Today students will look at different forms of graphs. All the graphs they have used to this point have been vertical bar graphs. Today we will turn the bar graph on its side and discuss interpreting the most common (mode) and the middle value (median) on the graph. Students will use this data to complete math problems with an unknown.
I begin by asking students to divide a piece of paper into 4 squares by folding the paper in half, and in half again. I ask them to label the first square "Pets", the second "Dogs", the third "Cats", and the last "Other". I ask them to write on the paper the number of pets they have in the first square. I tell them that zero is as important as 10 in this process. Next I ask students to write the number of dogs in the second square, the number of cats in the third and the number of other pets in the fourth. Now I ask students to cut the paper into the 4 squares and at their tables to put all "Dogs", "Cats" and "Others" into 3 separate piles. I ask them to hold onto the "Pets" piece.
I tell student that we are going to find the middle number of pets that students in our class have. The middle is the median, the number that separates our data into two groups where half will have more than the median and half will have less.
I ask students to hold up their papers and to line up against the bookshelf in order from least by the door to greatest by the board. I tell them there may be more than one of each number and that is ok, they will line up next to the people who have the same number. I tell them I want to see if they can find a way to do this silently. I give students a chance to line up. Next I go along the line reading the numbers aloud and asking students if they need to make any changes. Once we have a line that is in order, I tell them that we are going to find the middle number. but first we are going to find the most common number. I go down the line starting at zero and tell how many of each number. I record it on the board where students can see. I ask them which is the most common number of pets in our room. I tell them we call this the mode.
Now we will find the middle. I begin by asking 1 student at each end to go sit down. I repeat the process, taking 1 student from each end of the line, until the middle is reached. We look at whether the middle number and the most common number are the same. I say, "what was the middle number of pets?" (2). "What was the most common number of pets? "(2). "Are they the same number?" (yes). "Do you think they will always be the same number? Thumbs up if you think yes, or thumbs down if you think no." "Well, lets pretend that we only had 7 people in the room. Three people had 1 pet each, 2 had 2 pets, 1 had three and 1 had four pets." I write 1,1,1,,2,2,3,4 on the board. Now which number is the most common? (1) Which number is in the middle? Would someone like to cross out the end numbers and find the middle for us? (I have a volunteer come up and cross off the ends to find that 2 is in the middle.) Are the most common and the middle the same this time? (No). "Right, sometimes they are the same and sometimes they are not, but if you put the numbers in order you can check it out and see."
Next I tell students that at their tables they will find out the same thing about either dogs, or cats or other pets. They will arrange the papers in order on their desks, find and record the most common number and then find the middle number by taking away the ends until they reach the middle. They should record the middle number as well.
I give each table all of the papers for their category. I watch as they try to arrange and determine the mode and the median. We share this information and then I collect the small pieces of paper to use for graphing.
I give each child a piece of graph paper and ask children to hold it horizontally . I label the parts of my graph with the animal names down the side, and the numbers across the bottom. As I do each part, I have students do their own graphs. I check that students are doing the set up correctly. I ask them to fill in the graph to match the numbers I write on the board (I record the total number of pets, dogs, cats, and other from the papers the students filled out at the beginning of the lesson). Students now use the data I have presented to independently fill out their graphs.
Using the Data
Once the graphs are complete and we also have a record of the median and mode for each category that students discovered in their small groups, and I have posted on the board as they made their graphs, I pose some questions verbally that will require students to write an equation. I expect that students will write it in traditional (6 + 3 =) form , missing quantity or difference (6 + __ = 9), or mountain form (the 9 at the top of a triangle and 6 + __ at the bottom) and then solve the problem. I watch as students attempt to use the data to answer the questions I ask. I am interested in how they go about solving a missing quantity/difference problem. Students should be able to take apart the question, determine what is needed and solve the problem using the data they have in front of them. (MP1). One example is if there are 5 people with dogs and 12 people with cats or dogs, how many people have cats? 5 + __ = 12.