In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission again comes from my former student Emma White. Her topic, from Algebra: 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 + _HCl → _MnCl + _KCl + _Cl + _HO
In the blanks, a variable can be placed, such that:
aKMnO + bHCl → cMnCl + dKCl + eCl + fHO
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.
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