This blog post is the first for the Block Model Series. In this particular post, Math Trivia will be giving you an overview of what the Block Model is and how effective it is in enhancing the problem-solving skills of students. We will also take a quick look at the anatomy of a typical block model and what are the basic forms of block models.
The Block Model, also known as Singapore Math Model or Bar Model, is a promising way of doing Algebra and Arithmetic problem solving by transforming routine problems to nonroutine and rudimentary ones. This approach has earned international attention primarily because Singapore over the past years has always ranked top in Mathematics proficiency. Aside from the versatility of the technique, the Block Model provides a pictorial view of the relationship between quantities manipulated in the problem. This enables students to understand abstract concepts in a concrete perspective.
According to a conference paper presented by de Guzman and Belecina (2012), there was an observable improvement in the post-test performance of Grade 5 students who were taught using the Block Model concept compared to a controlled group which was taught using traditional routine problem-solving approach.
In the Block Model, a rectangle or rectangles called blocks are drawn and sometimes cut into sections to represent conditions stated in the problem. Arithmetic manipulations and simple filling up of values are done to give a view of the problem. Then, fundamental arithmetic operations are done to solve for the values of the missing parts. Lastly, the required value(s) is/are determined to arrive at the correct answer. An example problem is shown below.
The sum of two numbers is 25. One of the numbers exceeds the other by 11. What are the numbers?
Determine the quantities involved:
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Two numbers à two blocks (long block and short block)
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Define relationships and conditions:
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Sum of blocks is 25
One block exceeds the other by 11
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Draw blocks, relationships, and conditions:
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Derive (Solve) missing values:
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Step 1: Make identical blocks. On this case, if we extend smaller block, we must add 11 to the sum.
The lower block is equal to the one on top, so
Step 2: Solve for the value of the identical blocks
Step 3: From the diagram of step 1, we have that
Placing 18 as the value of the long block.
Step 4: Solve for the short block
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Deliver what is asked:
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The numbers are 7 and 18.
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Do check your answer:
*Use the defined relations and conditions |
Sum of blocks is 25 à 18 + 7 = 25
One block exceeds the other by 11 à 18 – 7 = 11
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The example shown utilizes pure block model in solving this problem technically without any Algebra involved. In this particular approach, no variables are used. Instead, the blocks are replaced with their actual values and if necessary are solved using any of the four fundamental operations. One important thing that matters in the example is how the blocks and conditions are illustrated. To give you some hints, here is a graphical summary of conventions that are used in block modeling.
Anatomy of a Typical Block Model
Shown is a typical block model with its parts. The sum is always placed on the right with a brace and an arrow is placed to signify the difference between quantities.
Three Basic Types of Block Model
1. Part-Whole Model
“The whole is greater than the sum of its parts” ~ Aristotle
While the famous expression above holds merit in most cases, it does not when we talk of Mathematics. On the contrary, Math tells us that the whole is equal to the sum of its parts. This is the idea of the Part-Whole Model. In this type of model, a larger block is partitioned into smaller blocks to represent values that when summed is equal to the whole. Problems involving single step arithmetic and with one quantity make use of this model.
2. Comparison Model
“If you compare yourself with others, you may become vain or bitter, for always there will be greater and lesser persons than yourself.” ~ Desiderata (Unknown Author)
Unlike in Desiderata, when you check side by side in the comparison model, you don’t have to be bitter. The Comparison Model is used when faced with problems with two or more quantities. Most algebraic and arithmetic problems use this model. For example, a Comparison Model was used to solve the sample problem above.
3. Before-After Model
“The future influences the present just as much as the past.” ~ Friedrich Nietzsche
Nietzsche was right when he said that the present relies upon the future and the past. The Before-After Model is used when solving problems that involves changing values of the quantities involved. In most cases, these problems involve 3 or more conditions and relationship. Most Age Problems in Algebra are solvable using this model.
Arithmetic and Algebra Problem Solving Made Easy
To most of you who are new to Block Model and who are very much comfortable with algebraic manipulations, using this method seems to be tedious and sometimes difficult. Most secondary school teachers would not include or even take a quick discussion about this concept when problem-solving surfaces during Algebra classes. While it somehow appears that the block model is designed to be a counter-intuitive method versus algebraic manipulations, it is actually a good foundation and supplement to the latter. Block model can actually help students visualize the problem and get a profound understanding of how various forms of representations (variables) can be used to determine unknown values.
The truth remains that Singapore Math is effective and one of its core is Block Model. Giving it a little space would surely make the mathematical learning experience more fun.
That would be all for now. Up next in our Block Model Series is an article about the Part-Whole Model dealing mainly with tons of examples and including free worksheets with step-by-step solutions. Adios!
Alfore is a reluctant blogger, a lazy poet, and a great daydreamer. He owns a Nobel, Pulitzer, Oscar and a bunch of weirdly named cats.
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