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I have been examining sites that illustrate the use of the block model for solving problems. I am an adult with a degree in math and I have an interest in seeing how math can be taught.

From what I have seen, block modeling can be a useful introduction to algebra. The blocks provide a visual way of working with unknown variables. However, there are cases where the illustrated use of block models requires more ingenuity and abstract reasoning than simple use of algebra.

I saw one problem given as follows: The ratio of money between A and B is $5:4.$ If each of A and B spends $\$45,$ then the ratio of the money they have left is $13:10.$ How much money did each have originally?

This is the solution that was given. To solve the problem, observe that $5$ is one greater than $4.$ If we multiply both $5$ and $4$ by $3,$ giving $15$ and $12,$ we now have the same ratio but now we have a difference of $3.$ This allows us to form $13$ out of the $15$ and $10$ out of the $12,$ with $2$ left over for each. This common difference of $2$ blocks represent $\$45,$ so each block is $\$22.50,$ so A and B started respectively with $15 \times \$22.50$ and $12 \times \$22.50.$

Anyone able to think of that would have no problem parameterizing the original values as \$5x and \$4x and solving (5x - 45)/(4x - 45) = 13/10.

Here is an example of the Singapore block model that I think works well. A person spends 1/6 of the money in their wallet and has \$15 dollars left over. How much did they start with? Create 6 blocks. Color one to represent the 1/6 that was spent. The remaining 5 blocks must add to \$15, so each block is \$3. Add the \$3 in the colored block to the \$15 to get \$18. Someone learning algebra may have a little trouble setting up the equation even though it is a relatively simple problem. The block model makes it easy to see how everything relates.

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    $\begingroup$ I will paraphrase my wise wife: "the block method is one of at least three methods the curriculum gives to solve problems and do algebra". She goes on to say, she doesn't bother belabor the method if the child doesn't understand it because it's just a tool to understand. The block method is not a replacement for the actual logic needed to solve problems. Furthermore, she is very suspicious of any sort of "block method" as a one size fits all to solve algebra problems. There are problems where the method fits and there are others where it is unnatural. You can see why this is not an answer. $\endgroup$ Jun 24 at 16:38
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    $\begingroup$ I agree with the point you appear to make in your last sentence, namely that (5x - 45)/(4x - 45) = 13/10 makes more sense for general teaching objectives than what you described. I fail to see what the learning objectives are for putting so much effort into the block method, since I don't see how those techniques are of much use for later and other work in mathematics. Translating proportionality statements into equations—yes; solving equations—yes; assuming cleverly chosen original numbers were used—no. This sounds like test prep for a very specific type of problem. $\endgroup$ Jun 24 at 18:12
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    $\begingroup$ There is no "block method", there is just one illustrative approach out of many that enterprising edu-consultants have borrowed from a coherent curriculum and have been selling as a silver bullet. The "block method" helps solving a problem arithmetically, which otherwise would be easier to solve algebraically. Writing the problem in a standard algebraic form as you did in the last paragraph is much easier than to come up with a different arithmetic solution each time. $\endgroup$
    – Rusty Core
    Jun 25 at 3:23
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    $\begingroup$ It's common in various Asian curricula to have problems at the upper elementary level that are solved with some ingenuity without algebra when they could be more easily solved with algebra. The hope is that students develop problem solving skills, not that they extensively study non-algebra methods for solving these problems. It's not clear whether they actually develop problem solving skills or instead discourage pupils without them (who then disappear into lower tracks and end up not being counted in the statistics). In either case, teaching the methods as rote methods defeats the point. $\endgroup$ Jun 25 at 4:27
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It's been many years since I looked at it, and I no longer have my copy available, but I believe that Elementary Mathematics for Teachers by Scott Baldridge and Thomas Parker was written to be used alongside the Singapore Math Curriculum. (In fact, a quick Google search shows that Baldridge and Parker's book is available for sale from Singapore Math, Inc, which would seem to support my memory.) I think it almost certainly discusses how to use and teach all of the various models that are used in Singapore Math, at some length.

(Just to add a brief, opinionated disclaimer: this is not an endorsement of that particular text, which I think is problematic in many respects. But it is almost certainly the correct answer to the OP's question.)

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This is a book I read years ago. It is quite general though, maybe you need something more practical https://www.amazon.com/Singapore-Model-Method-Learning-Mathematics/dp/B0033HXD1K#customerReviews

I don't personally endorse the method, nor the book. I think it is very specific and probably such detailed method wouldn't fit for students when they go on to secondary schools. I had myself this issue with students coming from Steiner's school.

Hope this helps.

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