So far, we’ve looked at how the visualization emphasized by Singapore Math allows young learners to develop the ability to group numbers and manipulate them in their minds. This ability equips them to tackle basic arithmetic, as students are able to break down and understand relationships between values.
As we progress students to more complicated problems, students should be able to duplicate that journey to visually representing problems and values in order to compare and contrast them to form a calculation.
As such, we introduce more nuanced problems and approach them with a visual problem-solving toolkit. We revisit to the PICTORAL stage of problem solving and develop another layer of complexity. The more confident and flexible that students become using visuals, the more the leap to instantaneous mental calculation will not be a far one to make.
Singapore Bar Models
We’ve adapted the bar models made famous by Singapore Math to effectively suit our instruction methods. There are two primary bar models we focus on:
Nearly every arithmetic question can be boiled down into the use of one or both of these models.
These models deal with a single subject associated with a total quantity that is broken up into parts. Arithmetically similar to a number bond, students may be asked to solve for one or more of the parts or the total.
Students can learn to recognize when this model is appropriate by looking for key words within the question. Words like “all together,” “in total,” “left over,” and so on indicate that they are solving for part of a whole or the whole itself.
Comparison models are used to represent two or more subjects with quantities that can be contrasted against each other.
These questions would use words such as “how many more” or “how much less”. Students can look for the key word “than” to identify when they are engaging in a comparison relationship.
Whole-Part Model Question: Sally has 18 hair clips. 6 are blue and the rest are red. How many are red?
Comparison Model Question: Sally has 18 hair clips, and Julie has 6 more than Sally. How many does Julie have?
Solving the Models
We’ve created a unique step-by-step approach to solving bar models so that students can recognize the patterns of bar model questions and eventually solve them without the use of the visual aid. The more these solutions become second nature, the more easily they will be able to hold all of the information in their minds for expeditious calculations.
Our Approach: IMMC
Our step-by-step process that works for any bar model they may face is as follows:
Students will identify the model they’ll need to use. As mentioned above, this strategy would include looking for key words in the question. Students then have an easy-to-grasp step into visualizing the type of relationship the subject or subjects have in the question.
M: More or Less
Now, students can take a closer look at the description of the problem. Are we trying to find a number that’s more or less than what we’re given? Should addition or subtraction be used? This will influence how we set up our model and will make the following step much easier, where we fill it in and hone in on the answer we need to solve.
Next, we can take the information given in the question and fill in a basic model. Every number should be accounted for, and this will help to reveal where the numbers are missing. Blank spots for numbers will usually either be the answer the question is looking for or a preliminary step to get there.
Note: It is important at this stage to identify where in the model lies the final answer the question is looking for.
The rest is simple arithmetic. We’ve identified the operation we need to use in Step 2 (More or Less), and we’ve isolated our numbers in Step 3 (Model - Fill In). Now, we take the foundational work we’ve already done and turn it into a formal number sentence and then into calculation.
Note: Writing the number sentence is critical at this stage.
A Model Foundation
As you can imagine, familiarity with these kinds of bar models sets up a student to solve all kinds of applications. Modelling problems in a systematic, visual way allows students to see patterns and easily pick out given and missing variables. They can expand on this ability by solving two-or-more step problems, and they can eventually layer in more challenging operations such as multiplication and division.
Bar models are a flexible enough tool to be used for a student’s very first math questions all the way up to complex, multi-step problems involving multiple operations. These models are the Singapore Math’s way of scaffolding the bridge between the concrete and the abstract for learners. As students practice and grow in confidence, they build a strong bridge themselves in their own minds.
Bar models bring abstract relationships into something that can be filled in and dissected. The more students practice that kind of organization of information and visualization of numerical relationships, the more students will start to organize the information they are encountered by in everyday life. They will begin to see models all around them and know exactly how to identify and solve for gaps.
This kind of confidence can make just about anything more approachable. And the more our students feel confident in approaching, the more they can take on and conquer—starting with the not-so-intimidating-after-all subject of mathematics.
Welcome to the team, parents—the Singapore Math team! We’re excited to have you on board and we look forward to helping you coach your little ones.
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