Directions: Using the digits 1 – 9, at most once each, what values of * a,b,c,d* and

*will make the following statement true?*

**x**

Will your values of a,b,c and d work for all values of x? Why or why not?

Directions: Using the digits 1 – 9, at most once each, what values of * a,b,c,d* and

Will your values of a,b,c and d work for all values of x? Why or why not?

OK, the MTBoS has really influenced the way I approach teaching and assigning problems. Coming back from Christmas Break, I needed a different way to get students thinking about math again, and found that thanks to Math = Love in the form of Gemini Puzzles.

What are Gemini Puzzles? In short, they are equality statements that are missing any mathematical symbols. Here is an example:

The thing I love about this is- I put these two on the whiteboard for students to see when they first walk in and instead of asking what we are doing today or talking about what they did last night, they started thinking about what was going on. The other great thing, I didn’t have to tell them what to do, they started playing around with the numbers to get it done.

When I asked my students what they were doing, there was a common theme:

Which is great until I ask students about the name of the problem, Gemini. I then get answers about twins and I tell them that’s relevant to the problem. They are given two equivalence statements, and they have to be twins. In other words, anything they add to one statement * has to be the exact same* as what they add to the second.

Then students start working in a stream of consciousness and they have this as an answer:

It doesn’t take too long for a student to say that there’s problem with this. Students are thinking “one plus one is two, times two is four”, but they are not properly showing that. Then the talk of Order of Operations hits, and students realize the first statement is only 3. By throwing in another set of symbols they find the correct solution:

By providing students with two equivalence statements that need the exact same symbols to make them true, you are giving students a ** reason** to think about the order of operations and how they interact. I have loved listening to student talk when working on these, and I even had students proudly come up to me with different solutions to a problem. They are amazed that there can be different ways to implement mathematical operations and get the same result (see if you can find which one they discovered!)

Gemini Puzzles are a great activity for the classroom, I know I’ll be using them from now on, I hope you enjoy them as well.

I currently have a range of students from 6th to 12th grade in my classes, with various mathematical skills ranging from 4th to 12th+, and these puzzles have engaged and challenged them all. It’s a great thing to see.

Using the Integers 0-9 (without duplication), provide four sets of points that represent two distinct lines. These lines can be written as two linear equations. Then provide a fifth point that represents the intersection (or solution) of those equations.

Line 1: (__, __) and (__, __)

Line 2: (__, __) and (__, __)

Solution (__, __)

Write the equation of both lines.

Using Integers (without repeating any number), fill in the boxes to create the following types of Linear Equations:

**Linear Equation with one solution:**

**Linear Equation with no solutions:**

**Linear Equation with infinite solutions:**

What is the smallest Integer interval that could produce all three equations?

In general, describe how to solve a liner equation that is in the slope-intercept form.

In general, describe how to solve a linear equation that is in standard form.