@John
I suppose you could use bars for that system, but I don’t know why you would want to. Algebra is much easier when the coefficients are not related at all.

The bar model can easily be used for systems of two variables where one of them drops out. This, for example, might be a 4th or 5th grade problem:

2x + 5y = 20
2x + 2y = 19

Systems with three variables are also used, but these are very simple. For instance, here is one from my Hobbit Math article:

x + y + z = 123
x = y + 15
z = y + 3

And in 5th-6th grade, just before switching to algebra, there are problems where the student needs a multiple of one of the equations, such as:

2x + 5y = 20
x + 3y = 11

Of course, these aren’t given as variables, but as word problems. Usually they have to do with purchasing an assortment of items, or fixing a variety of foods for a party, or collecting and trading stickers, etc. — so that the students have something concrete to imagine.

Thanks for that explanation. I played around with a Singapore bar model a bit last year, and was able to do systems questions like the one you described above. As long as one of the variables was the same in each equation, I could model it. Can you tell me if it is possible to model and solve using bars a system like:

2x + 5y = 20
3x + 2y = 19

But as Denise’s comment suggests, the only math skills you need to solve this problem are counting and pairing. You can do it on a tabletop with a jar of pennies: Take out 545 pennies. Separate them into two groups to represent the lids on the two dresses. Take 185 pennies out of the larger group. Equalize the two groups by pairing to correct for any error in your original separation. Then put the pennies you took out earlier back into the group representing the larger dress.

You now have two groups of pennies which contain the same number of pennies as the number of lids on the smaller and larger dresses, respectively. If you count the two groups of pennies, you’ll find that the smaller one has 180 pennies and the larger one has 365 pennies. That tells you how many lids are on each of the two dresses.

So you’ve solved the problem. And you didn’t need to subtract or divide. You didn’t need know how to use numerals, much less equations. Counting and pairing sufficed. (Of course, in the real world, with the two dresses at hand, one could solve the problem even more simply just by counting the lids themselves.)

So if the purpose of the problem is to get the answer, or to learn how to solve problems like it using manipulables, or to develop general number sense and mental math skills, then it might be considered a third grade problem. But does the fact that it can be solved in this way mean that the problem can’t (or shouldn’t) be used in higher grades to teach more advanced math skills, such as how to manipulate equations?

Adam and Carly have 135 baseball cards altogether. If Adam has 41 more baseball cards than Carly, how many baseball cards does Adam have?

The fastest way to solve it is by understanding the relationships, which the model method helps students see.

FYI, the math books used in Singapore include National Education in the Teacher’s Resource Pack. In the 3B Teacher’s book for Shaping Maths, the unit on Time is introduced using pictures of festivals and celebrations: Diwali, New Year, Christmas and Hanukkah. Here’s the note in the teacher materials from that page:

“Though we have many different races and religious groups in Singapore, we all still pursue the same dream – The Singapore Dream. NE message: We must preserve racial and religious harmony”

You don’t see that in the U.S. version of the curriculum.

This type of reasoning is very common in Singapore Math word problems. See examples in my Word Problems from Literature series. The problems are often modeled with bars (a sort of visual algebra), to make the number relationships easy to see.

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total = 545