In this Warm up students are given 4 identical pattern pieces. Each consist of 3 white tiles and one black tile. They look a little different because they are in different orientations, but I expect them to come to agreement that the ratio of black to white tiles is the same (1:3) for all four because in all of them there is one black tile for every 3 white tiles. They can also prove they are identical, because they are each given four pieces tiles for design with which to "build" their own tile floor. So, they can see by turning them that they are the same.
They are asked to "build" a floor with their four pieces that creates a different design from everyone else in their math family group and then figure out what the ratio of black:white tiles is for the floor they built. Students are expected to discuss with each other, which is a good way to encourage argumentation (MP3). I expect some kids to suggest that the ratio of their floor is 4:12 and I want other students in their group to share their ideas of simplifying.
As I circulate and begin to hear talk of simplifying I may display the sentence frame from the warm up ("for each ___ black there are ___ white") and ask students to use this to help support their argument and convince each other.
Students know where the grid paper is and some will want to extend their pattern. As I see this I encourage them to make this a part of the discussion as well. Does this larger design have the same or different ratio?
In this exploration students are asked to create their own tile Floor design using one of four pattern choices. If students are doing this correctly they should just be repeating the ratio of black to white tiles to reproduce the pattern over and over to create a design. The only trouble they may have is if they start their design in the middle of the space provided. If I see this I tell them to start at one of the edges or they may find their pieces overlapping and not fitting within the space.
I have set up the template so that all the pieces they have to choose from will fit when repeated. A student may think they are supposed to use all the pattern pieces to make their design, but if you make it clear that they are choosing a pattern to repeat over and over, I don't think they will make this mistake. I also encourage them to outline each of their pieces as they draw them so they don't lose track of the pattern. This also helps them think of the ratio as a unit. Some students may want to use the pattern they started in the warm up, which is fine. Allowing them to choose makes them more engaged and gives them more ownership.
Students will finish their floor designs for homework.