Translation to Old School Language: Understand Combining Like Terms

Goal: SWBAT Explain how two or more equivalent expressions can represent different or similar aspects of the same situation.

Standard 7.EE.2 “Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.”

Translation to old school language: Understand combining like terms.

What do they need to know or what will they learn?

Background ideas - they have seen these in the 6th grade core. They do not need to remember these words, but they need to know how to use these properties.

●  Commutative property

●  Associative property

●  Distributive property

●  Order of operations

These are the vocabulary words they will know how to use by the end of this lesson cycle.

●  Terms

●  Coefficient

●  Like-terms

●  Equivalent

●  Simplify

●  Expression

Develop

Regina’s Logo

1 2 3 4

How many squares will be in the 5th and 6th LOGO?

How many squares will be in the 100th LOGO?

How many squares will be in the nth LOGO?

Please, explain how your pattern works.

Goals:

●  recognizing patterns in the logos

●  finding a way to express those patterns

Justify choice of the task:

●  Open ended, not only one right answer. This allows us to discuss how different expressions can express the same rule

●  Low threshold – all students should be able to answer the first two questions

●  High ceiling – press students for more answers or more representations.

Anticipated responses

●  Table of values and then notice the always adding three to the number of tiles, so they will have that they always add 3. A possible formula might be recursive: next = previous + 3

●  Just come up with the formula

○  Tiles = 3x + 2 with no explanation

○  (press them to see that in the picture)

●  To get the 100th, they might multiply the 10th by 10.

1 2 3 4

●  formula is one x on the top, one x in the middle, one x on the bottom, and then +2 corners = 3x + 2

●  X + X + X + 2

1 2 3 4

●  This does show 3x + 2, but it views the logo as more of a line with 2 at the end.

Useful questions to use while monitoring

●  What is remaining the same from shape to shape? What is changing?

●  Can you show me where the +2 is in your picture?

●  Show me where the 3x is in your picture?

●  [to push the high end kids] Can you do it another way? Or can you find another way?

Solidify

Brigham’s Logo

1 2 3

John, Lexi, Taz, Jacob, and Emily came up with the following formulas. What were they thinking? How do you see each of their equations in the logo?

John: blocks = (2n - 1) + 4

Lexi: blocks = n + n + 3

Taz: blocks = 2n + 3

Jacob: blocks = 2n + 1 + 2

Emily: blocks = 2(n – 1) + 4

Goals:

●  evaluate student thought (putting themselves in someone else’s shoes)

●  making connections between multiple algebraic expressions and seeing that they are equivalent

●  Taz’s answer is “simplified”

Justify choice of the task:

●  It is another representation

●  Growth factor of 2 instead of 3

●  Forced them to try and explain someone else’s thinking

○  Deepen their understanding by going backwards

●  Emily’s answer was an error, how can we help people see their errors and establish that an error is o.k.

Anticipated responses:

●  I can’t see what they’re doing!?!

●  They want to do what they did before on Regina’s. Focusing on their own comfortable method that will match one of the formulas present, and then struggling on the others.

This one is possible for Lexi or Taz - it shows the n on top, n on the bottom, and 3 in the middle.

This is possible for Jacob. The + 2 is the constant both 2 on the ends. The one is in the middle, as are the 2 n’s.

This is possible for John. The (2n-1) [he just knew this was the formula for an odd number OR he saw that the middle was 2 of the n with one overlapped so it is a minus 1] is in the middle. The +4 are on the ends.

We think Emily just wrote the formula for (2n - 1) + 4 incorrectly as

2(n - 1) + 4.

Practice Task:

1. Design your own logo and then come up with two different expressions that represent your pattern.

______’s Logo

For the following problem, please explain why the expressions are equivalent and draw a picture. Simplifying the expressions and setting them equal to each other isn’t good enough! Convince me.

2. 2(x + 2) + 2 = 2x + 6

Simplify the following:

3. a + 3a + 3 + 1 4. b + 2(b + 1)

Goal:

●  gaining flexibility between seeing things as an expression or as a picture

●  leading to being able to see/do/understand simplifications or other representations without the picture

Justify:

●  Students should be somewhat familiar with what a pattern looks like which would allow them to design their own patterns.

●  Students should already be familiar with combining like terms from the develop and solidify tasks which will allow them to recognize the fact that the two expressions are equal-this shouldn’t need further explanation.

Anticipated Responses:

1. Scen’s Logo (Pronounced “Sken.” Short for Scott and Jen cuz we came up with this together.) **Note: Each logo starts with SIZE 1.

For the top logo, the expression is 2n+2n+3. The 2n+2n comes from looking at the top and bottom rows separately. If you look at, say, the bottom row, we split it up into groups of 2. There’s one group of 2 in size 1, there’s 2 groups of 2 in size 2, etc. So, an expression that represents those groups of 2 is 2n, or 2 times the size number. Since we have 2 rows (one on top and one on bottom), just add another 2n to the expression. The +3 part of the expression is in pink.

For the bottom logo, the expression is 4n+3. Instead of looking at the top and bottom rows separately, I looked at them as one group. So, each size was increasing by 4 blocks. An expression that represents that is 4n. The +3 part of the expression is in pink.

2 - 2(x+2)+2 is the same as 2x+6 because you have the same number of x’s in each equation. In my picture, the blue represents the 2(x+2) and the green represents the +2. The purple represents the +6 and the yellow represents the 2x.

Problem 3: a + 3a + 3 + 1 = 4a + 4

Problem 4: b + 2(b + 1) = 3b + 2