Engaging students: Slope-intercept form of a line

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 Fidel Gonzales. His topic, from Algebra I: the point-slope intercept form of a line.

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 advancing right in front of us. Using it in the classroom can be a tool that allows students to have a more hands on experience in the classroom. When I was in middle school, the only tool that we had to learn slope intercept form of a line was using a ti-inspire calculator. However, schools are receiving more funding and can provide students with tablets or computers to assist in their academic career. Gizmos is a website that contains many user-friendly programs that a student can use to learn a concept, or an educator can present to reinforce a skill. For the topic of slope intercept form of a line, the gizmo has two sliding parts that allows the user to change the values of the equation. One for the slope and one for the y- intercept. The student can adjust the values of both and observe the changes that occur to the line. This experience is more user friendly since it only allows the person to change those two aspects compared to having to input the equation each time into the graphing calculator. The reason that students would be more likely to be engaged is because they are already used to technology and there is still a need to incorporate technology into the classroom. So, students would prefer using a computer compared to the traditional paper and pencil. Imagine them having to graph by hand each graph to compare differences!

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

Slope intercept form is a way that data can be displayed. The data is usually continuously decreasing or continuously increasing. There is a magnitude of activities that can be used to help students gather a better understanding of the topic. As an educator, I would create a scavenger hunt that displayed either a word problem or a graph. Both will ask for the student to represent the information as slope intercept form. For each problem, there will be 4 answer choices that the student could choose for their answer. On their worksheet, there will be fill in the blanks that will be filled up from the letter that is in front of the correct answer. As the student progresses to the next problem, they will be filling out the letter blanks in a random order. So, if the person does the activity correctly, they should end up with the correct word phrase. The word phrase will be a math pun to add to the magic. This activity will allow students to switch from graph and word problems to slope intercept form.

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

As educators, we want to ensure that our students have the proper foundation to continue advancing their mathematic skills. Slope intercept form is an algebra base lesson. The skills that students used to reach this topic is addition. At a young age, students learn to count numbers in repeated increments. An example of this is when a student keeps adding 5 until they reach a certain number. Displaying this as slope intercept would be a line with no y intercept and a slope of 5. We have even used y intercepts in context to adding in past classes. An example of this would be a person wanting to sell 200 dollars’ worth of tickets that are worth 5 dollars each and they already started with 57 dollars. If they were to solve the problem using slope intercept form, they would put 200 as the y value and 57 as the y intercept of the problem. The slope would be 5. In the past, they would add 5 to 57 until they reach their goal. Slope intercept form is a way for students to display data with a constant increasing or decreasing value. It is more convenient for students to use slope intercept form compared to how they displayed the pattern in the past. They use it now since they learned why it works before they reach algebra.

References:

https://gizmos.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=166

https://www.geogebra.org/about

Engaging students: Finding the slope of a line

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 Bri Del Pozzo. Her topic, from Algebra: finding the slope of a line.

How could you as a teacher create an activity or project that involves your topic? As a teacher I would likely introduce a very popular and well-received project to my students, the project where students draw an angular image on a graph and then calculate the slope of 20 lines from their image. I love this project because it allows students to connect mathematics to art and encourages them to express themselves creatively. As a precursor to the project, I would introduce students to the types of slopes and their characteristics using a tool that I learned in my Algebra Class, Mr. Slope Guy (pictured below).

In the image the positive slope is indicated by the left eyebrow and above the plus-sign eye, the negative slope is the right eyebrow by the negative sign eye, the nose represents the undefined slope and is denoted by a vertical line and a “u” for undefined, and the mouth represents the zero-slope shown by a horizontal line and two zeros. The students could use this resource while completing their projects to serve as a reminder of the types of slopes. The main focus of the project, however, would focus on the process of how to find slope given two points on a line. (This project is based on an example from: https://kidcourses.com/slope/ #5)

How does this topic extend what your students should have learned in previous courses? In the grade 7 TEKS for mathematics, students are expected to “represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form .” This creates a foundation for finding the slope of a line by introducing students to multiple representations of what slope looks like. When discussing how to find the slope of a line, I think that the tabular representation is a great tool for students to visualize the meaning behind slope. In seventh grade math, students were able to conceptualize slope without using the formula. When finding slope in early algebra, I would encourage students to look at graphs from a new lens, noticing features such as the sign of the line, the steepness of the line, the difference in x’s and y’s at different points on the line, and the slope itself. When looking at a table, I would ask students to calculate the difference in x’s and y’s as they go down the rows of the table and have them compare those numbers to those that they saw in the graph.

How has this topic appeared in the news? As many of us know, over the past 18 months or so, the number of Covid-19 Cases in the United States has been on the rise. For a long time, the total number of cases in the United States was growing exponentially and very quickly. As more research has been done by the Centers for Disease Control and Prevention, we have learned that there is a way to flatten the curve and reduce the number of daily cases. This initiative to flatten the curve has resulted in the growth of cases to resemble liner growth rather than exponential growth. As mathematicians, we can calculate the slope of the line that represents the (linear) growth of Covid-19 cases per day. We can make comparisons between growth rates in different states and use that data to make predictions about effectiveness of Covid-19 prevention procedures.

Engaging students: Equations of two variables

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 Taylor Bigelow. Her topic, from Algebra: equations of two variables.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

This topic is perfect for word problems, you can make a lot of interesting word problems using 2 variables. Here are some examples of word problems.

● Sam is mowing lawns for money over the summer. They charge $10 an hour. They have a family discount of 20% per hour. If they mow non-family members laws for 10 hours this week and mowed family members laws for 3 hours, how much money did they make this week?
○ 10N+8F=?
○ N=10 and F=3
○ 10(10)+8(3)=124
○ So they made $124

● John is buying blue and yellow gummy bears at the store. He has $20 to spend on candy. Blue gummy bears come in bags of 20 for $1 each, and Yellow gummy bears come in bags of 50 for $3 each. He knows we want exactly 100 Blue gummy bears. How many yellow gummy bears can he buy?
○ B=Blue gummy Bears Y=Yellow gummy Bears
○ 20=B+Y
○ B= 100/20= $5 for 100 gummy bears
○ 20= 5+Y so Y=$15
○ With $15 he can buy 5 bags of yellow gummy bears. 5*50=250. So he can buy
250 yellow gummy bears

● Alex is building a fence for her backyard. She is building it in a rectangular shape, and she wants the length of the fence to be twice as long as the width of the fence. If the area of her backyard is 200 feet, how long is the width, and how long is the length?
○ L=length W=width
○ L*W=200
○ L=2W
○ 2W*W=200
○ 2W^2=200
○ W^2=100
○ W=10
○ So L=2(10)=20

These are just 3 examples I came up with on the spot. You can create a lot more, and
with a variety of difficulties.

How does this topic extend what your students should have learned in previous courses?
This topic builds on knowledge from elementary school and extends into almost all future math. It starts with kids understanding multiplication and addition, then to them being introduced to solving equations in middle school, and then is heavily used in high school math classes, and any math class that requires basic algebra skills in the future. I looked through some of the teks to find references to two-variable equations and found it only referenced in algebra 1 and 2. I also went back through 6th, 7th, and 8th grade and found where they were using one-variable equations since that is the prior knowledge that they are building onto with two-variable equations.

● 6th Grade
○ (9) Expressions, equations, and relationships. The student applies mathematical
process standards to use equations and inequalities to represent situations. The
student is expected to:
■ (A) write one-variable, one-step equations and inequalities to represent
constraints or conditions within problems;
■ (B) represent solutions for one-variable, one-step equations and
inequalities on number lines; and
■ (C) write corresponding real-world problems given one-variable,
one-step equations or inequalities.
○ (10) Expressions, equations, and relationships. The student applies
mathematical process standards to use equations and inequalities to solve
problems. The student is expected to:
■ (A) model and solve one-variable, one-step equations and inequalities
that represent problems, including geometric concepts; and
■ (B) determine if the given value(s) make(s) one-variable, one-step
equations or inequalities true.
● 7th Grade
○ (10) Expressions, equations, and relationships. The student applies
mathematical process standards to use one-variable equations and inequalities
to represent situations. The student is expected to:
■ (A) write one-variable, two-step equations and inequalities to represent
constraints or conditions within problems;
■ (B) represent solutions for one-variable, two-step equations and
inequalities on number lines; and
■ (C) write a corresponding real-world problem given a one-variable,
two-step equation or inequality.
○ (11) Expressions, equations, and relationships. The student applies
mathematical process standards to solve one-variable equations and inequalities.
The student is expected to:
■ (A) model and solve one-variable, two-step equations and inequalities;
■ (B) determine if the given value(s) make(s) one-variable, two-step
equations and inequalities true
● 8th Grade
○ Expressions, equations, and relationships. The student applies mathematical
process standards to use one-variable equations or inequalities in problem
situations. The student is expected to:
■ (A) write one-variable equations or inequalities with variables on both
sides that represent problems using rational number coefficients and
constants;
■ (B) write a corresponding real-world problem when given a
one-variable equation or inequality with variables on both sides of the
equal sign using rational number coefficients and constants;
■ (C) model and solve one-variable equations with variables on both
sides of the equal sign that represent mathematical and real-world
problems using rational number coefficients and constants
● Algebra 1
○ (2) Linear functions, equations, and inequalities. The student applies the
mathematical process standards when using properties of linear functions to
write and represent in multiple ways, with and without technology, linear
equations, inequalities, and systems of equations. The student is expected to:
■ (B) write linear equations in two variables in various forms, including y
= mx + b, Ax + By = C, and y – y1 = m (x – x1 ), given one point and the
slope and given two points;
■ (C) write linear equations in two variables given a table of values, a
graph, and a verbal description;
■ (H) write linear inequalities in two variables given a table of values, a
graph, and a verbal description
○ (3) Linear functions, equations, and inequalities. The student applies the
mathematical process standards when using graphs of linear functions, key
features, and related transformations to represent in multiple ways and solve,
with and without technology, equations, inequalities, and systems of equations.
The student is expected to:
■ (D) graph the solution set of linear inequalities in two variables on the
coordinate plane;
■ (F) graph systems of two linear equations in two variables on the
coordinate plane and determine the solutions if they exist;
■ (G) estimate graphically the solutions to systems of two linear
equations with two variables in real-world problems; and
■ (H) graph the solution set of systems of two linear inequalities in two
variables on the coordinate plane.
○ (5) Linear functions, equations, and inequalities. The student applies the
mathematical process standards to solve, with and without technology, linear
equations and evaluate the reasonableness of their solutions. The student is
expected to:
■ (C) solve systems of two linear equations with two variables for
mathematical and real-world problems.
● Algebra 2
○ (3) Systems of equations and inequalities. The student applies mathematical
processes to formulate systems of equations and inequalities, use a variety of
methods to solve, and analyze reasonableness of solutions. The student is
expected to:
■ (C) solve, algebraically, systems of two equations in two variables
consisting of a linear equation and a quadratic equation;
■ (D) determine the reasonableness of solutions to systems of a linear
equation and a quadratic equation in two variables;
■ (E) formulate systems of at least two linear inequalities in two variables;
■ (F) solve systems of two or more linear inequalities in two variables; and
■ (G) determine possible solutions in the solution set of systems of two or
more linear inequalities in two variables.

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

Algebra is a really old concept, dating back almost 4 thousand years ago. (So kids have been doing the same thing in classes for millennia.) The Babylonians were the first to use algebra in the 1900s. The Egyptians also used algebra around the same time, but they focused on linear algebra, while the Babylonians did quadratic and cubic equations. The ancient Greeks used geometric algebra around 300 BC. They solved algebra equations using geometry, and their methods are very different from the ones we use today. A thousand years later, around 800 AD, Muhammad ibn Musa al-Khwarizmi became the father of modern algebra. The middle east used Arabic numerals (the numbers 0-9 which we still use today). The word algorithm is even derived from his name. Algebra started thousands of years ago to solve problems and has been developed over time into what it is today.

Citations:
https://www.mathtutordvd.com/public/Who-Invented-Algebra.cfm
https://texreg.sos.state.tx.us/public/readtac$ext.ViewTAC?tac_view=4&ti=19&pt=2&ch=111

Engaging students: The quadratic formula

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 Sydney Araujo. Her topic, from Algebra: the quadratic formula.

D4. What are the contributions of various cultures to this topic?

The quadratic formula can be traced all the way back to the Ancient Egyptians. The ancient Egyptians knew how to calculate the area of different shapes but did not know how to calculate the length of the sides of a shape. Moving forward, it is speculated that the Babylonians developed the completing the square method to solve problems involving areas. The Babylonians used a more similar number system to the one we use today. Instead, they used hexagesimal which made addition and multiplication easier. We can also see a similar method used by the Chinese around the same time. Pythagoras and Euclid were some of the first to attempt to find a more general formula to solve quadratic equations, both using a geometric approach. They’re ideas differ slightly, Pythagoras observed that the value of a square root is not always an integer but he refused to allow for proportions that were not rational. Whereas Euclid proposed that irrational square roots are also possible. At the time, the ancient Greeks did not use the same number system that we use, so it was impossible to calculate square roots by hand. It wasn’t until the Indian mathematician, Brahmagupta, who came up with the solution to the quadratic formula. This is because Indian mathematics used the decimal system as well as zero which had a massive advantage over the Egyptians and Greeks. Brahmagupta was the one that recognized that there are two roots in the solution to the quadratic equation and described the quadratic formula.

A2. How could you as a teacher create an activity or project that involves your topic?

One of my fondest high school memories is from my junior year physics class. It was the famous Punkin’ Chunkin’ project. Students were put in groups and asked to build a trebuchet or catapult that could launch a pumpkin across a field. The only requirement was for the device to work, the distance was just fun extra credit. For this project we had to predict the pumpkins trajectory using different variables like the pumpkin’s weight, force, momentum, etc. However, by the time we were juniors, we had either taken Algebra 2 or were currently in it. So, our physics and algebra teacher were working together so that by the time this project came around we were working on quadratic equations in algebra. As the shape of the trajectory of a pumpkin was a parabola. Because of this experience, I can create an activity or even a similar project with the physics teacher. This way students see the different applications of quadratic equations and have a tangible real world math experience.

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

As the quadratic formula is taught in Algebra 1, students have only seen linear equations prior to that point. Students recognize that when they are solving these equations, they are looking for one solution, no solution, or infinitely many solutions. The one solution being a singular ordered pair and then they are done. What students then must extend on when they reach quadratic equations, and the quadratic formula is that they’re now looking for two separate solutions. So, at this point they know how to solve for x and understand inverses which is important when it comes to quadratic equations. During the solving process of a quadratic equation, students may have to take the square root of both sides of the equation which will give you a plus or minus sign in front of the square root. Which makes the connection on why there are two solutions to a quadratic equation and the quadratic formula, because a parabola has two roots.

Works Cited:

Brahambhatt, Rupendra. “Quadratic Formula: What, Why, and How It Changed Mathematics.” Interesting Engineering, Interesting Engineering, 16 July 2021, interestingengineering.com/quadratic-formula-what-why-and-how-it-changed-mathematics.

Engaging students: Absolute value

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 Ethan Gomez. His topic, from Pre-Algebra: absolute value.

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

This topic extends students understanding of distance relative to positive and negative integers. First, students learn the positive integers — the counting numbers. Then, students are introduced to negative numbers. Visually, positive integers are to the right of zero, and negative integers are to the left of zero; students understand that these numbers exist and where they lie relative to each other. Essentially, students start by having a directional sense of numbers. Also, students also have a good understand of distance. With the concept of absolute value, students are able to associate distance with positive/negative numbers. Negative numbers aren’t just randomly placed but are rather a certain unit away from the number zero. For example, the absolute value of -5 is 5. So, -5 is not just a number that happens to be to the left of zero, but it is also 5 units away from zero. We now have a spatial sense of integers along with the directional intuition, making the numbers feel a bit more tangible and less abstract.

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

Technology can be used to effectively engage students with the concept of absolute value since it allows students to explore its meaning. Students can discover the connection between distance and integers on their own, which reinforces the meaning-making process that teachers strive to provide students. For example, Gizmos has a wonderful tool that displays integers on a number line. On this gizmo, students are provided a visual that portrays the spatial and directional aspect of integers. This gizmo also makes students take note of the similarities between the absolute value of positive and negative numbers, forcing them to think about why they happen to be the same number sometimes.

https://gizmos.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=210

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

I could create a documentation sheet for students to keep track of what time they get home from school. They will keep track of this information for a week. The first time that they record will be the “reference time.” Every day after that, the students will document the time they get home, and how many minutes off it was from the first time, as well as if it was earlier or later than the first time. Having students think about “how many minutes off” they were from the first recorded time get them used to the idea of a magnitude, and how the number they are using tends to always be positive; the only difference is in the description of that number, which can be associated with the positive and negative characteristic of integers.

Engaging students: Adding and subtracting decimals

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 Sydney Araujo. Her topic, from Pre-Algebra: adding and subtracting decimals.

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

I have been riding horses since I was 5 years old, when I was around 12 years old I got into the equine sport called barrel racing. The sport is an equine speed event. Essentially horse and rider go through a clover leaf pattern as fast as possible. Placings are separated by 1000ths of a second. At competitions, there are different divisions, typically 4-5. These divisions are separated by half a second. For example, if the winning time of the barrel race was 15.536 seconds, then the winning times of the different divisions would be as follows, 16.036, 16.536, 17.036, and so on by simply adding half a second. It was always interesting to compare times and to see where I could possibly stand in different divisions based on my time and the winning time. I could see myself creating an activity that had my students be given different scenarios like being given a winning time and determining the winning times of the different divisions, determining which division a certain time would be in, how much faster or slower at time needs to be to place, and so on. This was an activity I did regularly at barrel races for myself and other people when watching.

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

One of the more popular movies I can think of is the movie Hidden Figures. The movie is about a team of African American women mathematicians who work for NASA to help launch an astronaut into orbit. There are several different scenes in the movie where math problems are being solved and this involves the adding and subtracting of decimals. It shows that doing math by hand and math itself is very important in the real world and has helped us make great discoveries and progress. Another movie where adding and subtracting decimals appeared is in the movie called Gifted, where an uncle of an extremely math gifted child suddenly becomes her guardian. She solves several advanced math problems and proofs throughout the movie. The topic also appears in the classic sci-fi TV show Star Trek. It is constantly brought up throughout the series, typically from the character Spock who will make calculations on the spot. As he is a very smart and logical character, he is often the one who must do the required math in the series.

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

Adding and subtracting decimals is constantly used in both mathematics courses and science courses throughout high school and eventually college. We see adding and subtracting decimals in some trigonometry concepts when solving for theta and using different trig functions. Students will also see this very often in algebra when dealing with real world situations that forces them to have to use decimals. It appears quite a bit when students approach quadratic equations as once, they learn the quadratic formula to solve quadratic equations that don’t have integers, they will run into many decimals and having to add and subtract. Looking even further into the future of student’s math courses, we often must add and subtract decimals when evaluating different limits and integrals. Adding and subtracting decimals also appears in physics courses. Students will often see many decimals in physics when solving problems using force, density, displacement, and so on. You often see more imperfect numbers and situations in physics as it is more often seen in the real world.

Engaging students: Powers and exponents

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 Ashlyn Farley. Her topic, from Pre-Algebra: powers and exponents.

One class activity that will engage students while reviewing and/or teaching Exponent/Power concepts is “Marshmallow and Toothpicks.” This activity can be used for teaching the basic of exponents, as well as exponent laws. The idea is that the toothpicks are different colors, and the different colors represent different bases, thus the same color means it’s the same base. The marshmallows represent the exponent, i.e. the number of times the student needs to multiply the base. By following a worksheet of questions, the students should be able to solve exponent problems physically, visually, and abstractly. This activity, I believe, is best done with partners or groups so that the students can discuss how they think the exponents/exponent laws work. After the activity, the students are also able to eat their marshmallows, which encourages the students to participate and complete their work.

Exponents are used in functions, equations, and expressions throughout math, thus having a deep understanding of exponents and their laws is very important. By fully mastering exponents and exponent laws, the students will be able to more easily grasp more difficult material that uses these concepts. Some specific ideas that use exponents and/or exponent laws in future math courses are: multiplying polynomials, finding the volume and surface area of prisms and cylinders, as well as computing the composition of two functions. Exponents are also used in many other situations than just math, such as in science or even in careers. Some careers that consistently use exponents and/or exponent laws are: Bankers, Computer Programmers, Mechanics, Plumbers, and many more.

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

An easy way to introduce students who have never seen exponents or exponential growth before is to use a graphing calculator. By plugging in an exponential function into the calculator and viewing the graph and zooming out, students can easily see how quickly numbers start to get The website Legends of Learning focuses on creating educational games for students in kindergarten through 9th grade. One game that goes over exponents, as well as the exponent laws, is Expodyssey. This game has the students solve problems to “fix” a spaceship to get back to Earth. The problems are built upon each other, so it starts by having the student answer what an exponent is, then what multiplying two exponents same base is, and keeps building from there. Each concept has multiple problems to be solved before moving on so that the students can show their mastery of the content. I believe that this game also helps improve cognitive skills by having the students do various activities simultaneously, such as calculating, reading, maneuvering elements and/or filling answers as required.

References:
Blog: Number Dyslexia
Link: https://numberdyslexia.com/top-7-games-for-understanding-math-exponents/

Engaging students: Finding prime factorizations

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 Bri Del Pozzo. Her topic, from Pre-Algebra: finding prime factorizations.

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

An activity that I would create for my students involving Prime Factorization is based on an example that I saw on Pinterest. I would prepare an activity where students would be given a picture of a tree and assigned a two-digit number. I would then have students decorate their tree and at the base of the tree, they would write their assigned number. Then, as the roots expand down, students would be able to write the factors of their number as a factor tree until they are left with only prime factors (based on the image from https://www.hmhco.com/blog/teaching-prime-factorization-of-36). In the example from Pinterest, the teacher focused on finding the greatest common divisors between two numbers and used the factors trees as guidance. For my activity, I would assign some students the same number and emphasize that some numbers (such as 24, 36, 72, etc.) can be factored in multiple ways, so the roots of the trees could look different depending on how the student decides to factor their number.

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

There are a few ways that Prime Factorization can be used in my students’ future math courses. Prime Factorization is incredibly useful when learning how to simplify fractions. By practicing Prime Factorization, students become more familiar with the factors of large numbers, which becomes helpful when simplifying fractions. In the instance that a fraction is not in its simplest form, students will have an easier time recognizing such and will feel more confident in simplifying the fraction. Additionally, Prime Factorization prepares students for finding Greatest Common Divisors. Knowing how to find Greatest Common Divisors can be useful when solving real-world problems as well as in simplifying fractions. At a higher level of math, Prime Factorization allows students to practice the skills needed to prepare themselves for factoring things more complicated than numbers. For example, the idea of factoring can be applied to factoring a common factor out of an expression, factoring quadratic equations, and factoring polynomials with complex numbers.

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.

Khanacademy.org would be a fantastic website to engage students in this topic because of the inclusion of multiple representations. This website allows students to work through multiple practice problems where they can find the Prime Factorization of a number. When the student gets the question correct, they can move on to the next question, or they have the option to view a brief explanation on how to arrive at the correct answer. If students get a problem incorrect, they can retry the problem or get help on the question. The “get help” feature also provides students with a brief explanation, with options in video form and picture/written form, of how to solve the problem. Another important feature of this website is the ability for students to write out their thoughts as they work through the problem. Khan Academy allows students the option to use an online “whiteboard” feature that appears directly below the problem. This “whiteboard” feature allows students to write out their work and also offers a walkthrough of how to draw a factor tree.

Resources:
https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples/pre-algebra-prime-factorization-prealg/e/prime_factorization
https://www.hmhco.com/blog/teaching-prime-factorization-of-36

Engaging students: Solving two-step algebra problems

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 Chi Lin. 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?

There is an interesting activity that I found online. It is called mini task cards. However, I want to rename this activity as “Find your partners” as an engage activity in this topic. I am going to create some two-step equations on the cards and give those cards randomly to the students at the beginning of the class. Each student has one mini card. The students will have 5 minutes to solve the equations and they will find the partners who have the same answers as them (there is 2-3 person in each group). The person who has the same answer with them will be the partner that they are working together with in the class. I will set up the answer as their group name (for example, if the answer is 1, then it means the group name is “Group One”). Here is an example that how the card will look like.

Reference:

12 Activities that Make Practicing Two-Step Equations Pop

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

Solving two-step equations is the foundation of solving multi-step equations. Solving two-step equations looks easy but it can become very hard. This topic can be applied in lots of areas such as high-level math classes, computer science, chemistry, physics, engineer, and so on. Most definitely, the students will see lots of problems about solving multi-step equations in different high-level mathematics courses in college, such as pre-calculus, calculus 1-3, differential equations, and so on. Also, the students will use the knowledge when they write the code in computer science class. For example, when they write down the code of two-step or multi-step algebra problems, they need to know which step goes first. If they do the step wrong, then the computer program will compute the wrong result. Moreover, the students will use solving two-step equations in chemistry class. For example, the students will apply this knowledge, when they write down the chemical equations and try to balance the equations.

How does this topic extend what your students should have learned in previous courses?
First, students should know what linear equations are and how to write down the linear equations. Second, students should know how to solve one-step algebra problems, such as $x+8=16$ or $x/8=16$ . Students should have learned that when they solve for the one-step equations (addition and subtract), whatever they do to one side of the equation, they need to make sure they add the same thing to the other side. For example, when they solve the equation $x+8=16$ , they can subtract 8 for both sides, which is $x+8-8=16-8$ . Therefore, x=8. Also, student should know that when they solve for the one-step equations (multiplication and division), they need to multiply both side by the reciprocal of the coefficient of the variable. For example, when they solve the equation $x/8=16$ , they need to multiply the reciprocal of $1/8$ for both sides, which is $x/8*8=16*8$ . Therefore, $x=128$ . Thus, when they learn to solve two-step equations, they need to combine these rules.

References:
https://www.khanacademy.org/math/algebra-home/alg-basic-eq-ineq/alg-one-step-mult-div-equations/a/one-step-equation-review

Solving Two-Step Equations

Engaging students: Solving one-step algebra problems

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: solving one-step algebra problems.

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

Solving one-step algebra problems strings into many future scenarios the student may (and will probably) encounter. One-step algebra problems infer that there must be two-step algebra problems and three-step algebra problems and so forth. As mathematicians, we know this to be true. While mathematics in my focus of study, I want to show the importance of learning this concept as it will aid in other classes. Stoichiometry is a concept taught in chemistry that has to do with the “relationship between reactants and products in a reaction” (Washington University in St. Louis, 2005). Chemical reactions require a balance. Essentially, once-step algebra expressions require just the same where both sides of the equations must be equal for the expression to be true. An example of a stoichiometry equation one may see in chemistry would be:

_KMnO ${}_4$ + _HCl → _MnCl ${}_2$ + _KCl + _Cl ${}_2$ + _H ${}_2$ O

In the blanks, a variable can be placed, such that:

aKMnO ${}_4$ + bHCl → cMnCl ${}_2$ + dKCl + eCl ${}_2$ + fH ${}_2$ O

Next, we would apply the Conservation of Mass. This concept deals with the number of atoms that must be on each side for the equation to be balanced. Writing the elements and their balanced equations with the variables, it follows:

K: a = d
Mn: a = c
O: 4a = f
H: b = 2f
Cl: b = 2c + d + 2e

As we can see, there is going to be more expressions and substitutions that must take place. That is something you can solve on your own if you wish. Overall, we see the importance of learning one-step algebra problems because this will be the foundation for solving more complex questions, even more so outside of the math classroom.

How has this topic appeared in high culture (art, classical music, theatre, etc.)?

Theatre is more than the actors on the stage. While the performance and show are the part most people acknowledge and enjoy, the technical part behind the performance is what allows the show to happen. Algebraic problems are often used in technical theatre, especially when it comes to building a set. A prime example is building a single foundation (usually used in One Act plays where the whole play takes place in one scene). Focusing on a rectangular foundation, if we know the amount of space the actors, set, and featuring décor need, we can use this in an algebraic expression. Furthermore, if we also know dimensions of one of the sides (length or width), a variable can be used for the unknown side (since the area of a rectangle is length times the width). If we want to take this a step further, multiple one-step algebraic expressions can be used when making the foundation. If we know the length and width of the foundation and the length and width of the sheet floorboards to be used, we can write various expressions to determine how many sheet floorboards need to be used lengthwise and widthwise (example shown below).

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

The use of technology is on the rise and the involvement of newer generations is greatly rising as well. Because of this, utilizing online resources is an effective way to capture the attention of the students and make math more engaging. Using algebra tiles is a perfect way to resemble this topic, even more so when it can be done online. Therefore, the teacher does not need to buy any materials and the students (especially high schoolers) don’t have to carry paper resources around or even home where, we all know, they will end up in the trash. Online algebra tiles provide a way to visually see the one-step algebra problem and work accordingly. Even so, these tiles can be an introduction and foundation on what is to come (these tiles are also a great source for solving two-step equations, distribution, polynomials, the perfect square, and so forth). Another insight for using online algebra tiles is in some schools where technology such as tablets/computers are provided, the students can share their screens to a projector (or whatever resources the classroom may have) and describe their thinking process to the class. This builds on the idea of students learning, processing, and being able to teach their peers what they learned as well.

References

End Result for x +4=8: https://technology.cpm.org/general/tiles/?tiledata=b5____g+afx__boy__aaapTtPhF%2B__qvtTauq7tSaurDtSaur5tSausBtSauwisCauwis4auwitAauwit2auwir6auwiuyauwiu0auwirEawq7ukawrDukawr5ukawsBukawwOrEawwOr6awwOsCawwOs4hFProblem%3A%20Solve%20for%20x.%20%20x%20%2B%204%20%3D%208__qPpBhFThere%20is%20one%20x%20left%20on%20the%20left%20sideand%20four%201s%20left%20over%20on%20the%20right.Therefore%2C%20x%20%3D%204.__v-wcgawWwOtBgawWwOt1gawWwOuzgawWwOu0

End Result for 4x=16: https://technology.cpm.org/general/tiles/?tiledata=b5____g+afx__boy__aaatCsnaatCtgaatCviaatCulauvIslauwaslauwGslauw8slauw6tjauwEtjauv8tjauvGtjauvFupauv7upauwDupauw5upauvDvlauv5vlauwBvlauw3vlhF%20%20%20%20%20Look%20at%20one%20x%20on%20the%20left%20side%20and%20seewhat%20it%20is%20paired%20with%20on%20the%20right%20side.We%20see%20that%20one%20x%20is%20paired%20with%20four%201s.Therefore%2C%20x%20%3D%204.__vuwngaqAr8vkgawZxzvnhFProblem%3A%20Solve%20for%20x.%204x%3D16__pEpz
https://chemistrytutor.me/balancing-chemical-equations-algebra/
http://www.chemistry.wustl.edu/~coursedev/Online%20tutorials/Stoichiometry.htm
https://technology.cpm.org/general/tiles/