Understand addition, and understand subtraction.

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Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

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Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

An infant's first understanding of the concept of number – a way of quantifying the world – comes upon realizing that not every object is unique, viz not one orange and one apple are two different things, while two apples are the same thing twice.  Only much later is this concrete representation abstracted to a "2" symbol.

All of the black blocks in MMB's are identical in shape, size and color. This teaser challenges the child who cannot use visual cues (shape, size and color) to know how the blocks differ. Using the balance clearly shows there is a difference. The child cannot rely on her first way of understanding number (counting objects) to solve this puzzle inherent in MMB. 

While the colored blocks are for multiplication, the black blocks are for addition. The only difference among the black blocks is weight, but this is a difficult one to detect precisely with the sense of touch. The balance allows the student to begin dealing with symbols as representations of exactly quantified reality – integers. Playing with the blocks to test which combinations balance is, in fact, an exercise in becoming adapt at understanding the symbolic representation of the first 10 integers and understanding how different integers combine to make larger integers.

As well as addition, subtraction is made real. For example, if a student places a "2" block on one pan and a "5" block on the other, trial and error will lead to seeing that 2 + 3 = 5. This can be conceptualized as "There is a 2 and a 3 inside of a 5." Rather than learning a meaningless (though easily memorized – and easily forgotten) algorithm, the student is prepared to understand the concept of subtraction as follows: "Since there is a 2 and a 3 inside a 5, then if I take a 3 out of a 5, a 2 will be left."

The student thus comes easily to grasp these building blocks of all future mathematical concepts.

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For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

The student sees in a concrete way that 10 can be decomposed (i.e. "is equal to") 1 & 9, 2 & 8, 3&7, and 4 & 6. The same can be "proven" "right before the student's eyes" that 9 can be decomposed into 8 and 1, and so on.

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Fluently add and subtract within 5.

In addition to addition, subtraction can be represented with Math Magic Blocks. Consider the case where a 5 is placed in one pan and 1 and 4 are placed in the other pan, but the 4 is placed number side down. As mentioned above (CCSS.Math.Content.K.OA.A1) if 1 + 4 =5, then in a sense, 1 and 4 are inside 5. The hidden block (4, in this case) is the other number, besides the visible 1, that is in 5. If you take that hidden number out of 5, you are left with 1. So the puzzle confronting the student is: what is the hidden number that, if taken away from 5, equals 1?

Playing with black blocks 1-5 fosters fluency in addition and subtraction.

Represent and solve problems involving addition and subtraction.

CCSS.Math.Content.1.OA.A.1

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.1

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Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Understand and apply properties of operations and the relationship between addition and subtraction.

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Apply properties of operations as strategies to add and subtract.2 Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

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Understand subtraction as an unknown-addend problem. For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.

Add and subtract within 20.

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Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

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Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Work with addition and subtraction equations.

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Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

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Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.

Represent and solve problems involving addition and subtraction.

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Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1

Add and subtract within 20.

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Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers.

Work with equal groups of objects to gain foundations for multiplication.

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Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

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Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Represent and solve problems involving multiplication and division.

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Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.

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Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

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Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1

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Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?

Understand properties of multiplication and the relationship between multiplication and division.

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Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

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Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Multiply and divide within 100.

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Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

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Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.3

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Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Use the four operations with whole numbers to solve problems.

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Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

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Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.1

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Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Gain familiarity with factors and multiples.

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Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

Generate and analyze patterns.

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Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

Write and interpret numerical expressions.

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Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

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Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

Analyze patterns and relationships.

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Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

Operations and Algebraic Thinking (5.OA)

Write and interpret numerical expressions.

1. Use parentheses, brackets, or braces in numerical  expressions, and evaluate expressions with these symbols.
   Both the associative and commutative properties of addition and multiplication become instantly obvious with MMB's:
   (1 + 2) + 3 = 1 + (2 + 3)    
   If the 1-block, 2-block, and 3-block are all placed in one pan, they always balance ("add up to") a 6-block, no matter whether which block is placed in the pan last. (Duh.)
   (2 × 3) × 4 = 2 × (3 × 4)
   Similarly, as long as all three blocks shown here are in a pan, they will balance a 24 block, no matter which order they are placed in the pan. (Duh, again.)
2. Express a whole number in the range 2-50 as a product of its prime factors. For example, find the prime factors of 24 and express 24 as 2x2x2x3.  Analyze patterns and relationships.

This is what MMB's were born to do!