# Engaging students: Equations of two variables

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 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

● 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.

○ (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.
○ (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
○ 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.

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