Students will create complex visual models of sums of fractions. These puzzled images will encourage students to use fraction-language to describe each other’s creations and to think about the order of operations that are needed to solve their classmates’ fraction puzzles.

*Complex Visual Modeling, Order of Operations, and Deeper Understanding of Rational Numbers*

**Expected Activity Time**

**Puzzling Pics **(20–40 minutes)

**Materials and Prep**

*Puzzling Pics*Student Sheets- iPad with Fraction Mash app
- Wifi access for sharing mashups

**Intro:** Have you ever seen the Rubik’s Cube or other puzzles that are seemingly simple to solve once you learn how they were created?

In this activity, you will create some puzzles with Fraction Mash that appear to be very complex but can be understood and solved in very simple ways when you see how they were put together.

Puzzling Pics: Start by creating a simple puzzle for your classmates. Also solve one of theirs. Then ramp up the complexity to create more puzzles and use your new skills to solve the tougher puzzles also created by your classmates.

**Puzzling Pics **(20–40 minutes):

Open the app, and select “Make A Mashup.”

- Have students start with four very simple images. For example, start with four basic colors: red, blue, yellow and green. Another option is to have one of the pictures be a face and the rest as solid colors.
- Have students create a simple puzzle with the four basic images using these rules: Mashup the first two images using vertical bars and a denominator no greater than 10. Save. Go back to Create, import that first mashup, and then remix with one of the other basic pictures. Change to using the horizontal bar grid and a denominator no greater than 10. Do this one more time using the last basic image and the pie slices grid with a denominator no greater than 10. Each mashup should have some of the components from each image included, and it also helps to keep the grid on at each mashup.
- Once they have created the puzzle that is the result of three remixes, have students exchange iPads and solve each other’s puzzles. A correct solution clearly describes the steps taken to create the image, using fractional sums in the solution.
- Be sure to have students save their puzzles, including the steps they took to make them.
- After the simple introduction, ramp-up the complexity and have students make and solve more fraction puzzles.

Students should be encouraged to describe the puzzles they created and the tricks they learned to reading and solving the puzzles.

Ask students:

- As you created your puzzles, did you intend to make them with a specific degree of difficulty? Was it difficult to find the balance between making the puzzle tough to solve but not so easy that it was boring?
- What were some things that made the puzzles especially difficult to solve?
- What happened if they you only used vertical columns? Or only horizontal rows?

These puzzles can get very complex, so one extension is to simply allow students to take their creations to the most extreme cases possible in the time allotted. Are they still solvable?

OR:

Explore the ways in which the creation of these visual models could be explained using other operations of fractions? Could some of the states of the creations of these puzzles be models for multiplication of two fractions? As students explore the grids and operations, delve deeper into the complexity that Fraction Mash can create with the combinations of different grids, denominators and remixes.

Start with four simple pictures and create a visual puzzle for your classmate to solve. Switch with a classmate and solve a puzzle that they designed.

**To Do:**

- Start with four very basic pictures in your camera roll. For example, pictures that are just solid colors work well for this activity. (Note: Black isn’t the best choice because the grid lines are hard to see with black.) Bright colors work well.

- Choose the first two pictures and import them into Fraction Mash. For the first mashup use vertical columns as the grid and choose a denominator that is less than or equal to 10. Tap Combine, keep the grid on, and save your first mashup to the camera roll.

- Go back to Create and import your newly saved mashup into the left frame. You will be mashing with another basic picture in the right frame. This time, set horizontal bars and a denominator that is less than or equal to 10. Tap Combine, keep the grid on, and again save your mashup.

- Go back to Create and import the last mashup you just made into the left frame and mash with the fourth basic picture on the right. This time, use pie slices as the grid and again a denominator less than or equal to 10. Combine, keeping the grid on, and save. That is your first puzzle to hand to a classmate for them to solve.

**Reflection Questions:**

- Describe your process for solving the first puzzle created by one of your classmates.

- Look at the first puzzle you made. Describe the puzzle using fractions to account for each piece in the picture. For example, 1/3 of Image 1 + 5/10 of Image 2 + 1/6 of Image 3 = The Puzzle. In your final puzzle, are there parts that are hard to describe?

On the number line from 0 to 1, make tick marks representing the amount of each image in your puzzle, and clearly label each tick.

- As you created more difficult puzzles, how did you strike the balance between making them hard but not too hard?

- How many remixes do you think could go into the creation of a puzzle if the goal was for someone to be able to solve it? Describe the ways in which you are able to deconstruct the puzzles and solve them. (Optional: What if the goal was just to create something beautiful or artistic. How many remixes do you think could take place?)