## 6th Grade and Pre-Algebra Standards

By now, you probably know that I am teaching all new courses this year. Over the last five years I taught only fifth grade, but this year I will be teaching three sections of sixth grade and one section of pre-algebra which is all seventh graders.

After using standards-based grading (SBG) with my fifth graders last year, I knew that it was going to take a lot of work to develop standards for two separate courses. Well, it is done! Ch..check!

I want to share those standards here in the hopes that others can make use of them or that teachers more experienced with these grade levels and this content can provide feedback. For reference, here is what a few of my 5th grade standards looked like last year -- my first year using SBG:

Standard ID |
Concept |
Standard |
Sample Problem |

A1 | Number System | Decompose numbers into expanded form according to place value. | Decompose 3,213 into expanded form. |

A2 | Number System | Recognize that in a multi-digit number, any digit represents 10 times as much as the same digit in the place to its right and 1/10 of what it the same digit represents in the place to its left. | In the number 23,488, how many times as large is the value of the first 8 as the value of the second 8? |

A3 | Terminology | Use appropriate language to describe parts of multiplication and division equations. | What are the terms for the different parts of a division problem? |

A4 | Whole Number Multiplication | Can complete 60 single-digit multiplication problems in 3 minutes. | 8 x 7 |

After one year using the standards, I had a two goals to keep in mind as I made my two new lists of standards:

**Simplify**- There was just too much information here for most fifth grader to parse. In writing these, I knew how important it was to be specific to ensure that it was clear when a standard was met. The result, however, was that most students had a difficult time knowing what each standard was all about. This year, I worked hard to declutter and simplify the language for each standard in order to make them more accessible to students**Shorten**- Man, there were a lot of fifth grade standards. 55 was too many for 5th graders to keep track of. I broke down concepts enough that it was very easy to see when students had met each standard, but the total number of standards was overwhelming to both students and to me. As I wrote my standards this year, I erred on the side of making each standard too broad, knowing that I could always split them into separate standards if needed during the year

As you look these over, keep in mind that I teach at an independent school that does not follow the common core state standards. So, if topics seem out of place or absent, that may be the cause. If you notice a glaring absence, though, I would still like to hear it. I can always add topics in if they are appropriate.

**Sixth Grade Standards**^{[6]}Just get in touch with me if you would like a copy of these in a more accessible format. Say, a spreadsheet.

Sixth grade covers decimal, fraction, and integer operations as well as geometry and measurement (both 2D and 3D). The course uses the Connected Mathematics Project 3 (CMP3) curriculum. Specifically, we use the *Let's Be Rational (Grade 6), Decimal Ops (Grade 6), Covering and Surrounding (Grade 6), Shapes and Designs (Grade 7), *and *Accentuate the Negative** (Grade 7)* modules to meet the content my school covers in grade 6. Here are my grade six standards:

ID |
Standard Content |
Standard Details |

R1 | Whole number computation | Add, subtract, multiply, and divide whole numbers |

R2 | Order rational numbers | Place fractions, decimals, and percents on a number line according to their value |

F1 | Add and subtract fractions | Add and subtract fractions with common denominators
Add and subtract fractions with different denominators Model addition and subtraction of fractions |

F2 | Multiply fractions | Multiply fractions
Model multiplication of fractions |

F3 | Divide fractions | Divide fractions and/or mixed numbers
Model division of fractions |

F4 | Solve story problems using fractions | Solve story problems that require a variety of fraction computations |

G1 | Area | Develop and apply formulas for finding the area triangles, rectangles, trapezoids, parallelograms, and irregular shapes |

G2 | Perimeter | Find the perimeter of 2D shapes by measuring
Find the perimeter of 2D shapes by reasoning about shape properties |

G3 | Circles | Develop a conceptual understand of π as a the relationship between a circle's circumference and diameter
Use radius, circumference, chord, diameter, segment, sector, arc, tangent, and secant to describe parts of a circle Develop formulas for the area and perimeter of a circle |

G4 | Angles | Measure angles using and angle ruler or similar tool
Classify the angles based on their size Identify and define complementary and supplementary angles |

G5 | Classify 2D and 3D shapes | Classify 2D shapes as regular or irregular
Classify 2D shapes as concave or convex Classify 3D shapes as prisms, pyramids, and spheres |

G6 | Volume | Find the volume of 3D shapes |

G7 | Surface Area | Find the surface area of 3D shapes
Construct nets for 3D shapes |

D1 | Add and subtract decimals | Add decimal numbers
Subtract decimal numbers Solve problems using addition and subtraction of decimal numbers |

D2 | Multiply decimals | Multiply decimal numbers
Solve problems using multiplication of decimals |

D3 | Divide decimals | Divide decimal numbers
Solve problems using division of decimals |

D4 | Convert between fractions, decimals, and percents | Generate equivalent forms of any fraction, decimal, or percent including negatives, improper fractions and mixed numbers
Use decimal long division to convert fractions into decimals |

D5 | Operations with percents | Find a percentage of a number
Find what percentage a number is of another number Solve problems using percents including percent discounts and percent change |

N1 | Add and subtract negative numbers | Add and subtract negative and positive numbers including fractions and decimals
Reason about adding and subtracting negative numbers with number line and chip models |

N2 | Multiply and divide negative numbers | Multiply and divide negative and positive numbers including fractions and decimals
Use patterns in the multiplication and division of positive numbers to reason about multiplying and dividing with negative numbers |

N3 | Arithmetic Properties | Define and provide examples of the associative, commutative, and distributive properties |

N4 | Order of Operations | Evaluate expressions while following order of operations |

The first thing to notice is that there are only 22 standards. Already this simplifies the process for students and parents. The downside, though, is that there are some standards that hold some pretty deep and rich ideas.

For example, "D4 - Convert between fractions, decimals, and percents" covers a lot of material. Some of this is review from their fifth grade year, and last year I had this set of content broken into at least five separate standards. I will have to be keep an eye on standards like this one to be sure that they are specific enough. If not, I can always rework them during the year after talking it over with students.

The second thing to notice is that I provided a very general overview of the standards (column 2) separate from a detailed description of what is covered (column 3). My hope is that students will have an easier time understanding what the standards cover by referencing column 2.

The CMP3 curriculum made writing these standards incredibly simple.^{[7]}Just on the incredibly unlikely chance that someone who helped author the CMP3 texts reads this, I want to say **"Thank you!"** The CMP3 teacher's guides are incredibly helpful. The simple but detailed analysis of the mathematics covered in each module is amazing. When teaching a new unit, I have two go-to references for the content: the CMP3 teacher's guide and *Math Matters.* I know some studies have shown that TG's aren't always used, but I DEFINITELY use these, even when I am not using the curriculum. The front of the Teacher's Guide has a list of the ideas covered in the text as well as the common core standards that are covered. I just combed through those and combined related topics into individual standards. It helps that I have taught a lot of this before (albeit when I was a young noob teacher).

**Pre-Algebra Standards**^{[8]}Just get in touch with me if you would like a copy of these in a more accessible format. Say, a spreadsheet.

Creating the pre-algebra standards for the seventh grade course was a much more involved process. The text for the course is Larson Pre-Algebra seen to the right.^{[9]}Not the Florida version shown here. We use the regular version which has a female soccer player on the cover.

As I have noted before, this text is about as traditional as they come. Each section introduces a super-specific skill, teaches an algorithm in one or two examples, then has students practice applying that algorithm multiple times. There is no big list of ideas or content to cover in the front of the text -- this may be because I only had access to the student edition -- so I had to pull the big content ideas from the text.

I started by making a list of each section in the book and what algorithm or content each section addressed. From there, I started combining related sections into groups. Related ideas were sometimes multiple chapters apart. For example, "solving one-step equations by multiplying both sides" was section 2.6. But, if the value you needed to multiply by was a fraction or decimal, you had to wait until chapter 5. It seems insane to me to separate those ideas at all, since they are not really different conceptually. So I combined them into a single standard.

After getting a rough list of standards from the text, I reorganized them and added detail about what meeting them would look like. In the end, I ended up with what I think is a pretty good list. Having never taught this course, though, I am sure that there will be some changes as the year progresses.^{[10]}The last four standards are topics that the course does not need to cover, but that ARE in the text. I added them there for reference, but I will likely not get to them.

ID |
Standard Content |
Standard Details |

1 | Properties and Identities of Addition and Multiplication | Define and provide examples of the associative, commutative, and distributive properties
Define and provide examples of the additive and multiplicative identities |

2 | Operations with integers | Add, subtract, multiply and divide positive and negative integers
Reason about adding and subtracting positive and negative numbers using a number line |

3 | Fraction operations | Add and subtract fractions and/or mixed numbers that have common or uncommon denominators
Multiply and divide fractions and/or mixed numbers |

4 | Algebra terminology | Define and identify the following terms: coefficients, terms, constants, variables, expression, and equation
Define: Solve, simplify, substitute, evaluate |

5 | Evaluate expressions | Evaluate expressions with exponents, absolute value, and square roots while following order of operations |

6 | Write algebraic expressions and equations | Write algebraic expressions and equations that describe a variety of situations using variables |

7 | Solving one-variable equations | Solve equations by operating on both sides of the equation
Solve equations with variables on both sides and by combining like terms |

8 | Exponents, Square Roots, and Scientific Notation | Evaluate integer exponents
Use exponent properties to simplify expressions Approximate the values of square roots Write numbers in scientific notation and use exponent properties to operate on numbers written in scientific notation |

9 | Ordering Real Numbers on a number line | Place any real number (integers, fractions, decimals, percents, and irrationals) on number line according to value |

10 | Simplify and operate on fractions with variables | Simplify fractions with variables
Add and subtract fractions with variables where finding common denominators is required |

11 | Proportions | Write proportions to describe situations
Solve proportions using properties of equality |

12 | Percents | Convert percents to and from fractions and decimals
Find percent change Use equations to solve problems involving percentages |

13 | Probability and combinatorics | Calculate probability using a fraction of form (favorable outcomes) / (all possible outcomes)
Calculate expected value of a situation |

14 | Graphing | Use equations, t-charts, and graphs to represent 2-variable equations and functions |

15 | Slope | Understand that slope is a proportional representation of a rate of change |

16 | Linear Equations | Identify linear equations in comparison to quadratics, exponential, and other equations
Write linear equations in slope-intercept form and identify the slope and y-intercept Graph linear equations using the y-intercept and slope Find the x-intercept of a linear equation using substitution |

17 | Functions | Represent functions using function notation
Understand functions as "rules" or "machines" that generate a single output for each unique input |

18 | Graphs of inequalities | Graph inequalities with appropriate shading and dashing of lines |

19 | Pythagorean formula and trig | Use the pythagorean theorem to find the missing sides of right triangles
Use the pythagorean theorem to find the distance between two points on a coordinate plane Identify when the pythagorean theorem applies and when it does not |

20 | Classifying shapes | Classify shapes based on the their properties (angles, sides, concavity) |

21 | 2D Measurement | Reason about the area and perimeter of two-dimensional shapes |

22 | 3D measurement | Reason about the volume and surface area of 3-dimensional shapes |

23 | Statistical displays | Represent data using a variety of statistical displays |

24 | Population sampling | Collect samples of a population
Identify common issues that can affect the accuracy of a data sample |

25 | Transformations | Dilations
Translations Rotations Reflections |

Since I will not be following the (illogical) progression of the text, I have officially decided that my pre-algebra course will not be organized into units.

At first I was pretty nervous about the idea of having no larger organized unit structure. When will I have tests? How will I pace myself? But then I remembered that I have been trying (for years) to have fewer and fewer big tests. A unit-less structure will support that as there won't be any predetermined time for a unit test.

Going unit-less is also more respectful of the nature of mathematics. If mathematics is a web of ideas that are connected in amazing ways, adding arbitrary units will only disguise this nature. Solving 1-step and 2-step equations should not be separated into different chapters or units. The ideas in pre-algebra build on each other, and I want my practice to reflect that.

**Closing Thoughts**

Creating lists of standards is exhausting. But, I cannot think of a better way to prepare to teach a new course. When I first got the news that I was teaching pre-algebra in July, I was pretty overwhelmed. I am comfortable with algebraic content, but I was not even sure what content was covered.

Taking the time to outline what ideas I want my students to master forced me to engage deeply with the course content. It helped me to see connections between ideas as I grouped algorithms, skills, and concepts into standards. And, it helped build my confidence as I prepared for the course.

With the start of the school year only a few days away, I am excited to help students, new and old, discover the beauty and interconnectedness of mathematics. I cannot wait!