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I actually prefer a manipulative like Digi-Blocks Each box can hold only ten units. Dumping the box to simulate “borrowing” makes the trade crystal clear. I have seen even junior high students incredulous that after dumping the box, the total number of packed and unpacked units did not change. These students have done trading activities with base ten flats, rods, and cubes without ever acquiring true conservation of number. I also like the use of the term “equivalent” as opposed to “equal” because the form of 3 tens 2 ones is not identical to the form 2 tens 12 ones. Carefully distinguishing the difference implicitly anticipates “equivalent” fractions, where two fractions of differing appearance are equivalent because the underlying value is equal. The use of “equivalent” helps build consistency, for example, in geometry, when students must differentiate equal measure as opposed to identical and/or congruent. Perhaps the “equal” sign should be renamed the “equivalent” sign because equivalent is what we usually mean. Another valuable way to exploit differing representations is to use different ways to record the model. What the authors call equivalent “representation” is actually equivalent variations of the model, in this case, base ten blocks.

2r12c, where r stands for rods and c stands for cubes. Older students also benefit from using various types of representation. Grades 1-2, Subtraction Is More Than Take Away: I like the discussion of the different meanings of subtraction. Grades 1-2, Modeling Addition and Subtraction: Of course I like this page if only for the reference to Digi-Blocks. Grades 3-4, Helping Facts: Students who are acquiring profound understanding of fundamental mathematics still need fluency with facts. This page contains useful tips for recalling and reconstructing multiplication facts. Grades 3-4, Meaning of Division is a good explanation of the various types of division. The authors did a good job with Remainders, even providing a nice segue into bases. I also liked the Multiplication Menu, and the discussion of the meaninglessness of “gozinta” and misconceptions inherent in the long division algorithm. Grades 3-4: Finding Parts and Making Wholes contains a nice list of misconceptions. Grades 3-4: Parts of a Group: American egg cartons are very useful for modeling fractions. Instead of putting any old counters in the egg cups, it is better to use plastic eggs in up to six colors.

Then the ribbons are unnecessary and the egg cartons can be used to play fraction games with even first and second graders. As an aside, Japanese egg cartons hold ten eggs, making them ideal for place value lessons. Grades 5-6 Greatest Common Factors and Least common Multiples:I like the Venn diagram for finding common factors. Adding and Subtracting Fractions with Pattern Blocks, good explanations and activities. Modeling Multiplication of Fractions, good activities. Modeling Division of Fractions with Pattern Blocks, avoids multiplying by the reciprocal. Grades 5-6, Estimating Decimals: I like the emphasis on the significance of zero “placeholders” as indicators of precision because of the connection to measurement and data recording in science. The authors also point out the problems with “context-free” computation. Real math occurs in a context. Real math always has a story. Grades 7-8, Analyzing Change: The story graphs nicely anticipate the early topics of physics. Grades 1-2, Connecting Representations: I would have liked the confusion over the difference between number and numeral or other representations explicitly stated, however this major misconception is implied in the text and diagram.

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