Download Unit 1, Concept 1 – Number Sense, Fractions, and Algebraic

Unit 1, Concept 1 – Number Sense, Fractions, and Algebraic Thinking
Instructional Resources: Carnegie Learning: Bridge to Algebra
Standards:
7NS 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational
numbers to whole-number powers.
7AF 1.2 Use the correct order of operations to evaluate algebraic expressions such as 3(2x + 5)².
7AF 1.3 Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative,
commutative) and justify the process used.
7AF 2.1 Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or
multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents.
Concept
(and related skills)
Understand the composition of
whole numbers
• Represent algebraic situations
using pictures
• Use the order of operations
• Relate factors, multiples, and
divisibility
• Identify numbers as prime or
composite
• Use powers and exponents to
write repeated multiplication
• Recognize and use the
commutative and associative
properties of multiplication
• Recognize and use the
multiplicative identity
Textbook Connections
Vocabulary
Lessons
1.1 Introduction to Picture Algebra
1.2 Factors and Multiples
1.3 Least Common Multiple
1.4 Prime and Composite Numbers
1.5 Prime Factorization
1.6 Powers and Exponents
1.7 Greatest Common Factor
Associative Property
of Multiplication
Base
Common factor
Common Multiple
Commutative
Property of
Multiplication
Composite factor
Composite Number
Divisible
Exponent
Expression
Factor
Factor pair
Greatest common
Factor
Least common
Multiple
Multiple
Multiplicative
Identity
Order of operations
Power
Prime factor
Prime factorization
Prime number
Variable
Variable expression
86
Unit 1, Concept 2 – Number Sense, Fractions, and Algebraic Thinking
Instructional Resources: Carnegie Learning: Bridge to Algebra
Standards:
Note: This concept does not explicitly address a seventh grade standard, but it is necessary for the development of future concepts.
Concept
(and related skills)
Understand the composition of
fractions
• Use fractions to represent parts of a
whole
• Use fractions to divide more than
one whole into equally sized parts
• Write equivalent fractions
• Write fractions in simplest form
• Compare and order fractions
Textbook Connections
Vocabulary
Lessons
2.1
2.2
2.3
2.4
2.5
2.6
Dividing a Whole into Fractional Parts
Dividing More Than One Whole into Parts
Dividing Groups into Fractional Parts
Equivalent Fractions
Simplifying Fractions
Comparing and Ordering Fractions
Completely simplified
Denominator
Equation
Equivalent fractions
Fraction
Greater than
Least common denominator
Less than
Numerator
Reasonable solution
Simplest form
Simplest terms
87
Unit 1, Concept 3– Number Sense, Fractions, and Algebraic Thinking
Instructional Resources: Carnegie Learning: Bridge to Algebra
Standards:
7NS 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational
numbers to whole-number powers.
7NS 2.2 Add and subtract fractions by using factoring to find common denominators.
Concept
(and related skills)
Textbook Connections
Vocabulary
Lessons
Understand operations on
fractions and mixed numbers
• Add and subtract like fractions
• Add and subtract unlike fractions
• Convert between improper
fractions and mixed numbers
• Multiply and divide fractions
• Add, subtract, multiply, and divide
mixed numbers
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Adding and Subtracting Fractions with Like
Denominators
Adding and Subtracting Fractions with
Unlike Denominators
Improper Fractions and Mixed Numbers
Multiplying Fractions
Dividing Fractions
Adding and Subtracting Mixed Numbers
Multiplying and Dividing Mixed Numbers
Working with Customary Units
Capacity
Cup, pint, quart,
Gallon
Improper fractions
Inch, foot, yard, mile
Least common
Denominator
Length
Like fractions
Metric system
Mixed number
Multiplicative
Identity
Multiplicative
Inverse
Ounce, pound, ton
Reciprocal
Remainder
Unlike fractions
Weight
88
Unit 2, Concept 1– Decimals, Ratios and Proportions, and Percents
Instructional Resources: Carnegie Learning: Bridge to Algebra
Standards:
7NS 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational
numbers to whole-number powers.
7NS 1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications.
Concept
(and related skills)
Understand the composition of
decimals and operations on
decimals
•
•
•
•
•
•
•
Textbook Connections
Vocabulary
Lessons
4.1 Decimals as Special Fractions
4.2 Place Value and Expanded Form
4.3 Decimals as Fractions: Comparing and
Write decimals as fractions
Represent decimals using a placeRounding Decimals
value chart, in word form, and with 4.4 Adding and Subtracting Decimals
expanded form
4.5 Multiplying Decimals
4.6 Dividing Decimals
Compare and order decimals
4.7 Working with Metric Units
Round decimals
Add, subtract, multiply, and divide
decimals
Represent decimals using base-ten
pieces
Use metric units to measure length,
mass, capacity
Decimal
Dividend
Divisor
Expanded form
Gram
Liter
Meter
Metric system
Place-value
Product
Quotient
Rounding off
Standard form
90
Unit 2, Concept 2 – Decimals, Ratios and Proportions, and Percents
Instructional Resources: Carnegie Learning: Bridge to Algebra
Standards:
7AF 4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation.
7MG 1.1 Compare weights, capacities, geometric measures, times, and temperatures within an between measurement systems (e.g.,
miles per hour and feet per second, cubic inches to cubic centimeters).
7MG 1.3 Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve
problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer.
Concept
(and related skills)
Understand ratio and proportion
Write ratios as fractions
Compare ratios
Write proportions
Solve proportions
Find the means and extremes of a
proportion
• Find unit rates
• Solve problems using proportions
•
•
•
•
•
Textbook Connections
Vocabulary
Lessons
5.1
5.2
5.3
5.4
Ratios and Fractions
Writing and Solving Proportions
Using Ratios and Rates
Using Proportions to Solve Problems
Extremes
Means
Proportion
Rate
Ratio
Unit Rate
Variable
91
Unit 2, Concept 3 – Decimals, Ratios and Proportions, and Percents
Instructional Resources: Carnegie Learning: Bridge to Algebra
Standard:
7NS 1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications.
7NS 1.6 Calculate the percentage of increases and decreases of a quantity.
7NS 1.7 Solve problems that involve discounts, markups, commissions, and profit, and compute simple and compound interest.
Concept
(and related skills)
Understand percents
• Write percents as decimals and
fractions
• Write decimals and fractions as
percents
• Use benchmark percents (1% and
10%) to find the percent of a
number
• Use a proportion to find the
percent of a number and to find a
number that corresponds to 100%
• Use a proportion to find percent,
given two numbers
• Find simple interest
• Find the percent increase or
decrease of a quantity
Textbook Connections
Vocabulary
Lessons
6.1
6.2
6.3
6.4
6.5
6.6
Percents
Making Sense of Percents
Finding the Percent of a Number
Finding One Whole, or 100%
Finding Percents Given Two Numbers
Percent Increase and Percent Decrease
Benchmark percent
Commission
Discount
Markup
Percent
Percent decrease
Percent increase
Principal
Simple interest
92
Unit 3, Concept 1 – Integers, Algebraic Problem Solving and Linear functions (Including Graphing)
Instructional Resources: Carnegie Learning: Bridge to Algebra
Standards:
7NS 1.1 Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10) with approximate numbers
using scientific notation.
7NS 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational
numbers to whole-number powers.
7NS 2.1 Understand negative whole-number exponents. Multiply and divide expressions involving exponents with a common base.
7NS 2.5 Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero
on a number line; and determine the absolute value of real numbers.
7AF 2.1 Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or
multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents.
Concept
(and related skills)
Understand integers and
operations on integers.
• Write integers to represent reallife situations
• Graph integers on a number line
• Compare integers
• Add, subtract, multiply, and divide
integers
• Write the absolute value of a
number
• Represent numbers using powers
of 10
• Multiply and divide by powers of
10
• Read and write numbers using
scientific notation
Textbook Connections
Vocabulary
Lessons
7.1
7.2
7.3
7.4
7.5
7.6
7.7
Negative Numbers in the Real World
Adding Integers
Subtracting Integers
Multiplying and Dividing Integers
Absolute Value and Additive Inverse
Powers of Ten
Scientific Notation
Absolute value
Additive inverse
Difference
Expanded form
Exponent
Integer
Integer addition
Integer subtraction
Negative exponent
Negative integer
Number line
Opposites
Positive integer
Power
Power of ten
Product
Quotient
Scientific notation
Sum
94
Unit 3, Concept 2 – Integers, Algebraic Problem Solving and Linear functions (Including Graphing)
Instructional Resources: Carnegie Learning: Bridge to Algebra
Standards:
7AF 1.1 Use variables and appropriate operations to write an expression, and equation, an inequality, or a system or equations or
inequalities that represents a verbal
description (e.g., three less than a number, half as large as area A).
7AF 4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in
the context from which they arose, and verify the reasonableness of the results.
7AF 4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation.
Concept
(and related skills)
Textbook Connections
Understand algebraic problem
Lessons
solving (linear applications) and
plotting sets of points to
8.1 Picture Algebra
represent solutions
8.2 Writing, Evaluating, and Simplifying Expressions
8.3 Solving One-Step Equations
• Use picture algebra to represent 8.4 Solving Two-Step Equations
and solve problems
8.5 Plotting Points in the Coordinate Plane
• Evaluate expressions
• Solve one-step equations
• Solve two-step equations
• Identify and graph points in the
coordinate plane
• Make tables of values
• Create a graph of ordered pairs
• Use equations, tables, and
graphs to solve problems
Vocabulary
Algebraic expression
Cartesian coordinate system
Coordinate plane
Equation
Evaluate
Inverse operations
One-step equation
Ordered pair
Origin
Solve
Two-step equation
Variable
x-axis and y-axis
x-coordinate and y-coordinate
95
Unit 3, Concept 3 – Integers, Algebraic Problem Solving and Linear functions (Including Graphing)
Instructional Resources: Carnegie Learning: Bridge to Algebra
Standards:
7AF 1.1 Use variables and appropriate operations to write an expression, and equation, an inequality, or a system or equations or
inequalities that represents a verbal description (e.g., three less than a number, half as large as area A).
7AF 1.5 Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation
represented by the graph.
7AF 3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is
always the same and know that the ratio (“rise over run”) is called the slope of a graph.
7AF 3.4 Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference
to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the ratio of the quantities.
Concept
(and related skills)
Understand linear functions
and graphing lines
•
•
•
•
•
•
•
Textbook Connections
Vocabulary
Lessons
8.7
Use tables and graphs to
13.1
represent functions
13.2
Use function notation
13.3
Make input-output tables for
13.4
linear functions and use these to 13.5
graph those functions
Find the slope of a line as a
ratio and as a rate of change
Find the x- and y-intercepts of a
line
Graph lines using slopes and
intercepts
Use the slope-intercept form of
the equation of a line to graph
the line
Solving Problems Using Multiple Representations
Relations and Functions
Linear Functions
Slope and Rates of Change
Finding Slope and Y-Intercepts
Using Slope and Intercepts to Graph Lines
“Rise” and “Run”
Dependent variable
Domain and range
Function
Function notation
Graph
Independent variable
Input-output table
Linear equation
Linear function
Multiple representation
Rate of change
Relation
Slope
Slope-intercept form
Table
x-intercept and y-intercept
96
Unit 4, Concept 1 – Probability and Statistics, Number Systems, Area, and the Pythagorean Theorem
Instructional Resources: Carnegie Learning: Bridge to Algebra
Standards:
7NS 1.5 Know that every rational number is either a terminating or a repeating decimal and be able to convert terminating decimals into
reduced fractions.
7NS 2.1 Understand negative whole-number exponents. Multiply and divide expressions involving exponents with a common base.
7NS 2.3 Multiply, divide, and simplify rational numbers by using exponent rules.
Concept
(and related skills)
Understand rational and
irrational numbers
• Use the number line to compare
and order rational numbers
• Learn about and classify
different types of numbers
• Find powers of rational
numbers
• Multiply and divide powers of
rational numbers
• Identify decimals as
terminating or repeating
• Write repeating decimals as
fractions
• Use and apply the distributive
property
Textbook Connections
Vocabulary
Lessons
14.1
14.2
14.3
14.4
14.5
Rational Numbers
Powers of Rational Numbers
Irrational Numbers
Real Numbers and their Properties
The Distributive Property
Distributive property
Integer
Irrational number
Natural number
Power
Rational number
Real number
Repeating decimal
Terminating decimal
Venn diagram
Whole number
98
Unit 4, Concept 2 – Probability and Statistics, Number Systems, Area, and the Pythagorean Theorem
Instructional Resources: Carnegie Learning: Bridge to Algebra
Standards:
7 SDAP 1.1 Know various forms of display for data sets, including a stem-and-leaf or box-and-whisker plot; use the forms to display a
single set of data or to compare two sets of data.
7 SDAP 1.3 Understand the meaning of, and be able to compute, the minimum, the lower quartile, the median, the upper quartile, and the
maximum of a data set.
Concept
(and related skills)
Understand elements of
probability and statistics
• Find the probability of an event
• Distinguish between
independent and dependent
events
• Find the probability of a
compound event
• Find the mean, median, mode,
and range of a set of data
• Create and interpret frequency
tables and histograms
• Create and interpret stem-andleaf plots
• Create and interpret box-and
whisker plots
• Create and interpret circle
graphs
Textbook Connections
Vocabulary
Lessons
11.1
11.2
11.3
11.4
11.5
11.6
11.7
Finding Simple Probabilities
Finding Probabilities of Compound Events
Mean, Median, Mode, and Ranges
Histograms
Stem-and-Leaf Plots
Box-and Whisker Plots
Circle Graphs
Box-and-whisker plot
Circle graph
Complementary events
Compound event
Event
Frequency table
Histogram
Independent and dependent events
Mean
Median
Mode
Outcome and favorable outcome
Probability of an event
Random
Range
Sample space
Stem-and-leaf plot
Theoretical and experimental probability
Upper and lower quartiles
99
Unit 4, Concept 3 – Probability and Statistics, Number Systems, Area, and the Pythagorean Theorem
Instructional Resources: Carnegie Learning: Bridge to Algebra
Standard:
7MG 3.3 Know and understand the Pythagorean Theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths and the
lengths of other line segments and, in some situations, empirically verify the Pythagorean Theorem by direct measurement.
Concept
(and related skills)
Understand areas and the
Pythagorean Theorem with
applications
• Find squares of numbers and
find and estimate the square
roots of numbers
• Prove the Pythagorean
Theorem and use it to solve
problems
• Use the converse of the
Pythagorean Theorem
• Find Pythagorean triples
Textbook Connections
Vocabulary
Lessons
10.4
10.5
10.6
Squares and Square Roots
The Pythagorean Theorem
Using the Pythagorean Theorem
Converse
Hypotenuse
Leg
Perfect square
Pythagorean theorem
Pythagorean triple
Radical sign
Radicand
Square
Square root
100