Engaging students: Multiplying binomials

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Emma White. Her topic, from Algebra: multiplying binomials.

How could you as a teacher create an activity or project that involves your topic?

When it comes to multiplying binomials, there are various activities that can make this topic interesting and fun. Furthermore, I believe this activity will make the topic of multiplying binomials stick in the students’ heads. For those reading, note that when I refer to FOIL, this is a method that lets you multiply two binomials in a particular order. It stands for: First, Outer, Inner, and Last (for more information on this concept, “Multiplying Binomials by the FOIL Method” by Professor Dave Explains on YouTube does a wonderful job of explaining the concept. The link is down below and skip to time stamp 1:00 for binomials). One resource that makes multiplying binomials more tangible is “FOIL Bingo”. In the resource provided below, a teacher took the time to create various bingo cards with two binomials in each square. The students would have to solve the binomials and when the teacher calls out the product of the binomials, the students would cover that spot and so forth. It is like regular bingo where you want to get a certain amount in a row, blackout the card, get a certain design, etc. The choice is up to the teacher. Another way to do this game (if you’re wanting to conserve time), if give the students a bingo sheet for homework the day before or even as an entry ticket the day of. Then the student could solve the binomials prior to playing the game and will have the answers in front of them instead of having to wait for each student to solve the problem during the game. This could eliminate the risk of going too slow and having students get bored or going too fast and having those who need more time to solve left behind. Lastly, the layout provided in Excel can be altered. Therefore, you could change the values if you wanted to do this activity with your class more than once.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Around 600-700 AD, the Hindu mathematicians had taken the Babylonia methods of approaching equations a step further when it came to introducing unknowns, sometimes more than one unknown in a single problem. It wasn’t until the Medieval times did the Islamic mathematicians discuss the variable x and how important it was. This is when the binomials theorems evolved. Furthermore, the Islamic mathematicians were able to use many operations on polynomials and soon binomials, such as multiplication, division, finding roots, and more! One thing I find highly fascinating is the Islamic mathematicians advanced the study of algebra, which “flourished during the golden age”. Evermore so, private collections were found in a lost Islamic library, which was destroyed in the 13^th Century. These private collections “altered the course of mathematics.” An example of a concept that was furthered studied was the Fibonacci sequence (which is, in my opinion, one of the most fascinating things in math history and how it relates to the world and finding mathematics around us, but that is for another time…). All I can say is the Babylonians, the Hindu and Islamic mathematicians were a driven and mathematically inclined people and it blows my mind how far these people brought the world of mathematics.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

When it comes to finding ways to use technology for multiplying binomials, I truly believe visuals are essential. I’m a little biased since I was introduced to a way of multiply binomials just last semester in one of my teaching classes and it BLEW MY MIND. I wish I knew how to do this earlier in high school! Essentially, this online source allows the student to use algebra tiles without having them physically in front of them. Therefore, they can use this source if they have technology capable of doing so (such as a phone, computer, tablet, etc.). This source is visual and easy for students to understand and manipulate. The student starts by placing the corresponding tiles for one binomial across the top like a table (would be 4 x-tiles and 2 1-tiles). Along the left side, the other binomial is represented (long ways/up-and-down). You then multiply corresponding values and where they meet in the open area (example: where an x-tile and another x-tile meet, it would become since x times x is ). Algebra tiles can also be used for upcoming topics the students would learn, such as completing the square. For a student who may have trouble grasping the idea of multiplying binomials and struggling to understand the concept of abstracts, using algebra tiles will hopefully help with the misunderstandings and confusion. All I’m saying is if this concept of online algebra tiles assisted a college student and made the topic MUCH easier to visualize and explain, I’m sure most high school students will find the use of technology in their math class interesting. Who knows, some students may come to love math more because of it!

Reference(s):

“Multiplying Binomials by the FOIL Method” by Professor Dave Explains:

https://www.youtube.com/watch?v=RTC7RIwdZcE

“FOIL Bingo”: http://www.teachforever.com/2009/03/two-review-games-multiplying.html

“FOIL Bingo” Direct Link to Download: http://teachforever.googlepages.com/bingoalg1foil.xls

“History of Polynomials”: https://polynomialshistory.weebly.com/history.html

“How modern mathematics emerged from a lost Islamic libray”: https://www.bbc.com/future/article/20201204-lost-islamic-library-maths

Algebra Tiles: https://technology.cpm.org/general/tiles/

Engaging students: Multiplying binomials

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Cire Jauregui. Her topic, from Algebra: multiplying binomials.

How can technology be used to effectively engage students with this topic?

Khan Academy has a whole series of videos, practice problems, and models to help students learn about multiplying binomials. The first in this series is a video visualizing the problem (x+2)(x+3) as a rectangle and explains that multiplying the binomials would give the area taken up by the rectangle. This would help students connect multiplying binomials to multiplying numbers to find area. This can also help students who learn better with visual examples by giving them a way to show a picture demonstrating the problem they are multiplying. Khan Academy then moves from using a visual representation to a strictly alpha-numerical representation so students can smoothly transition from having the pictures drawn out to just working out the problem. The first video in the series of pages at Khan Academy can be found at this link: https://tinyurl.com/KhanAcademyBinomials

How does this topic extend what your students should have learned in previous courses?

Multiplying binomials extends on two-digit times two-digit multiplication that students learn and practice in elementary and middle school courses. This video from the platform TikTok by a high school teacher Christine (@thesuburbanfarmhouse) shows the connection between vertical multiplication of two numbers and the multiplication of binomials together: https://tinyurl.com/TikTokFOIL By showing students that it works the same way as other forms of multiplication that they have already seen and hopefully mastered, it sets the students up to view the multiplication of binomials and other polynomials in a way that is familiar and more comfortable. This particular video is part of a miniature series that Christine recently did explaining why slang terms such as FOIL (standing for “first, outside, inside, last” as a way to remember how to multiply binomials) which many classrooms have used (including my own high school teachers), which are helpful when initially explaining multiplication of binomials, ultimately can be confusing to students when they move on to multiplying other polynomials. I personally will be staying away from using terms like FOIL because as students move on to trinomials and other larger polynomials, there are more terms to distribute than just the four mentioned in FOIL.

How can this topic be used in your students’ future courses in mathematics or science?

As I mentioned in the last question, learning to multiply binomials can lead students to success in multiplying polynomials. This skill can also help students factor polynomials in that it can help them check their answers when they are finished. It can also help them recognize familiar-looking polynomials as having possible binomials as factors. If a student were to see 12x²-29x-8 and couldn’t remember how to go about factoring it in other ways, a student could use a guess-and-check method to factor. They might try various combinations of (Ax+B)(Cx-D) until they find a satisfactory of A, B, C, and D that when the binomial is multiplied, creates the polynomial they were trying to factor. Without solid skills in multiplying binomials, a student would likely be frustrated in trying to find what A, B, C, and D as their multiplication could be wrong and seemingly no combination of numbers works.

Engaging students: Adding and subtracting polynomials

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Enrique Alegria. His topic, from Algebra: multiplying polynomials.

How can this topic be used in your students’ future courses in mathematics and science?

This topic can be used in students’ future courses in mathematics by simplifying expressions of increasing degree. In Algebra II students are expected to simplifying polynomials of varying degrees as they move on to multiplying and dividing polynomials. From there determining the factors of a polynomial of degree three and degree four. Real-world problems can be solved through the simplification of several like terms. Each term representing a specific part of the problem. We can even compare the addition and subtraction of polynomials to runtime analysis in Computer Science. Measuring the change in the degree and how that affects the output. In a way, this can translate to the runtime of a program. For example, a chain of commands with a constant time is run. A loop is nested in another loop that is placed after the first expressions. This has changed the overall runtime of the program from constant time to quadratic because of the degree of the nested loops. The overall time would be the addition of the expressions and their corresponding times.

How does this topic extend what your students have learned in previous courses?

This topic extends from the early concept, ‘Combining Like Terms.’ Starting with adding and subtracting items of similar groupings such as 8 apples and 4 apples altogether are 12 apples. Bringing students to place value such as adding 3 ones and 2 ones to adding multi-digit numbers. We then leap towards Algebra introducing expressions and equations. Learning about linear and quadratic equations and graphing them. Students should have learned about monomials in correspondence with coefficients and exponents. From there, students are familiar with algebraic terms. Those are the building blocks that we are going to be expanding upon. Once students familiarize themselves with several terms in an expression, they will focus on adding or subtracting like terms by focusing on both the coefficient, term, and exponents on the variables. Shortly after the students can continue to be challenged by using terms such as $6xy$ or $3a^2b^3+4a^2b^3c^2$ to focus on the terms and confirm if they are ‘like’ to be combined or just notice the fact that they have some common variables with the same exponents but with a slight difference other than the coefficient, the expression cannot be simplified as one may think.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Adding and subtracting polynomials can be engaging to students with the help of Brilliant. This site starts with helping students identifying polynomials and their degrees to help students understand how to describe them. Then moving to the arithmetic of polynomials performing addition and subtraction operations on the polynomial numbers. This source goes through polynomials through challenging and insightful exercises. For example, a quadrilateral of sides such as $5$ , $3x+4$ , $4x+1$ , $17x-10$ , and from there simplifying the expression. Students would be able to substitute values and determine if a specific quadrilateral has been made. I can have students go through a few exercises as a class or on their own and then they can come up with a problem on their own that would be posted to the ‘public’ (which would be only their class) so that the students will be able to have classroom interaction and grow as they challenge each other. Students can apply this concept by creating a large polynomial expression and then simplifying it and lastly graphing the equation.

References:

Polynomials. Brilliant.org., from https://brilliant.org/wiki/polynomials/

Simplifying Expressions. Brilliant.org., from https://brilliant.org/wiki/simplifying-expressions/

Engaging students: Adding and subtracting polynomials

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Christian Oropeza. His topic, from Algebra: multiplying polynomials.

How could you as a teacher create an activity or project that involves your topic?

The activity would consist of each student being given a bowl with 20 pieces of candy, which has multiple colors (e.g., Skittles or M&M’s) and a worksheet, which by the end will show students how to add and subtract polynomials(Reference 1). The objective for each student is to group all of the pieces of candy by the same color. Once this has been completed, the students will write down on the worksheet for “Part 1”, how many pieces of candy are in each group. Next, the students would be given 10 more pieces of random colored candy. Then, the students will regroup the new pieces of candy and write down the new number of candies in each group for “Part 2”. For “Part 3”, students will eat(or put away) 10 of their candies randomly. Finally, the students will write down the new number of candies in each group. Then the students would be asked, “What did each one of you do to put the candies in groups?”, “what operation was used for Part 2 of the worksheet”, and “what operation was used for Part 3 of the worksheet”. The students’ responses should be somewhere along the lines of “group the candies by the same color”, “addition”, and “subtraction”. Then the students would be told to relabel each group of colored candies into a different variable. For example, green=x, red=x², yellow=k, blue=y, etc. Knowing the previous information, the students will next repeat the Part 1, 2, and 3, but using the assigned variables instead of the colors. The purpose of this activity is to show students that each variable in a polynomial must be grouped by like terms when performing addition or subtraction.

How does this topic extend what your students should have learned in previous courses?

This topic relates to previous math classes by activating students’ prior knowledge on the concept of adding and subtracting integers. This means knowing the rules of addition and the rules of subtraction. For example, students should know that a 3+2=5=3+2, but 3-2=1 $\ne$ 2-3 (i.e., commutative property). Students should also know that the when subtracting a negative integer, the signs cancel out and all that is left is the addition of a positive integer (e.g., -(-2)=2). Students should also be familiar with grouping anything into specific groups. For example, if students were given colored tiles, then the students should be able to group the tiles into different colored groups. The distributive property is a topic the students should have covered before, which helps out when trying to simplify an expression involving parenthesis (e.g., 2(3+a)=6+2a. The idea of closure for integer properties and operations is the key to adding and subtracting polynomials, so students must have understood this concept prior in order to use the operation of addition and subtraction on like terms.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Technology is always a great way to engage students especially with the newer generation of students where technology is part of their everyday life. The website mathisfun.com (Reference 2) is an excellent piece of technology to introduce this topic to the students because the website breaks down the idea of adding and subtracting polynomials piece by piece in easy manner that will help students see patterns and activate prior knowledge. With the inclusion of examples and non-examples students will learn where to minimize their potential errors. Some of the examples are animated with colors to help the more visual students understand and recognize the pattern for each problem. Another example of effective technology is the website Khan Academy (Reference 3,4,5). Khan Academy has great videos that thoroughly explains this topic. Reference 3 defines the word “polynomial” in math language by breaking the word into two words, which will help students remember and recognize this topic more easily. Also, Reference 2 goes over the vocabulary associated with adding and subtracting polynomials (e.g., coefficients, monomial, binomial, trinomial, and degree). Reference 4 goes over an example of adding a polynomial by going through step by step procedures. Reference 5 does the same thing as Reference 4, but over an example of subtracting polynomials.

References:

Engaging students: Multiplying binomials

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Sarah McCall. Her topic, from Algebra: multiplying binomials.

B2. How does this topic extend what your students should have learned in previous courses?

My hope is that this topic may be easier to understand if student’s can first recall an easier concept that they have already mastered, and then build upon that foundation to learn new skills. For example, at this point students should have already learned the distributive property. To introduce this new concept, I would begin by writing 4(x-5)=4 on the board and asking students what the very first step would be to solve for x. They should know to start by distributing the four to both x and -5, to get 4x-20=4. After completing a few similar examples as a class and/or in groups, then the idea of multiplying binomials would be introduced. This way, students are less intimidated when presented with new material, and they will have a good understanding of how to distribute to each term.

D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Teaching students some of the history behind what they are learning can be a great engaging tool. In this case it is helpful to know where the foil method first originated. I would incorporate this by discussing how it first was used in 1929; in William Bentz’ Algebra for Today. In Algebra for Today, Bentz was the first person to mention the “first terms, outer terms, inner terms, last terms” rule. Students should be knowledgeable about the history behind the math they are using, so that they realize the importance of this method. I also believe that it will be cool for students to see how a method developed is still relevant 88 years later. This technique was created in order to provide a memory aid, or “mnemonic device” to help students learn how to multiply binomials. The fact that it is still being used even today proves what an influential concept it was at its time, and throughout the years.

E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

I am a huge fan of incorporating technology in the classroom, and YouTube is especially great because most students already use YouTube outside of school. The following clip (stopped at 1:48) provides a clear, concise explanation and demonstration of the FOIL method for multiplying binomials. It explains how factoring and foiling are related, and shows students which order to distribute in (first, outer, inner, last). The acronym FOIL is easy for students to remember, and gives them something that they can write down each time they complete a problem to help them distribute properly. Additionally, the clip is just under two minutes, which is the perfect time to ensure that students don’t zone out or lose interest before the end of the video. I would choose to follow up this video by completing a few examples as a class, emphasizing the four steps of foiling as mentioned in the video and how to use them.

References

http://pballew.blogspot.com/2011/02/origin-of-foil-for-binomial.html

Engaging students: Adding and subtracting polynomials

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Kelly Bui. Her topic, from Algebra: multiplying polynomials.

A2. How could you as a teacher create an activity or project that involves your topic?
The main idea of the activity will be finding an expression to represent the area of the border given the dimensions of the outer rectangle and the inner rectangle. The students will need to know how to multiply binomials and add or subtract polynomials. Therefore, this activity would be towards the end of the unit. Students will be asked to roam around the classroom or the hallway in search of items that already have dimensions labeled. For example, in the hallway there may be a bulletin board with the dimensions (2x² – 7) for the length and (3x – 4) for the width. Inside of the bulletin board, there will be the dimensions of a smaller rectangle. The question will be asked: What expression will represent the area I want to cover if I want to cover the only the border with paper?
Students may work in partners or groups to put minds together to solve this problem. Every object labeled with dimensions will be in the shape of a rectangle and the math involved will require students to multiply binomials and subtract polynomials.

B2. How does this topic extend what your students should have learned in previous courses?
Before students learn to add and subtract polynomials, they learn how to combine like terms such as 3x and 5x. When we add and subtract polynomials, it is very similar to combining like terms in algebraic expressions. Students will need be familiar with the concept of combing like terms before they add or subtract polynomials. To introduce the topic of combining polynomials, it can be set up horizontally.

Such as: (3x² – 5x + 6) – (6x² – 4x + 9)

By setting it up this way, students can determine which terms can be combined and which terms need to be left alone. Additionally, students will build on the concept of combining like terms as it applies to this process as well. Setting it up horizontally will also increase the chance of preventing the mistake of forgetting to distribute the negative sign throughout the second polynomial. Once students are comfortable doing it this way, the addition and subtraction can be set up as a vertical problem where students must now take the step to align the like terms together in order to add or subtract. By taking the step to set up the polynomials horizontally before vertically, it will give the students a deeper understanding of what concept is actually behind adding and subtracting polynomials.

E1. How can technology be used to effectively engage students with this topic?

The website http://www.quia.com provides a multitude of activities relating to different subjects. The game I chose to correlate adding and subtracting polynomials is identical to the actual game Battleship. The game can be played by anyone with access to the internet and Adobe. This game is interactive because you won’t have to perform math on every single shot fired at the enemy. If the student does hit one of the vessels, in order to actually “hit” the enemy’s ship, the student must successfully add or subtract two polynomials. If a student hits a vessel but is unable to solve the polynomial correctly, the game will highlight the hit area so that the student can try again. This game can either engage the students to see who can sink all of the enemy’s ships first, or it can be assigned as a homework assignment that requires showing work and screenshotting the end result of the game. Lastly, you can choose the level of difficulty of the game. For example, on the hard level, you must determine the missing addend or minuend to the expression, or add or subtract polynomials of different degrees.
The website also offers an option to create your own activities, so if Battleship isn’t panning out as desired, it is possible to create your own game for your students.
Game: https://www.quia.com/ba/28820.html

References:

Area of the Border: https://www.sophia.org/concepts/adding-and-subtracting-polynomials-in-the-real-world

Combining Like Terms: https://courses.lumenlearning.com/boundless-algebra/chapter/introduction-to-polynomials/

Battleship: https://www.quia.com/ba/28820.html

Engaging students: Multiplying binomials

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Lucy Grimmett. Her topic, from Algebra I: multiplying binomials like $(a+b)(c+d)$ .

How could you as a teacher create an activity of project that involves your topic?

There are tons of activities that could be created with this topic. The first thing that came to mind was giving each student a notecard when they walked in the room. Each notecard would have a binomial on it. Students would be asked to find a partner in the classroom and multiply their binomials together. They would be able to assist one another, discuss possible misconceptions, and ask questions that they might not want to ask in front of an entire class. This could be a quick 5-minute warm up at the beginning of class, or could turn into a longer activity depending on how many partners you want each student to have. This wouldn’t involve much work on the teacher’s part; all you would have to do is create 30 differing binomials. If you feel the need to create a cheat sheet with answers to every possible pair you can, but that would involve more work then necessary.

How does this topic extend from what your students should already have learned in previous courses?

In previous courses and chapters in algebra, students are set up with knowledge of combining like terms. The most common idea of combining like terms is adding or subtracting, for example 2-1=1 or 2+1=3. Students don’t realize that in the elementary school they are combining like terms. This is a key tool used when multiplying binomials. As future math teachers, we know that when we see 2x + 3x we can quickly combine these numbers to get 5x. This simplifies an equation. Students will struggle with this at first because they will not be used to having a variable, such as x, mixed into the equation, literally. This will be a similar issue when discussing multiplying binomials. Students will have to get used to seeing (4x+1)(3x-8) and turning it into the longer version $12x^2+3x-32x-8$ and then finding the like terms to simplify again, creating the shorter version $12x^2-29x-8$ . This is an extension of like-terms.

How can technology be used to effectively engage students with this topic?

Algebra tiles are a great tool for students and teachers to use. Even better is an online algebra tile map. This allows a teacher to show students how to use algebra times from a main point, such as a projector, rather then walking around the room and individually showing them. Teachers can have students work individually with their iPad’s (if they have them) or use actual algebra tiles. This would be a great engagement piece for a day when students are recapping distributing or “FOIL” as many teachers like to call it. This can also be a great discovery lesson when students are learning how to multiply binomials. This all depends on if students have used algebra tiles before, and how comfortable the teacher is with implementing a lesson like this in the classroom. Another idea is pairing students and giving them binomials to multiply, which they will present to the class in a short presentation using their online algebra tile tool.

Here’s the link for the online algebra tiles:

http://technology.cpm.org/general/tiles/

Engaging students: Multiplying polynomials

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Daniel Herfeldt. His topic, from Algebra: multiplying polynomials.

How could you as a teacher create an activity or project that involves your topic?

Activities for multiplying polynomials are endless. An activity that I would do with my students is a game called polynomial dice. To do this, you would first is to get several blank dice and write random polynomials on each side of the dice. Then in class, divide the students into groups of no more than three. Each group will get a pair of dice. Have the students roll the dice and they should have two different polynomials. Once they have rolled, have them multiply the polynomials together. This is best done with groups so that the students can share their work with their partners to see if they both got the same answer. If they did not get the same answer, they can go back through each other’s steps to see where they went wrong. If you want to make the game a bit harder, you can add more dice to make them multiply three polynomials, or maybe even more. This is a great game because it can be used for multiplying polynomials, as well as dividing, adding and subtracting. It could be a great review game before a major test to have students remember how to do each individual property. For example, have the students roll the dice, then with the two polynomials they get, they first add the polynomials, followed by the difference, then the product, and finally the quotient.

How can this topic be used in your students’ future courses in mathematics or science?

Multiplying polynomials is used all over mathematics. It is first introduced in Algebra I and Algebra II. Multiplying polynomials can be very difficult for students and make them not want to do the work. This is due to there being so much work for one problem. Since there is so much work, there is a lot of room for mistakes. This topic is used is Algebra I, Algebra II, Algebra III, Pre-Calculus, Calculus and just about every higher math course. If a student is looking to go into an architecture or engineering field, they will have to apply their knowledge of polynomials. Due to this, the topic is one of the most important topics that students need to understand. Knowing how to multiply polynomials also makes it easier to divide polynomials. If a student is struggling with dividing polynomials, you can go back to showing them how to multiply them. Once a student sees the pattern of multiplying polynomials, they are more likely to get the hang of dividing them.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

I believe this video would be a great engage for the students when you, as a teacher, are teaching the students how to multiply polynomials for the first time. This video helps students remember what exactly is a polynomial. Although there is only three types of polynomials in the video (monomial, binomial, and trinomial), it uses the three main types that students will be using in a high school level. Another great thing in the video is that it shows how to tell the degree of the polynomial. Although it seems easy to just say the power of x is the same as the degree, students still might forget how to do it. For example, a student might think that a digit by itself and with no variable has a degree of one, but is really a degree of zero. The final point that is key to this video is that it shows students how to line up the terms. Some students might put 6+x^2+3x, and although that is still correct, it will be better written as x^2+3x+6.

Thoughts on the Accidental Fraction Brainbuster

I really enjoyed reading a recent article on Math With Bad Drawings centered on solving the following problem without a calculator:

I won’t repeat the whole post here, but it’s an excellent exercise in numeracy, or developing intuitive understanding of numbers without necessarily doing a ton of computations. It’s also a fun exercise to see how much we can figure out without resorting to plugging into a calculator. I highly recommend reading it.

When I saw this problem, my first reflex wasn’t the technique used in the post. Instead, I thought to try the logic that follows. I don’t claim that this is a better way of solving the problem than the original solution linked above. But I do think that this alternative solution, in its own way, also encourages numeracy as well as what we can quickly determine without using a calculator.

Let’s get a common denominator for the two fractions:

$\displaystyle \frac{3997 \times 5001}{4001 \times 5001} \qquad$ and $\displaystyle \qquad \frac{4001 \times 4996}{4001 \times 5001}$ .

Since the denominators are the same, there is no need to actually compute $4001 \times 5001$ . Instead, the larger fraction can be determined by figuring out which numerator is largest. At first glance, that looks like a lot of work without a calculator! However, the numerators can both be expanded by cleverly using the distributive law:

$3997 \times 5001 = (4000-3)(5000+1) = 4000\times 5000 + 4000 - 3 \times 5000 - 3$ ,

$4001 \times 4996 = (4000+1)(5000-4) = 4000\times 5000 - 4 \times 4000 + 5000 - 4$ .

We can figure out which one is bigger without a calculator — or even directly figuring out each product.

Each contains $4000 \times 5000$ , so we can ignore this common term in both expressions.
Also, $4000 - 3\times 5000$ and $5000 - 4 \times 4000$ are both equal to $-11,000$ , and so we can ignore the middle two terms of both expressions.
The only difference is that there’s a $-3$ on the top line and a $-4$ on the bottom line.

Therefore, the first numerator is the larger one, and so $\displaystyle \frac{3997}{4001}$ is the larger fraction.

Once again, I really like the original question as a creative question that initially looks intractable that is nevertheless within the grasp of middle-school students. Also, I reiterate that I don’t claim that the above is a superior method, as I really like the method suggested in the original post. Instead, I humbly offer this alternate solution that encourages the development of numeracy.

Your Plan is Foiled

I normally prefer the phrase “by using the distributive law” instead of the more colloquial “by FOIL,” but I’ll make an exception in this case.