Category: exponents and exponential functions

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Simplify Rational Monomial Expressions

This post provides details about a handout for simplifying rational monomial expressions. It incorporates a couple strategies to make the simplification of rational monomial expressions more accessible. The strategies include address prerequisites skills ahead of time, chunking, and scaffolding. This incrementally walks the students through the steps.

The Pages of the Handout

The handout has 3 pages.

  • Page 1 is an initiation with two parts. There is a review of prerequisite skills aligned with the new topic. There is also a preview of the new topic with scaffolding to separate the factors into individual fractions.
  • Page 2 provides a Before and Now to draw upon student prior knowledge of simplifying using exponents rules. This is followed by scaffolded steps to separate the expression into individual fractions for each type of term (e.g., Xs). This provides a load reduction for what the student has to focus on.
  • Page 3 involves negative and 0 exponents with an additional step to address each.

Access to Handout

Here is a link to the handout.

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0 and Negative Exponents – Scaffolded

This post presents a scaffolded and meaning making approach to exponents that are 0 or negatives.

Handout

The slide show below presents all 4 pages.

  • The handout starts with an initiation to preview the prerequisites for what is presented in the lesson. It also introduces a chart that will be used for discovery.
  • Page 2 presents a discovery activity of following a pattern of dividing by 2 down to the 0 exponent. The concept of exponents is presented as the number of occurrences of the base. This leads to the idea of a 0 exponent indicating the base is no longer present, but there is still 1.
  • Similarly, on the 3rd page the pattern of dividing continues into negative exponents to show the resulting fractions. The negative exponents are then presented as reciprocals.
  • For terms with multiple factors (e.g., 5x vs just x) the students are presented steps to write the factors separately. This unpacks the reason why the negative exponent acts only on one of the factors (unless both are grouped with parentheses).

Access to Handouts

Here is a link to the student handout, and a link to the teacher handout.

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Factoring Quadratics – Loh Method

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.

Dr. Loh’s Website

Here is a link to a page on his site that provides his explanation.

Factoring Background

In the expression below, we know we want binomials with constants that are factors of 12 and sum to 8. The factors could be complex, making this challenging.

A Visual Presentation

Below shows how I made sense of out his method, in simpler terms with visuals. These are images from a Jamboard. Here is a link to a FB Reel presentation and a YouTube presentation.

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.

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Power Rules on a Google Jamboard

Exponents and Basic exponent rules are challenging. The Power Rules add another layer of challenge. This post outlines an instructional approach. The original problem is decomposed and then recomposed to show how the underlying concepts of the Power and Power of a Product Rules.

Overview

The Jamboard is configured in similar fashion as the Jamboard used for the Product and Quotient Rules. The exponential terms and variables are moveable parts. The background is a scaffolded to guide the decomposition. Here is a FB Reel and a YouTube video showing how it works. NOTE: I decompose the expression down to individual X values in lieu of using the Product Rule because I want them to see how many Xs there are. Also, the Product would be relatively new to them, I wanted to reduce the task demand placed on the working memory.

Jamboard Access

Here is a link to the Google Jamboard. To get access, you must make a copy.

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