I submitted these problems toOpen Middle. Check out their site for many more.

# About

Dan Meyer introduced us to the idea of “open middle” problems during his presentation on “Video Games & Making Math More Like Things Students Like” by explaining what makes them unique:

- they have a “closed beginning” meaning that they all start with the same initial problem.
- they have a “closed end” meaning that they all end with the same answer.
- they have an “open middle” meaning that there are multiple ways to approach and ultimately solve the problem.

Open middle problems require a higher depth of knowledge than most problems that assess procedural and conceptual understanding. They support the Common Core State Standards and provide students with opportunities for discussing their thinking.

Some additional characteristics of open middle problems include:

- They often have multiple ways of solving them as opposed to a problem where you are told to solve it using a specific method. Example
- They may involve optimization such that it is easy to get an answer but more challenging to get the best or optimal answer. Example
- They may appear to be simple and procedural in nature but turn out to be more challenging and complex when you start to solve it. Example
- They are generally not as complex as a performance task which may require significant background context to complete. Example

We hope you enjoy these problems. Please be sure to send us any ideas for problems we can add.

**Pythagorean Theorem:**

Fill in the empty blanks so that you create two points equidistant from (4,-1). You can use whole numbers 1 through 9, but can only use a number once.

( __ , __ ) and ( __, __ )

**Positive Correlation:**

Create a table of 10 ordered pair that creates a positive correlation (that is not linear):

X:_________________________

Y

Graph your data:

Is your correlation strong or weak? Explain how you know.

Give an example of what this data might represent.

**Negative Correlation:**

Create a table of 10 ordered pair that creates a negative correlation (that is not linear):

X:_________________________

Y

Graph your data:

Is your correlation strong or weak? Explain how you know.

Give an example of what this data might represent.

**Parallel Lines:**

Directions: Fill in the empty spaces so that you create two distinct parallel lines. You can use whole numbers 1 through 9, but can only use a number once.

___ x + ___ y = ___

___ x + ___ y = ___

**Perpendicular Lines:**

Directions: Fill in the empty spaces so that you create two distinct perpendicular lines. You can use whole numbers 1 through 9, but can only use a number once.

___ x + ___ y = ___

___ x – ___ y = ___

It’s sad to admit it but after getting an Engineering degree many years ago, I’ve forgotten most of these concepts and would be challenged to get even half of these problems correct. This clearly demonstrates the “you’ll lose it if you don’t use it” saying.