# Engaging students: Solving two-step algebra problems

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 comes from my former student Jessica Williams. Her topic, from Pre-Algebra: solving two-step algebra problems.

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

Learning two-step algebra problems can difficult for students at first glance of the equation; therefore I believe using a hands on manipulative to demonstrate is a big help, especially for your visual/kinesthetic learners. I’ve recently helped demonstrate this lesson I’ve seen online to my sister, who is in 7th grade. It worked marvelously with her; therefore I would definitely do it in my actual classroom. To teach this lesson, I would bring in cups and colored chips for each student to use to demonstrate the equation given. For starters, present the students with an equation to solve. (2x+3=9) Next, present the students of the guide lines/rules of the cups and chips. Let them know that if the variable is a positive number, to place the cup facing upwards. Similarly, if the variable is a negative number, tell them to place the cup facing down. Let the students know that the coefficient of the variable is what lets you know how many cups to use. Next, you would guide the students with questioning but asking them to display what 2x is using their cups. They should each have two cups facing upwards. Next, they will place 3 chips next to their cups to represent the +3 and have an equal sign with 9 chips on the other side. This would lead into asking the students what they could do to get rid of the 3 chips on one side, which results in having to get rid of 3 on the side with 9 as well. This will lead the students to 2x=6, and you can ask the students if 2 cups equals 6 chips, then how many does only one cup equal. They should get to x=3, with enough scaffold questioning. Then the teacher could provide multiple more examples to do on their own with the objects in front of them. This allows for the students to visual see why solving the two-step equations work the way they do. It shows students how you have to “do to one side what you do to another.”

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

Prior to learning about two-step equations the students will have worked with one step equations. They are used to seeing simple equations, such as x-2=6 or 2x=10, where they only have to complete one step. Before teaching the students two-step equations, the teacher should allow a couple practice problems to access the student’s prior knowledge. The two-step algebra problems are only a slight extension to what they have practiced. It also extends on basic addition, subtraction, multiplication, and division. They know how to do all of these things; however adding a variable to the mix is quite an extension in the perspective of the students. They have to be taught the meaning and definition of a variable and how it has a specific value that needs to be solved for in order for the equation to be correct. Learning this topic will also help prepare the students for more difficult math such as solving quadratic equations, word problems, etc.

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

Technology can always be made fun for the kids, however it can be very good for extra practice as well. For example, http://www.math-play.com/Two-Step-Equations-Game.html is an amazing way to practice and have fun at the same time. The game requires player one and player two to go against each other. Each player has to answer a two-step equation correctly in order to shoot his or her ball at the basketball hoop. The player at the end with the most points wins the game! This game is extremely engaging for the students because it involves competition. What does every student love to do? WIN! It boosts their confidence. If the student would rather work alone, that’s fine as well. It still benefits every student why keeping their mind in the game and focusing on answering correctly. The students can also answer question on Khan academy or watch videos for refreshment before the next class. There are so many ways technology can be beneficial. In previous lessons, I have used Kahoot and plickerz. Both require 100% engagement from each student and they both require individual accountability.

References:

http://www.math-play.com/Two-Step-Equations-Game.html

Annenberg Learner: https://www.learner.org/workshops/algebra/workshop1/lessonplan2.html

# Engaging students: Order of operations

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 comes from my former student Megan Termini. Her topic, from Pre-Algebra: order of operations.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

The order of operations appears in pop culture in many different ways. An example is the song “Cupid Shuffle” by Cupid. There are certain steps that you do in a specific order. If you do not follow the order, then it is no longer the cupid shuffle. An activity would be incorporating the order of operations into the “Cupid Shuffle”. For example, the chorus is,

“Parentheses, Parentheses, Parentheses, Parentheses,

Exponents, Exponents, Exponents, Exponents,

Now Mult. or Div., Now Mult. or Div.

There are certain dance moves to go along with each step in the song. Here is a video of some students doing the song and dance (Reference A). This is a very effective way of teaching the students the order of operations(PEMDAS) because many students love music and dancing, and they are more likely to remember the song and dance moves than just memorizing the order itself.

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

There are tons of activities that you could do that involve the order of operations. As the teacher, you would want to create an activity that is fun and engaging for the students. Something that involves everyone in the class and not just a few students. One activity that would-be fun is Order of Operations War. Many students love playing the card game war. Now it is the same game just involving the order of operations. Each student will get a deck of cards and evenly deal them. Then they will get note cards with each of the operations on it. They will each flip 3 cards, arrange them with the operations and try to get as close to the target number as they can. The person who gets the closest is the winner of the round. This game would be a great way of getting all the students involved and a good way of learning the order of operations. (Reference B)

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

Learning the order of operations is very important for the students to learn, especially for their future courses in mathematics or science. The order of operations is used is almost every mathematics course from then on and most of the science courses. That is why is it very important to understand how it works. You know that you will use them in math and science course, but also you will use the idea of order of operations in computer sciences courses. When programming, the code has to be in a specific order to work. Just like a math problem, if you don’t apply the operations in the correct order, then you won’t get the correct answer.

References:

A. (2014, March 11). Retrieved September 01, 2017, from https://www.youtube.com/watch?v=EfgtWthLvk4

B. Order of Operations War With Just A Deck of Cards. (n.d.). Retrieved September 01, 2017, from http://us9.campaign-archive2.com/?u=3c5f5b9960a466398eccb35f8&id=cf58289e69&e=c87fd3cb28

# Engaging students: Dividing fractions

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 comes from my former student Kerryana Medlin. Her topic, from Pre-Algebra: dividing fractions.

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

One of the more practical uses of dividing fractions is cooking. Anybody who has baked in the past will know that sometimes one does not possess the proper measuring cup for the job and that they have to crunch some numbers. (This happens a lot when in college.)

The basic idea behind the activity is to ask the students to follow a recipe using a 1/3 cup measuring cup and a teaspoon. This will also allow them to practice dividing whole numbers by fractions, which strengthens to concept as well. They will be reminded that a whole number can be expressed as the number over one.

The ingredient list would be as follows:

Treats:

5-6 cups of rice cereal

1 cup of marshmallow fluff

1/3 cup of sprinkles

Buttercream:

½ cup unsalted butter

1 ½ cups powdered sugar

1 ½ teaspoons of vanilla extract

1-3 teaspoons of milk

They would be asked to figure out how many 1/3 cups each component would take. This would also help the students to use the skill of adding fractions (1 and ½ being 3/2) before dividing. The recipe would ultimately make rice cereal treats with icing on top (enough for the entire class). This is envisioned as an activity in which the students work either individually or in small groups to do the calculations and then come together as a class to provide answers and give me the proper amount of ingredients to put into the recipe.

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

Dividing fractions involves prior knowledge from fractions, generally. If dividing by flipping the dividend and then multiplying the resulting two fractions, the student must use their knowledge of multiplication of fractions and inverses, assuming that they have learned anything about inverses at this point. If the student is taught to find the greatest common denominator first, then they will use their knowledge of greatest common denominators and basic division to find the quotient. They will also be reminded of the concept of whole numbers being expressed as fractions in this topic.

How did people’s conception of this topic change over time?

Originally, division of fractions would have been thought of in terms of practical use only and was likely conceptual since the symbolism of fractions was not the clearest. An example of fraction systems that were more difficult to comprehend, would be the Egyptian system, since they would add together unit fractions to represent non-unit fractions, unless it was fraction that had a repeating unit fraction, such as 2/7 = 1/7 + 1/7 (Weisstein). When symbols became clear, the division was done by taking the fractions, finding their common denominator, then dividing the numerators and denominators, leaving the quotient. The Babylonians mostly used the method of taking the inverse of the divisor and then multiplying by the dividend (O’Connor and Robertson, 2000). This is still a popular method. Today we can do either, but some believe that doing this operation algebraically might be better for students because thinking about division of fractions in only a practical sense will stifle their imagination (Ahia and Fredua-Kwarteng, 2006).

References:

Ahia, Francis and Fredua-Kwarteng, E.. (2006) Understanding Division of Fractions: An Alternative View.

O’Connor, J. and Robertson E.. (2000). An overview of Babylonian mathematics. Retrieved from

http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_mathematics.html

http://mathworld.wolfram.com/EgyptianFraction.html

# Engaging students: Ratios and rates of change

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 Kelly Bui. Her topic, from Algebra: ratios and rates of change.

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

The activity I created would involve having the entire class make Rice Krispies treats as either groups or partners. The recipe I linked below calls for 6 cups of Rice Krispies, but for the sake of the activity, each table will receive 1 ½ cups of the cereal. Every table will receive the original recipe and determine how many large marshmallows they will need and how many tablespoons of butter they will need for the recipe to be modified to using only 1 ½ cups of cereal. This activity will allow students to use the ratios to convert measurements, such as 40 marshmallows / 6 cups of Rice Krispies. After a group finishes their calculations and finds the ratio of each ingredient in respect to the amount of cereal, they can begin making their Rice Krispies treats. To extend this to a project, the Rice Krispies Treats activity can be done in class and students will be assigned to find a recipe which involves either using the ratio to create a smaller or larger serving of the recipe.

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

Ratios are introduced in middle school when we compare a part to another other part or part to whole. Students also begin to grasp that 12 inches / 1 foot is a relationship between two quantities because there are 12 inches per foot. We also see the use of ratios in high school chemistry when converting units. A simple ratio we first learn is that density is the ratio of mass to volume. This can then be extended, for example, when students begin to solve for the number of moles of an element given its mass in grams. Before teaching a chemistry class that 1 mole = 6.022×10^23, instructors could begin with simple conversions of the length of a state in miles and converting that length into inches. Once students understand the process and the concept that we are taking one unit and converting it to another unit, it will be easier to apply it to more complex situations in chemistry.

As a class, to get into the process of using ratios to convert units, the students can make their own conversion ratios with different objects to model this relationship. For instance, 4 fire extinguishers are the length 1 lab table and 8 lab tables are the length 1 school bus, and based on these ratios, students must find the length of a school bus in terms of fire extinguishers. This activity will allow the students to use objects they see every day and create a relationship among them.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Shark Tank is a show which involves 5 or 6 sharks (investors) and entrepreneurs that go into the “shark tank” to pitch their ideas seeking one or more partners who will invest in their business. Most entrepreneurs seek a money amount for an amount of stake in their company or business. If the entrepreneurs are lucky, they will get a deal with one or more of the sharks. In the video below, Aaron Krause pitches his product, the “Scrub Daddy” in which he asks for a \$100,000 investment for a 10% equity in his company. We see the topic of ratios appear in this business-related show because 10% equity of \$100,000 means he values his company at \$1 million, in other words our ratio is 10% / \$100,000. This ratio can be used to find the value of the company at 100%. In addition, the sharks also like to know the breakdown of the cost per unit. In this video, Mr. Krause states that it takes \$1.00 to create a scrub daddy and he sells it for \$2.80 wholesale. This gives the sharks the knowledge of how much they would earn for 1 Scrub Daddy. Given the sharks are willing to negotiate, like in the video, Lori gets 20% equity of the company. For each \$2.80 / 1 Scrub Daddy, she will earn \$0.56.

# Engaging students: Adding a mixture of positive and negative numbers

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 comes from my former student Rachel Delflache. Her topic, from Pre-Algebra: adding a mixture of positive and negative numbers.

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

An activity that could be helpful for engaging students in engaging in addition and subtraction would be Snakes and Humans. The activity is done using red and black counting tiles—the red representing humans (positive integers) and the black representing snakes (negative Integers). The activity begins by letting the students know that whenever one snake meets one human they leave together (i.e. cancel each other out). After the introduction is given, a series of addition problems can be given. After the student are comfortable with the addition problems, more challenging problems can be given such as 5- (-3) or 5 humans minus 3 snakes. From this point, the students have to figure out how to take away three snakes when they are only given 5 humans to begin with. The trick is that they have to add three human/snake pairs to the original group of humans before they can take away the three snakes, which results in them ending with 8 humans. This activity is beneficial in engaging students because it allows them to explore addition and subtraction of negative and positive integers without the anxiety that seeing traditional math problems may cause students.

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

This topic builds on students prior understanding of addition and subtraction of positive integers. Adding a negative integer can be introduced as subtracting a positive integer, which is something students should already be comfortable with. By equating it to something the students already know, it allows the students to have more confidence in their abilities going into the lesson After the students have mastered adding a negative number, the lesson would be able to move onto subtracting a negative number, a more unfamiliar topic to the students. For this part of the lesson, an activity like the one above could be use to allow the students to discover that subtracting a negative integer is the same as adding a positive integer and why. The benefit to building on a procedure that the students are already comfortable with is that it allows the students to be more comfortable going into the lesson.

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

One website that can be used to help engage students is http://www.coolmath-games.com/. While this website does not have instructional aspects, it does have games that are centered around math. One such game was Sum Points, in which the player tries to make the total points on the board equal to zero by adding and subtracting different numbers. The benefit of this website is that it allows students to sharpen their abilities in adding and subtracting integers without feeling like they are doing math. Students tend to enjoy using computers, and playing games on the computer tends to be a favorite for students. This tool gives them the pleasure of playing on the internet, while also allowing them to stay on task with learning.

# Engaging students: Solving one-step algebra problems

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 Deetria Bowser. Her topic, from Algebra: solving one-step algebra problems.

What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

To create a successful word problem that would both interest, and engage students, the teacher must “know his class.” Knowing one’s class involves knowing the many different students your students have. For example, if one knows that there are a lot of baseball players in the classroom, then creating word problems that involve baseball would be engaging for these students.

Additionally, to benefit all students you could do problems that involve finances. Including more “finance problems” will help students realize the importance of math, and how they can apply it in everyday life. An example of such problem would be “Damon’s earnings for four weeks from a part time are shown in the table. Assume his earnings vary directly with the number of hours worked. Damon has been offered a job that will pay him \$7.35 per hour worked. Which job is better pay (Tucker, A.)? Including word problems that students can relate to now or in the future can help students stay engaged while learning, and answer the question that is most commonly asked by students: “When will I ever use this in real life?”

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

As a teacher, creating engaging activities and/or projects can prove to be quite difficult for word problems that are one- or two-step algebra problems, due to the fact that most students completely shut down once a word problem is presented to them. To combat this I have found that making it into a cooperative game can help soothe the anxiety caused by word problems. One game that is great to play with one or two step algebra problems is called rally coach. In this game, students are paired off. Student A is expected to work on solving the problem, while Student B is expected to watch, listen, check, and praise just as a coach would. Once the students think they have the correct answer, they will raise their hand so that the teacher may check it. If they get the answer correct, then the teacher will give them another problem (this time Student A and Student B switch roles). If the answer is incorrect, they must continue working on the problem. The end goal of the game is to answer as many questions as possible before time runs out. By playing this game students are able to help each other solve one or two step word problems.

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

In future courses many problems will involve one or two step algebra problems. For instance, in science courses like chemistry and physics, one will need to know how to solve for different variables of equations. For example, if one is in a chemistry course and is given a word problem (i.e If a 3.1g ring is heated using 10.0 calories, its temperature rises 17.9°C. Calculate the specific heat capacity of the ring) that provides heat energy (Q) mass of a substance (m) and change in temperature (deltaT), but is asked to solve for the specific heat, students will need to know how to solve for the specific heat either by isolating the variable in the beginning (Cp=Q/mdeltaT) or plugging in the givens and isolating the variable (Daniell, B).

References

Daniell, B. (n.d.). Energy Slides 3 [Powerpoint that contains Specific Heat problem].

Tucker, A. (2016). Direct Variation. Retrieved September 01, 2017, from

http://www.showme.com/sh/?h=PQvPbm4

# Engaging students: Negative and zero exponents

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 Austin DeLoach. His topic, from Algebra: negative and zero exponents.

B1. How can this topic be used in your students’ future courses in mathematics or science?

The topic of negative and zero exponents is very important when or if the students get to calculus. Although that will be several years down the line, having a solid fundamental grasp on the idea of negative and zero exponents will help them understand derivatives a lot better. Because derivatives of “simple” functions just multiply the coefficient by the exponent and then subtract one from the exponent, it is important for the students to have a good understanding of what negative and zero exponents are. If they do not understand already, they will be confused about why, for example, the derivative of 3x is just 3. It also greatly simplifies derivatives of things like 4/x2, as the students will simply be able to recognize that that is the same thing as 4x-2 and follow standard rules instead of needing to think about the quotient rule and waste time with that. It will also help them in the more near future when they work with simplifying expressions with the exponents written in different terms (i.e. with a positive exponent or with a negative exponent in the denominator), as it will help them recognize what simplifications mean the same thing. Explaining that understanding negative exponents will thoroughly help them in the future may be enough for some students to want to solidify their grasp on the topic.

Although this is not about the early adoption of negative and zero exponents in the mathematical community, Geoffrey D. Dietz points out more recent bias for or against the use of negative exponents in textbooks in his Journal of Humanistic Mathematics (linked at the bottom of this answer). Dietz brings up the idea of what is considered “simplified” when it comes to negative exponents vs exponents in denominators. He rated over 20 mathematics textbooks from 1825 to 2012 from “very tolerant” of negative denominators in simplified answers to “very intolerant”. Interestingly, his first encounter with an “intolerant” textbook was not until the 20th century, and textbooks began getting more polarized as very tolerant or very intolerant closer to the end of the 20th century and getting closer to today. This is interesting when it comes to adoption by the mathematical community, as there is a significant inconsistency, even today, about whether negative exponents can be considered “simplified” or not. It will be important to point this out to your students so they can be prepared for their future teachers who may have different preferences on simplification from you, as that will help them understand the polarity in the mathematical community on this topic, as well as hopefully make them want to understand what negative exponents really mean. Dietz recommends giving your students practice with not only converting negative exponents to positive exponents, but also from positive to negative, in order to make sure they are prepared for whatever preferences come up as well as solidifying their understanding of what negative exponents mean.

E1. How can technology be used to effectively engage students with this topic?
This video from Khan Academy does a good job at explaining why negative and zero exponents are what they are. Although Khan Academy videos will likely not be the most engaging for all students, this video is short enough to maintain the attention of the class, and it the logic in it is helpful for the students who don’t understand how the definition of negative and zero exponents was decided on. The presenter does well explaining the idea of “going backwards” and dividing by the number when you decrease the exponent. It’s a good way to explain the “why” for students who ask about it, and it also is a good way to change up the pace for students, as playing videos during class could prevent it from becoming stale for the students, keeping them engaged for longer.

# Thoughts on Silly Viral Math Puzzles

I’ve seen silly math puzzles like this one spawn incredible flame wars on social media, and for months I’ve wanted to write an article about how much I’ve grown to loath these viral math posts.

Of course, after months of dilly-dallying, someone else beat me to it: http://horizonsaftermath.blogspot.com/2017/08/sick-of-viral-math.html. I encourage you to read the whole thing, but here’s the post’s outline of the myths perpetuated by these puzzles:

1. Math is just a bag of tricks.
2. Math is memorizing a set of rules.
3. Math problems have only one right answer.
4. Being smart means solving problems quickly.
5. Math is not for you.

# Engaging students: Finding the domain and range of a function

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 comes from my former student Brittany Tripp. Her topic, from Precalculus: finding the domain and range of a function.

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

One of my favorite games growing up was Memory. For those who haven’t played, the objective of the game is to find matching cards, but the cards are face down so you take turns flipping over two cards and have to remember where the cards are so when you find the match you can flip both of the matching cards. To win the game you have to have the most matches. I think creating an activity like this, that involves finding domain and range, would be a really fun way to get students’ engaged and excited about the topic. You could place the students in pairs or small groups and give each student a worksheet that has a mixture of functions and graphs of functions. Then the cards that are laying face down would contain various different domains and ranges. In order to get a match you have to find the card that has the correct domain and the card that has the correct range for whatever function or graph you are looking at. You could increase the level of difficulty by having functions, graphs, domains, and ranges on both the worksheet and the cards. This would require the students to not only be able to look at a graph of a function or a function and find the domain and range, but also look at a domain and range and be able to identify the function or graph that fits for that domain and range.

These pictures provide an example of something similar that you could do. I would probably adjust this a little bit so that the domain and ranges aren’t always together and provide actual equations of functions that the students’ must work with as well.

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

Finding the domain and range of a function is used and expanded on in a variety of ways after precalculus. For instance, one way the domain and range is used in calculus is when evaluating limits. An example is the limit of x-1 as x goes to 1 is equal to zero, because when looking at the graph when the domain, x, is equal to 1 the range, y, is equal to zero. Finding domain and range is something that is applied to a variety of different type of functions in later courses, like when looking at trigonometric functions and the graphs of trigonometric functions. You look at what happens to the domain of a function when you take the derivative in calculus and later courses. You work with the domain and range of different equations and graphs in Multivariable calculus when you are switching to different types of coordinates such as polar, rectangular, and spherical. There are also multiple different science courses that use this topic in some way, one of those being physics. Physics involves a lot of math topics discussed above.

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

I found a website called Larson Precalculus that technically is targeted toward specific Precalculus books, but exploring this website a little bit I found that is would be a super beneficial tool to use in a classroom. This website has a variety of different tools and resources that students could use. It has book solutions which if you weren’t actually using that specific textbook could be a really helpful tool for students. This would provide them with problems and solutions that are not exactly the same to what they are doing, but similar enough that they could use them as examples to learn from. This website also includes instructional videos that explain in depth how to tackle different Precalculus topics including finding domain and range. There are interactive exercises which would give the students ample opportunities to practice finding the domain and range of graphs and functions. There are data downloads that give the students to ability to download real data in a spreadsheet that they can use to solve problems. These are only a few of the different resources this website provides to students. There are also chapter projects, pre and post tests, math graphs, and additional lessons. All of these things could be used to engage students and help advance and deepen their understanding of finding domain and range. The only downfall is that it is not a free resource. It is something that would have to be purchased if you chose to use it for your classes.

References:

http://esbailey.cuipblogs.net/files/2015/09/Domain-Range-Matching.pdf

http://17calculus.com/precalculus/domain-range/

http://www.larsonprecalculus.com/pcwl3e/

# Finding the Regression Line without Calculus

Last month, my latest professional article, Deriving the Regression Line with Algebra, was published in the April 2017 issue of Mathematics Teacher (Vol. 110, Issue 8, pages 594-598). Although linear regression is commonly taught in high school algebra, the usual derivation of the regression line requires multidimensional calculus. Accordingly, algebra students are typically taught the keystrokes for finding the line of best fit on a graphing calculator with little conceptual understanding of how the line can be found.

In my article, I present an alternative way that talented Algebra II students (or, in principle, Algebra I students) can derive the line of best fit for themselves using only techniques that they already know (in particular, without calculus).

For copyright reasons, I’m not allowed to provide the full text of my article here, though subscribers to Mathematics Teacher should be able to read the article by clicking the above link. (I imagine that my article can also be obtained via inter-library loan from a local library.) That said, I am allowed to share a macro-enabled Microsoft Excel spreadsheet that I wrote that allows students to experimentally discover the line of best fit:

http://www.math.unt.edu/~johnq/ExploringTheLineofBestFit.xlsm

I created this spreadsheet so that students can explore (which is, after all, the first E of the 5-E model) the properties of the line of best fit. In this spreadsheet, students can enter a data set with up to 10 points and then experiment with different slopes and $y$-intercepts. As they experiment, the spreadsheet keeps track of the current sum of the squares of the residuals as well as the best guess attempted so far. After some experimentation, the spreadsheet can also provide the correct answer so that students can see how close they got to the right answer.