In this lesson students learn what proportional growth looks like in a table. Students use their knowledge of scaling ratios to calculate the number of each color tile needed to cover certain areas using a given ratio. They are then asked to organize that data in a table that makes sense to them. Students then share the way they set up their table with the class. Having students create the table themselves and see how others organized the data differently helps them make sense of and interpret the data in any table. (mp1) This is also a good way of familiarizing students with the mathematical tools that are available to them. (mp 5)
warm up setting up a table.docx This warm up shows students a black and white tile design made up of four tiles (three white and one black). They are told that it takes all four tiles to cover 1 square foot and are asked to find the dimensions of each single tile.
Some scaffolding may be needed to clarify the question. See the notes warm up setting up a table notes.docx on clarifying the warm up. Students may get stuck on the term "dimensions" and not understand that the question is asking for the side measurements. They may give an answer of 1/4 because they are thinking of the area rather than side length. It is really helpful to show on the diagram what is being asked for. After this has been clarified they may have an answer of 1/2 of a foot. Asking if we could call 1/2 of a foot something else may get them to say 6 inches. I might just be silly and say "I don't go around saying that something is half of a foot long...what would I say instead?"
This warm up helps reinforce the idea of proportional growth by showing students the relationship between the area being covered and the repeating pattern of tiles.
In this exploration warm up setting up a table notes.docx students "collect" or generate some data by extending the tile design to larger areas and figuring out the number of black, white, and total tiles needed to cover 2 square feet and 3 square feet.
Once students have determined the numbers, ask them to organize the information in a table that makes sense to them. Having students come up with an organizational system themselves gives them more familiarity with the purpose of a table and helps them develop a better sense of how it can be used as a tool. They have seen tables before and I don't show them one here. I really want to see what they come up with on their own. Of course, they can look at what the other members of their math family group are doing as well. I may start by showing what the information looks like all mixed up and disorganized and ask how we could organize it to better keep track and make sense of what the numbers represent.
Students come up and share how they organized the information with the class. Make sure that a variety of methods are shared. Some students may put headings, others may not. Some students may use pictures, some may use numbers. Some students will organize them vertically and others horizontally. As I circulate I like to make a note of students who have created a certain type of graph. That way if no one has shared this type of graph I can ask them to come up. I like to jot down a couple of people in case one is too shy to share, but I try to encourage them by saying things like "Molly came up with an interesting way that's different from the ones we've seen so far". However, if she still doesn't want to come up and explain I can put her table under the document camera and ask the class if anyone can see how Molly organized her data and explain it. Having students explain each others work can be equally beneficial in helping them make sense of the table.
Ask students to figure out how many black, white, and total tiles would be needed to extend this design to cover floors with different areas. Wrap up table.docx Tell them to show the information either in the table they made or one of the other tables they saw. Allowing them to choose helps them evaluate which tool (table) is the clearest or most useful.