Download Q2 - FCCSC

Franklin County Community School Corporation - Brookville, Indiana
Curriculum Map
Course Title: 6th Math
Quarter: 2
Academic Year: 2011-2012
Essential Questions for this Quarter:
1.
2.
3.
4.
5.
6.
How do we use algebra?
What is the difference between expressions and equations?
How do you solve one-step equations and equalities?
What is a linear function and how do you graph it?
What is the relationship between ratios and fractions?
What are ratios and proportions and why do we use them?
Unit/Time Frame
Expressions and
Equations
5-1 Write &
Evaluate
Expressions
5-2 Properties
6-1 Add & Subt.
Equations
6-2 Mult. & Divide
Equations
6-3 Two-Step
Equations
7-1 Relations &
Functions
7-2 Inequalities
7-3 Integers &
Coordinate Plane
Standards
6.3.1a
6.3.1b
6.3.1c
6.3.1d
6.3.2a
6.3.2b
6.3.3a
6.3.4a
6.3.4b
6.3.4c
6.3.5a
6.3.6a
(and 6.3.3)
6.3.6b
(and 6.3.3)
6.3.6c
6.3.7a
6.3.7b
6.3.7c
6.3.7d
6.3.8a
6.3.8b
Content
Numerical Expression
Order of Operations
Variables
Algebraic expressions
Commutative Properties
Associative Properties
Identity Properties
Distributive Property
Equations
Linear function/equation
Inequalities
Functions
Coordinate Plane
Skills
…Translate a verbal situation into a
one-step equation with one variable.
…Solve one-step linear equations
with one variable algebraically.
…Translate a verbal situation into a
one-step inequality with one variable.
Assessment
Textbook
assignments
Worksheet
assignments
Quizzes
...Solve one-step inequalities with
one variable algebraically.
Tests
…Write an equation based on a
formula to solve a problem situation.
Oral Responses
…Solve an equation (based on a
formula) for an unknown quantity.
Observations
…Interpret mathematical expression
that use grouping symbols (e.g.,
parentheses).
…Place parentheses in a given
expression to equal a particular
value.
…Create expressions placing
parentheses to indicate which
operation to perform first.
1
Resources
Textbook: Glencoe
McGraw-Hill
Math Connects
Course 1
(2012 edition)
ConnectED.mcgrawhill.com
Myskillstutor.com
Mathnook.com
Teachers Helper
Skills Tutor
Saxon Math 7/6
Acuity
Mailbox
Star Math
Hands on Equations
Placement Test
Edhelper.com
ISTEP
Franklin County Community School Corporation - Brookville, Indiana
Curriculum Map
Course Title: 6th Math
Quarter: 2
Academic Year: 2011-2012
Essential Questions for this Quarter:
1.
2.
3.
4.
5.
6.
How do we use algebra?
What is the difference between expressions and equations?
How do you solve one-step equations and equalities?
What is a linear function and how do you graph it?
What is the relationship between ratios and fractions?
What are ratios and proportions and why do we use them?
Unit/Time Frame
Standards
6.3.8c
6.3.9a
6.5.8a
6.5.8b
6.5.8f
Content
Skills
…Write an expression from a
situation and place parentheses to
indicate which operation to perform
first.
…Describe geometric quantities
using variable expressions. Ex: Given
the length of the side of a square (x),
write an expression to represent the
perimeter. Given a rectangle with a
length of 5 and a width of y, write an
expression to represent the area of the
rectangle. Write an expression to
represent the perimeter of the rectangle.
…State or write the purpose and
process of the order of operations.
…Evaluate expressions by applying
the correct order of operations.
…Generate equivalent expressions
by applying the properties of
operations.
…Define quadrants and label on the
coordinate plane.
2
Assessment
Resources
Glencoe
Mathematics Study
Guide & Practice
Workbook (Previous
textbook adoption)
Brain Pop
Acuityathome.com
Scholastic Math
Daily Math Practice
Grades 6+
Franklin County Community School Corporation - Brookville, Indiana
Curriculum Map
Course Title: 6th Math
Quarter: 2
Academic Year: 2011-2012
Essential Questions for this Quarter:
1.
2.
3.
4.
5.
6.
How do we use algebra?
What is the difference between expressions and equations?
How do you solve one-step equations and equalities?
What is a linear function and how do you graph it?
What is the relationship between ratios and fractions?
What are ratios and proportions and why do we use them?
Unit/Time Frame
Standards
Content
Skills
…Identify which quadrant an ordered
pair will be graphed based on the
(positive or negative) sign of its
coordinates.
…Write the ordered pair for a given
point in a coordinate plane.
…Graph ordered pairs in the four
quadrants of the coordinate plane.
…Define linear functions.
…Write solutions of linear functions
with integer values as ordered pairs
with the use of function tables.
…Write a linear equation from a
situation and graph the resulting
ordered pairs of integers on a
coordinate plane.
…Investigate and state observations
of how a change in one variable of a
linear function relates to a change in
the second variable.
3
Assessment
Resources
Franklin County Community School Corporation - Brookville, Indiana
Curriculum Map
Course Title: 6th Math
Quarter: 2
Academic Year: 2011-2012
Essential Questions for this Quarter:
1.
2.
3.
4.
5.
6.
How do we use algebra?
What is the difference between expressions and equations?
How do you solve one-step equations and equalities?
What is a linear function and how do you graph it?
What is the relationship between ratios and fractions?
What are ratios and proportions and why do we use them?
Unit/Time Frame
Standards
Content
Skills
…Compute the surface area of right
prisms using appropriate units.
…Apply these techniques in the
context of solving real-world and
mathematical problems. Note: Ensure
that the dimensions of the prisms
include fractional lengths.
…Compute the volume of right
prisms using appropriate units. Apply
these techniques in the context of
solving real-world and mathematical
problems.
…Compute the volume of a cylinder
using appropriate units. Apply these
techniques in the context of solving
real-world and mathematical
problems.
4
Assessment
Resources
Franklin County Community School Corporation - Brookville, Indiana
Curriculum Map
Course Title: 6th Math
Quarter: 2
Academic Year: 2011-2012
Essential Questions for this Quarter:
1.
2.
3.
4.
5.
6.
How do we use algebra?
What is the difference between expressions and equations?
How do you solve one-step equations and equalities?
What is a linear function and how do you graph it?
What is the relationship between ratios and fractions?
What are ratios and proportions and why do we use them?
Unit/Time Frame
Ratios and
Proportional
Relationships
3-1 Ratios & Rates
3-2 Ratio Tables
3-3 Solve Ratio &
Rate Problems
Standards
6.1.6a
6.2.6a
6.2.6b
CC.6.RP.2
6.2.7a
6.2.7b
6.2.7c
CC.6.RP.3b
CC.6.RP.3d
Content
Ratio
Equivalent ratios
Proportion
Skills
Assessment
…Model ratios using objects,
drawings, or other concrete or
pictorial representations.
Textbook
assignments
…Define ratio using the notations:
a/b, a to b, a:b.
Worksheet
assignments
…Interpret ratios to represent the
relative size of two quantities. Note:
Quizzes
Include situations involving part to part
(girls to boys) and part to whole (girls to
total students). Connect part to whole
situations to prior work with fractions.
Tests
…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
Observations
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." Note: this
is an extension of IAS 6.2.6.
Acuity
…Define proportion.
ISTEP
Description: Connect proportions to
prior work with equivalent fractions.
5
Resources
Textbook: Glencoe
McGraw-Hill
Math Connects
Course 1
(2012 edition)
ConnectED.mcgrawhill.com
Myskillstutor.com
Oral Responses
Mathnook.com
Teachers Helper
Skills Tutor
Saxon Math 7/6
Mailbox
Star Math
Hands on Equations
Placement Test
Edhelper.com
Franklin County Community School Corporation - Brookville, Indiana
Curriculum Map
Course Title: 6th Math
Quarter: 2
Academic Year: 2011-2012
Essential Questions for this Quarter:
1.
2.
3.
4.
5.
6.
How do we use algebra?
What is the difference between expressions and equations?
How do you solve one-step equations and equalities?
What is a linear function and how do you graph it?
What is the relationship between ratios and fractions?
What are ratios and proportions and why do we use them?
Unit/Time Frame
Standards
Content
Skills
…Write proportions based on
problem situations, using a variable
as the unknown.
…Solve problems involving
proportions.
Assessment
Resources
Glencoe
Mathematics Study
Guide & Practice
Workbook (Previous
textbook adoption)
Description: Solve proportions mentally
(connecting to the concept of equivalent
fractions) as well as solving for the
unknown using an equation.
Brain Pop
…Solve unit rate problems including
those involving unit pricing and
constant speed. For example, if it took
Scholastic Math
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?
…Use ratio reasoning to convert
measurement units; manipulate and
transform units appropriately when
multiplying or dividing quantities.
6
Acuityathome.com
Daily Math Practice
Grades 6+
Franklin County Community School Corporation - Brookville, Indiana
Curriculum Map
Course Title: 6th Math
Quarter: 2
Academic Year: 2011-2012
Essential Questions for this Quarter:
1.
2.
3.
4.
5.
6.
How do we use algebra?
What is the difference between expressions and equations?
How do you solve one-step equations and equalities?
What is a linear function and how do you graph it?
What is the relationship between ratios and fractions?
What are ratios and proportions and why do we use them?
Unit/Time Frame
Standards for
Mathematical
Practice
Standards
6.7.1a
6.7.2a
6.7.3a
6.7.4a
6.7.5a
6.7.6a
6.7.7a
6.7.8a
6.7.9a
6.7.10a
6.7.11a
SMP1.
SMP2.
SMP3.
SMP4.
SMP5.
SMP6.
SMP7.
SMP8.
Content
Skills
Assessment
…Analyze problems by identifying
relationships, telling relevant from
irrelevant information, identifying
missing information, sequencing and
prioritizing information, and observing
patterns.
Textbook
assignments
…Make and justify mathematical
conjectures based on a general
description of a mathematical
question or problem.
Quizzes
…Decide when and how to divide a
problem into simpler parts.
…Apply strategies and results from
simpler problems to solve more
complex problems.
…Express solutions clearly and
logically by using the appropriate
mathematical terms and notation.
Support solutions with evidence in
both verbal and symbolic work.
…Recognize the relative advantages
7
Worksheet
assignments
Resources
Textbook: Glencoe
McGraw-Hill
Math Connects
Course 1
(2012 edition)
ConnectED.mcgrawhill.com
Tests
Myskillstutor.com
Oral Responses
Mathnook.com
Observations
Teachers Helper
Skills Tutor
Saxon Math 7/6
Acuity
Mailbox
Star Math
Hands on Equations
Placement Test
Edhelper.com
ISTEP
Franklin County Community School Corporation - Brookville, Indiana
Curriculum Map
Course Title: 6th Math
Quarter: 2
Academic Year: 2011-2012
Essential Questions for this Quarter:
1.
2.
3.
4.
5.
6.
How do we use algebra?
What is the difference between expressions and equations?
How do you solve one-step equations and equalities?
What is a linear function and how do you graph it?
What is the relationship between ratios and fractions?
What are ratios and proportions and why do we use them?
Unit/Time Frame
Standards
Content
Skills
of exact and approximate solutions to
problems and give answers to a
specified degree of accuracy.
…Select and apply appropriate
methods for estimating results of
rational-number computations.
…Use graphing to estimate solutions
and check the estimates with analytic
approaches.
…Make precise calculations and
check the validity of the results in the
context of the problem.
…Explain whether a solution is
reasonable in the context of the
original situation.
…Note the method of finding the
solution and show a conceptual
understanding of the method by
solving similar problems.
8
Assessment
Resources
Glencoe
Mathematics Study
Guide & Practice
Workbook (Previous
textbook adoption)
Brain Pop
Acuityathome.com
Scholastic Math
Daily Math Practice
Grades 6+
Franklin County Community School Corporation - Brookville, Indiana
COMMON CORE AND INDIANA ACADEMIC STANDARDS
INDIANA ACADEMIC STANDARDS
Standard 1… Number Sense
Students compare and order positive and negative integers*, decimals, fractions, and mixed numbers. They find multiples* and factors*.
6.1.1
6.1.2
6.1.3
Understand and apply the basic concept of negative numbers (e.g., on a number line, in counting, in temperature, in “owing”).
Example: The temperature this morning was -6º and now it is 3º. How much has the temperature risen? Explain your answer.
Interpret the absolute value of a number as the distance from zero on a number line and find the absolute value of real numbers.
Example: Use a number line to explain the absolute values of -3 and of 7.
Compare and represent on a number line positive and negative integers, fractions, decimals (to hundredths), and mixed numbers.
Example: Find the positions on a number line of 3.56, -2.5, 1 5 6 , and -4.
6.1.4
Convert between any two representations of numbers (fractions, decimals, and percents) without the use of a calculator.
Example: Write 5 8 as a decimal and as a percent.
6.1.5
Recognize decimal equivalents for commonly used fractions without the use of a calculator.
Example: Know that 1 3 = 0.333… , 1 2 = 0.5, 2 5 = 0.4, etc.
6.1.6
Use models to represent ratios.
Example: Divide 27 pencils to represent the ratio 4:5.
Find the least common multiple* and the greatest common factor* of whole numbers. Use them to solve problems with fractions (e.g.,
to find a common denominator to add two fractions or to find the reduced form for a fraction).
Example: Find the smallest number that both 12 and 18 divide into. How does this help you add the fractions 512 and 718 ?
6.1.7
*
*
*
*
*
positive and negative integers: …, -3, -2, -1, 0, 1, 2, 3, …
multiples: e.g., multiples of 7 are 7, 14, 21, 28, etc.
factors: e.g., factors of 12 are 1, 2, 3, 4, 6, 12
least common multiple: e.g., the least common multiple of 4 and 6 is 12
greatest common factor: e.g., the greatest common factor of 18 and 42 is 6

9

Franklin County Community School Corporation - Brookville, Indiana
COMMON CORE AND INDIANA ACADEMIC STANDARDS
Standard 2… Computation
Students solve problems involving addition, subtraction, multiplication, and division of integers. They solve problems involving fractions, decimals,
ratios, proportions, and percentages.
6.2.1
6.2.2
6.2.3
6.2.4
6.2.5
6.2.6
6.2.7
6.2.8
6.2.9
6.2.10


Add and subtract positive and negative integers.
Example: 17 + -4 = ?, -8 – 5 = ?.
Multiply and divide positive and negative integers.
Example: Continue the pattern: 3  2 = ?, 2  2 = ?, 1  2 = ?, 0  2 = ?, -1  2 = ?,
-2  2 = ?, etc.
Multiply and divide decimals.
Example: 3.265  0.96 = ?, 56.79  2.4 = ?.
Explain how to multiply and divide positive fractions and perform the calculations.
Example: Explain why 5 8  1516 = 5 8  1615 = 2 3 .
Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation
was used for a given situation.
Example: You want to place a towel bar 9 3 4 inches long in the center of a door 27 1 2 inches wide. How far from each edge should you
place the bar? Explain your method.
Interpret and use ratios to show the relative sizes of two quantities. Use the notations:
a/b, a to b, a:b.
Example: A car moving at a constant speed travels 130 miles in 2 hours. Write the ratio of distance to time and use it to find how far the
car will travel in 5 hours.
Understand proportions and use them to solve problems.
Example: Sam made 8 out of 24 free throws. Use a proportion to show how many free throws Sam would probably make out of 60
attempts.
Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips.
Example: In a sale, everything is reduced by 20%. Find the sale price of a shirt whose pre-sale price was $30.
Use estimation to decide whether answers are reasonable in decimal problems.
Example: Your friend says that 56.79  2.4 = 2.36625. Without solving, explain why you think the answer is wrong.
Use mental arithmetic to add or subtract simple fractions and decimals.
Example: Subtract 1 6 from 1 2 without using pencil and paper.

10

Franklin County Community School Corporation - Brookville, Indiana
COMMON CORE AND INDIANA ACADEMIC STANDARDS
Standard 3… Algebra and Functions
Students write verbal expressions and sentences as algebraic expressions and equations. They evaluate algebraic expressions, solve simple linear
equations, and graph and interpret their results. They investigate geometric relationships and describe them algebraically.
6.3.1
Write and solve one-step linear equations and inequalities in one variable and check the answers.
Example: The area of a rectangle is 143 cm2 and the length is 11 cm. Write an equation to find the width of the rectangle and use it to solve the
problem. Describe how you will check to be sure that your answer is correct.
6.3.2
Write and use formulas with up to three variables to solve problems.
Example: You have P dollars in a bank that gives r% simple interest per year. Write a formula for the amount of interest you will receive in one
year. Use the formula to find the amount of interest on $80 at 6% per year for one year.
6.3.3
6.3.4
6.3.5
6.3.6
6.3.7
Interpret and evaluate expressions that use grouping symbols such as parentheses.
Example: Find the values of 10 – (7 – 3) and of 2(10 – 7)(3 + 1).
Use parentheses to indicate which operation to perform first when writing expressions containing more than two terms and different
operations. Example: Write in symbols: add 19 and 34 and double the result.
Use variables in expressions describing geometric quantities.
Example: Let l, w, and P be the length, width, and perimeter of a rectangle. Write a formula for the perimeter in terms of the length and width.
Apply the correct order of operations and the properties of real numbers (e.g., identity, inverse, commutative*, associative*, and
distributive* properties) to evaluate numerical expressions. Justify each step in the process. Example: Simplify 3(4 – 1) + 2. Explain your method.
Identify and graph ordered pairs in the four quadrants of the coordinate plane. Example: Plot the points (3, -1), (-6, 2) and (9, -3). What do
you notice?
6.3.8
Solve problems involving linear functions with integer* values. Write the equation & graph the resulting ordered pairs of integers on a
grid. Example: A plant is 3 cm high the first time you measure it (on Day 0). Each day after that the plant grows by 2 cm. Write an equation
connecting the height and the number of the day and draw its graph.
6.3.9
Investigate how a change in one variable relates to a change in a second variable. Example: In the last example, what do you notice
about the shape of the graph?
*
*
commutative property: the order when adding or multiplying numbers makes no difference
(e.g., 5 + 3 = 3 + 5), but note that this is not true for subtraction or division
associative property: the grouping when adding or multiplying numbers makes no difference
(e.g., in 5 + 3 + 2, adding 5 and 3 and then adding 2 is the same as 5 added to 3 + 2), but note
that this is not true for subtraction or division
*
*
distributive property: e.g., 3(5 + 2) = (3  5) r (3  2)
integers: …, -3, -2, -1, 0, 1, 2, 3, …
11
Franklin County Community School Corporation - Brookville, Indiana
COMMON CORE AND INDIANA ACADEMIC STANDARDS
Standard 4… Geometry
Students identify, describe, and classify the properties of plane and solid geometric shapes and the relationships between them.
6.4.1
6.4.2
6.4.3
6.4.4
6.4.5
6.4.6
6.4.7
Identify and draw vertical*, adjacent*, complementary*, and supplementary* angles and describe these angle relationships.
Example: Draw two parallel lines with another line across them. Identify all pairs of supplementary angles.
Use the properties of complementary, supplementary, and vertical angles to solve problems involving an unknown angle. Justify
solutions.
Example: Find the size of the supplement to an angle that measures 122º. Explain how you obtain your answer.
Draw quadrilaterals* and triangles from given information about them.
Example: Draw a quadrilateral with equal sides but no right angles.
Understand that the sum of the interior angles of any triangle is 180º and that the sum of the interior angles of any quadrilateral is 360º.
Use this information to solve problems.
Example: Find the size of the third angle of a triangle with angles of 73º and 49º.
Identify and draw two-dimensional shapes that are similar*.
Example: Draw a rectangle similar to a given rectangle, but twice the size.
Draw the translation (slide) and reflection (flip) of shapes.
Example: Draw a square and then slide it 3 inches horizontally across your page. Draw the new square in a different color.
Visualize and draw two-dimensional views of three-dimensional objects made from rectangular solids.
Example: Draw a picture of an arrangement of rectangular blocks from the top, front, and right-hand side.
*
*
*
*
*
vertical angles: angles 1 and 3 or 2 and 4
adjacent angles: angles 1 and 2 or 2 and 3, etc.
complementary angles: two angles whose sum is 90º
supplementary angles: two angles whose sum is 180º (angles 1 and 2)
quadrilateral: a two-dimensional figure with four sides
*
similar: the term to describe figures that have
the same shape but may not have the same size
12
1
2
4
3
Franklin County Community School Corporation - Brookville, Indiana
COMMON CORE AND INDIANA ACADEMIC STANDARDS
Standard 5… Measurement
Students deepen their understanding of the measurement of plane and solid shapes and use this understanding to solve problems. They calculate
with temperature and money, and choose appropriate units of measure in other areas.
6.5.1
6.5.2
6.5.3
6.5.4
6.5.5
6.5.6
6.5.7
6.5.8
6.5.9
6.5.10
Select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles.
Example: A triangular sheet of metal is about 1 foot across. Describe the units and tools you would use to measure its weight, its angles,
and the lengths of its sides.
Understand and use larger units for measuring length by comparing miles to yards and kilometers to meters.
Example: How many meters are in a kilometer?
Understand and use larger units for measuring area by comparing acres and square miles to square yards and square kilometers to square
meters.
Example: How many square meters are in a square kilometer?
Understand the concept of the constant π as the ratio of the circumference to the diameter of a circle. Develop and use the formulas for
the circumference and area of a circle.
Example: Measure the diameter and circumference of several circular objects. (Use string to find the circumference.) With a calculator,
divide each circumference by its diameter. What do you notice about the results?
Know common estimates of π (3.14, 227 ) and use these values to estimate and calculate the circumference and the area of circles.
Compare with actual measurements.
Example: Find the area of a circle of radius 15 cm.
Understand the concept of significant figures and round answers to an appropriate number of significant figures.
Example: You measure the diameter of a circle as 2.47 m and use the approximation 3.14 for π to calculate the circumference. Is it
reasonable to give 7.7558 m as your answer? Why or why not?
Construct a cube and rectangular box from two-dimensional patterns and use these patterns to compute the surface area of these objects.
Example: Find the total surface area of a shoe box with length 30 cm, width 15 cm, and height 10 cm.
Use strategies to find the surface area and volume of right prisms* and cylinders using appropriate units.
Example: Find the volume of a cylindrical can 15 cm high and with a diameter of 8 cm.
Use a formula to convert temperatures between Celsius and Fahrenheit.
Example: What is the Celsius equivalent of 100ºF? Explain your method.
Add, subtract, multiply, and divide with money in decimal notation.
Example: Share $7.25 among five people.
*
right prism: a three-dimensional shape with two congruent
ends that are polygons and all other faces are rectangles
13
Franklin County Community School Corporation - Brookville, Indiana
COMMON CORE AND INDIANA ACADEMIC STANDARDS
Standard 6… Data Analysis and Probability
Students compute and analyze statistical measures for data sets. They determine theoretical and experimental probabilities and use them to make
predictions about events.
6.6.1
Organize and display single-variable data in appropriate graphs and stem-and-leaf plots*,
and explain which types of graphs are appropriate for various data sets.
Example: This stem-and-leaf diagram shows a set of test scores for your class:
Stem
6
7
8
9
6.6.2
6.6.3
Leaf
2 3 7
1 5 5 6 8 9
0 1 1 2 3 3 5 7 8 8
1 2 2 3 3 4
Find your score of 85 in this diagram. Are you closer to the top or the bottom of the class
on this test?
Make frequency tables for numerical data, grouping the data in different ways to investigate how different groupings describe the data.
Understand and find relative and cumulative frequency for a data set. Use histograms of the data and of the relative frequency
distribution, and a broken line graph for cumulative frequency, to interpret the data.
Example: A bag contains pens in three colors. Nine students each draw a pen from the bag without looking, then record the results in the
frequency table shown. Complete the column showing relative frequency.
Color
Frequency
Red
2
Green
4
Purple
3
Relative
Frequency
2
9
Compare the mean*, median*, and mode* for a set of data and explain which measure is most appropriate in a given context.
Example: Twenty students were given a science test and the mean, median and mode were
as follows:
mean = 8.5, median = 9, mode = 10.
What does the difference between the mean and the mode suggest about the twenty quiz scores?
14
Franklin County Community School Corporation - Brookville, Indiana
COMMON CORE AND INDIANA ACADEMIC STANDARDS
6.6.4
6.6.5
6.6.6
Show all possible outcomes for compound events in an organized way and find the theoretical probability of each outcome.
Example: A box contains four cards with the numbers 1 through 4 written on them. Show a list of all the possible outcomes if you draw
two cards from the box without looking. What is the theoretical probability that you will draw the numbers one and two? Explain your
answer.
Use data to estimate the probability of future events.
Example: Teams A and B have played each other 3 times this season and Team A has won twice. When they play again, what is the
probability of Team B winning? How accurate do you think this estimate is?
Understand and represent probabilities as ratios, measures of relative frequency, decimals between 0 and 1, and percentages between 0
and 100 and verify that the probabilities computed are reasonable.
Example: The weather forecast says that the chance of rain today is 30%. Should you carry an umbrella? Explain your answer.
*
*
*
*
stem-and-leaf plot: the example under 6.6.1 shows 62, 63, 67, 71, 75, 75, 76, etc.
mean: the average obtained by adding the values and dividing by the number of values
median: the value that divides a set of data, written in order of size, into two equal parts
mode: the most common value in a given data set
Standard 7… Problem Solving
Students make decisions about how to approach problems and communicate their ideas.
6.7.1
6.7.2
6.7.3
Analyze problems by identifying relationships, telling relevant from irrelevant information, identifying missing information, sequencing
and prioritizing information, and observing patterns.
Example: Solve the problem: “Develop a method for finding all the prime numbers up to 100.” Notice that any numbers that 4, 6, 8, …
divide into also divide exactly by 2, and so you do not need to test 4, 6, 8, … .
Make and justify mathematical conjectures based on a general description of a mathematical question or problem.
Example: In the first example, decide that you need to test only the prime numbers as divisors, and explain it in the same way as for 4, 6,
8, … .
Decide when and how to break a problem into simpler parts.
Example: In the first example, decide to find first those numbers not divisible by 2.
Students use strategies, skills, and concepts in finding and communicating solutions to problems.
6.7.4
Apply strategies and results from simpler problems to solve more complex problems.
Example: In the first example, begin by finding all the prime numbers up to 10.
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6.7.5
6.7.6
6.7.7
6.7.8
6.7.9
Express solutions clearly and logically by using the appropriate mathematical terms and notation. Support solutions with evidence in
both verbal and symbolic work.
Example: In the first example, use a hundreds chart to cross off all multiples of 2 (except 2), then all multiples of 3 (except 3), then all
multiples of 5 (except 5), etc. Explain why you are doing this.
Recognize the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy.
Example: Calculate the perimeter of a rectangular field that needs to be fenced. How accurate should you be: to the nearest kilometer,
meter, centimeter, or millimeter? Explain your answer.
Select and apply appropriate methods for estimating results of rational-number computations.
Example: Measure the length and height of the walls of a room to find the total area. Estimate an answer by imagining meter squares
covering the walls.
Use graphing to estimate solutions and check the estimates with analytic approaches.
Example: Use a graphing calculator to estimate the coordinates of the point where the straight line y = 8x – 3 crosses the x-axis. Confirm
your answer by checking it in the equation.
Make precise calculations and check the validity of the results in the context of the problem.
Example: In the first example, check whether some of the numbers not crossed out are in fact primes.
Students determine when a solution is complete and reasonable and move beyond a particular problem by generalizing to other situations.
6.7.10
6.7.11
Decide whether a solution is reasonable in the context of the original situation.
Example: In the first example, decide whether your method was a good one — did it find all the prime numbers efficiently?
Note the method of finding the solution and show a conceptual understanding of the method by solving similar problems.
Example: Use a hundreds chart to find all the numbers that are multiples of both 2 and 3.
Ratios and Proportional Relationships RP
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Common Core Standards
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.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.”
6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b is not equal to 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.”
6.RP.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.
1 Expectations for unit rates in this grade are limited to non-complex fractions.
The Number System NS
Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
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 ¾ 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.
Compute fluently with multi-digit numbers and find common factors and multiples.
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.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 1-100 with a common factor as a multiple of a sum
of two whole numbers with no common factor.
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Apply and extend previous understandings of numbers to the system of rational numbers.
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.
6.NS.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.
6.NS.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 oC > –7 oC to
express the fact that –3 oC is warmer than –7 oC.
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.
6.NS.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 EE
Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.
6.EE.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.
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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.
6.EE.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.
6.EE.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.
Reason about and solve one-variable equations and inequalities.
6.EE.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.EE.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.
6.EE.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.
6.EE.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.
6.EE.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 G
Solve real-world and mathematical problems involving area, surface area, and volume.
6.G.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.
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6.G.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.
6.G.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.
6.G.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 SP
Develop understanding of statistical variability.
6.SP.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.
6.SP.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.
6.SP.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.
6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
6.SP.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 in which the data
were gathered.
d. 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.
Standards for Mathematical Practice
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STANDARDS for MATHEMATICAL PRACTICE
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their
students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education.
1. Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They
analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution
pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the
original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students
might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get
the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and
graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on
using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems
using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving
complex problems and identify correspondences between different approaches.
2. Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities
to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically
and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to
contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative
reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of
quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
3. Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing
arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze
situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others,
and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from
which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish
correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct
arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even
though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies.
Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve
the arguments.
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4. Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In
early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional
reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem
or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know
are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are
able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs,
flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical
results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
5. Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper,
concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry
software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of
these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school
students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using
estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the
results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade
levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve
problems. They are able to use technological tools to explore and deepen their understanding of concepts.
6. Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their
own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are
careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately
and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give
carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of
definitions.
7. Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven
more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later,
students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 +
9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and
can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can
see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can
see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x
and y.
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8. Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary
students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a
repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope
3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x
+ 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to
solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the
reasonableness of their intermediate results.
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