## Exploding Dots

### 4.1 Common Core State Standards

Here is a list of some overt intersection points between EXPLODING DOTS and the Common Core State Standards in mathematics. Many aspects of this course meet other standards to some partial or tangential degrees too.

High School

A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Comment: BINGO! This is the delight and the surprise of lesson 1.7. One can look at the remaining standards in this cluster and see how to use lesson 1.7 as a springboard to them.

A.SSE.4  Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

Comment: Although lesson 1.9 attends to the infinite geometric series formula, the supplemental material in the companion guide attend to the finite version.

8NS.1   Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Comment: This is the content of lessons 2.2 and 2.4

6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

Comment: The model presented In lesson 1.5 connects with this.

5.NBT.1  Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

Comment:  This is lesson 2.1.

5.NBT.2  Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Comment: We’ve walked right up to this point in this course. See lesson 2.5 on the  web (lesson 1.10 in these notes) to explore multiplication by ten, in particular.

5.NBT.3  Read, write, and compare decimals to thousandths.

a)      Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392=3×100+4×10+7×1+3×(1/10)+9×(1/100)+2×(1/1000).

b)    Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

Comment:  This is lesson 2.1 (and one can go the extra step to part b) of the standard with ease).

5.NBT.4  Use place value understanding to round decimals to any place.

Comment: We haven’t rounded in this course, but we are right there for this work.

5.NBT.6  Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Comment: The conceptual understanding for finding quotients is spelled out in lesson 1.6.

5.NBT.7  Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Comment: The work of lessons 1.4, 1.5, and 1.6 applies equally well to the decimals of lesson 2.1.

4.NBT.2  Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

Comment: Lessons 1.1 and 1.2 do this.

4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place.

Comment:  It is easy to take this extra step in lesson 1.2.

4.NBT.4  Fluently add and subtract multi-digit whole numbers using the standard algorithm.

Comment: Lessons 1.4 and 1.5 completely explain the algorithm.

4.NBT.5   Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Comment: We do these one-digit multiplications in lesson 1.4. One can go the extra step and look at two-digit multiplications too.

4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Comment: Lesson 1.6 attends to the first part of this standard.

Cluster 3.OA.A  Represent and solve problems involving multiplication and division.

Comment: Exploding dots well on the way to address a number of standards under this listing.

3.NBT.3  Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9×80, 5×60) using strategies based on place value and properties of operations.

Comment: Exploding dots lesson 1.4 is right at this point.

2.NBT.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

1. 100 can be thought of as a bundle of ten tens–-called a “hundred.” (see illustrations)
2. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

Comment: This is the content of lessons 1.1 and 1.2 of the course.

2.NBT.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Comment: Although we haven’t made direct comparisons like this  in this course, lesson 1.2 takes us right to this point.

2.NBT.7            Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Comment: This is lessons 1.4 and 1.5.

2.NBT.8 Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.

Comment: Exploding Dots has brought us to this point.

2.NBT.9 Explain why addition and subtraction strategies work, using place value and the properties of operations.

Comment: This is lessons 1.4 and 1.5.

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