Movement and Kisses for Beginning Algebra Students

Hoy, Barbara

My beginning algebra students benefit greatly from non-lecture styles of teaching. They generally dislike mathematics and have not been successful learning math in the past. Doing something unusual and fun releases their fear and facilitates their receptivity. Instead of just watching and listening to me, kinesthetic learning often proves better for their needs. Therefore, I get them moving and I activate them with “kisses.”

Movement

One of the activities that I use with my classes involves moving their bodies as if they are on a number line. When introducing addition of integers, not only do I draw a number line on the board, but also I point out how a number line can be pictured on the classroom floor. The floors in our building consist of foot-square tiles. I ask the students to pick a line where the tiles meet and use it as a number line. They are to choose a particular intersection of tiles as a zero point and lay their pencil on the floor to mark their zero. When I ask them to stand on the zero of their number line, the response is mostly disbelief. They have a hard time realizing that I am actually asking them to stand in class instead of just letting them sit at their seats and take notes.

After they are all eventually standing, we begin adding signed numbers on the floor. For 3 + (-2), we each begin by standing on our designated zero, step 3 lines (separating tiles) in the positive direction to the right, then step 2 lines in the negative direction to the left. We each end up 1 line to the right of where we started. Each student can feel with their bodies that 3 + (-2) = 1. We go through numerous problems, each time moving along our floor number lines. Each problem involves fairly small numbers due to maneuverability constraints in the room. The last problem I present is something like -1 + (-100). They laugh a bit and say they would end up outside in the quad. They are also able to tell me that the answer is -101.

Subtraction problems can also be performed by students moving along a floor number line. To evaluate -4 – (-2), we stand on our zero points and move 4 in the negative direction to the left. If the problem were to add -2, we would move two more lines to the left. But since it is to subtract -2, we reverse direction and move to the right. They incorporate through movement that subtraction is adding the opposite.

A similar activity can be used for an introductory lesson on graphing. On the floor, we designate lines between tiles for the x-axis and y-axis. We stand on the origin and move right or left, forward or backward, according to the numbers in each ordered pair. Many students say that these movements help them tremendously with their directions.

Chips

Besides offering whole-body movement, I help my students learn addition of integers through an exercise involving game chips. Chips are needed in two colors, white for positive integers and blue for negative integers. Students will need about 10 of each, depending on the numbers the professor chooses for the problems. When a white chip and a blue chip meet, we pretend they disintegrate like matter and antimatter. Thus to add +3 + (-5), each student gathers three white chips and five blue chips. The three white chips wipe out three of the blue chips, leaving two blue chips, or -2. To add -3 + (-4), all the chips are blue and combine to -7. Students get the feel of adding with opposite signs and with like signs, which helps when they are using only pencil and paper and without chips.

Once, and only once, I attempted to have a class use chips to subtract integers. Switching colors of chips was too confusing for my math-anxious students and did not formulate the point of adding the opposite very well. For evaluating subtraction problems using movement, reversing direction walking along a number line works better than using game chips.

Kisses

Another kinesthetic activity involves hands-on learning and “kisses.” In junior high school, I remember exchanging love notes signed with X’s and O’s for kisses and hugs. Now, older and wiser and teaching at a community college, when I see an “X,” I think of an algebraic variable (besides still thinking of kisses). Connecting kisses and variables led me to use Hershey’s kisses when I introduce equation solving in my beginning algebra classes.

Supplies needed for hands-on equation solving are bags of Hershey’s kisses (enough for about 10 chocolates per student) to use for the x-variables, game chips of one color (again enough for about 10 per student) to use for the constants, and handouts with a picture of a balance scale. A real balance scale can also be brought in for demonstration.

The idea is to figure out the weight of one Hershey’s kiss (solve for x) by keeping the imaginary balance scale balanced. We can add or remove the same weight from each side (addition principle) or cut the weights on each side in half, thirds, etc. (multiplication principle). If a problem is to solve for x in the equation x + 2 = 5, we know that x + 2 has the same value (weight) as 5. Students put one kiss and 2 chips on the left side of the paper balance scale and 5 chips on the right side.

To solve for x, they need to get the kiss alone. The single kiss will weigh the same as the number of chips on the other side of the balance scale. They can remove the 2 chips from the left side. Then 2 chips also need to be removed from the right side to keep the scale in balance. The students physically remove 2 chips from both sides of their paper scales. They will understand that the fulcrum of the balance scale corresponds to the equal sign in the equation and that the same operation needs to be done to both sides of the scale and thus to both sides of the equation.

Problems can be presented with variables on both sides of the equation. For the problem of solving for x in 3x + 1 = 2x + 4, students put 3 kisses and 1 chip on the left side of their scale and put 2 kisses and 4 chips on the right side. They remove 2 kisses from both sides and remove 1 chip from both sides. They can then see that x = 3.

The concept of combining like terms can be readily understood. An example is to solve for x in the equation 2x + 3+x + 2 = x + 1 + x + 2 + 2x. When all the kisses and chips are put on their respective sides of the scale, students see that the equation is the same as 3x + 5 = 4x + 3. They remove 3 kisses from both sides and remove 2 chips from both sides. Then 2 = x.

If there is still time, if the students are still enjoying the activity, and if there are still enough uneaten kisses, multiplication/division may be attempted. Solve for x: 2x = 6. The students put 2 kisses on the left side and 6 chips on the right. They can remove half the weight from each side and keep the scale balanced (multiply each side by ½ or divide by 2). Then x = 3. Addition and multiplication properties can be combined as in solving the equation 4x + 1 = 2x + 5.

Similar to using game chips for demonstrating addition but not subtraction, using only positive constants in hands-on equations is advisable. Since I devote only one 50-minute class period to this exercise, it works best when kept uncomplicated. Many more varied examples are done algebraically during subsequent classes once the students have the concept of keeping equations balanced.

At the end of class, students comment on how much fun it was. They feel good about their abilities and about math. They can actually solve simple equations. They also can tell their friends that their math professor gave them kisses (Hershey’s, that is!).

By Barbara Hoy, Onondaga Community College

Barbara Hoy is an Assistant Professor in the Mathematics Department at Onondaga Community College in Syracuse, New York.

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