Creating Effective Math Problems

 What Makes a Good Math Problem?

This week in class we continued to talk about creating math problems and what characteristics contribute to an effective problem. Some of these characteristics we discussed include:

• The problem is relevant to the students
• There are multiple approaches and methods to solving the problem
• Students can use various manipulatives
• Wide base where everyone can join in
• The problem uses soft language 
• The problem creates discussion
• Students have the option to work together and collaborate
• There is a "high ceiling", which means that there is the potential to increase difficulty and challenge students

Wrapping up Joel's Cat Food Problem

We wrapped Joel's Cat Food problem this week with a Math Congress. In large groups, we were asked to discuss the different approaches we used to solve Joel's Cat Food problem. A few students were assigned leader roles where they explained how their group solved the problem and made sure that every student in the group understood the approach. After our group discussed three different approaches to solving the problem, we went on a Gallery Walk where we got to see how other groups solved the problem. Here are some pictures of the different approaches that students took. Our group discussed how some of the other approaches were more visually geared and how our approaches could seem difficult to understand for some students. One aspect of the solutions that all of us determined to be important was that the information given was shown. Knowing that Bob's store sells 12 cans for $15 and Maria's store sells 20 cans for $23 was shown on almost every piece of chart paper. Even though this information was provided on the slideshow, most students found it helpful to re-write this information on the chart paper.

Proportional Thinking

We continued the lesson by discussing the concept of proportional thinking and how we think about proportions on a daily basis. Proportional Thinking is the ability to compare quantities multiplicatively and ties into our previous lessons on Ratios and Fractions. Some of the ways we think proportionately can be as obvious as baking or as subtle as thinking about how much gas you need to get from Point A to Point B. Fractions and Ratios is not a foreign concept and plays an important role within life, so it is extremely important that students learn about proportions and how to compare them.

Integers, Gizmos, and CLIPS

We wrapped up the class this week by discussing integers. One of the biggest problems with integers is trying to apply "rules" that students have learned to a problem where the rule does not apply. There are many different ways to develop an understanding of integer size and operations and creating rules can limit a students understanding of how integers work. Gizmos is a website we discussed briefly discussed in class that can help students understand integers. Gizmos are interactive math and science simulations that help students understand concepts that they struggle with. There are several Gizmos that are aimed towards Integers that allow students to develop their own understanding as opposed to a definition created through rules. The term "integers" may be foreign to students, but the actual concept is more understood than a student might think. The Critical Learning Paths Supports website, or CLIPS, is an educational website that provides students with videos and activities to help understand math. One of the areas it focuses on is integers and how integers are embedded in daily tasks. Outside temperature will usually determine how someone dresses. When students are told this, they begin to understand how integers can be thought of as a whole numbers. This website allows students to explore the concept of integers and see how integers are present in their daily routine.

Fractions; The Other Half (part 2)

This week in class we continued our discussion on fractions. With last week's intro to fractions, we continued to develop multiple perspectives of how we see fractions.



Racing With Fractions!

One of the opening activities reinforced the idea that others may look at a fraction and see it differently.  Several volunteers were asked to go up to the front of the classroom and play a game of Red Light/ Green Light, which set the stage for the introductory activity. After a few minutes of students being sent back to the beginning of the race, we were asked to go back to our seats. A slide similar to this picture was shown to us.
Although the actual race results were not similar to the results we were shown, we began to think of the race in fractions. Henna finished one quarter of the race while Christian finished three quarters of the race. We were told to think of these fractions as the amount of race that the student finished. We were then asked to place them in order of race finished. With the understanding of fractions I gained from last week's class, I wasn't surprised to see the different ways that my peers had answered the question. I tried to visualize the fractions and place the racers along a line. I noticed other students were doing the same, while some were converting the fractions into percentages and listing the racers according to their percentage. This activity was fun and engaging as it got students out of their seats for some friendly competition and allowed us to approach the question using our own methods.

Dividing Fractions - Old vs. New


When I was learning the concept of dividing fractions, I was told to use a specific formula. I would start by flipping the second fraction. I would then change the division sign to a multiplication sign and multiply across the formula. The answer I got was right, but I completely avoided the concept of division and was confused as to why I got the right answer. The way I learned how to divide fractions in class is clear and makes more sense to me. Instead of flipping the second fraction and replacing the division sign with a multiplication sign, I just divided through. I get the same answer, which is the right answer, and I use less steps to get there. With this new understanding, I can divide fractions without getting lost. I will certainly not continue to teach the "cross-multiplication" approach and ensure that every student is exposed to this way of approaching division with fractions.


Mr. Tan's Tangram


One activity that I was familiar with was Mr. Tan's Tangram. Each student was given a set of seven shapes and was told the story of Mr Tan. With the mixture of shapes, we were told that it was possible to make a square. I personally struggled to find a way to make a square, even though I had come really close on a few tries. After being shown the solution, I saw that I was really close to finding the answer. I liked this activity as I got to explore the way shapes fit together and come up with different layouts, even though I could not find the answer. This activity was brought up in the fractions lesson because it deals with ratios and fractions. The large triangles are equivalent to one quarter of the square each. Each of the smaller shapes can be placed within on of these triangles and be given a fraction corresponding to it value within the whole square. This activity is a great hands-on activity and allows students to learn by themselves not only how to make a square with the tangram shapes, but also grapple with the topic of fractions and how each of these shapes work within the whole square.

Fractions In Everyday Life

Students are taught about fractions within a short period of time. There is not a lot of focus within this topic of math, even though it is extremely important. I was taught to use a formula, as I discussed earlier, so I could quickly arrive at an answer without actually understanding the process. Fractions play an important role in the world and in many day to day activities. Whether you are baking, paying taxes, or building a house, fractions encompass our lives in many different forms. Without learning how fractions work, we miss out on an opportunity to learn and knowing how to perform day-to-day tasks.

Teaching Fractions

The way I will teach students fractions will be a lot different from the way I was taught. Much like the way I am learning about fractions now, I will use games and classroom activities to engage students with the material and concepts. Tug Team Fractions is a fun game that students can play to learn about comparing fractions. I would use this game along with the Race Activity we worked on at the beginning of class to teach students how fractions relate to one another and how you can compare them. As I continue this course, I am relearning essential math skills and learning how I could approach teaching these concepts through a fun and engaging way.

Introduction to Fractions!

This week's class covered the topic of fractions. We started the class by choosing any fraction that we wanted. I decided to use the fraction 3/4 as it is an easy fraction to work with. We were then asked what is important about our fraction and why is it a fraction? The fraction 3/4 is important to me because it is an easy fraction to work with and can help teach students about fractions. It is also easy to visualize as you can imagine a pizza cut into four and one slice is eaten. You can also visualize this fraction using quarters.

When we were asked why the numbers we chose were fractions, I tried to create a general definition that could be applied to every fraction. There is a numerator (the number on the top), a denominator (the number on the bottom), and a line between the two. This seemed to be the general definition for most students in the class.

However, most of the fractions created in the class were different. I started to notice that some students had created fractions that were a lot different than the one I created. The lesson then progressed into the types of fractions.


The first type of fraction that we looked at is called the Proper Faction. This fraction equals less than one. The fraction 3/4 is an example of a proper fraction.



Another type of fraction is the Mixed Numbers Fraction. This fraction equals more than 1. The fraction 1 and 2/4 is an example of a mixed numbers fraction.



The third type of fraction is called a Unit Fraction. This fraction is similar to the proper fraction as it equals less than 1. The difference is that the numerator is the number one.

One of the tools used to teach us about fractions is a book called "The Hersey's Milk Chocolate Fractions Book". This book uses a Hersey's milk chocolate bar as a manipulative to help students grapple with the understanding of fractions. Each table was given a Hersey's chocolate bar and was asked to break up the bar into twelve pieces. One of the reasons why I really liked this activity is because I think it will be effective with Junior/Intermediate students. Most students, if not all, know what a chocolate bar is and dividing it into twelve can help them understand how a portion of something (such as the chocolate bar) can be represented by a fraction. As the book was read to us, we were asked to group the chocolate in several ways that represented fractions. The book also uses pictures to help demonstrate how the chocolate pieces should be divided.


It was also beneficial to understanding fractions when we were asked to fill out a graphic organizer on fractions. This organizer was a chart we had to fill in that helped us to define what a fraction is and give examples and non-examples of fractions. This chart could be filled out by students as an activity either before they are introduced to fractions or after they are taught. It can be used to help the teacher understand what students think of when asked to define a fraction and guide students.

I can see the various manipulatives used in class being extremely beneficial to teaching students about fractions. While the chocolate bar and book are extremely helpful, we were also given various shapes, egg cartons, and several informative sheets that broke down fractions. As I continue this class, one of the key elements that I am noticing is the use of several manipulatives. One type of activity may help some students to understand what a fraction is, but others may need a different manipulative to help them. Activities that engage students and promote collaboration is another important element to teaching. The Hersey's chocolate bar activity allowed us to work in groups and got us excited. The presentations that followed the lesson also promoted the use of manipulatives and collaboration amongst students. 

How to Subtract!

This week in class we learned about the importance of different algorithms and why it is important to promote several approaches to answering a question. With the EQAO assignment in mind, I have tried to not only see the importance of different algorithms but also attempted to answer math-based questions using different methods. We got to play around with some simple math expressions in class using different approaches. It was surprising to see that many students take different approaches that may have seemed confusing to me before this class, but now appear to be useful and easy ways to approach math. Understanding this will help me to become an effective teacher as I plan to implement several algorithms throughout my math courses.

387-146=241

Having developed my math skills over the years, it didn't take me long to figure out that
387-146=241. My approach to answering this question, however, is a way that can be extremely confusing to students who have a different learning process. I learned that it can be extremely effective to start with a number line, even if some students don't find it as useful as others.


By breaking up the equation into steps, students can understand the role of numbers and how they can be broken up into smaller numbers in order to make math easier. By starting on the right side with 387, students can move to the left (addition would move to the right) by blocks of numbers that they understand. In the picture to the right, 146 is broken up into three parts: 100, 40, and 6.  As students subtract the numbers and move along the line to the left, they arrive at 241. This method is known as the Partial Subtraction, which is subtracting from the greatest to the least place value.

Compensation Subtraction
Another method of approaching this problem is known as the Compensation Subtraction method. This method involves making a friendly number. By adding 4 to 146, students might have an easier way of solving the problem. 387-150 may appear less daunting than 387-150 and students may prefer this method. It is important to note that the 4 added to 146 needs to be subtracted at the end to balance the equation; when you borrow, you must return.

Constant Difference Subtraction
Much like the compensation method, the Constant Different Subtraction method involves making a friendly number. Unlike the compensation method, however, the equation isn't balance out at the end. If you add 4 to 146, you need to minus 4 from 387. The equation will read 383-150, but the answer will still be the same. Some students may prefer this over the compensation method, but both are equally acceptable.

During my placement I have been given the opportunity to watch students during their math class to see how they answer questions. While I helped students get to the right answer, I was surprised to see many using methods that I was never taught. The class I have been placed in is a grade eight elementary class and they are just beginning to learn the basics of algebra. Understanding the role of symbols within math can be challenging to many students. I was surprised to see that visual manipulatives formed the basis for many students' understanding. Breaking down the problem in front of the students is an extremely effective method and can be used at any grade. Another thing I noticed is that while my teaching associate promoted several approaches, he made it necessary for students to understand more than one method and use several on tests and quizzes.