In this lesson students are asked what information they are given by a graph. They determine what each point represents and what information it gives them. It is important for students to see how to use the structure of the graph for information. It is also important for students to make connections between the graph and the data table. Asking them where the information is in the graph and then where the same information is (or how it shows up) in the table and visa versa can be really helpful. Making these connections is really helpful for ELL students because there is so much visual prompting. We can point to something and name it or ask the question to help them develop language with the content.
Students are given a graph with several points representing black and white tiles in a particular floor design. They are also given a blank ratio table. Before I ask students what each point represents we do a little orienting. I start by telling them that the graph is giving us information about two things and ask what are those two things? I ask a student to come up and show us on the screen where he/she found those two things on the graph (axes). Then I point out that each point on the graph gives us specific information about those two things. Students are asked what information is given at each point. "What does each point tell us?"
As I circulate I am listening for students who are still having trouble getting started. I may have them put their finger on one of the points and ask what that is telling us about the number of black tiles (there are 2 of them), and the number of white tiles, etc. As students begin to decipher the information I ask them to draw What the floor might look like at each point.
As students start to point out that there is a "point missing" on the graph I ask them to try to draw what they think the floor might look like at the point that is missing. This helps them "find" the numbers for that point. This is especially helpful for my ELL students as they can transfer between the drawing of the tile floor and the numbers on the axes of the graph. I find asking "quanto blanco?" very helpful as we indicate the number of white tiles in the drawing and then repeating the question while pointing to the point and the axis.
Once students have figured out the numbers (coordinates) for the missing point it is easy for them to record the coordinates for the given points into the ratio table. They also notice that there are still empty spaces on the table. I look for a student who is filling those with more numbers and ask how he/she figured them out (from the ratio or from the line on the graph). I point this out to the class by saying "Jake has noticed that the graph is giving us information other than just the points. What Other information can we get from the graph?" They may suggest continuing the ratio of 2 black tiles for every 3 white tiles or that the "pattern" tells us more. They may suggest that the line crosses the grid at the "corners". Some may point out that we can get the simplified or "broken down" ratio from the graph as well. It is important to validate all observations because it helps them learn to persevere and keep looking.
I start this next section by placing those "missing" points on the graph and asking students what information they give. I know they will quickly tell me, but I have them come up and show me on the graph where they can find that information. I want them to get used to using the information on the axes. Next I place a point at (1, 1.5) and ask them what information this gives us about the numbers of black and white tiles. This is a little tricky for them, because at first they come to think that points can only be placed at the "corners" of the grid. I want them to understand that they can get information anywhere along that line. I expect them to quickly figure out that it represents 1 black tile, but they may have more trouble with one and one half white tiles. If someone figures it out I ask them to come up to the board and show the class where those numbers are on the graph and how they relate to the point. I ask students to discuss in their group and then ask someone to come up and see if they can use the axes to figure how many black and white tiles are represented by that point.
I ask them to go back to their drawing of the floor pieces and see if they can figure out why 1 black to 1 and a half white tiles makes sense. Some students may wonder if it is possible to cut a tile in half. Though this may seem like an off task question it may be worth a tiny detour if it helps them make sense of the possibility. There may be a student with a parent who cuts tile or who recently tiles a floor.
Next I place a new point on a graph and display it under the document camera asking students what information it gives us. I continue placing points (one at a time) that maintain the same ratio as the last and ask what information each gives. When a student suggests that they all follow the same pattern I ask them how they can tell. They may explain that they all line up on the graph or that the ratio is being maintained with the same "up and over" pattern on a graph, or they may point out that all the numbers simplify to the same ratio. I suggest we put the information into a ratio table for this pattern. I ask students to come up and place points to challenge their peers.
Next I place a point that does not follow the same ratio and ask students to discuss briefly in their math family groups whether it belongs in the same table or if we should start a new one. I ask them to explain how they can tell. If they don't mention both the graphical and ratio evidence I will ask
I want them to see the connection between the different representations as well as to practice articulating their observations and explanations.
Students have a few minutes to start their homework graphing a ratio table.docx which is to create a graph from an incomplete ratio table.