Dr. Po-Shen Loh shared a possible new method for factoring a quadratic. This post provides a layman’s attempt to share the steps that teachers may find intriguing and possibly useful, especially for complex roots.
The premise of Loh’s method is that in lieu of considering two factors separately, you can focus on the following:
Average of the two factors which is the coefficient of x (linear coefficient) divided by 2. In the case below, that is 8/2 = 4.
The common distance of each factor from the average, d. In the case below, the factors are converted into the expressions 4-d and 4+d because both are d units away from 4.
This results in a single unknown, the distance d.
The aforementioned expressions with d replace the factors. Now we have an easy quadratic equation to solve using square root. Once d is determined, the factors of 12 are now known and we are on our way.
This method works for complex| factors. This makes Loh’s method less time intensive than using the Quadratic Formula, and there is no formula to memorize.
The post shows a lesson to introduce factoring out a GCF. Factoring (and multiplication proficiency) are challenging for many students.
The handout is chunked to incrementally add task demands. A Jamboard is used as a complement to flesh out the concept of a factoring by allowing engaging the student kinesthetically, with scaffolding as guidance. (See image at bottom for how to make a copy to access it.)
The sequence starts with a review of factoring and distributive property. The boxes are a preview of the scaffolding for the full factoring to be presented in subsequent chunks.
The Jamboard slides aligned with the first chunk has students pull apart the variables with exponents to help them understand the meaning of the exponents during factoring. For example, there are two Xs overlaid on the x squared. Similarly, there are two 5s, one overlaying the other to allow them to be distributed.
The next chunk in the lesson is factoring out with the terms written with the multiplication symbol written in between. The problems are aligned with the distributive problems to help with the concept of “reverse distribution.” The same problems are presented again in box form as the boxes will be used to support the full factoring. Using the same problems helps the students understand how the boxes work. The Jamboard reinforces the distribution by allowing the movement of the 5s out of the binomial, including with the box method.
The last chunk combines the factoring of individual terms and the factoring out. This is conducted with the box methods for support, and then the boxes are faded. The scaffolding starts by writing the expression with parentheses because that step is easy to overlook when explaining it to students and can be confusing for students.
Here is what the final version looks like.
Note: the handout has blank templates for copying and pasting if you want to create your own handout.
Factoring, as used in elementary school, is the act of changing a number into numbers that multiply to produce the original number. For example, 12 can be factored to 3×4. 3 and 4 are called factors of 12.
In the photo above, a “factor tree” is used to help identify the factors. I have often seen the factor tree used as the initial approach to teaching factors. I’ve also seen it used as the primary means of providing intervention for students struggling with factoring. Think about that. Students who didn’t understand the initial instruction that likely involved the mathy approach shown above were provided the SAME approach.
Math topics can be presented with a more concrete introduction which can allow for more in-depth understanding. The photo below shows a CRA approach to factoring. This approach can be used as part of UDL or as an instructional strategy for intervention.
Factors are “things being multiplied”, e.g. 2 and 3 are factors of 6.
Factoring polynomials like 5×2 + 15 above is one of the most challenging algebra topics, especially for students with special needs. To make these problems more accessible a more concrete approach is possible. The photo above shows how finding a greatest common factor (GCF) is possible using a hands on, visual approach. Click on this link to a folder with a document explaining this approach and with a document that is a master for these card cutouts.
Typically, multiplying two binomials is presented with FOIL. This approach is problematic for two reasons: it is a mneumonic for a purely symbolic representation and it is also an isolated strategy that does not connect well to prior knowledge.
Another approach is to use a graphic organizer and context. In the photo below I presented students with the scenario of expanding my patio (under the guise of being so popular I needed more space to entertain). The top figure shows the expansion of length alone. This allows for a simple distributive problem and (and eventually factoring out the GCF). The bottom figure shows an expansion of length and width which leads to multiplying binomials. Students would see that the area of the new patio is computed with (6 + x)(5 + x) which is the same as finding the area of each individual rectangle (presented in photo at the bottom). (This is useful for factoring trinomials as well.)
This scaffolded approach is effective because it presents the concept in a different representation, it connects to prior knowledge of distributive property (useful for memory storage) and it is connected to prior knowledge of area of rectangles so it has meaning.
Subsequent problems would use the boxes as a graphic organizer (see photo below).