6th Grade Math: Number Operations

• Mathematics in Grade 6 covers a range of topics that build upon foundational concepts and prepare students for more advanced mathematics.
• Key areas include number operations, ratios and proportional relationships, algebraic expressions, geometry, and statistics and probability.

Number Operations

• Focuses on fluently dividing multi-digit numbers and performing all operations with multi-digit decimals.
• Division problems may include dividing by numbers with up to two digits.
• Decimal operations include addition, subtraction, multiplication, and division to the hundredths place.
• Emphasizes the application of these skills in real-world problem-solving contexts.
• Utilizes the standard algorithm for division and operations with decimals.
• Understanding the concept of factors and multiples is crucial.
• Identifying prime and composite numbers is important.
• Finding the greatest common factor (GCF) and least common multiple (LCM) of two whole numbers is a key skill.
• The distributive property may be used to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor.
• Example: 36 + 8 = 4(9 + 2)

Ratios and Proportional Relationships

• Understanding the concept of a ratio and using ratio language to describe a ratio relationship between two quantities is vital.
• Examples: "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."
• Understanding the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship is neccesary.
• Example: "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."
• Using ratio and rate reasoning to solve real-world and mathematical problems is essential.
• Using tables to compare ratios is a common strategy.
• Solving unit rate problems, including those involving unit pricing and constant speed is required.
• Finding percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity) is an important application.
• Using ratio reasoning to convert measurement units; manipulating and transforming units appropriately when multiplying or dividing quantities is necessary.

Algebraic Expressions

• Writing and evaluating numerical expressions involving whole-number exponents is needed.
• Writing, reading, and evaluating expressions in which letters stand for numbers is essential.
• Evaluating expressions at specific values of their variables is a key skill; includes expressions that arise from formulas used in real-world problems.
• Performing arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).
• Applying the properties of operations to generate equivalent expressions is important.
• Identifying when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them) is needed.
• Example: The expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.

Equations and Inequalities

• Understanding solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?
• Using substitution to determine whether a given number in a specified set makes an equation or inequality true is necessary.
• Using variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set is needed.
• Solving real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers is required.
• Writing an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem is needed.
• Recognizing that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams is important.

Geometry

• Finding the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; applying these techniques in the context of solving real-world and mathematical problems is required.
• Finding the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism is necessary.
• Applying the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems is needed .
• Drawing polygons in the coordinate plane given coordinates for the vertices; using coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate is required.
• Applying these techniques in the context of solving real-world and mathematical problems is important.
• Representing three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures is needed.
• Applying these techniques in the context of solving real-world and mathematical problems is essential.

Statistics and Probability

• Recognizing a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers is required.
• Understanding that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape is needed.
• Recognizing that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number is important.
• Displaying numerical data in plots on a number line, including dot plots, histograms, and box plots is necessary.
• Summarizing numerical data sets in relation to their context, such as:
• Reporting the number of observations.
• Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
• Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations (outliers) from the overall pattern with reference to the context in which the data were gathered.
• Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.