2, left 1 to plot other points on the line. I connected the points to draw the graph of the line.

I graphed the yintercept, (0,0) and counted up 2, right 1 and down 2, left 1 to plot other points on the line. I connected the points to draw the graph of the line.

form of a line 
 (y = mx + b), the y coefficient of x, which is m, indicates the slope. 
x x

Changing the sign of m changes the slope, or the steepness, of the line. When a line has a positive slope, its height increases from left to right. When a line has a negative slope, its height decreases from left to right. 

I graphed the yintercept, (0,0) and counted down 2, left 1 and up 2, right 1 to plot other points on the line. I connected the points to draw the graph of the line.

* How did Alex graph the line given by his equation? How did Morgan graph the line given by her equation? * Can you think of another way that Alex and Morgan could have used to find the graphs of their lines? * What are some similarities and differences between Alex’s and Morgan’s problems? * What are some similarities and differences between Alex’s and Morgan’s graphs? * How does changing the sign of m affect the graph of a line?

* How did Alex graph the line given by his equation? How did Morgan graph the line given by her equation? * Can you think of another way that Alex and Morgan could have used to find the graphs of their lines? * What are some similarities and differences between Alex’s and Morgan’s problems? * What are some similarities and differences between Alex’s and Morgan’s graphs? * How does changing the sign of m affect the graph of a line?

4.5.5

4.5.5