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 again comes from my former student Chelsea Hancock. Her topic, from Pre-Algebra: dividing fractions.

Applications (A1)

Students can encounter the division of fractions in a variety of places outside of the classroom. Some of these instances could even happen in your own home! When using fractions, the most common examples include slicing pizza or pie into equal slices. Here is one of those problems:

Assume you have seven-eighths of a whole pizza left. Three of your friends walk into the kitchen and ask for one-fourth of the whole pizza each. If you wanted to share with your friends, will you have enough pizza for each friend to get the amount they want? (Divide 7/8 by 1/4 and see if it’s bigger than three).

$\displaystyle \frac{7/8}{1/4} = \displaystyle \frac{7 \times 4}{8 \times 1} = \displaystyle \frac{28}{8} = 3 \frac{1}{2}$

It is bigger than three, therefore there is, in fact, enough pizza left for all three of your friends to get the amount they wanted.

Other problems might involve finding a fraction of a fraction of a whole. Here is an example of this:

I have a giant cookie jar with 36 cookies in it. My family comes over and eats some of the cookies. If 1/3 of the cookies are eaten and 3/4 of the eaten cookies had frosting, how many of the eaten cookies had frosting? (Multiply 36 by 1/3 to get 12. Then multiply 12 by 3/4).

$\displaystyle 36 \times \frac{1}{3} = \displaystyle \frac{36}{3} =12$

$\displaystyle 12 \times \frac{3}{4} = \displaystyle \frac{12 \times 3}{4} = \frac{36}{4} = 9$ .

Nine of the eaten cookies had frosting.

Curriculum (B2)

In previous mathematics classes, students have obtained a wide variety of skills which can be used when dividing fractions. These skills include the multiplication of whole numbers, the division of whole numbers, and how to reduce fractions to their simplest form. Dividing fractions is an extension of these skills. It can also be said that students already understand what a fraction is. On a separate note, we will discuss how many students relate to fractions and how they think of fractions when confronted with them.

Many students find fractions difficult and intimidating, often freezing when they see a fraction. Involve more than one fraction in a problem and students will get easily frustrated and give up. This can be caused by the way a student perceives fractions. Many students are taught that a fraction is simply part of a bigger whole number. While this is true, many students lose focus on the big picture and get caught up on the fact that a fraction is less than 1 whole unit. In order to help avoid this, teachers could instead try explaining fractions in a slightly different way: a fraction is just a number written like a division problem. The video found at http://www.youtube.com/watch?v=3xwDryouw6o can help to provide a more in-depth explanation about this new perspective on fractions.

By thinking of a fraction as simply a division problem, students automatically incorporate their previous knowledge on dividing whole numbers. When students work through a problem with dividing fractions, they will go through the steps of “keep, change, and flip.” Once they have changed the division symbol to a multiplication symbol and flipped the second fraction, the students will be ready to use their previous knowledge on multiplying whole numbers. After the numerators and denominators are multiplied respectively and the new fraction is obtained, the students must recall previous knowledge on the reduction of fractions to their simplest form.

Technology (E1)

A video can be used to engage students and give them a foundation for dividing fractions. The video I chose, which can be found at http://www.youtube.com/watch?v=uMz4Hause-o, is an excellent example of an acceptable engagement tool. In the video Flocabulary uses music and repetition to describe how to perform the task of dividing fractions. This will help the students be able to recall the information about dividing fractions later on when they need to. Flocabulary explains the process step-by-step and then demonstrates the method in action, using two different fractions to help students understand how it works. Then the video goes on to explain why we flip the second fraction in a division problem, which is vital for ensuring that actual learning is taking place and not simple memorization. Students need to know why they perform certain steps and why the trick works. While the cartoon animations are meant to target a younger audience, this clip is easy to follow and the repetitious nature of the music puts an interesting spin on learning mathematics.

Engaging students: Determining the largest fraction

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 Shama Surani. Her topic, from Pre-Algebra: determining which of two fractions is largest if the denominators are unequal.

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

An activity that involves students to determine which of two fractions is greatest is called Compare Fractions, which is a two-player math game found at the website http://www.education.com. The objective of the game is to work together to determine who has created the largest fraction. The materials necessary is a deck of cards with the face cards removed, notebook paper, and a pencil. Below are the directions of this game:

Review the concepts of numerator and denominator.
Decide on a dealer and have him/her shuffle the cards.
Divide the deck evenly among the players.
Have the players place their cards face down in a pile in front of them.
To begin playing, have players turn over two cards from their respective decks and place them in front of themselves.
Players can then decide which card they want to be in the numerator and which card they want to be the denominator.
Now the players have to calculate who has the largest fraction. There are a variety ways this can be done. Encourage different methods in determining which fraction is larger. One way is to multiply the numerator and denominator of each fraction by the denominator of the other fraction. For example, with the fractions 5/6 and 4/7, compute 5/6 x 7/7 = 35/42 and 4/7 x 6/6 =24/42. The largest fraction is 35/42 so 5/6 must be greater than 4/7.
The player who has the largest fraction wins all of the cards played in the round. For the instance of a tie (when the both students have equivalent fractions), split the cards evenly among the players.
The game is over when the players have accumulated all of the cards.
Have the players count their cards. Whoever has the most cards, wins.

I believe this activity will be fun for the students because they are creating their own fractions with the cards. Once the students are comfortable with determining which of the two fractions is greatest, the teacher can start timing the students if he/she wants to.

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

In previous courses, students should have learned how to draw a number line and determine where on the number line two natural numbers are located. They would have known how to compare numbers or order the numbers from least to greatest or greatest to least. Then, the students were exposed to fractions as being a part of whole, and being called rational numbers. This concept is then extended to ordering fractions with equal denominators with visual diagrams and using the number line. In a visual illustration, the students can be exposed to two circles of the same size but divided into the same amount of sections. For example, both circles can be divided into four equal sections, but one can have two sections filled in while the other has three sections filled in. Students then can determine which circle is larger. In this case, the circle with three sections filled in is larger. Then this concept is extended to be written in fraction format where the first circle is 2/4 and the other circle is 3/4. When the students have fractions with equal denominators, they look at the numerator to see which fraction is larger or smaller. Determining which of two fractions is greatest if the denominators are not equal extends off this previous concept. The best way is to show the students visually how different shapes such as a square or a circle can be divided equally into different sections. For example, the first circle might be divided into four sections with three sections shaded while the other circle can be divided into eight sections with seven shaded. In fraction form, the first circle is 3/4 while the other circle is 7/8. Here the students will notice that the denominators are different but by looking at the shaded circles, they can see that 7/8 is larger than 3/4.

E1. How can technology be used to effectively engage students in this topic?

Technology is increasing day by day, and in many respects, technology can be the tool for aiding learning in the classrooms. One way that technology can be used to effectively engage students in determining which of two fractions is greatest when the denominators are unequal by playing simple online games. Since several schools are distributing i-Pads to their students, I have found an i-Pad application called “Fraction Monkeys” that the students can download for free for this lesson. This application is a wonderful tool in demonstrating how fractions with same or different denominators are located on the number line. The objective of this game is that a monkey with a fraction will appear on the screen. The student will have to place the monkey correctly on the number line. Sometimes the card the monkey holds up is in reduced form, so the student will have to think about how that reduced form relates to the number line.

For example, below is a picture of a number line with the denominator being 16. When the student is finished placing the monkeys on the correct location, they will notice that the monkeys were placed differently depending on what fraction they received.

By providing the students with a guided worksheet, the students will be able to compare which fractions are greater and which fractions is less than the other by viewing the number line. For example:

$\displaystyle \frac{7}{8} ~~ ? ~~ \frac{3}{4}$

The student will answer that 7/8 is greater than 3/4 since 3/4 comes before 7/8 on the number line. I believe this activity will help the students conceptualize how to compare fractions. In addition, in case when the student incorrectly places a monkey on the number line, a hint with little squares pops up where the student can visually see how their fraction relates to the number line. Below is a picture demonstrating this:

Another computer game that involves comparing fractions is named “Balloon Pop Math.” This is also a good resource to use because it shows balloons with fractions with a visual of a circle divided in equal sections. The idea of this game is to pop the balloon with the smallest fraction to the largest fraction with different denominators. Below is a picture from the game demonstrating the fractions 7/8 and 4/5. The students will be able to see that 4/5 is less than 7/8 by looking at the circle so the student will pop the balloon that contains 4/5. This game is also wonderful to use because it contains three levels. The first level allows the students to compare two fractions. The next one allows the student to compare three fractions, and the last level allows the students to compare four fractions. This will be a good engagement activity to allow the students to do before teaching about how to compare which two fractions is greater than the other.

References:

http://www.fractionmonkeys.co.uk/activity/

http://www.sheppardsoftware.com/mathgames/fractions/Balloons_fractions1.htm

http://www.education.com/activity/article/capture-that-fraction/

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

Engaging students: Probability and odds

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 Michelle Nguyen. Her topic, from Pre-Algebra: probability and odds.

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

As a teacher, I would place 100 red marbles and 25 blue marbles in a bag and have each group of students draw a marble each time from a bag for five times. After drawing a marble, the student would put the marble back and then redraw. After five times, the class would come together and the students would compare how many red marbles to how many blue marbles they have. The students will compare the ratios and guess if there are more red marbles or blue marbles in the bag given. By doing this, the students will see whether there is a big chance of drawing a red or blue marble. After doing the activities, I would ask questions that will scaffold the students into saying that there is a higher probability in picking a red marble than a blue marble because the red marbles are picked more often when compared to the blue marbles that got picked.

D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

With the popularity of gambling rising in the French society, mathematical methods were needed for computing chances. A popular gambler named De Mere talked to Pascal about questions about chances. Therefore, Pascal talked to his friend Fermat and they began the study of probability. The created the method called classical approach which is the probability fractions we use today. In order to verify the results of the classical approach, Fermat and Pascal used the frequency method. During this method, one would repeat a game a large number of times with the same conditions. Bernoulli wrote a book named Ars Conjectandi in 1973 to prove the classical approach and the frequency method are consistent with another one. Later on Abraham De Moive wrote a book to provide different examples of how the classical methods can be used. As time passed by, probability moved from games of chance to scientific problems. Laplace wrote a book about the theory of probability but he only considered the classical method. After the publication of this book, many mathematicians found that the classical method was unrealistic for general use and they attempted to redefine probability in terms of the frequency method. Later on, Kolmogorov developed the first rigorous approach to probability in 1933. There are still researches going on about probability in the mathematical field of measure theory.

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

In the movie “21” there is math problem that is similar to the popular Monty Hall problem. In the movie, a kid is given the chance to pick one out of three doors with a car in it in order to win. Once a door is chosen, the announcer will open a door without a car. Therefore, the start off is 33% of a car existing and 66% with an empty door. Since a door was open, the chance of switching your choices gives you a higher winning percentage because the one you chose at the beginning will still be 33% while switching will change your chances to 66%. This youtube video is a clip from the movie:

References:

http://www.math.wichita.edu/history/activities/prob-act.html#prob1

http://staff.ustc.edu.cn/~zwp/teach/Prob-Stat/A%20short%20history%20of%20probability.pdf

http://www.examiner.com/article/21-and-the-monty-hall-problem

Engaging students: 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 again comes from my former student Michael Dixon. His topic, from Pre-Algebra: prime factorizations.

A1. What word problems can your students do now?

One word problem that is easily relatable would be something involving food!

For instance: “Don loves peanut butter and jelly sandwiches. One day he noticed a jumbo jar of peanut butter has 72 servings and a jar of jam only has 40 servings. If he opened the [first] jars on the same day and used exactly one serving each day, how many days until he emptied a peanut butter jar and a jam jar on the same day? Use prime factorization to solve.”

Obviously, this involves finding the least common multiple of 72 and 40. I would introduce this problem at the beginning of class, after my students have already been introduced to the idea of prime factorizations. I do not expect that my students would know how to calculate the lcm using prime factorizations, rather I would want to strike up a class discussion asking students to explore what they know about factorizations and see if they can find any patterns that would lead to the solution. I want to lead them to the idea that prime factorizations make finding the lcm far easier than listing the multiples of each number, especially when large numbers are involved.

B1. Future Curriculum

As mentioned in the previous paragraph, students can learn to use prime factorizations to calculate the greatest common factor or the least common multiple of numbers easily. To take this quite a bit further, we can introduce students to the idea of using factorizations, gcd, and lcm in formal abstract proofs. We would ask them to actually prove anything, just think about the ideas. Ask students how they know that the math that they use everyday actually works. Why does every number have a unique factorization? Why can I calculate the gcd and lcm of any two numbers, and know that that answer is the only answer? Then explain that later on, in higher level math classes, we actually flawlessly prove why our number system works, and how and why primes are important, such as in the Euler Phi function. Without prime factorizations, we would be unable to prove quite a lot of the math that we take for granted.

E1. How can technology be used to engage students?

After your students have been working with prime factorizations for a while and they are getting more proficient, what’s an obvious escalation? Make the numbers larger! Ask your students to factor numbers like 198 and 456. See how long it takes them to work through these. Then, ask them how long it would take to factor numbers like 2756 or even 12857. How could they do these? Is it even reasonable to try? What about 51,234,587 (this is actually prime)?

Here we can introduce using a computer, and using a computer to do the calculations for us. Just a simple website is adequate to show them just how useful computers are when doing large calculations. A website such as Math is Fun is an excellent tool to demonstrate the magnitude of some prime numbers and composite numbers, and show that even as numbers get very, very large, they are not divisible by any numbers other than themselves and one.

References

www.mathsisfun.com/numbers/prime-factorization-tool.html

http://tulyn.com/wordproblems/prime_factorization-word_problem-7928.html

Engaging students: Solving one-step linear equations

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 Jessica Trevizo. Her topic, from Pre-Algebra: solving one-step linear equations.

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

Many students have played “Around the World” at one point in their elementary childhood, or have at least heard of the game. Around the World is an activity that is commonly used by elementary school teachers when they are teaching multiplication. Students are supposed to sit in the form of a circle. One person is chosen to attempt to go around the world. He/she will stand behind a student and will compete against the student that is sitting down. Once both students are ready the teacher holds up a multiplication card. The student who responds with the correct answer first gets the chance to move on to the next person. If the student who is standing up loses then he/she gets to sit down while the other student who obtained the correct answer advances. Every person has to attempt the problem on a sheet of paper, but they are not allowed to call out the answer. The student who “goes around the world” first is the winner. If a student is not able to complete the entire circle then the student who advanced the farthest is the winner. The same idea will be used after the students have learned how to solve one step linear equations. After having a deep conceptual understanding of the topic it is very important for the students to keep practicing problems. Around the World allows the students to keep practicing in an entertaining way. The students should be able to solve the equations within 30 seconds since it only requires one step to solve. The ability to use calculators with this activity will vary depending on the level of difficulty of the problems as well as the teacher.

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

Being able to solve one step linear equations is an important skill that every student should acquire. After the students learn how to solve one step linear equations they are expected to be able to solve multi-step equations, solve absolute value equations, solve inequalities, finding the side lengths of a shape given a certain area in geometry, etc. If the students are not able to master solving one step linear equations then they will have a very difficult time in other math courses.

In geometry the Pythagorean Theorem requires the skill to solve one step equations. Students are expected to solve for the missing variable in order to find the missing side length of a right triangle. In Algebra II the students are required to manipulate equations in order to solve systems of linear equations through substitution. Also this basic skill is necessary when finding the inverse of a function. This topic is also used in physics. For example, if the student is asked to find the acceleration of an object given only the force and the mass, then it involves using Newton’s second law which states Force=mass*acceleration.

E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

This website is an amazing tool that allows the students to visualize how to solve linear equations using algebra tiles. If the teacher decides to teach this lesson using algebra tiles in the classroom, then this website will allow the students to continue to practice at home. Also, the website automatically lets the student know if he/she responded correctly. Obtaining quick results allows the student to know whether or not they truly understand how to solve the equations as opposed to having a worksheet with 50 problems for homework and not knowing if the same mistake was repeated. Also, by using the online algebra tiles the students are able to understand the zero pair concept and see how it is being applied. This website can also be used for other algebra topics such as factoring, the distributive property, and substitution.

http://illuminations.nctm.org/Activity.aspx?id=3482

Engaging students: Order of operations

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 Dorathy Scrudder. Her topic, from Pre-Algebra: order of operations.

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

Students should know how to add, subtract, multiply, divide, and use exponents by the time we cover this topic. To begin the class, I will have students split into two groups. Both groups will be given a series of one-step equations that correspond to a multi-step equation; however, one group will be given the steps out of order. We will then discuss why the two groups were working on the same multi-step equation but have different answers. The students should find it interesting and ask a few follow up questions such as, a) how do we know which answer is correct – it is correct by use of the order of operations which was decided on by mathematicians in the 1600s; b) how do we know what order to do the operations in – we use the acronym PEMDAS which stands for Parenthesis, Exponents, Multiplication/Division, Addition/Subtraction; c) how are we supposed to remember the acronym – we can either pronounce it pem-das or use the saying Please Excuse My Dear Aunt Sally.

C2. How has this topic appeared in high culture (art, classical music, theatre, etc.)?

The order of operations is also utilized in many areas outside of math. Take a play for example. To begin, an actor or actress must first audition for the desired role. Once they have been hired, the actor/actress must learn their lines and then rehearse with the other actors and actresses before the opening night of the play. The actor/actress cannot perform the opening night if they have not learned their lines yet. The set designers must also follow the order of operations. They must first design what they want the set to look like and then decide what materials they need and how much to buy. Once they have the materials, they cannot start painting intricate details until they have constructed the set. Following the order of operations is an important concept and hopefully these examples will help the students understand why we need to follow the steps in the given order.

D2. How was this topic adopted by the mathematical community?

Multiplication before addition has been a common practice since before algebra was written, however, it was always an assumption and mathematicians never felt that it had to be proved. The earliest printing that we have where multiplication comes before addition is from the early 1600s. Dr. Peterson, from Ask Dr. Math, has stated that he believes the term “order of operations” has only just come into common use within the past century by textbook authors. Sarah Sass, from University of Colorado in Denver, has found that students have trouble when it comes to the multiplication and division step and again at the addition and subtraction step of the order of operations. She suggests that instead of using “please excuse my dear aunt sally,” in which students often assume all multiplication comes before division and all addition comes before subtraction, Sass suggests that we teach “Pandas Eat: Mustard on Dumplings, and Apples with Spice.” This allows the students to understand that the mustard and dumplings, or the multiplication and division, go together at the same time, while the apples and spice, or addition and subtraction, are completed at the same time, all from the left to the right.

References:

http://www.math.ucdenver.edu/~jloats/Student%20pdfs/4_Order%20of%20OperationsSass.pdf

http://mathforum.org/library/drmath/view/52582.html

Engaging students: Solving for unknown parts of rectangles and triangles

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 student Brittney McCash. Her topic, from Pre-Algebra: solving for unknown parts of rectangles and triangles.

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

As a teacher, I want to do activities that the students would enjoy as much as possible. In doing so, I came up with a festive idea to incorporate my concept. Gingerbread houses. They are fun to build, while at the same time your thinking mathematically without realizing it. My job would be to bring these concepts forth. My engagement for the activity would probably be video on the shapes it takes to build a gingerbread house. Then I would pass out a blueprint of a gingerbread house that has missing angles or sides and have the students solve for them. This allows them to either set up proportions and see the similarities, or to solve for the sides using the characteristics of the shapes given. After the exploration of the blueprint, would come the construction part. I would have pre-cut pieces of graham crackers or other materials I would use, and have the students pick the pieces that match their blueprint; not every student will have the same. This is where the fun part would come. They would get to construct their gingerbread house, but if they made mistakes during their blueprint, their gingerbread house wouldn’t look right. Shapes wouldn’t fit, or maybe the gingerbread house wouldn’t stand because it didn’t have the right support. As these issues come up, I would be there to guide them in their discovery of “What went wrong.” This leads them to see how important having the corrects measurements truly are and how major they can effect the outcomes of things. Depending on the length of class time you have, this would probably be a two day activity.

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

To engage the students with this topic, I would pose a question asking the students, “What would happen if the Eiffel Tower wasn’t congruent on all four sides?” This question alone opens the floor for many different discussions to take place, but my main goal would be to establish what congruent is by definition, and how does that effect shapes and their placement. Through this question we would come to the conclusion that the tower would either lean, not be sturdy, or maybe not even stand at all if the sides of the Eiffel Tower were not congruent. This shows how important measurements are when building buildings. My next step would be to go over how to solve for sides of triangles or squares if they are congruent. Once this is established, I can pose the question, “Now what if we were not given any angles or measurements? How could we tell if triangles are congruent?” This opens the room up for ideas how this would be done, and I would introduce the Theorems of Side-Angle-Side, Side-Side-Side, Angle-Side-Angle, and Angle-Angle-Side. Without going to extreme detail, I would express how important it is for them to grasp the concepts of solving for unknowns on triangles so that they are able to later, in Geometry, understand and utilize the idea for the theorems.

E1. How can technology, be used to effectively engage students with this topic?

No matter where you go these days, technology is everywhere, so why not embrace it? There are two ways that technology can be useful in the classroom. One with websites or activities online that shed new light to a topic that is being taught, and also by helping students learn skills on technology that they will need later on. There are not many jobs out there, if any that do not use technology, so helping students get a grasp on it sooner rather than later may help them later on. My engagement for this aspect on my topic would be to do an online activity. Depending on the school, this will either be done in the classroom or a computer lab. I’ll have the students log on and open up this website: Cool Math . This website would be terrific in opening up this subject. I believe this because it doesn’t just jump right in to solving for unknowns. It gives you a quick overview of the relationships certain shapes have, then it gives you an odd geometric figure to find the perimeter of. This figure only has so many measurements given to them, and they have to solve for the rest using the relationships and definitions of the shapes involved. Another really interesting attribute I liked about this website, was that each shape had its own color. When it came time to solve for the big oddly shaped geometric figure, each shape involved was colored differently. This is great because I know how hard it is for some students to distinguish shapes from one another, and this might be a way for them to better visual the shape and its encountering partners to help tell what the relationship may be.

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 Andrew Wignall. His topic, from Pre-Algebra: absolute value.

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

In some sense, absolute value has been with us for a long time, but it’s also relatively recent. Distances have always been measured as a positive value – Denton and Dallas are 39 miles apart, for instance. It’s not that one is 39 miles away, and the other is -39 miles away – they’re both the same distance apart. We take negative numbers for granted in our lives now, and have learned to accept them relatively early in our advancing math education in schools. Absolute value developed as a way to “remove” the negative from negative numbers for calculation and discussion.

In fact, mathematicians didn’t discuss absolute value much until the 1800s. Karl Weierstrass is credited with formalizing our notation for absolute value in 1841! However, this is because negative numbers were not given serious consideration by mathematicians until the 19^th century, when the concept of negative numbers was more formally defined. With negative numbers, mathematicians needed a way to talk about the magnitude of the negative numbers – and so entered absolute value!

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

The concept of absolute value is used in many places in many math and science classes. In geometry, volume and area are almost always positive – if you are dealing with figures of variable size, you’ll need to use an absolute value to ensure the volume/area is positive. When dealing with square roots of squared figures, we often have to deal with two possible answers, positive and negative – but absolute value simplifies this complication in many calculations. In physics, time and distance are always positive, so we again need absolute value. In chemistry and statistics, percentage error is often expressed as a positive value. Calculus uses absolute value when dealing with derivatives and logarithms.

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

It’s important to address absolute value as not just removing the negative sign from negative numbers, but also that it functions as a measurement of magnitude, or distance from zero. Springboard Mathematics with Meaning suggests an activity where a number line is placed on the floor and students are lined up along the number line. Students record their position, and then measure their distance. Their position is positive or negative, but their distance from 0 is always positive – the absolute value of their position!

Students can also work backward, and place two students so they are each a distance of 4 from 0. Students can also express inequalities, with any students more than 5 away from 0, or any students less than 3 units from 0.

By having students on the positive and negative side of the number line, they can see how absolute value is calculated:

$|x| = x$ if $x \ge 0$ ;

$|x| = -x$ if $x < 0$ .

There are several benefits to this activity. First, it is a physical activity, which gets students out of their chairs and physically active and awake. Second, it can be used to demonstrate how absolute value is distance from zero (by measuring distance), the magnitude (length of distance), and students can derive a formal definition for how absolute value is determined analytically. It allows students to think about absolute value abstractly, concretely, or theoretically. The activity can be referenced any time in the future curriculum when absolute value is required for a quick refresher.

References

Barnett, B. (2010). Springboard algebra I: Mathematics with meaning. New York: CollegeBoard. http://moodlehigh.bcsc.k12.in.us/pluginfile.php/8095/mod_resource/content/1/1.7%20Absolute%20Value.pdf

Rogers, L. (n.d.). The History of Negative Numbers. : NRICH. Retrieved January 22, 2014, from http://nrich.maths.org/5961

Tanton, J. (2009). A brief guide to ‘absolute value’ for high-school students. Thinking Mathematics. Retrieved January 22, 2014, from http://www.jamestanton.com/wp-content/uploads/2009/09/absolute-value-guide_docfile.pdf

The Loneliest Number

Courtesy of Math with Bad Drawings: http://mathwithbaddrawings.com/2014/02/12/the-loneliest-number/