Engaging students: Solving word problems of the form “a is p% of b”

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 Christian Oropeza. His topic, from Algebra: solving word problems of the form “a is p% of b.”

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.)

Students would be able to answer word problems that involve real world applications. For example, a student could be asked: “Sam went to Academy to buy clothes, sports equipment, and fishing gear. At the register the total of Sam’s transaction before tax is $141.32. Given that the sales tax is 8.25%, what would Sam’s total be after tax?” These type of word problems would be relatable to students, which would show them the importance of this topic in life. Students always ask the question, “how is this used in everyday life?”, and with these type of word problems students may be able to generalize the concept more easily. When students cannot relate to a topic in math they become easily discouraged, give up, and stop paying attention in class, but with problems like these the students would be able to incorporate the topic into their own lives. Some other problems that students could be asked could involve any type of scenario where there is a percentage to be found between two numbers (Reference 1 & 4).

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

This topic can be used in different scenarios for math and science, but Chemistry is an excellent example. In chemistry, there is a topic that covers calculating percent composition. The basic idea of this topic is to calculate the percentage of each element’s mass in regard to a molecule’s total molecular mass. An example would be, “Calculate the mass percent composition of each element in a potassium ferricyanide, K₃Fe(CN)₆ molecule.” (Reference 2). These types of problems would help students understand how much a certain element or compound is in a particular molecule. Another example of how this topic can be used, is in math when a student has to convert between fractions, decimals, and percentages in a word problem. An example could be, “Mia has a basket full of fruit. In this basket she has 1/5 apples, 2/3 oranges, and 2/15 bananas. What percent of each fruit does she have in relation to the basket?” Students would be able to work on their converting skills to enhance their understanding of multiple representations of the same number (Reference 3).

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

Technology is always a great way to engage students especially with the newer generation of students where technology is part of their everyday life. The website mathisfun.com (Reference 4) is an excellent piece of technology to introduce or review this topic to the students because the website goes through visual representations of how a percentage of a whole looks like. Also, the website has a section where a student can input a number and a slider that allows the student to move it around to see what number would represent a certain percentage of the number inputted. Another example of effective technology is the website Khan Academy (Reference 1) because it has real world problems that are relatable. The website also gives hints and step-by-step solutions for each question in case a student is stuck and does not know what to do next. The use of multiple websites is good for students to have a variety to choose from in case one is easier to understand than another.

References:

Engaging students: Expressing a rate of change as a percentage

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 Peter Buhler. His topic, from Pre-Algebra: expressing a rate of change as a percentage.

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

As a teacher, one activity that could be used to engage students would be to use a real world application. This topic is unique, as it can be applied directly to shopping at a store. This activity could include having students bring in a catalog of a sale (either from a grocery store or department store) to the classroom to use. Then students would be encouraged to calculate percent discounts based on markdowns, or they could use a fixed percent discount (ex: 30% off everything) and calculate the new prices of various items from the store.

This activity is not only effective for teaching the topic, but also engages students since this is a topic that everyone deals with on a regular basis. Also, allowing students to bring in catalogs gives the students the freedom to operate within the classroom, as opposed to being given a generic worksheet and asked to solve those problems. An extension of this could be to introduce exponential growth (which is still rate of change and uses percentages) and can be applied to banking, credit, mortgages, and other applications that students may know little about.

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

Although the rate of change and percentages may be introduced at the junior high level, students will continue to use various aspects of these topics even into college level math courses. Derivatives are a huge part of calculus, and it is a known fact that derivatives are simply the rate of change of the original function. On the other hand, percentages can also lead to discussions around probability, chemical compositions within a compound, or even calculating grades for a certain class. All of these deal with using rate of change or percentages in classes outside of pre-algebra.

One application of this could be to introduce derivatives in a class outside of calculus and in a way that students would easily understand. If a student is able to understand the idea behind the rate of change, then they can understand a derivative. Likewise, the teacher can introduce certain applications of percentages outside of mathematics in order to tie in other topics.

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

As mentioned previously, one method to engage students is through real world applications. Both rate of change and percentages can be found in compound interest. There is a link to a video on YouTube which illustrates how powerful compound interest really can be. The use of graphics and other visuals within the video would allow for student to grasp how large the rate of change is, even after starting with small numbers.

Another useful tool that could be used in the classroom is an online calculator to observe the rate of change. If students have the ability to access the internet, then they could access the URL listed below. The website allows for students to put in different dollar amounts to observe the rate of change in regards to investment. While there is certainly a time to teach students how to calculate this without the website, this could be something that the students use to gain insight into how quickly compound interest can occur. It also gives students the opportunity to observe how different values change the final total and therefore make observations about how compound interest works. The link is: https://www.calculatestuff.com/financial/compound-interest-calculator.

References:

https://www.youtube.com/watch?v=immQX0RKFY0

https://www.calculatestuff.com/financial/compound-interest-calculator

Engaging students: Fractions, percents, and 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 again comes from my former student Kim Hong. Her topic, from Pre-Algebra: fractions, percents, and decimals.

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

I think making the students create a foldable, a short and quick project, would be a good and concrete activity for teaching fractions, decimals, and percents. Each flap is a topic. There is a definition and example. On the back of the foldable the students could create a table going between fractions, decimals and percents with many “harder” values.

The foldable is portable and quick, and can be a helpful and quick resource.

The students can also draw pictures inside the flaps. E.g A pizza and its slices to show fractions.

http://smithcurriculumconsulting.com/m4m_foldable/

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

This topic can be used in a students’ future course when they come across proportions and rates. They could see proportions when it appears in physics such a changes in time and speed. They could see rates of change when it appears in calculus involving derivatives. These values are factions that can be changed to decimals and percents because everything is a part of a whole.

Also, fractions, which are numbers over a whole, are the same as the term rational quantities. Rational quantities are numbers that can be written as a ratio that is a fraction. There is a subset of the Reals that are called the Rationals. In advanced logic and math courses, students will be able to work with this subset of the Reals.

I found this really awesome website the students could play around with for the first minutes of class to get their juices flowing. Basically the objective of the game is to group the equal values in circles. There is a check answer option as well.

It starts off very simple with very easy mental math and then with each level, the difficulty increases.

http://www.mathplayground.com/Decention/Decention.html

Linus Pauling and the Golden Rule

Source: http://www.azquotes.com/quote/378980

Averages and percentages

Source: http://www.xkcd.com/937/

Percentage points

Engaging students: Expressing a rate of change as a percentage

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 Samantha Smith. Her topic, from Pre-Algebra: expressing a rate of change as a percentage.

A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

The TLC show Extreme Cheapskates follows the lives of Americans who are very frugal with their money. In this clip, a man takes his wife to the movies and does everything he can to save money. Expressing a rate of change as a percentage is most commonly associated with spending money, such as a sign in a store saying “50% off all merchandise.” Using this clip as an introduction, I can have my students practice calculating how much money they are saving on buying certain items. I can bring in a catalog and coupons and have my students “buy” 3 items and calculate how much they saved. This is a real world application that students will use for the rest of their lives. Looking back on the video, students may notice that the man had a rate of change of 100%. Instead of paying full price for the drink and popcorn, he saved 100% of his money (or paid 0%). Even though his wallet was happy, I’m sure his wife wasn’t after seeing this on TV.

C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Facebook is the largest social networking site on the internet. There are many high school students that constantly check their Facebook and most of them post to get attention from their followers. The article link below gives 7 pieces of advice on how to get more attention on Facebook. For example, number 1 says “Photo posts get 39% more interaction.” As I am introducing the topic of changing rates as a percentage, I can have my students try to analyze what these numbers really mean. The important thing to stress about this article is not the just the numbers themselves, but the verbs attached to the percentages such as “increasing.” This shows the rate is changing. Combining this topic and a website the students use every day is sure to grab their attention.

http://blog.bufferapp.com/7-facebook-stats-you-should-know-for-a-more-engaging-page

C3. How has this topic appeared in the news?

I know, for myself, that I love eating fast food, and I’m sure I am not the only one. However, after New Year’s Resolutions are made, many people choose to give up the glorious taste and convenience of fast food for options that are healthier. This trend causes many fast food chains, such as McDonalds, to lose customers. As mentioned in the article below, McDonald’s guest counts have fallen 16% in the U.S. in 2013. This causes the company to make changes to attract more customers. Rates of change expressed as percentages are very common in the analysis of businesses. Students will perk up when they hear this topic because it is interesting to see how their personal diet choices effect major restaurants.

http://abcnews.go.com/Business/wireStory/mcdonalds-profit-fewer-customers-21634926