> y bjbj 7{{5LLLLL````dL`$8 \9LLLR"LLv3M!h0F!F!F!LTF!: Grade 6How will we assess this?Related textbook pages Related additional resources and activitiesRatios and Proportional Relationships (6.RP)Understand ratio concepts and use ratio reasoning to solve problems.1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, 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.
2. Understand 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. For 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.
3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
a. Make tables of equivalent ratios
relating quantities with whole number
measurements, find missing values in
the tables, and plot the pairs of values
on the coordinate plane. Use tables to
compare ratios.
b. Solve unit rate problems including
those involving unit pricing and
constant speed. For example, if it took 7
hours to mow 4 lawns, then at that
rate, how many lawns could be mowed
in 35 hours? At what rate were lawns
being mowed?
c. Find a percent of a quantity as a rate
per 100 (e.g., 30% of a quantity means
30/100 times the quantity); solve
problems involving finding the whole,
given a part and the percent.
d. Use ratio reasoning to convert
measurement units; manipulate and
transform units appropriately when
multiplying or dividing quantities.The Number System (6.NS) Apply and extend previous understandings of multiplication and division to divide fractions by fractions.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?Compute fluently with multi-digit numbers and find common factors and multiples.
2. Fluently divide multi-digit numbers using the standard algorithm.
3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).Apply and extend previous understandings of numbers to the system of rational numbers.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.
6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
a. Recognize opposite signs of numbers as
indicating locations on opposite sides of
0 on the number line; recognize that
the opposite of the opposite of a
number is the number itself, e.g., (3)
= 3, and that 0 is its own opposite.
b. Understand signs of numbers in
ordered pairs as indicating locations in
quadrants of the coordinate plane;
recognize that when two ordered pairs
differ only by signs, the locations of the
points are related by reflections across
one or both axes.
c. Find and position integers and other
rational numbers on a horizontal or
vertical number line diagram; find and
position pairs of integers and other
rational numbers on a coordinate
plane.
7. Understand ordering and absolute value of rational numbers.
a. Interpret statements of inequality as
statements about the relative position
of two numbers on a number line
diagram. For example, interpret 3 > 7
as a statement that 3 is located to the
right of 7 on a number line oriented
from left to right.
b. Write, interpret, and explain
statements of order for rational
numbers in real-world contexts. For
example, write 3 C > 7 C to express
the fact that 3 C is warmer than 7C.
c. Understand the absolute value of a
rational number as its distance from 0
on the number line; interpret absolute
value as magnitude for a positive or
negative quantity in a real-world
situation. For example, for an account
balance of 30 dollars, write |30| = 30
to describe the size of the debt in
dollars.
d. Distinguish comparisons of absolute
value from statements about order. For
example, recognize that an account
balance less than 30 dollars represents
a debt greater than 30 dollars.
8. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.Expressions and Equations (6.EE) Apply and extend previous understandings of arithmetic to algebraic expressions.
1. Write and evaluate numerical expressions involving whole-number exponents.
2. Write, read, and evaluate expressions in which letters stand for numbers.
a. Write expressions that record
operations with numbers and with
letters standing for numbers. For
example, express the calculation
Subtract y from 5 as 5 y.
b. Identify parts of an expression using
mathematical terms (sum, term,
product, factor, quotient, coefficient);
view one or more parts of an
expression as a single entity. For
example, describe the expression 2 (8 +
7) as a product of two factors; view (8 +
7) as both a single entity and a sum of
two terms.
c. Evaluate expressions at specific values
of their variables. Include expressions
that arise from formulas used in real-
world problems. Perform 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). For
example, use the formulas V = s3 and A
= 6 s2 to find the volume and surface
area of a cube with sides of length s =
1/2.
3. Apply the properties of operations to
generate equivalent expressions. For
example, apply the distributive property
to the expression 3 (2 + x) to produce the
equivalent expression 6 + 3x; apply the
distributive property to the expression
24x + 18y to produce the equivalent
expression 6 (4x + 3y); apply properties of
operations to y + y + y to produce the
equivalent expression 3y.
4. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y
are equivalent because they name the same number regardless of which number y stands for.Reason about and solve one-variable equations and inequalities.5. Understand 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? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6. Use 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.
7. Solve 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.
8. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions;
represent solutions of such inequalities on number line diagrams.Represent and analyze quantitative relationships between dependent and independent variables.9. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
Geometry (6.G)Solve real-world and mathematical problems involving area, surface area, and volume.1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
2. Find 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. Apply 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.
3. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
4. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
Statistics and Probability (6.SP)Develop understanding of statistical variability.1. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, How old am I? is not a statistical question, but How old are the students in my school? is a statistical question because one anticipates variability in students ages.
2. Understand 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.
3. Recognize 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.
Summarize and describe distributions.
4. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
5. Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations.
b. Describing the nature of the attribute
under investigation, including how it
was measured and its units of
measurement.
c. 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 from the overall
pattern with reference to the context
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