Playing Math Games with older students

I currently teach remedial math students on the college level. These are the students who fail to pass the math placement test to enroll in College Algebra - that dreaded class that everyone must pass to graduate. The math curriculum at our community college starts with Basic Math, moves to Fractions, Decimals and Percents, and then to Basic Algebra Concepts. Most of my students are intelligent and want to learn, but they are deeply afraid of math. I refer to them as mathphobics.

We all have this type of student in our classrooms, whether it is middle school, high school, or college. When working with this type of student, it is important to bear in mind how all students learn. I always refer back to the Conceptual Development Model which states that a student must first learn at the concrete stage (use manipulatives) prior to moving to the pictorial stage, and in advance of the abstract level (the book). This means that lessons must include the use of different manipulatives. I use games a great deal because it is an easy way to introduce and use manipulatives without making the student feel like “a little kid.” I can also control the level of mathematical difficulty by varying the rules; thus, customizing the game to meet the instructional objectives my students are learning. However, as with any classroom activity, teachers should monitor and assess the effectiveness of the games.

When using games, other issues to think about are:

1) Excessive competition. The game is to be enjoyable, not a “fight to the death”.

2) Mastery of the mathematical concepts necessary for successful play. Mastery should be at an above average level unless teacher assistance is readily available when needed. A game should not be played if a concept has just been introduced.

3) Difficulty of the rules. If necessary, the rules should be modified or altered in order that the students will do well.

4) Physical requirements (students with special needs). These should be taken into account so that every player has an opportunity to win.

In addition to strengthening content knowledge, math games encourage students to develop such skills as staying on task, cooperating with others, and organization. Games also allow students to review mathematical concepts without the risk of being called “stupid”. Furthermore, students benefit from observing others solve and explain math problems using different strategies.

Games can also….

1. Pique student interest and participation in math practice and review.
2. Provide immediate feedback for the teacher. (i.e. Who is still having difficulty with a concept? Who needs verbal assurance? Why is a student continually getting the wrong answer?)
3. Encourage and engage even the most reluctant student.
4. Enhance opportunities to respond correctly.
5. Reinforce or support a positive attitude or viewpoint of mathematics.
6. Let students test new problem solving strategies without the fear of failing.
7. Stimulate logical reasoning.
8. Require critical thinking skills.
9. Allow the student to use trial and error strategies.

Mathematical games give the learner numerous opportunities to reinforce current knowledge and to try out strategies or techniques without the worry of getting the “wrong” answer. Games provide students of any age with a non-threatening environment for seeing incorrect solutions, not as mistakes, but as steps towards finding the correct mathematical solution.

$3.50

One math game my students truly enjoy playing is Bug Mania.  It provides motivation for the learner to practice addition, subtraction, and multiplication using positive and negative numbers. The games are simple to individualize since not every pair of students must use the same cubes or have the same objective. Since the goal for each game is determined by the instructor, the time required to play varies. It is always one that my students are anxious to play again and again!

A Negative number times a Negative Number Equals a Positive Number? Are You Kidding?

Have you ever wondered why a negative number times a negative number equals a positive number? As my mathphobic daughter would say, "No, Mom. Math is something I never think about!" Well, for all of us who tend to be left brained people, the question can be answered by using a pattern. After all, all math is based on patterns!

Let's examine 4 x -2 which means four sets of -2. Using the number line above, start at zero and move left by twos, four times. Voila! The answer is -8. Locate -8 on the number line above.

Now try 3 x -2. Again, begin at zero on the number line, but this time move left by twos, three times. Ta-dah! We arrive at -6. Therefore, 3 x -2 = -6.

On the left is what the mathematical sequence looks like. Moving down the sequence, observe that the farthest left hand column decreases by one each time, while the -2 remains constant. Simultaneously, the right hand answer column increases by 2 each time. Therefore, based on this mathematical pattern, we can conclude that a negative number times a negative number equals a positive number!!!!

Isn't Mathematics Amazing?

Algebra - Using Two-Sided Colored Beans to Add and Subtract Positive and Negative Numbers

When it comes to adding and subtracting positive and negative numbers, many students have great difficulty. In reality, it is a very confusing and abstract idea; so, it is important to give the students a concrete visual to assist them in seeing the solution. This idea is based on the Conceptual Development Model which is important to use when introducing new math concepts. (See the July 26, 2023 for more details about this learning model.) As a result, when teaching the concept of adding and subtracting positive and negative numbers, what would fall into each category?

When using the two-sided colored beans, the concrete stage of the Model would be where two-sided colored beans are used as an actual manipulative that can be moved around or manipulated by the students. There are a few rules to remember when using the beans.

1. The RED beans represent negative numbers.
2. The WHITE beans represent positive numbers. 
3. One RED bean can eliminate one WHITE bean, and one WHITE bean can cancel out one RED bean. 
4. All problems must be rewritten so that there is only one sign (+ or -) in front of each number.

Sample Problem

1) The student is given the problem - 5 + 2.

2) Since -5 is negative, the student gets out five red beans, and then two white beans because the 2 is positive.

3) Since some of the beans are red and two are white, the student must match one red bean with one white bean. (I tell my students that this is barbaric because the red beans eat the white beans. They love it!)

4) Because three red beans have no partner (they're left over) the answer to – 5 + 2 = - 3. (See example above.)

After mastering the concrete stage of the Conceptual Development Model, the students would move on to the pictorial stage. Sketching a picture of the beans would be considered pictorial. Have students draw circles to represent the beans, leaving the circles that denote positive numbers white and coloring the circles that represent negative numbers red.

As an example, let’s do the problem 3 - +5. First, rewrite the problem as 3 - 5. Now draw three white beans. Draw five more beans and color them red to represent -5. Match one white bean to one red bean. Two red beans are left over; therefore, the answer to 3 - +5 is -2.

3 - +5 = 3 – 5 = -2 

When students understand the pictorial stage, then abstract problems such as the ones in textbooks can be presented. (Notice, the textbook is the last place we go for an introduction.) I have found that most of my remedial college students move straight from the concrete stage (beans) to the abstract stage without any problem. Many put away the beans after two or three lessons. What works best for your students as they master this algebraic concept is something you will have to determine.

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If you would like a resource that gradually goes through these lessons, you can purchase it on Teachers Pay Teachers. It introduces the algebraic concept of adding and subtracting positive and negative numbers and contains several integrated hands-on activities. They include short math lessons with step-by-step instructions on how to use the beans, visual aids and illustrations, four separate and different practice student worksheets with complete answers in addition to detailed explanations for the instructor.

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The Disturbing Saga of Math Cannibals!!!

I have just started to teach Basic Algebra Concepts to my college mathphobics.  This is where the "rubber meets the road" as they say.  The biggest hurdle for my students is understanding positive and negative numbers.  Multiplying and dividing seem to be no problem, but addition and subtraction are another story.  To state that subtracting a positive number is the same as adding a negative number is considered hieroglyphics to many.  Since many of my students are visual/kinesthetic learners, I needed a strategy that would connect the abstract to the concrete. 

I took film canisters (a Trash to Treasure idea!) and filled them with two sided beans. One side of the bean is red (negative), and the other side is white (positive). Suppose the students have the problem -5 + 2.  They would get out five red beans and two white ones as illustrated on the left. Then the fun begins because suddenly the beans become "cannibalistic".  The red ones begin to "eat" the white ones and vice versa. (In reality, the students are matching each red bean with a white one and moving them aside; see illustration on the right.)  After each bean has been “eaten” by the opposing color, three red beans remain.  As a result, the answer to the problem of -5 + 2 is -3.

Don't stew; study!

If the problem were -2 - 6, the students would lay out two red beans and six red beans.  Since all the beans are the same color and no bean desires to "eat" anyone on their team, the student simply counts all of the red beans.  So  -2 - 6 = - 8.

What happens with a problem such as 5 + ^-3?  At the beginning, I have the students get out five white beans and three red ones; then match them resulting in the answer of 2.  Unfortunately, in our Algebra book, the double signs vanish by about the third page of the chapter; so, the students must recognize what to do. 

 

The first option is to insert a + sign such as in the problem – 4 – 2 = -4 - ⁺2.  This allows them to see that, in reality, they are subtracting a positive number. 

However, what do they do with -4 - ^-2?   I instruct them to circle the two signs, and use the multiplication rule for a negative times a negative to change the double minus signs into a plus sign as seen in the illustration on the left. They can then proceed to use their beans to solve the problem.  This may seem unusual, but it makes sense to my mathphobics.

You might ask, "How long do the students use the beans?"  It’s interesting, but all of my students put them away, just at different times.  A few only need them for the first assignment whereas others need them for many.  I once had a special education student who was mainstreamed into my regular PreAlgebra class.  He was the last one to rely on the beans, but he did eventually put them away.  The important thing was he had a picture in his head that he could use over and over again.  Incidentally, he passed the class with a “C”, completing all of the same work the other students did.

Need a game instead of a worksheet to practice adding and subtracting positive and negative numbers?  Try Bug Mania or Roll and Calculate.  Just click on the name of the game.