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While students tend to show fluency in using operations with algebraic expressions, there is some confusion when demonstrating algebraic expressions as fractions. Most students are aware of how to add, subtract, multiply, and divide when using fractions of integers, but are easily intimidated by fractions of algebraic expressions. Students understand that algebraic expressions expressed as fractions can be separated in the numerator, i.e (x+5)/6 = x/6 +5/6. However many students will attempt to separate the denominator of an algebraic expression expressed as a fraction and wind up with the wrong answer, i.e. 4/(x+5) does not equal 4/x + 4/5.

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Post the question "How can we rewrite (x+4y-7)/(2x-y)" on the center, top of the board. Then the students will write down their ideas on a sticky note, and then post the sticky note on the board in one of the three columns under the question. The first column would be labeled "very sure", the second "pretty sure", and the third and final column being "unsure." Students won't be required to write their name on the sticky note, only the rewritten expression.

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I would start by dividing the class into groups of 3-4 students and hand out a set of cards (see above) with examples and non-examples of separating algebraic expressions that are expressed as fractions. Each group of students will work through the set of cards dividing them into two groups - the correct examples of separating fractions and the incorrect examples. While dividing the cards into their group, the students should be discussing the rationale for why a problem is correct or incorrect with one another and writing down their final group reasoning. While moving through the cards, students should be actively comparing the cards to the group they belong in and contrasting them against the other group. The goal is to have the students correctly sort all the cards into groups of examples and non-examples despite the various forms of presentation in the problems.

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Having the students discuss their rationale for each example and non-example and then writing them down should fulfill this step. To further assess their current understanding I could ask the groups of students to craft their own example and non-example and compare them to the ones I handed out.

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As students are working, I can walk around and see how the groups are answering. Once all the groups have finished sorting every card, the whole class will reconvene to discuss the results. The groups will take turns sharing where they sorted the card in question and why and any group that had a conflicting answer will be given a chance to discuss why they chose a different answer. Once the class has come to a consensus on where a card belongs, I will display it on the board in the corresponding example or non-example group. During this time each group will have a chance to discuss their answers and ideas and I can follow up with student answers on how they reached that answer to understand their new conceptions. The end goal will be for everyone in the class to agree with where each card sits on the board.

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Conceptual conflict will first occur within the student groups as the students discuss and sort each card from the set. Students may disagree during the sorting process and have to discuss their rationale with one another until the group can come to an agreement. During this the students will have to consider the opposing opinion of their classmate and evaluate their own reasoning to come to an understanding. Conceptual conflict will next occur when we've reconvened as a class and are discussing where each card should be sorted. As the students share their reasonings, they will have to defend their ideas against classmates that have different reasonings and choose whether or not they should adapt their reasoning.

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After we discuss each card in class, the students will have a chance to restructure their thinking and resort their own set of cards. Students could even take a look at the other cards with their new thinking to see if they incorrectly sorted any other cards. The groups should end with a new, correctly sorted set of cards and a better conceptual understanding of how to properly separate fractions.

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https://www.ocf.berkeley.edu/~reinholz/ed/resources/algebraic_equations.pdf