Beginner's Algebra

Guess How Many is Under Cup X

Poddle Weigh In - find the missing weight

Pennies and Rocks as Manipulatives

Ethan had trouble understanding how to work out equations like 3X +2 = 15.  He couldn't see why we would take the 2 away from both sides so I tried to make it more visual and make more sense to him.  This is what we did.

This time I decided to make a rock represent X and I hid 4 pennies under the cup.  Ethan had to figure out what my mystery rock was worth.  I then set up the problem 2X + 3 ==11.  I used 2 rocks to be 2X and pennies for the numbers.  Pennies made sense to him to use since we all know that one penny equals 1 cent.  Now to find out what the rocks were worth. 

 

We wanted just rocks on one side and pennies on the other side so we took three pennies from the rock side and took three pennies from the penny side.  That left us with 2X = 8.  It was then easy for him to see that X = 4.  then he peeked under the cup to see if he was right.

 

Game of Missing Addends

Kisses (I love this lady's idea)

A 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.

Read her other great ideas here.

Algebra Planet Blaster

Easy Algebra Lessons

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