The first is in a Grade 6 class and we wanted to specifically give them a multiplicative comparison problem. These problems tend to be more difficult for students compared to equal groups problems; I think this may be because students have a hard time visualising what is happening. Bar models can certainly help here.
Our question was:
Mr. Jacobs eats some candies. Mrs. Delaney eats four times as many candies as him. Together, they eat a total of 260 candies. How many candies did each of them each?A typical response is shown below.
By using bars to represent how many candies Mr. Jacobs has and how many candies Mrs. Delaney has, the calculations needed to be performed are clearer to see: the combined five bars must represent the total 260 candies. Thus one bar must be 260÷5=52.
In another example (this time from a Grade 2/3 class) the student uses a bar model to visualise this part-part-whole problem:
A school has 83 water bottles. If 29 are filled, how many are not filled?
A typical response is shown below:What I like about this, is that the student drew a bar model first and used this to help decide what number sentence could be written. Once this was done, the student decided to use an empty number line to calculate the difference. So whilst this might be thought of as a subtraction problem (when the unknown isolated, you get 83-29) it can be solved by addition. The beautiful thing about bar models is that they allow students to visualise both ways.
I'm not sure if students will become experts at using bar models to solve problems unless we as teachers model them effectively and consistently. In many ways a whole school approach is necessary. In primary, teachers should model how to use bar models to solve additive thinking problems. In junior grades, teachers need to model how to use bar models to solve multiplicative thinking problems. In intermediate grades, teachers need to model how to use bar models to solve proportional reasoning problems.
The pay off will be massive.