A + B = 26
B + C = 45
C + A = 33
How would you go about it? Would you begin by combining equations? Or would you start by making substitutions? Is this a problem a 5th grader could solve? Without guessing and checking?
What if the problem looked like this?
Is it easier to solve? It's exactly the same problem, isn't it? The visual representation of the problem makes a huge difference, though. Now it's obvious that the two green blobs are the same size even though they exist on different scales (or different equations). And the two blue blobs are the same. Aha, the two red ones as well.
I have prealgebra and algebra students who ask if the A in one equation has the same value as the A in another, whether the two equations are part of a system or not. Imagine if those students had been solving math puzzles like this one throughout elementary school. Would their concept of variable be more clear?
Elementary age students at the math center think problems like this are great fun. They have no idea they are exploring linear functions or algebraic relationships. All they know is that these problems make them think. Algebraic reasoning problems give young students a chance to apply their knowledge of basic math facts within fairly complex scenarios. Problems like this one inspire young minds and satisfy their need for a greater challenge. Our students are incredibly proud when they are able to solve one of these math problems successfully.
How would a young student solve such a problem? We ask our students to compare any two scales and find what they have in common. Let's take the first two scales. Students will point out that the blue blob is common to both. Then we have them look for differences. Students notice that the scale weights differ and the partner blobs are different. We ask them what they think might be causing the weight on the second scale to be greater. It's obvious to students that the bigger red blob on scale two is causing an increase in weight.
Once they understand the effect of changing the partner from a small green blob on the first scale to a bigger red blob on the second, we can look at the quantitative aspects of the problem. We then ask what is the difference in weight between the two scales. Students will do the computation and find the difference is 19. That's the difference between scale one and scale two. What else does this number mean? What other difference does it describe? Students will relate 19 to the difference in weight of the green blob and the red blob. The red blob weighs 19 more weight units than the green blob.
We write this as:
R = G + 19
Now we have a relationship between the red blob and the green blob. This relationship tells us that if we replace a red blob with a green blob plus 19 weight units the scale reading will stay the same. So let's do it.
We head over to third scale. The equation there is:
G + R = 33
We replace the red blob. We get:
G + G + 19 = 33
What if we take 19 weight units off the scale?
What will the scale read then?
33-19 = 14
Our new equation is:
G + G = 14
At this point, students recognize this as a doubles problem and easily find the value of G to be 7. They then use this value to find the weights of the other blobs.
Modeling this problem with young students as a whole group activity is a very powerful. They excitedly share their insights and answers. We'll do several of these together before they work independently to solve similar problems. Eventually we make our way toward the original abstract problem. We replace the green blob with the letter G, then the blue blob disappears and gives way to the letter B, and finally we part ways with the red blob and bring out the letter R. The letters remain on the scale however so the context is reserved. Once the students are comfortable working with letters, we then remove the scale. Students solve systems of three equations by the end of 5th grade. More importantly, students learned how to think through abstract problems, a skill that will forever be of value.