Practical Steps for Problem Sum Solving with Models

Tackling problem sums using models can be a hazardous affair for the uninitiated. Some people are put off by those usually wordy Math questions, which seem to be one hurdle to applying models. Here, a poor command of English can trip up the student trying to understand what is being asked.

How to picture the problem and draw the correct model diagram seems to be another hurdle. Failure to pick an appropriate model++ means relationships between items given in the problem sum can’t be shown correctly and in full detail. This often leads to wrong answers.

Fortunately, there is a method to this madness, so to speak. The recommended solution is to adopt a systematic approach.

++See the cautionary note in “Step 2. Use An Appropriate Model”. Click this link to view it.

How to Solve with Models: My Step-by-Step Guide

In this No Problem Sums article, I share my thoughts on how to use models to present the problem and get to the answer. This may not be exactly what is taught in the schools; I certainly would like to understand better with your feedback on this.

Suffice it to say, here is a practical way to solve problem sums with models.

Step 1. Start With Analysis

Proper analysis of the statements given in a problem sum is required to ferret out data and relationships about items being discussed in the Maths question. This need not be complicated though, when you follow a simple plan of attack.

Collect Data

  • Look for quantities* and know what they represent.
    Examples: a total; an item in a group; etc.
  • Look out for comparisons, noting the differences that are indicated.
    Example: item A is more than item B; by how much?
  • Watch out for before-and after-scenarios, paying attention to what items have changed and their old and new quantities.
    Examples: some units of item C were used up and 7 were left; or 4 units were added to item D and now D has twice as many as before; etc.
  • Note other quantities or factors, if any, that relate items to one another.

In the Answer Sheet for each worked example, I underline the data collected.

Know What Is Being Asked

  • Understand what answers the problem sum is seeking and which of the data collected earlier lead to them. Examples: what the grand total is; how much more/less an item is when compared to another; etc.
  • When the problem sum asks multiple questions, pay attention to whether a question relies on an earlier answer.

To ensure that the gist of questions aren’t missed, I use bold red text in the Answer Sheet to underscore such details. If permitted in their schools, students could use a (yellow) highlighter pen as a way to emphasize the same.

*Quantities refer to numbers, amounts, sizes, etc.

Step 2. Use An Appropriate Model

After analyzing the problem sum, the next major step is to select a model to use. As a visual representation of the Maths question and its associated facts, a model uses bars to depict quantities of and express relationships between items.

Models generally use one of two concepts: Comparison (aka Change) and Part-Whole. These provide different ways to look at the problem and how answers can be found from them. Variations of these 2 themes can be applied to solve different types of problem sums.

Comparison Model Basics

In the Comparison Model, bars are laid out in a vertical stack, as shown:

Comparison Model basic diagram

The long and short (pun intended ;-)) of it is the relationships between items are immediately obvious from the model diagram. Visually, this aids in comparison; hence the name.

Part-Whole Model Basics

As the concept name suggests, the Part-Whole Model revolves around the sum of parts making up the whole. Thus, in this model diagram, bars are laid side-by-side, such that their aggregate is depicted by the full-length bar which spans end-to-end.

Part-Whole Model basic diagram

The Part-Whole Model is useful when looking at the proportionate quantities.

Both model concepts are used to solve Problem Sum #1. One is seen as more apt than the other for answering this Primary 2 level Maths question.

Click hereto take a quick look at Problem Sum #1.

When a model cannot fully represent the problem to be solved, extra steps may be needed. One way is to draw additional models utilizing intermediate results as new data.

Another is to rely on other mathematical methods to compute the answer; however, this defeats the use of models.

Selecting the appropriate model to use is key to the smooth flow from understanding the question to arriving at the answer. It is prudent to pay attention to this initial step and not jump straight in to try to find the answer.

Step 3. Answer In An Orderly Manner

There are a few steps to take to get to the answer, so these should be written out properly. They follow a logical sequence, like this:

  • As mentioned earlier, select the appropriate model to use, based on what the question is asking and the type of data supplied.
  • Draw the model diagram and label it appropriately. Ensure that the relationships between items are correctly depicted.
  • From the diagram, derive intermediate answers that lead to the final one. Show the working for each of these steps.
  • Present the answer in a statement; or fill in the blank if the answer statement is provided.

Practise, Practise, Practise!

There you have it, a step-by-step approach to using models to solve problem sums. The trick is in doing the first two steps right: making sense of the data and selecting the correct model to use. This requires practice, so keep working on it until you are familiar with the flow.

Don’t forget to browse the No Problem Sums library of worked examples. That is a good way to get your Maths problems answered 😉

Comments are closed.