Download Problem Solving Alg Prep

Willmar Public Schools
Curriculum Map
Note: Problem Solving Algebra Prep is an elective credit. It is not a math credit at the high school as its intent is to help students prepare for Algebra by
providing students with the opportunity to develop the skills necessary to be successful in a high school algebra course. For this reason, it addresses many of the
7th and 8th grade standards and benchmarks.
Subject Area
Mathematics—Senior High
Course Name
Problem Solving Algebra Prep
Unit
Content
Standards Addressed
Skills/Benchmarks
12
days
Arithmetic with
Letters
Grade 7 Number & Operation
7.1.1 Read, write, represent
and compare positive and
negative rational numbers,
expressed as integers,
fractions and decimals.
7.1.1.3: Locate positive and
negative rational numbers on the
number line. Understand the
concept of opposites.
Arithmetic and
Algebra
Representing
Numbers Using
Letters
Integers on the
Number Line
Adding Integers
Subtracting
Integers
Multiplying
Integers
Dividing Positive
and Negative
Integers
Simplifying
Expressions—one
variable
Simplifying
Expressions—
several variables
Positive
Exponents
Formulas with
Variables
7.1.2 Calculate with positive
and negative rational numbers,
and rational numbers with
whole number exponents, to
solve real-world and
mathematical problems.
7.1.2.1: Add, subtract, multiply, and
divide positive and negative
integers.
7.2.4 Represent real-world and
mathematical situations using
equations with variables.
Solve equations symbolically,
using the properties of
equality. Also solve equations
graphically and numerically.
Interpret solutions in the
original context.
7.2.4.1: Represent relationships in
various contexts with equations
involving variables and positive and
negative rational numbers. Use the
properties of equality to solve for
the value of a variable. Interpret the
solution in the original context.
7.1.2.5: Use the symbol for absolute
value. Demonstrate understanding
of absolute value with respect to
distance.
For example: Solve for w in the
equation P = 2w + 2ℓ when P = 3.5
and
ℓ = 0.4.
Another example: To post an
Internet website, Mary must pay
$300 for initial set up and a monthly
Date
June 2009
Essential Questions
Assessments
What is the difference
between numerical and
algebraic expressions?
Daily Homework
Quizzes
Chapter Review
Chapter Exam
Informal Assessments
to include observations,
review of student notes,
individual conferences,
small group work &
daily check for
understanding
How can unknown quantities
be expressed?
How are arithmetic
operations performed with
positive and negative
numbers?
When and how can variable
expressions be simplified?
How are variable expressions
used to solve real world
problems?
Willmar Public Schools
Curriculum Map
fee of $12. She has $842 in savings,
how long can she sustain her
website?
8th Grade Algebra
8.2.4 Represent real-world and
mathematical situations using
equations and inequalities
involving linear expressions.
Solve equations and
inequalities symbolically and
graphically. Interpret solutions
in the original context.
8.2.4.2: Solve multi-step equations
in one variable. Solve for one
variable in a multi-variable equation
in terms of the other variables.
Justify the steps by identifying the
properties of equalities used.
Students will be able to:
1.
2.
3.
4.
5.
6.
7.
12
days
The Rules of
Arithmetic
Commutative
Property of
Addition
Commutative
Property of
Multiplication
Associative
Grade 7 Algebra
7.2.3 Apply understanding of
order of operations and
algebraic properties to
generate equivalent numerical
and algebraic expressions
containing positive and
negative rational numbers and
grouping symbols; evaluate
such expressions.
Recognize numerical and
algebraic expressions.
Understand the use of variables
in algebraic expressions.
Understand positive and
negative integers, opposites,
and absolute value.
Discover and use rules related
to adding, subtracting,
multiplying, and dividing
integers.
Simplify expressions with one
or more variables.
Read and write exponents.
Use formulas with variables.
7.2.3.1: Use properties of algebra to
generate equivalent numerical and
algebraic expressions containing
rational numbers, grouping symbols
and whole number exponents.
Properties of algebra include
associative, commutative and
distributive laws.
Why did the calculator give
me the wrong answer?
Why does the order in which
operations are performed
often change the outcome?
How does the use of
mathematical properties
simplify or make some
Daily Homework
Quizzes
Chapter Review
Chapter Exam
Informal Assessments
to include observations,
review of student notes,
individual conferences,
small group work &
daily check for
Willmar Public Schools
Curriculum Map
7.2.3.3: Apply understanding of
order of operations and grouping
symbols when using calculators and
other technologies.
Property of
Addition
Associative
Property of
Multiplication
The Distributive
Property—
Multiplication
The Distributive
Property—
Factoring
Properties of Zero
Properties of 1
Powers and Roots
Order of
Operations
1.
2.
3.
4.
5.
6.
8.
Linear Equations
with One Variable
Writing Equations
Solving Equations:
x-b=c
Solving Equations:
x+b=c
Solving
Multiplication
Equations
Solving Equations
with Fractions
Solving Equations
with More than
understanding
Without guessing, how can
the solution to an equation be
found using concepts of
algebra?
Daily Homework
Quizzes
Chapter Review
Chapter Exam
Informal Assessments
to include observations,
review of student notes,
individual conferences,
small group work &
daily check for
understanding
Students will be able to:
7.
18
days
problems easier to solve?
8th Grade Algebra
8.2.4 Represent real-world and
mathematical situations using
equations and inequalities
involving linear expressions.
Solve equations and
inequalities symbolically and
graphically. Interpret solutions
in the original context.
Use the Commutative Property
of Addition & Multiplication.
Use the Associative Property
for Addition & Multiplication.
Use the Distributive Property
over multiplication and to
factor.
Apply the Property of Zero.
Apply the Property of 1
Raise expressions to integer
powers.
Find square root of perfect
squares.
Apply Order of Operations to
simplify and/or evaluate
expressions.
8.2.4.3: Express linear equations in
slope-intercept, point-slope and
standard forms, and convert
between these forms. Given
sufficient information, find an
equation of a line.
How can the sides of a right
triangle be found?
For example: Determine an
equation of the line through the
points (-1,6) and (2/3, -3/4).
How is and interval
expressed using
mathematical symbols?
8.2.4.4: Use linear inequalities to
represent relationships in various
contexts.
What real world applications
ask for an interval solution?
Willmar Public Schools
Curriculum Map
For example: A gas station charges
$0.10 less per gallon of gasoline if a
customer also gets a car wash.
Without the car wash, gas costs
$2.79 per gallon. The car wash is
$8.95. What are the possible
amounts (in gallons) of gasoline that
you can buy if you also get a car
wash and can spend at most $35?
One Step
Equations Without
Numbers
Formulas
The Pythagorean
Theorem
Inequalities on the
Number Line
Solving
Inequalities in One
Variable
Using Equations
8.2.4.5: Solve linear inequalities
using properties of inequalities.
Graph the solutions on a number
line.
For example: The inequality -3x < 6
is equivalent to x > -2, which can be
represented on the number line by
shading in the interval to the right
of -2.
Students will be able to:
1.
2.
3.
4.
5.
6.
14
days
Applications of
Algebra
7th Grade Algebra Standard
7.2.2 Recognize proportional
Solve one-step equations
involving addition, subtraction,
or multiplication.
Solve multi-step equations
involving the four basic
operations.
Solve for a variable.
Use the Pythagorean Theorem
to solve for an unknown side.
Represent inequalities on a
Number Line.
Solve inequalities in one
variable.
7.2.2.1: Represent proportional
How can percentages be
estimated without the use of a
Daily Homework
Quizzes
Willmar Public Schools
Curriculum Map
Writing
Equations—Odd
and Even Integers
Using the 1%
Solution to Solve
Problems
Using the Percent
Equation
Solving Distance,
Rate, & Time
Problems
Using a Common
Unit--Cents
Calculating
Simple Interest
Deriving a
Formula for
Mixture Problems
Ratio and
Proportion
Using Proportions
relationships in real-world and
mathematical situations;
represent these and other
relationships with tables,
verbal descriptions, symbols
and graphs; solve problems
involving proportional
relationships and explain
results in the original context.
relationships with tables, verbal
descriptions, symbols, equations
and graphs; translate from one
representation to another. Determine
the unit rate (constant of
proportionality or slope) given any
of these representations.
For example: Larry drives 114
miles and uses 5 gallons of gasoline.
Sue drives 300 miles and uses 11.5
gallons of gasoline. Use equations
and graphs to compare fuel
efficiency and to determine the
costs of various trips.
7.2.2.2: Solve multi-step problems
involving proportional relationships
in numerous contexts.
For example: Distance-time, percent
increase or decrease, discounts, tips,
unit pricing, lengths in similar
geometric figures, and unit
conversion when a conversion
factor is given, including conversion
between different measurement
systems.
Another example: How many
kilometers are there in 26.2 miles?
7.2.2.3: Use knowledge of
proportions to assess the
reasonableness of solutions.
For example: Recognize that it
would be unreasonable for a cashier
to request $200 if you purchase a
$225 item at 25% off.
calculator?
How do percents relate to a
portion of the circle?
A trip takes more time to
return than it did to get to the
destination. How is the
average rate for the entire trip
calculated?
What is meant by simple
interest, and how is it
calculated?
Three different dried fruits
are mixed to make a snack
pack. How should the price
to sell the snack pack be
determined?
Chapter Review
Chapter Exam
Informal Assessments
to include observations,
review of student notes,
individual conferences,
small group work &
daily check for
understanding
Willmar Public Schools
Curriculum Map
7.2.4 Represent real-world and
mathematical situations using
equations with variables.
Solve equations symbolically,
using the properties of
equality. Also solve equations
graphically and numerically.
Interpret solutions in the
original context.
7th Grade Data Analysis and
Probability
7.4.2 Display and interpret
data in a variety of ways,
including circle graphs and
histograms.
7.2.4.2: Solve equations resulting
from proportional relationships in
various contexts.
For example: Given the side lengths
of one triangle and one side length
of a second triangle that is similar to
the first, find the remaining side
lengths of the second triangle.
Another example: Determine the
price of 12 yards of ribbon if 5
yards of ribbon cost $1.85.
7.4.2.1: Use reasoning with
proportions to display and interpret
data in circle graphs (pie charts) and
histograms. Choose the appropriate
data display and know how to create
the display using a spreadsheet or
other graphing technology.
Students will be able to:
1. write an algebraic equation for
a number sentence.
2. identify formulas to use in
specific types of problems.
3. write problems using algebraic
formulas.
4. solve problems by applying
algebraic equations.
10
days
Data, Statistics, &
Probability
Organizing Data
Range, Mean,
Median, & Mode
Box-and-Whiskers
7th Grade Data Analysis &
Probability
7.4.1 Use mean, median and
range to draw conclusions
about data and make
predictions.
How is data used to make
decisions?
7.4.1.1: Design simple experiments
and collect data. Determine mean,
median and range for quantitative
data and from data represented in a
display. Use these quantities to draw
What does the center of the
data mean?
How are probabilities
Daily Homework
Quizzes
Chapter Review
Chapter Exam
Informal Assessments
to include observations,
review of student notes,
Willmar Public Schools
Curriculum Map
Plots
The Probability
Fraction
Probability and
Complementary
Events
Tree Diagrams and
Sample Spaces
Dependent and
Independent
Events
The Fundamental
Principle of
Counting
Multistage
Experiments
conclusions about the data, compare
different data sets, and make
predictions.
For example: By looking at data
from the past, Sandy calculated that
the mean gas mileage for her car
was 28 miles per gallon. She
expects to travel 400 miles during
the next week. Predict the
approximate number of gallons that
she will use.
7.4.3 Calculate probabilities
and reason about probabilities
using proportions to solve
real-world and mathematical
problems.
7.4.3.2: Calculate probability as a
fraction of sample space or as a
fraction of area. Express
probabilities as percents, decimals
and fractions.
For example: Determine
probabilities for different outcomes
in game spinners by finding
fractions of the area of the spinner.
7.4.3.3: Use proportional reasoning
to draw conclusions about and
predict relative frequencies of
outcomes based on probabilities.
For example: When rolling a
number cube 600 times, one would
predict that a 3 or 6 would be rolled
roughly 200 times, but probably not
exactly 200 times.
Students will be able to:
1.
2.
organize data into graphs.
read and interpret graphic
representations
calculated?
What is the difference
between the probability of
blue and green and the
probability of blue or green?
How is probability used to
make decisions?
individual conferences,
small group work &
daily check for
understanding
Willmar Public Schools
Curriculum Map
3.
4.
9 days
Linear Equations and
Inequalities in the
Coordinate plane
The Coordinate
System
Graphing
Equations
Intercepts of Lines
Slopes of Lines
Writing Linear
Equations
Lines as Functions
Domain and
Range of a
Function
7th Grade Algebra Standard
7.2.2 Recognize proportional
relationships in real-world and
mathematical situations;
represent these and other
relationships with tables,
verbal descriptions, symbols
and graphs; solve problems
involving proportional
relationships and explain
results in the original context.
determine range and measures
of central tendency.
compute probabilities and
complementary event involving
statistics.
What is meant by slope?
7.2.2.4: Represent real-world or
mathematical situations using
equations and inequalities involving
variables and positive and negative
rational numbers.
For example: "Four-fifths is three
greater than the opposite of a
number" can be represented
as 54 =−n + 3 , and "height no bigger
than half the radius" can be
represented as h ≤ 2r .
8th Grade Algebra:
8.2.1 Understand the concept
of function in real-world and
mathematical situations, and
distinguish between linear and
nonlinear functions.
Another example: "x is at least -3
and less than 5" can be represented
as −3 ≤ x < 5 , and also on a number
line.
8.2.1.1: Understand that a function
is a relationship between an
independent variable and a
dependent variable in which the
value of the independent variable
determines the value of the
dependent variable. Use functional
notation, such as f(x), to represent
such relationships.
For example: The relationship
between the area of a square and the
side length can be expressed as
What is a function?
What does domain and range
have to do with independent
and dependent variables?
What does a shaded region in
the coordinate plane mean?
What might the graph of
distance traveled with respect
to time over a week-long trip
look like?
Daily Homework
Quizzes
Chapter Review
Chapter Exam
Informal Assessments
to include observations,
review of student notes,
individual conferences,
small group work &
daily check for
understanding
Willmar Public Schools
Curriculum Map
f ( x) = x2 . In this case, f (5) = 25 ,
which represents the fact that a
square of side length 5 units has
area 25 units squared.
8.2.1.2: Use linear functions to
represent relationships in which
changing the input variable by some
amount leads to a change in the
output variable that is a constant
times that amount.
For example: Uncle Jim gave Emily
$50 on the day she was born and
$25 on each birthday after that. The
function f ( x=) 50 + 25x represents the
amount of money Jim has given
after x years. The rate of change is
$25 per year.
8.2.1.3: Understand that a function
is linear if it can be expressed in the
form f ( x=) mx + b or if its graph is
a straight line.
For example: The
function f ( x) = x2 is not a linear
function because its graph contains
the points (1,1), (-1,1) and (0,0),
which are not on a straight line.
8.2.2 Recognize linear
functions in real-world and
mathematical situations;
represent linear functions and
other functions with tables,
verbal descriptions, symbols
and graphs; solve problems
involving these functions and
8.2.2.1: Represent linear functions
with tables, verbal descriptions,
symbols, equations and graphs;
translate from one representation to
another.
8.2.2.2: Identify graphical properties
of linear functions including slopes
Willmar Public Schools
Curriculum Map
explain results in the original
context.
and intercepts. Know that the slope
equals the rate of change, and that
the y-intercept is zero when the
function represents a proportional
relationship.
8.2.2.3: Identify how coefficient
changes in the equation f (x) = mx +
b affect the graphs of linear
functions. Know how to use
graphing technology to examine
these effects.
Students will be able to:
1.
2.
identify the parts of a graph.
locate and plot points in the
coordinate system.
3. solve equati0ns for ordered
pairs and graph a line.
4. find the x-intercept and yintercept of a graph.
5. determine the slope of a line
6. write and solve an equation of
a straight line.
7. identify and evaluate functions.
8. determine the range of a
function with a given domain.
9. graph inequalities.
10. interpret and create graphs
without numbers.
10
days
Geometry
Angles and Angle
Measure
Pairs of Lines in
Planes and in
Space
7th Grade Algebra Standard
7.2.1 Understand the concept
of proportionality in realworld and mathematical
situations, and distinguish
between proportional and
other relationships.
7.2.1.2: Understand that the graph
of a proportional relationship is a
line through the origin whose slope
is the unit rate (constant of
proportionality). Know how to use
graphing technology to examine
How is geometry used in
carpentry or architecture?
How are angles labeled to
avoid confusion where more
than one angle is pictured?
Daily Homework
Quizzes
Chapter Review
Chapter Exam
Informal Assessments
to include observations,
review of student notes,
Willmar Public Schools
Curriculum Map
Angles Measures
in a Triangle
Naming Triangles
Quadrilaterals
Congruent and
Similar Triangles
7.3.2 Analyze the effect of
change of scale, translations
and reflections on the
attributes of two-dimensional
figures.
8th Grade Geometry and
Measurement
8.3.1 Solve problems
involving right triangles using
the Pythagorean Theorem and
its converse.
what happens to a line when the unit
rate is changed.
How can triangles be
classified?
7.3.2.1: Describe the properties of
similarity, compare geometric
figures for similarity, and determine
scale factors.
What is needed for two
triangles to be congruent?
For example: Corresponding angles
in similar geometric figures have
the same measure.
8.3.1.1: Use the Pythagorean
Theorem to solve problems
involving right triangles.
For example: Determine the
perimeter of a right triangle, given
the lengths of two of its sides.
Another example: Show that a
triangle with side lengths 4, 5 and 6
is not a right triangle.
Students will be able to:
1.
2.
3.
4.
5.
6.
name and determine the
measure of angles.
identify how lines are related in
planes and space.
use theorems to help solve
problems involving triangles.
name triangles by their
characteristics.
determine the measures of
angles in quadrilaterals.
use theorems to determine
whether triangles are congruent
or similar.
What is meant by similar
geometric figures?
How can a quadrilateral be
divided into two triangles?
individual conferences,
small group work &
daily check for
understanding