Multiply polynomials
Students will apply the understanding of multiplying polynomials to determine the appropriate area of a picture frame to fit a specific object.
Essential Question:
How much backing is needed for this mural and frame?
Many of us have tried to "solve" a Rubik's cube. In fact, they have machines that can do it for us. In the video above, a group of people tried to work with the cubes a little bit differently. They un-solved the Rubik's cubes to create an impressive mural of Albert Einstein. These folks didn't just solve the cube - they built a frame that fit it perfectly. The question is, how did they know that the mural would fit perfectly inside the frame (without measuring with a ruler ahead of time)? Let's find out!
Warm-Up Activity
Sketch the frame that you think would fit the mural in the picture. Knowing that there are 999 Rubik's cubes being used, as well as context clues from the video, estimate the exterior dimensions of the frame. Each dimension of a Rubik's cube is 5.7 cm. For a little bit more information, click here. Here's an example of what your image may look like (Just don't make fun of the drawing!).
Determining the exterior dimensions of the frame
Before we determine the exterior dimensions of the frame, we need to find out the area and dimensions of the mural itself. To do this, we need to do some backwards math. For the sake of this problem, we will assume that the width of the frame is 5 cm.
- Knowing that there are 999 Rubik's cubes, we need to find out how many cubes are along the base and height
- Doing this, you should get these dimensions
- Now that we know the amount of cubes being used for each dimension, we can set up our equation.
- 999x^2 represents the area of the mural itself
- The 640x represents the area of the strips of frame along each side of the mural
- The 100 represents the 4 corners of frame that do not have a side adjacent to the mural
- Using Desmos, plug the value of x into the equation
What your solution means
Interpreting our solutions and justifying are two very important aspects of solving a problem. Keep in mind that your solution is in square centimeters (or centimeters squared).
- What would the area be in square meters?
- What does it mean to have that much area? It's a large number, but does that mean it will be a large backing?
- For a challenge, convert the area to square inches. Check your work here.
Individual Practice
Click here for additional practice in multiplying polynomials. This is a matching activity that reinforces the concept of multiplying. For each problem, use the method that you feel is most appropriate.
Additional Practice (Extension)
Now that you've determined the area needed to provide backing for the Albert Einstein mural, let's step it up a little bit. Below is a time-lapse video of a group who created a Martin Luther King, Jr. mural out of 3x3x3 Rubik's cubes. Use the estimation that the frame width is 12cm. What is the area needed to provide the backing for this incredible mural?