Category: Polynomials

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Complete the Square for Vertex Form

This post provides a handout that guides students through the various steps for completing the square to transform an equation into vertex form. Students are guided through each step in isolation.

Overview

Students are presented each step in a separate chunk of the lesson. Then the steps are chained together, with scaffolding that is faded. This is a different approach than presented in a previous post. The chunks, examples, and scaffolding help make students more independent in completing the work. This frees up the teacher to provide more 1 on 1 support.

Chunks of the Lesson

The initiation addresses prerequisite skills: factoring, perfect squares, fractions, and doubles. In lieu of having students divide by 2, I focus on identifying fractions that add to the linear coefficient as you will see in the second page.

Desmos Activity to See Completing the Square

To introduce completing the square, I recommend a visual activity like this one from Desmos.

The students identify the constant that results in a perfect square. They do so by identifying doubles that result in the linear coefficient (e.g., 6 = 3 + 3). The examples help guide them through this process. This section could be presented after a hands on activity on

Students are then tasked with factoring perfect squares in isolation, including those with fractions. The doubles are modeled for whole numbers first, generalized to fractions.

At this point, the students have identified the constant to complete the square and then factored expressions. The next sections have students complete the square and then factor in equations. Note that the equations are structured as a step after the students would have subtracted the original constant, leaving the quadratic and linear terms on the right.

The last section chains all the steps together, first with scaffolding then without. Additional practice would be generated with other handouts that have problems in isolation.

Access to the Handout

Here is a link to the handout.

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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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Exponent Rules with Jamboard

The Product and Quotient Rules are challenging largely because there are few ways to present them in an accessible format. This post outlines an approach using manipulatives on a Google Jamboard.

Jamboard

The Jamboard activity involves using moveable variables as manipulatives to provide a hands on approach to learning the rules. The first slide presents the idea of exponents as repeated multiplication. The Product and Quotient Rules use the manipulatives approach to unpack the underlying concepts of the rules. Note: there are set problems followed by blank templates. Here is a FB Reel and a YouTube video showing how this works.

Accessing Jamboard

Make a copy to access the Jamboard

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Speeding Up and Slowing Down on a Graph looking at Velocity and Acceleration

We all understand speed intuitively. Velocity and acceleration are more mathy and harder for many students to comprehend, especially on a graph. Even harder is the idea of when an object is slowing down or speeding up. In this post I attempt to unpack both.

Velocity is speed with a direction. Negative in this case does not indicate a lower value but simply which way an object is traveling (think of backing up). Both cars below are traveling at equivalent speeds.

Velocity can be graphed (the red curve below). Here is information found in the graph.

  • Where the graph is above the x-axis (positive) the car is traveling to the right. Below is negative which indicates the car is traveling to the left.
  • The 2 points on the x-axis indicate 0 velocity meaning the car stops (no speed).
  • The blue lines are tangent lines with slope equal to acceleration (the derivative of velocity). If the line is going down from left to right, the acceleration is negative. Up, positive.velocity graph with tangents

Below is an example of using prior knowledge and visuals to help make sense of speeding up and slowing down.

As stated previously, the points on the x-axis indicate 0 velocity – think STOP sign. When the car is above the x-axis, it has positive velocity. Below, negative. When the car is headed down, the acceleration is negative. Up, positive.

Now for speeding up or slowing down.

  • Think of a bank account balance.
    • If you have a positive balance and a transaction that is negative, Your balance is moving towards 0.
    • Suppose you have a negative balance in your bank account and you have a positive transaction. Your balance is moving towards 0.
  • As the car moves towards a stop sign it will slow down. The speed is getting closer to 0.
    • Car at top left has positive velocity and negative acceleration so the change is going opposite of the direction. The car is moving towards 0, like the bank balance analogy.
    • Car at top left has positive velocity and negative acceleration so the change is going opposite of the direction. The car is moving towards 0.
  • As the car moves away from a stop sign it speeds up. The speed is moving away from 0.
    • Positive bank balance, positive transaction means balance is moving away from 0.
    • Negative bank balance, negative transaction means balance is moving away from 0.
speeding up and slowing down
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Intro to Factoring Out GCF – Scaffolded and Jamboard

The post shows a lesson to introduce factoring out a GCF. Factoring (and multiplication proficiency) are challenging for many students.

Overview

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.)

Initiation

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.

Jamboard

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.

Handout

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.

Access to Handout and Jamboard.

Make a copy of the Jamboard in order to access it. Here is a link to the handout.

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