The following reflects an outline for an Algebra unit on factoring that was developed using CBCI philosophies as outlined in Concept-Based Curriculum and Instruction for the Thinking Classroom by Lynn Erickson, Lois Lanning, and Rachel French.
Unit Title: Factoring: aka – Expression Makeovers
Conceptual Lens: Transformations
Time Allocation: Approximately 8 days in a high school Algebra course
Unit Strands:
- Identifying and using Greatest Common Factor
- Factoring Trinomials
- Factoring Difference of Squares
- Using Factoring to Solve Equations
Unit Overview: There are many ways to transform an expression into an equivalent expression that looks different. How many ways can you think of to rewrite 10? What about 3(x+2)? Factoring is a common method for rewriting a sum of terms as a product of polynomials, which can be useful in solving quadratic equations.
What we would like students to…
- Know: How to identify the greatest common factor (GCF) (where GCF is a number, a variable, or a binomial); the definitions of factor, greatest common factor, perfect square trinomial, and difference of squares; zero factor theorem; standard form a quadratic equation.
- Do: Factor out a GCF; factor a trinomial; factor a difference of squares; solve a quadratic equation by factoring.
- Understand: Factoring is writing an equivalent expression; factoring is the reverse process of multiplying polynomials; factoring can be used to solve equations.
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Guiding Questions | ||
| 1. Factoring is writing an equivalent expression, transforming the expression from a sum of terms to a product of polynomials | Factual Questions:
What is a factor? What is a greatest common factor (GCF)? What is factoring by grouping? How can identifying the GCF help us transform an expression into an equivalent expression? How are your two equivalent expressions similar? How are they different? |
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| 2. Factoring is the reverse process of multiplying polynomials |
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| 3. Transforming an expression on one side of an equation may help in solving the equation |
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Two suggested Learning Experiences:
Lesson 1:
This would be the first day of the unit. This lesson introduces students to greatest common factor, as well as reminds students of previous experiences that involved transformations of expressions, setting the foundation for the transferable macro-concept of transformations.
Objectives:
- Students will make connections to previous experiences in writing equivalent expressions.
- Students will identify common factors, and the greatest common factor, of expressions and write the factored form of these expressions.
Standards:
- CCSS.Math.Content.HSA.SSE.A.1.a – Interpret parts of an expression, such as terms, factors, and coefficients.
- CCSS.Math.Content.HSA.SSE.A.2 – Use the structure of an expression to identify ways to rewrite it.
- CCSS.Math.Practice.MP7 – Look for and make use of structure.
Resources and Materials:
Teacher – White board (or poster paper) and marker
Student – Pencil and paper
Learning Tasks:
- Warm-up Discussion: When students enter the room, have the following task on the board/projector, “Write at least two different expressions that are equivalent to 10 but look different.” Students may work individually or with a neighbor. (Potential answers: 2*5, 3+7, 11-1, 3*2+4, etc.) After a minute, ask students to share some of their expressions and record them beside “10” on the board/projector. Repeat the process with 3(x+4), which will likely trigger students to apply the familiar distributive property, giving 3x+12.
- Review of key features/terms: Using the lists of equivalent expressions from the warm-up, have students identify expressions that display a sum vs. a product, recall the definition of factor, and identify the factors in the product expressions. Ask students to speculate about (greatest) common factors in variable expressions based off their understanding of (greatest) common factors in numbers.
- GCF Maze: For practice, have students do a GCF maze, like this one available on Pintrest, https://www.pinterest.com/pin/415034921893301951/.
- The Transformation: Now that we have the GCF, let’s actually transform the expression to an equivalent, but different looking, one. This Khan Academy video, https://www.youtube.com/watch?v=FvS7v6KM1ig, has a good explanation of “undistributing” this GCF, starting around the 2.5 minute mark.
Assessment:
Teacher will informally assess prior knowledge and understanding based off discussions. The warm-up for Day 2’s lesson will be a sum of terms where students are asked to identify the GCF and rewrite the expression as a product of the GCF and the remaining factor(s).
Lesson 2:
This will be day 3 of the unit. Students will have discussed factoring by grouping in the context of polynomials that are already set into 4 terms. Today’s lesson will center around rewriting trinomials into 4 terms as a precursor to factoring by grouping.
Objectives:
- Students will transform trinomials into expressions with 4 terms using both a “trial and error” method and an area method.
Standards:
- CCSS.Math.Content.HSA.SSE.A.2 – Use the structure of an expression to identify ways to rewrite it.
Resources:
Teacher: Computer with projector, term tiles and practice problems for area activity (see explanation below)
Students: pencil and paper
Learning Tasks:
- Warm-up: Review factoring by grouping from an expression with four terms. i.e. – “Use the information on GCF from last class to rewrite this expression as a product of two binomials: xy + 2x + 3y + 6.” (Answer: (x+3)(y+2))
- Discussion: Ask students to discuss with a neighbor or two the following question, “How might someone say that FOIL connects the two expressions in the warm-up?” If necessary, guide students toward the idea that the four terms of the initial expression are the F, O, I, and L if you “FOIL” the product.
- Rewriting trinomials. Using an example like (x + 7)(x – 4), remind students that in binomials with only one variable, after we “FOIL,” we can collect like terms, giving us a trinomial instead of an expression with 4 terms. Let this discussion lead to the idea of transforming trinomials into expressions with 4 terms, and then attempting factoring by grouping. Students should quickly see that this trial and error method is not always effective and, therefore, not the most efficient.
- Use the area activity at http://courses.wccnet.edu/~rwhatcher/VAT/Factorin to demonstrate an alternative to “trial and error.” This activity also serves the purpose of helping students to visualize the connection between product and sum via the concept of area. Note: This app is clunky and not very user friendly. To save time (and avoid the need for a class set of computers), create several sets of terms tiles. Have students work in small groups on problems together, using the same method and process as the app but at their desk with the paper tiles.
- Exit ticket: “Summarize a key finding from the area activity just completed. Is there a pattern or process you can summarize that would help a fellow student quickly split a trinomial into a set of four terms that would allow factoring by grouping?”
Assessment:
Teacher will informally assess knowledge and understanding based off discussions, as well as the information students provided on the exit tickets. For the warm-up the next day, project a couple of the exit ticket ideas (or an amalgamation of ideas). Ask students to apply it to a practice problem, and then answer “was the method(s) helpful? Would you like to tweak the process/hint in any way?”
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Hi Angie,
Your lesson is laid out very clearly with what you want the learners to know, understand and be able to do. The format is very readable and easy to understand.
Beth
Thanks, Beth! I appreciate that!
Angie,
Great lesson plan! As othere have said, making the connection between different ways to write the number “10” (which should feel like a comfortable activity for them) and connecting it to a new topic is a great way to introduce the lesson. I like how you are building up their confidence in topics that they know to establish a solid groundwork in what is to come. Well done!
Thanks, Kristyn! In rewriting 10, I was envisioning the benefit of students easily understanding what we mean by equivalent expressions. I hadn’t even thought about the confidence building side of it, but you are absolutely correct. Boost morale right from the start. Great point!
Angie,
I also appreciated the title. Sound positive and fun.
In the first lesson, going back to finding the GCF of a monomial as a lead in to factoring them out of a binomial is a great way to tap prior knowledge and make them comfortable with the process, since they have seen it before. Using the data they generated and you recorded as a segue to discussing some vocabulary was also interesting. Math vocabulary is something we all need to stress more often.
In the second lesson, I have never tried having them rewrite a 4 term polynomial as a lead in to them turning a trinomial into a 4 term polynomial to achieve factoring. I think it’s a great connection and plan to use that next year with my students.
Thanks.
Chris –
Thanks for the positive feedback! Warm-ups are one of my favorite ways to connect prior knowledge to current tasks. Teaching at both the high school and the college levels, I started each day with a Warm-Up task on the board. I like that this gives the students something to start on right away, whether they’re early to class or running in as the bell rings. Plus it gives us, as teachers, time to do administrative tasks, talk to the student who wants to explain why their homework isn’t done, return homework, etc. Beyond the logistical advantages, it can be used to refresh students on a topic we’ll be building off of and/or review information from a recent class.
I’m glad to hear, too, that you’re going to try a new approach after reading the lesson. One of the textbooks I used in an Introductory Algebra course at a community college taught that approach. When Audrey suggested the area connection in the discussion board, I was reminded of it, and thought they would go well together. I hope it goes well for you!
Thanks!
Angie
First of all, I think you have a great unit title. “Expression Makeovers” makes me interested to find out more about what I (as a student) would be learning. The question “What would an ‘expression makeover’ look like immediately pops into my head. Your title is short yet full of potential! Good work.
Secondly, I like your question in the unit overview: “How many ways can we rewrite 10?” What a great way to introduce the concept of factoring polynomials! I’m sure some students would say ‘5 times 2’ and ‘5 plus 5’, but there are so many other possibilities out there! You may even catch a student re-writing 10 as a power of 10 (scientific notation… see my blog!)
The previous question about 10 could be turned into a WODB (one of the activities in our MTH 607 course this summer). WODB stands for ‘Which One Doesn’t Belong’. You can check out these WODB’s here: http://wodb.ca/
Lastly, I think you have a great ‘debatable’ question: “Is there a best way to factor a trinomial”. This will open up a class discussion that will allow students to argue and justify for the method that they think is ‘best’. This will definitely spark a debate!
Thanks for the feedback! I wasn’t sure if the title was too kitschy, so I’m glad to read it elicited a positive reaction. Thanks, too, for the introduction to WODB. I love the idea of there being multiple correct ways to chose which one doesn’t belong. As long as you can justify why it’s the odd one out, it can be correct. (In addition to being able to say “yes, that’s correct!” more often, requiring justification is great practice for students at articulating their reasoning, as well as a foray into informal proofs. Winning all around!) I will have to add this activity to my bag of tricks, if you will. Thank you!
(And yes, your scientific notation does fit right in as an option for rewriting 10. I hadn’t thought of that, but I’m sure some student would. That’s the fun part about that; students will wow you with what they come up with!)