Beginner's Algebra

Update: New Idea - click here to see our recent fun algebra discovery or scroll down.

Guess How Many is Under Cup X

Poddle Weigh In - find the missing weight

New Idea:  I saw this idea on Nurtured By Love and thought it was a great idea.  It's a program called Hands on Equations.  I didn't buy the program but just used the idea myself out on Ethan today and he so "got it".  It's a good visual.  Better than the one I've done with him in the past, see below with rocks and pennies.

If you look on the Hands on Equations website you see that he has the kids visualize that the math equation is balanced on a scale so that the numbers on each side of the equals sign "balance"  I had Ethan balance the ruler on his finger and I pretended to place the chips on the ruler and asked what would happen if I took one chip off of one side and he knew that the whole thing would crash down.  So I asked what would happen if I took one chip from each side and he knew that it would stay in balance.  So he could now visually understand why we could take chips off from one side as long as we took them off of the other.  Make sense?

So we started with this equation: x + 8 = 3x and set it up on the ruler like this.  I used a paper clip to stand in for his finger in the center of the balance.  I used poker chips to be the Xs and regular math counting manipulatives to represent the numbers in the equation.

I asked Ethan what we could take away from each side to keep the ruler balanced and he knew right away that it was one of the Xs (chips), so we took one chip away from each side.  (I wrote in tiny letters under the equation -x on both sides to show in writing what I had done, along with removing the chips visually.)  That left us with 8 = 2x and he could tell then that x must be 4.

So we tried another, 2x + 4 = 3x + 1

He took away 2x's from each side and then 1 block from each side.  He was so happy afterwards and felt like he was so smart (yippee!) and so we continued with other equations and then moved onto to doing the equations without the use of the ruler and manipulatives.

Try this way out.  It's a great visual!  I listed some of them because they are hard to make up on the spot sometimes.

5x + x = 8 + 4x

4x + 5 = 2x + 13

3x + 4 = 6x + 1

4x + 4 = 5x = 1

simple algebra worksheets

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.

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