# Identifying Median and Mode #WithMathICan

#### 1 Hook

Have each student play the NASA Aeronautics Memory Game and record their time (in seconds) on a Google Doc (for a low-tech option, have students record their times on the white board/chalkboard).

*Here you'll want to quickly assess the data and add in some extra data points if necessary. For example, if the dataset has no mode (because no two times are the same) add in some repeating numbers, or if there is an even number of data points you can add in another so that the median is a whole number. You could say these other data points represent other teachers in the school.

Tell students that all together these numbers are a set of data that represents the memory match times of the class. Ask students if there is any particular order in which they think the data should be arranged and why. They will likely suggest low-high pretty immediately (which is how it should ultimately be arranged). However, discuss any and all options students present, asking questions like:

"What are the benefits of arranging the data this way?" "What are some potential drawbacks of arranging the data this way?"

#### 2 Introduction to Median and Mode

Ask students which data point they would choose to represent the class memory match times to the whole school. Have them write down their answer and their reasoning. Be sure to stress that there is no wrong answer here, as an argument could be made for any data point. Give students a couple minutes to think through this.

Have the class come back together and begin calling on students to share out the data point they picked and their reasoning. Some example questions to help guide the discussion here could be: "Why did you choose that?" "Did anyone else pick the same number but for a different reason?" "Would it be incorrect to pick _______? Why or why not?"

Hopefully during this portion students will identify the mode and the median. Once the discussion wraps up, circle back to the mode and introduce it as a mathematical concept. For example, "some of you said you would choose ____ to represent the class because it is the number that appears most frequently. In math, we call this the mode." *If no student identified the mode during the discussion, that's okay! You can introduce it by saying "What if I picked ____? Why might I do that?"

Here it could be helpful to do a bit of guided practice. Have some datasets on hand to show students and as a class identify the mode in each one. Perhaps use some datasets that have more than one data point that is repeated so as to stress that the mode is the number that occurs *most* frequently. This could be a good place for students to make some safe mistakes - for example: "I like that you chose ____ because it occurs repeatedly, but is there another number that is repeated even more?"

Once you feel students have a pretty good handle on the mode, return to the original class dataset and identify the median in the same way - "some of you picked ______ to represent the class because it's exactly in the middle. In math we call this the median."

Here it could also be helpful to do some guided practice using the other datasets you have on hand.

#### 3 Independent Practice

Give students a few datasets and have them identify the median and mode of each on their own. Remind students that it is absolutely okay to make mistakes here and that mistakes are a crucial part of learning any new concept. Once they are all finished, have them check their answers with a partner.

As students are working independently, circle around and keep an eye out for students who are making common errors (i.e. identifying the mode as the second most frequent number or leaving out a data point when ordering them from low-high, therefore resulting in an incorrect median). Remember which students made these errors and at the beginning of the wrap up discussion you can celebrate them for making your "favorite mistakes" - mistakes that are common and that help us learn by reminding us to attend to precision. When presenting these "favorite mistakes" you can ask the class "what do I like about what _____ did here? What would I change?" This can draw attention to easy errors to avoid while also normalizing mistakes as part of the mathematical process and necessary for learning.

#### 4 Wrap Up Discussions

Bring students back together for a full class discussion about finding the median and the mode. Some example questions could be:

"Which problem was the most difficult? Why? Can someone explain how they thought through it? Did anyone approach this differently?"

"Which do you think is easier to find? The median or the mode? Why? Does anyone disagree? Why?"

"Is it possible for the median and the mode to be the same number? Why or why not?"

"Did anyone make a mistake during independent practice that they caught or that their partner caught? Can you explain what you did?"

"When might you want to use the median to represent a dataset instead of the mode? Why? When might you want to use the mode to represent a dataset instead of the median? Why?"

Finally, direct the attention back to the original class dataset of memory match times. Ask the students if they would prefer to have their class data represented by the median or by the mode and call on students to explain. This could lead to a great discussion about data manipulation and representation - perhaps the median is a much lower time than the mode and would therefore make the class look more impressive, or vice versa. Remember to stress that there is no wrong answer here!