# Mathematics

## What Do Students Learn in 6th Grade Math?

In sixth grade, students learn key concepts along the progression toward middle school algebra. Ratios and proportions emerges as a new domain of study, where students explore and reason with ratios and rates in order to solve problems. Sixth graders will also investigate negative numbers for the first time and round out their study of the rational number system before operating with all rational numbers in seventh grade. Work with numerical expressions extends to algebraic expressions, which sets students up to solve one-step equations and inequalities. Students will also continue their study of area and volume of geometric shapes, and will learn how statistics can be used to better understand data about our world.

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## Units

Unit 1

18 Lessons

### Understanding and Representing Ratios

Students are introduced to the concept of ratios, learning ratio language to describe the association between two or more quantities and different strategies to solve ratio problems.

Unit 2

14 Lessons

### Unit Rates and Percent

Students investigate rates and percentages by identifying the rates associated with a ratio, defining a percent as a rate per 100, and applying strategies to solve rate and percent problems.

Unit 3

17 Lessons

### Multi-Digit and Fraction Computation

Students extend their understanding of multiplication and division to divide fractions by fractions, and develop fluency with whole number and decimal operations.

Unit 4

13 Lessons

### Rational Numbers

Students are introduced to integers and rational numbers, extending the number line to include negative values, understanding the order of rational numbers, and interpreting them in context.

Unit 5

12 Lessons

### Numerical and Algebraic Expressions

Students venture into the Expressions and Equations domain, using variables to represent unknown or changing quantities, and using properties of operations to investigate equivalent expressions.

Unit 6

14 Lessons

### Equations and Inequalities

Students discover how to use equations and inequalities to model relationships between quantities, and investigate the meaning of having a solution to an equation or an inequality.

Unit 7

17 Lessons

### Geometry

Students explore measurements of geometric figures in two-and three-dimensions, finding area, surface area, and volume in mathematical and real-world problems.

Unit 8

14 Lessons

### Statistics

Students get their first experience of statistics in this unit, defining a statistical question and investigating the key concepts of measures of center and measures of variability.

## How Do Students Build 6th Grade Math Skills Across Units?

Sixth-grade students start their year with a unit on ratios. In Unit 1, Understanding and Representing Ratios, students have the opportunity to study a concept that is brand new to them, while leaning on reasoning skills around multiplicative comparisons learned in prior grade levels. Students learn both concrete and abstract representations, including double number lines and tables, which they will be able to use throughout the year.

In Unit 2, Unit Rates and Percent, students continue their study of ratios by extending the concept to rates and percentages. Students use the representations they learned in Unit 1 to reason through more complex ratio, rate, and percent problems. Later in Unit 6, students will revisit solving percent problems when they study solving equations.

In Unit 3, Multi-Digit and Fraction Computation, and Unit 4, Rational Numbers, students focus on the number system, honing skills they’ve been developing in previous grades with fluency, applying understandings to new computations with fractions, and expanding their understanding of the world of numbers to include negatives. Including these units at this point in the year offers opportunity to remediate any related previous grade-level skills and concepts early on while also allowing time for spiraling and integration of these skills into future units.

Unit 5, Numerical and Algebraic Expressions, and Unit 6, Equations and Inequalities, prepare students for future work with more complicated equations in seventh and eighth grades. Students lean on their work with the number system from Unit 3 to support their work with numerical expressions and solving equations. In Unit 6, students revisit ratio concepts from the first two units by representing relationships in the coordinate plane and with equations. Students also apply their equation skills to percent problems as another method to solve problems.

In Unit 7, Geometry, students learn how composing and decomposing unfamiliar shapes into familiar ones can extend their ability to find area and volume. Students draw on knowledge and skills from major work of the grade covered in previous units of the year in order to determine measurements, understand formulas, and represent 2-dimensional shapes in the coordinate plane. In Unit 8, Statistics, the last unit of the year, students are introduced to the study of statistics. They learn how to represent sets of data and how using different measurements about the data set can be used to analyze the information and answer the statistical question. By studying numbers in statistical contexts, students are able to expand and solidify their understanding of the number system.

Note that this course follows the 2017 Massachusetts Curriculum Frameworks, which include the Common Core Standards for Mathematics.

### What math skills and concepts do students learn in 6th grade?

In sixth grade, students set the foundations for middle school algebra as they use ratios and proportions to solve problems, extend the number system to include negative numbers, and extend their work with numerical expressions to include algebraic expressions.

### What Common Core Math Standards do students focus on in 6th grade?

The major work of the course focuses on the following Common Core State Standards:

• 6.RP.A Understand ratio concepts and use ratio reasoning to solve problems.
• 6.NS.A Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
• 6.NS.C Apply and extend previous understandings of numbers to the system of rational numbers.
• 6.EE.A Apply and extend previous understandings of arithmetic to algebraic expressions.
• 6.EE.B Reason about and solve one-variable equations and inequalities.
• 6.EE.C Represent and analyze quantitative relationships between dependent and independent variables.

### What are the fluency expectations and culminating standards for 6th grade math?

• 6.NS.2: Fluently divide multi-digit numbers using the standard algorithm.
• 6.NS.3: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
• 6.NS.1: Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?. (Culminating)

### How do students build upon their elementary math skills in 6th grade?

6th grade students build upon their elementary school math skills as they extend the number system to include negative numbers, apply new computations to fractions, extend their understandings of arithmetic to algebraic expressions, and extend their work with measuring geometric figures.

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