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Misconceptions surrounding percents are prolific in math. Students interact with percents in their daily lives and will most likely have formed their own ideas about percents. Moreover, students are often taught the most you can have of a quantity is 100 percent because 100 percent of a quantity implies you have the whole. Unfortunately, this idea, "The most you can have is 100 percent of something" expires when students begin interacting with ratios, proportional relationships, commissions,  etc. This rule also undermines students' ability to conceptualize a percent, and recognize the relationship between percents, fractions, and decimals.

Citations:
Bush, Sarah, Karp, Karen, and Dougherty, Barbara. 2015. "12 Math Rules that do Expire." Mathematics Teaching in the Middle School 15 (November): 208-15.

“Common Misunderstandings - Levels 5–9 Sense of Percent Tool.” Victoria State Government Education and Training , State of Victoria (Department of Education and Training), 27 Aug. 2017, www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/assessment/Pages/senseper.aspx.

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Give the following statement to students: A percent is a part of a whole. Through think, pair, share have the students consider if this is an adequate definition of percents. If they agree it is, explain why, but if not, have students explain why it is insufficient. During the share part of this activity I will be adding student responses to a t-chart on the whiteboard. One side will be reasons why the statement is a sufficient definition for percentages and the other side will be reasons why it is not. Through this activity I'll be able to see students' perceptions of percents. While it isn't wrong, to think of a percent as a part of a whole in some cases, if that is your only way to understand percents, how would you reconcile 120% or 1/20%--quantities that exceed 1 or do not reach 1. I'm not going to tell students whether I think that statement is sufficient immediately after the activity, but we will return to our t-chart at the end of class.

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Using the Concept Cartoon formative assessment (FACT from Keeley’s 75 formative assessments book), I would give student's the cartoon below. By themselves, students would respond to Mr. Faherty.

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Giving students the opportunity to write down a response to Mr. Faherty's comment gives them the opportunity to address their own understanding of percents. This  cartoon is an especially effective tool because in context 120% will make intuitive sense to some students immediately and they'll recognize that Mr. Faherty just used a lot more water than he did last month. The students who agree with Mr. Faherty, that you can't have 120% of a quantity obviously still use that misconception to explain percents, but this cartoon allows for a segue into a tangible example of when/how percents exceed 100 or exceed the whole.

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After students have had time to write their responses to Mr. Faherty, I would give them time to discuss in groups their responses. While talking with their groups, they'll use a different colored pen/marker to change their responses to Mr. Faherty if they would like.

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I would present students with the following fact: Mr. Faherty was out of town during the month of June, so he only used 2 gallons of water. In the month of July, Mr. Faherty used 4.4 gallons of water. Is Mr. Faherty's water bill correct? I will give students a moment to think about the above statement and question. Then I'll ask them to share their thoughts with me. Some things that I need to come out of the discussion are: there was an increase in the amount of water used from June to July and is the increase from 2 gallons to 4.4 gallons an increase of 120%? After they share, I'm going to have the students calculate the percent increase independently.

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We're going to return to our t-chart. The students that concluded the statement, a percent is a part of a whole, was an adequate definition for a percent may revise their thinking. Ultimately, I'm looking for every student to come to the conclusion, a percent could be a part of a whole but it could be more than the whole quantity as in the case of Mr. Faherty.