Visualizing and Factoring Polynomials

https://www.oercommons.org/courses/grade-3-module-4-multiplication-and-area

Grade 3: Module 4 – Multiplication and Area

·         Focus on Topic C (three lessons at 1 hour each)

·         Could include Topic B – Lessons 6 & 7 if need additional background support on area and area models

https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Factorize/

Factorize

“This interactive applet allows a student to visually explore the concept of factors by creating different rectangular arrays for a number. The user constructs the array by clicking and dragging on a grid. The length and width of the array are factors of the number. A student can elect an option of a randomly selected number or the student selects his own number between 2 and 50.”

Notes: Instructions provided below the interactive grid are easy to follow. Students will need to know the terms “factors” and “factorizations” before using this resource. The resource is best viewed on a computer or tablet but can be used on a smartphone with some navigating and resizing of the screen with each move.

One way to make this more interactive for a whole class activity is to have someone offer a number and then ask students to reason through the number of factors that number will have. Then, enter the number into the “Use Your Own Number” box to confirm how many factors are to be found (and drawn). Make sure students understand the factors ultimately are the dimensions for the area of the rectangle drawn.

https://phet.colorado.edu/en/simulation/area-model-introduction

https://phet.colorado.edu/en/simulation/area-model-multiplication

Area Model Introduction

Learning Goals:

·         Recognize that area represents the product of two numbers.

·         Develop and justify a strategy that uses the area model to simplify a multiplication problem.

·         Represent a multiplication problem as the proportional area of a rectangle.

·         Looks for patterns in the total area calculation.

Area Model Multiplication

Learning Goals:

·         Recognize that area represents the product of two numbers and is additive.

·         Represent a multiplication problem as the area of a rectangle, proportionally or using generic area.

·         Develop and justify a strategy to determine the product of two multi-digit numbers by representing the product as an area or the sum of areas.

Use this online resource to practice the foundational understanding about area models and multiplication. It’s particularly useful for working on partial products (finding the missing area of a section when given dimensions and total area).

Use this online resource to extend the learning about area models and multiplication. It’s particularly useful for working with multi-digit numbers and identifying partial products (finding the missing area of a section when given dimensions and total area).

https://www.oercommons.org/courses/area-maze 

Area Maze

This resource introduces people to the area maze puzzles of Naoki Inaba.Use it to determine if students can apply the concepts explored with area models and multiplication to composite diagrams made up of rectangle areas. Students are asked to calculate the missing measurement of these composite diagrams by finding the value of the question mark in the diagrams. All of the shapes are rectangles but are not drawn to scale, so students must rely on reasoning, problem solving, and patterns in relationships to solve the puzzle.

Be sure to review the instructions at the bottom of the site with the students as a class. You might work through a few problems as a class, discussing students’ strategies along the way, and identifying places where students struggle. Encourage students to create their own sketches of the composite shape. Grid paper and rulers may be of use for this activity.

After a few problems have been explored and discussed, partner students to work on any remaining problems. If a computer lab is available, students can do the work directly on the site. If not, then you might use a “snipping tool” on the computer to capture individual puzzles to print and hang around the room to make this into a partnered station activity. Alternatively, you can provide students with the link to the site and have them work through remaining problems as homework or additional practice, if needed.

https://www.oercommons.org/courses/area-maze-2  

Area Maze

This resource is similar to the Transum Mathematics Area Maze puzzles with the same concepts and skills. Students can pick any problem on which to work unless it is “locked”. Locked puzzles are unlocked once a student successfully completes the puzzles that are unlocked.

Use this as additional practice with understanding the relationship of dimensions and area of rectangles. Students should have strong understanding of decomposing and composing side lengths and multiplication of length and width to produce a specific area.

An unrelated app named “Area Maze” also is available for free for students who want to practice on their smartphones. The app is available for both iOS and Android.

https://www.oercommons.org/courses/the-cuny-high-school-equivalency-curriculum-framework

The CUNY HSE Curriculum Framework SECTION 4 - Math: Problem-Solving in Functions & Algebra

·         Unit 9 (pp. 201-221)

“The math section of the Framework, focuses on problem-solving in functions and algebra. It integrates problem-solving strategies, productive struggle, perseverance and mathematical discussion into content learning. This section includes a curriculum map, model lessons, rich engaging math problems, samples of student work, powerful routines for math classrooms, classroom videos, and more.”

 Notes: This is a strong resource that is filled with rich instructional strategies, open-ended activities, and tons of support for the teacher. It is the last unit of an entire curriculum guide, which can be viewed with greater detail about what is covered and where on pp. 27-29. Below are some summaries about what’s included in Unit 9 and are taken directly from the material.

Each Unit is centered around either a Lesson or Teacher Support. The 5 Lessons are full-descriptions of activities, from launch to reflection, with step-by-step teacher notes, student handouts and samples of student work. The 5 Teacher Supports are each organized around a core problem and contain a list of skills, key vocabulary, an overview of the problem, and suggestions for how to teach and process the problem. We also suggest supplemental problems which expand on the core problems and explore other important content within the unit.” (pp. 25-26)

https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Algebra-Tiles/

·         Use the Expand mode to show how to take the dimensions (factors) to build the area model from the outside first. Students can then solve the problem by multiplying each of the dimensions.

·         Use the Factor mode to work in the opposite direction of what was done in Expand mode. Here students are given a polynomial and must first build the area model (in the large square space). (Note: It doesn’t state this explicitly, but students should be reminded that the area model should be in the shape of a rectangle or square. This should have been observed early on the work with other area model activities, too.) Once the area model is built, the student then must find the dimensions (factors) that made the area. The factor tiles are placed in the narrow spaces outside the area model, which shows students how each tile matches the exact length or width of the tile inside the area model. This is how they can visualize factorization of polynomials.

Notes: At first glance, the outline from which students “Build the Model” appears to be setup as an area model cut into four sections. However, the narrow strip down the left-hand side and top actually are place holders for the dimensions. The entire area is built inside the largest square space. This might be confusing for students who have been building area models on grid paper and/or with sketches drawn without grids. There are lots of detailed instructions below the workspace that should be reviewed and practiced with students before they work on this independently. It is suggested that you review one instruction and then model what it means in the workspace before moving forward to avoid confusion, as some of the actions and features may be very unfamiliar to students who are uncomfortable with technology.

Solve mode can be useful in seeing how the tiles can be setup to reflect a given equation; however, teachers should be mindful that the line down the middle represents the equals sign. This might be confusing for students, so explicit instruction about it is needed. Likewise, the “zero pair” function can be very misleading if students ignore the line down the middle. For example, given 3 = 2x-5, students might want to drag a negative five from the right to the left-hand side of the line (equals sign) to match up with the positive three. This would then look like three sets of zero pairs can be removed, which would lead to an incorrect value. When the negative five was moved to the left side of the equals sign, it had to be added to itself on the right side first. This would have made it into a positive five on the left, i.e. no zero pairs would be needed since everything would be positive. This kind of confusion could be avoided by having students use the workspace on the screen to model what is happening on each side of the equals sign (line) at every step and then connecting it back to why this is needed (balance). Students can use the tiles for each move on both sides of the equals sign and then use the zero pair function to eliminate things that are no longer needed in the workspace because they have equaled zero. Be careful when students end up with a negative x value in the end. The only way to show that the positive value of x is the inverse of the negative x value is to use the Flip feature. Make sure students understand why this feature works (not just that it flips). Model this on the board by dividing by the negative 1 coefficient, for example, so students see that each side still had something done to it besides simply being flipped. Before students click the “Check” to see if they correctly solved for x, have them substitute the solution back into the original equation to check for balance themselves. This will help them to better understand why no other value but the one they found will work for x on both sides of the equation.

Use this resource to allow students to visualize polynomials using the now-familiar area model. Having *manipulatives in the classroom is preferred; however, this online tool is a very useful back-up or supplementary resource for students to use outside of class.

Learning Goals:

·         Develop and justify a method to use the area model to determine the product of a monomial and a binomial or the product of two binomials.

·         Factor an expression, including expressions containing a variable.

·         Recognize that area represents the product of two numbers and is additive.

·         Represent a multiplication problem as the area of a rectangle, proportionally or using generic area.

·         Develop and justify a strategy to determine the product of two multi-digit numbers by representing the product as an area or the sum of areas.

Notes: This is a great online tool; however, it needs to be demonstrated to students on how to use the settings and features (partial products, area model calculation), especially with variables. It does allow students to input the values they want to use and see how those values connect to the total dimension and area of a given section. It also shows how those solutions are calculated procedurally. It also gives students a chance to explore how to find the dimension (factor) when given the area of a section. This leads to factorization of polynomials. The downside of the resource is that it lacks the visual context that algebra tiles can bring, so it is recommended to use a set of algebra tiles BEFORE using this resource, if possible.