Advanced Bar Models Problems

We call this an advanced bar model problem. How would you solve this 5th-grade question using bar models?

Mr. Lim had some stickers and stamps for his class. The number of stickers was half as many as the number of stamps. After giving each student in the class 3 stickers and 8 stamps, Mr Lim had 39 stickers and 2 stamps left. How many stickers and stamps did Mr Lim had at first?

This was an answer by one of our 5th grader:

Explanation

As the number of stamps is twice as many as the number of stickers, the bottom bar model is twice as long as the top one. Dividing the bottom bar by 2, we get 4 units + 1, which should be equal to the top bar model’s 3 unit + 39. Equating these two quantities, we can solve for the quantity for 1 unit, which is the number of students in the class. Once we know the number of students, we simply multiply the number of stickers and stamps each student receives and add the remaining quantities to get the total.

Why is this an advanced bar model?

The key to solving this question is to first choose the bar model unit to represent the number of students. This is an advanced bar model because it is less intuitive than normal word problems as we usually see examples where the unknown units are representing quantities of physical objects, e.g. stickers and stamps. However, in the multiplication sentence, n students x m stamps = mn stamps, we need to first find out what the number of students is.

How to spot these questions?

How should students know what quantity to represent with the unknown unit? Looking at the two bar models above again, we see that all three quantities – the number of stickers, the number of stamps and the number of students, are all unknown. However, the number of students is the only quantity that is the same for both cases. Hence, in order for the two models to use the same unknown unit, we have to use the number of students as the fixed quantity to solve first.