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Solid understanding of numbers (Base Ten) is key to success.

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Children should know that they can break up tens or add to them in order to solve problems. Ex, adding groups of 12's

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Invented algorithym provides students different ways to solve for answers using their math background knowledge and skills

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Direct modeling with 10's can be broken into the following subcategories:  multiplication, measurement division and partitive division.  Similarly, adding strategies for multiplication and division problems can be broken down into the same subcategories:  multiplication, measurement division and partitive division.

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it is extremely important for students to be exposed to the representation of numbers/strategies at an early age... this will lead to really understanding and being able to solve problems. this leads into remembering that all students come knowing something and it is important to challenge their thinking.

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I do see that there are similar strategies for multiplication/division as for addition/subtraction.  I've also found that my students do need to become more comfortable with breaking tens apart or seeing numbers differently

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Students were solving the problems by what made sense to them. There was a lot of decomposing.

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Children need to be flexible in understanding that tens can be broken apart.

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Measurement division and partitive division will push students to do different strategies.

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The development of multiplication and division strategies are just like addition and subtraction strategies with some "added twists". Therefore my students must have a strong add/sub foundation to move into mult/div with multi digits numbers in fourth grade. I think those holes or gaps that some students have lead to frustration in the upper grades.

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Students need to have a strong knowledge of base ten to be challenged.

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Students start exploring division with Direct Modeling strategies.  They use individual counter with smaller numbers, but when the numbers get larger they tend to use base ten blocks.  In order to use the base ten blocks to help them they need to know that 1 ten is the same as 10 ones and not see the ten as a 1.  Students need to have place value understanding and grouping ideas.  1 ten is 1 of this group (of tens) but also 10 of this group (of ones).

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Students need to have a strong base ten understanding to solve multiplication and division problems.  They need to be able to break numbers apart and understand that numbers can be represented in different ways.  For example, 53 can be represented as 50+3 but also as 40+13.

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Being able to decompose numbers aids in solving multiplication/division problems

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Decompose numbers to make it easier for them

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children see numbers and groupings differently just as we did with the choral counting patterns. students will often rely on the number patterns and groupings they feel most confident with

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students can use direct modeling to organize ideas into groups with more than ten objects in each group. They can break tens apart or trade tens for ones to form groups. This will challenge their knowledge of base-ten concepts.

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I realized while reading that how truly important it is to give students the time and space to struggle and solve these problems. Additionally, it drove home that it is so important to listen to them explain their thinking. Those students came to the solutions in different ways, but their solutions were accurate.

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Children use modeling with tens, counting strategies, and invented algorithms to solve for multiplication and division problems. I think this is important because it shows the development of student thinking when applied to multi-digit multiplication and division. We need to understand student thinking progression, so that we can continue to challenge their thinking.

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"Children need to be flexible in understanding that tens can be broken apart"

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Doubling Strategies are common when students are inventing multiplication and division algorithms

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Children use their foundational knowledge of breaking up 10's to progress through strategies and continue into making their invented algorithms.  It is important that they have the basics of working with directing modeling, breaking down 10's etc. to be able to grow.

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Students need to be able to see that 42 is 4 tens 2 ones, but also that it's 3 tens and 12 ones. They need to know that there is more than one way to see units in numbers. Showing their work gives us insight to their thinking.