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CAHSEE Prep Instructional Guide 2011-12
1
Subject: CAHSEE Prep
Benchmark Assessments and Instructional Guide
Instructional Guides are provided as a resource for the Alliance classroom teacher. They identify the high-priority standards to be taught during each quarter of
instruction in the context of proposed units. High priority standards are assessed on quarterly benchmark exams.
In 1999, California enacted a law requiring that every California public school student pass an examination to receive a high school diploma. The primary purpose
of the California High School Exit Exam (CAHSEE) is to significantly improve pupil achievement in public high schools and to ensure that pupils who graduate from
public high schools can demonstrate grade level competency in reading, writing and mathematics. The CAHSEE is administered over two days. On the first day,
students will take the English-language arts portion of the test; on the second day, they will take the mathematics portion. All of the questions on the CAHSEE are
based on California’s academic content standards in English-language arts and mathematics. The focus of this instructional guide will be on the 7 math strands
tested: 1) Measurement; 2) Geometry; 3) Mathematical Reasoning; 4) Number Sense; 5) Statistics, Data Analysis and Probability; 6) Algebra and Functions; and
7) Algebra 1
Unit
Unit One: Measurement
Rational numbers are used as the basis for developing
the concepts of ratio, proportion, and percent. Ratios
and rates are defined algebraically, and introduced in a
real world context. It is emphasized that a rate is a special case of a ratio where the numerator and denominator are in different units. The concept of equivalence is
explored in terms of equivalent ratios and unit rates.
This naturally leads to a discussion of proportion, which
is an equation that states the equivalence of two ratios.
The cross multiplication property is used to solve proportions, and problems involving proportions are solved in a
real world context. Once the concept of a proportion is
established, the definition of a percent is given in terms
of proportion. The percent equation,
a
p
, is intro=
b 100
duced as a way to write other rational numbers as percentages, and to solve different problems about percent,
including finding the percent of a number ( p), finding
the base (b), and finding part of a base (a). Another
important aspect of the study of percent is the concept
of percent describing change between quantities.
Namely, the percent change between two numbers is
found, and the percent change is used to change a
number. The concepts of percent, ratio, and proportion
appear in a wide variety of real world situations. Prob-
High Priority Standards
And
Learning Targets*
Measurement and Geometry:
1.1 Compare weights, capacities,
geometric measures, times,
and temperatures within and
between measurement systems (e.g., miles per hour and
feet per second, cubic inches
to cubic centimeters).
# Q1
Items
2
Learning Targets
1P Describe a ratio in different ways (i.e.
a b
1.2 Construct and read drawings
and models made to scale.
Learning Targets
1B Explain the meaning of two ratios or rates
being equivalent.
Learning Targets
1C Write a rate as a unit rate.
1D Solve real world problems involving dis-
a : b,
a
,
b
to )
1Q Explain the meaning of two ratios or rates being
equivalent.
Learning Targets
1A Write a ratio and proportion based on a
verbal description.
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.
Supporting Medium/
Low Priority Standards
& Learning Targets
Number Sense:
1.0 Students know the properties of,
and compute with, rational numbers expressed in a variety of
forms.
2
1.1 Use variables and appropriate operations to write an expression, an
equation, an inequality, or a system of equations or inequalities
that represents a verbal description (e.g., three less than a number, half as large as area A).
Learning Targets
1R Read a completed solution to an equation or
inequality, find any errors, and rewrite the solution, justifying each step in the process.
1S Solve word problems involving area and perimeter.
1T Find the surface area of cylinders, prisms, and
cones.
1.2 Use the correct order of operations
to evaluate algebraic expressions
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
Unit
lems involving distance, discounts, markups and commissions, and the construction of scale models are covered.
High Priority Standards
And
Learning Targets*
tance, discounts, markups, and commission, and the construction of scale
models.
2.1 Compare weights, capacities,
geometric measures, times,
and temperatures within and
between measurement systems (e.g., miles per hour and
feet per second, cubic inches
to cubic centimeters).
# Q1
Items
2
Learning Targets
1V Explain how the concepts of ratio, proportion,
and percentage are related.
Algebra and Functions:
1.4 Use algebraic terminology (e.g.,
variable, equation, term, coefficient, inequality, expression, constant) correctly.
Learning Targets
1W Write a ratio and proportion based on a verbal
description.
1X Find the percent of a number.
4.0 Students solve simple linear equations and inequalities over the rational numbers.
2
Learning Targets
1L Find the area of more complex polygons
and irregular polygons in the coordinate
plane.
2.3 Compute the length of the
perimeter, the surface area of
the faces, and the volume of
a three-dimensional object
built from rectangular solids.
Understand that when the
Learning Targets
1U Simplify both sides of an equation before solving, including using the distributive property
and combining like terms.
1.3 Convert fractions to decimals and
percents and use these representations in estimations, computations, and applications.
Learning Targets
1E Use the multiplicative inverse property to
solve more complex proportions and
explain how cross multiplication is a
process used to solve these proportions.
1F Find the area and perimeter of a triangle
or quadrilateral.
1G Find the area and perimeter of more
complex or irregular polygons.
1H Solve word problems involving area and
perimeter.
1I Find the volume of cylinders, prisms, and
cones.
1J Find the surface area of cylinders, prisms,
and cones.
1K Solve word problems involving surface
area and volume.
2.2 Estimate and compute the
area of more complex or irregular two- and three- dimensional figures by breaking the figures down into
more basic geometric objects.
CAHSEE Prep Instructional Guide 2011-12
Supporting Medium/
Low Priority Standards
& Learning Targets
such as 3(2x + 5)2.
2
Learning Targets
1Y Use cross multiplication to solve proportions
1Z Find the area and perimeter of a triangle or
quadrilateral.
Mathematical Reasoning:
1.0 Students make decisions about
how to approach problems
2.0 Students use strategies, skills, and
concepts in finding solutions.
3.0 Students determine a solution is
complete and move beyond a particular problem by generalizing to
other situations.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
2
Unit
High Priority Standards
And
Learning Targets*
lengths of all dimensions are
multiplied by a scale factor,
the surface area is multiplied
by the square of the scale
factor and the volume is multiplied by the cube of the
scale factor.
# Q1
Items
CAHSEE Prep Instructional Guide 2011-12
Supporting Medium/
Low Priority Standards
& Learning Targets
Learning Targets
1M Solve word problems using equations
and inequalities.
1N Find the surface area of cylinders,
prisms, and cones.
2.4 Relate the changes in measurement with a change of
scale to the units used (e.g.,
square inches, cubic feet)
and to conversions between
units (1 square foot = 144
square inches or [1 ft2] = [144
in2], 1 cubic inch is approximately 16.38 cubic centimeters or [1 in3] = [16.38 cm3],
2
Learning Targets
1O Solve word problems involving area and
perimeter.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
3
CAHSEE Prep Instructional Guide 2011-12
Unit
Unit Two: Geometry
The concepts of plane geometry are the foundation for
exploring geometric objects in the coordinate plane.
Geometric objects are plotted on the coordinate plane,
and letters are used to represent the objects. The concept of area and length is revisited in terms of the coordinate plane, including the areas and perimeters of rectangles and triangles.
A transformation is then defined as an action that
somehow moves an object throughout the plane. The
image of an object under a transformation is the new
shape after applying the transformation. The image of
a geometric object is then found by translation and reflection. This concept is expanded to identify the conditions that indicate two geometric objects are congruent
and the meaning of congruence in terms of the sides
and angles of two objects
High Priority Standards
And
Learning Targets*
Measurement and Geometry:
3.2 Understand and use coordinate graphs to plot simple figures, determine lengths and
areas related to them, and determine their image under
translations and reflections.
Learning Targets
2A Plot shapes and polygons in the coordinate plane.
2B Explain the meaning of a transformation,
and an image under a transformation.
2C Find the image of an object under translation.
2D Find the image of an object under reflection.
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
of other line segments and, in
some situations, empirically
verify the Pythagorean theorem by direct measurement.
Learning Targets
2E Describe the Pythagorean theorem, and
explain why the theorem is true.
2F Find the lengths of a missing side of a right
triangle using the Pythagorean theorem.
2G Explain the converse of the Pythagorean
theorem and what information it provides
about a right triangle
3.4 Demonstrate an understanding
of conditions that indicate two
geometrical figures are congruent and what congruence
means about the relationships
between the sides and angles
of the two figures.
# Q1
Items
2
2
Supporting Medium/
Low Priority Standards
& Learning Targets
Algebra and Functions:
1.2 Use the correct order of operations
to evaluate algebraic expressions
such as 3(2x + 5)2.
1.4 Use algebraic terminology (e.g.,
variable, equation, term, coefficient, inequality, expression, constant) correctly.
4.0 Students solve simple linear equations and inequalities over the rational numbers.
Measurement and Geometry:
1.1 Compare weights, capacities, geometric measures, times, and
temperatures within and between
measurement systems (e.g., miles
per hour and feet per second, cubic inches to cubic centimeters).
2.1 Use formulas routinely for finding
the perimeter and area of basic
two-dimensional figures and the
surface area and volume of basic
three-dimensional figures, including rectangle, parallelograms,
trapezoids, squares, triangles, circles, prisms, and cylinders.
Learning Targets
2J Find the area of triangles and quadrilaterals in
the coordinate plane.
2
Learning Targets
2H Determine when two objects are congruent.
2I Explain the meaning of congruence.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
4
CAHSEE Prep Instructional Guide 2011-12
Unit
Unit Three: Mathematical Reasoning
This unit explores specifically the concept mathematical reasoning in the context of problem solving.
Steps to problem solving are shared and students
are encouraged to personalize and prioritize based
on their perspectives. Analyzing and determining if
information is relevant typically begins the problem
solving process followed by identifying patterns and
the eventual solution. Sometimes a conjecture needs
to be made when solving a problem. Students practice making conjectures and develop the skills to justify their thoughts. Academic language is the foundation used when conjectures are justified.
Estimation skills are introduced as a method for
checking whether answers are reasonable. Two
estimation strategies that prove useful are rounding
and using compatible numbers. Estimation helps
provide a quick way to check an answer to a problem.
High Priority Standards
And
Learning Targets*
Mathematical Reasoning:
1.1 Analyze problems by identifying
relationships, distinguishing
relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns.
1.2 Formulate and justify mathematical conjectures based on a
general description of the
mathematical question or problem posed.
2.1 Use estimation to verify the reasonableness of calculated results.
2.3 Estimate unknown quantities
graphically and solve for them
by using logical reasoning and
arithmetic and algebraic techniques
2.4 Make and test conjectures by
using both inductive and deductive reasoning.
3.3 Develop generalizations of the
results obtained and the strategies used and apply them to
new problem situations.
# Q1
Items
2
2
2
2
2
2
Supporting Medium/
Low Priority Standards
& Learning Targets
Number Sense:
1.0 Students know the properties of,
and compute with, rational
numbers expressed in a variety
of forms
1.2 Add, subtract, multiply, and divide rational numbers (integers,
fractions, and terminating decimals) and take positive rational
numbers to whole-number powers.
Algebra and Functions:
1.1 Use variables and appropriate
operations to write an expression, an equation, an inequality,
or a system of equations or inequalities that represents a verbal description (e.g., three less
than a number, half as large as
area A).
1.2 Use the correct order of operations to evaluate algebraic expressions such as 3(2x + 5)2.
1.4 Use algebraic terminology (e.g.,
variable, equation, term, coefficient, inequality, expression,
constant) correctly.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
5
CAHSEE Prep Instructional Guide 2011-12
Unit
Unit Four: Number Sense
This unit begins the study of algebra by introducing
the real number line and how to place numbers on
the line. The rational numbers are defined as the set
of numbers that can be expressed as a fraction
where the numerator and denominator are integers.
Addition and subtraction of rational numbers are
viewed on the real number line. Multiplication and
division of two integers is also studied. Rules for multiplying and dividing integers are then developed.
Next, the absolute value of a number is then defined
as the distance a number is from 0 on the real number line. Absolute values of both positive and negative rational numbers are found.
The different forms of a rational number are then explored, including a decimal and a fraction written in
lowest terms. Rational numbers are classified as either a terminating or repeating decimal, and the difference between the terminating and repeating decimals is explored. The concept of the least common
denominator is then reviewed, and used to rewrite
multiple fractions so that there is a common denominator. This leads to a discussion of the addition and
subtraction of rational numbers. Multiplication of fractions is explored, and the reciprocal is defined and
used to divide fractions. Operations on decimals are
also mastered and used in a real world context.
High Priority Standards
And
Learning Targets*
Number Sense:
1.1 Read, write, and compare
rational numbers in a scientific notation (positive and
negative powers of 10) with
approximate numbers using
scientific notation.
# Q2
Items
2
Learning Targets
4EE Explain the relationship between the rational
numbers and the number line.
4FF Identify properties of rational numbers.
4GG Describe a ratio in different ways (i.e. , , to )
4HH Explain the meaning of two ratios or rates being
equivalent.
Learning Targets
4A Write numbers in scientific notation, and
explain the need for scientific notation.
1.2 Add, subtract, multiply, and
divide rational numbers (integers, fractions, and terminating decimals) and take
positive rational numbers to
whole-number powers.
2
Learning Targets
4K Convert between fractions and decimals.
4L Explain how to convert between the decimal and fraction forms of a rational
1.4 Differentiate between rational and
irrational numbers.
Learning Targets
4II Explain the difference between a rational and irrational number.
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.
Learning Targets
4B Multiply and divide fractions.
4C Explain the meaning of division by a
fraction.
4D Add, subtract, multiply, and divide decimals.
4E Evaluate expressions with rational numbers.
4F Simplify both sides of an equation before
solving, including using the distributive
property and combining like terms.
4G Use the multiplicative inverse property to
solve more complex proportions and
explain how cross multiplication is a
process used to solve these proportions.
4I Evaluate the power of a rational number.
4H Write powers from a verbal description.
4J Solve word problems involving exponents.
1.3 Convert fractions to decimals
and percents and use these
representations in estimations, computations, and applications.
Supporting Medium/
Low Priority Standards
& Learning Targets
Number Sense:
1.0 Students know the properties of, and
compute with, rational numbers expressed in a variety of forms
Learning Targets
4JJ Solve equations of the form and explain why the
solution is +/- a.
4KK Solve equations of the form .
Algebra and Functions:
1.1 Use variables and appropriate operations to write an expression, an
equation, an inequality, or a system
of equations or inequalities that represents a verbal description (e.g.,
three less than a number, half as
large as area A).
2
Learning Targets
4LL Solve equations of the form and explain why the
solution is +/- a.
4MM Solve equations of the form .
1.2 Use the correct order of operations
to evaluate algebraic expressions
such as 3(2x + 5)2.
1.4 Use algebraic terminology (e.g., vari-
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
6
Unit
High Priority Standards
And
Learning Targets*
# Q2
Items
number.
4M Explain how the concepts of ratio, proportion, and percentage are related.
1.6 Calculate the percentage of
increases and decreases of a
quantity.
2
Learning Targets
4N Find the percent of a number.
4O Find percent change (increase/decrease), or use percent to
change a number, and explain the difference.
1.7 Solve problems that involve
discounts, markups, commissions, and profit and
compute simple and compound interest.
Learning Targets
4NN Write a ratio and proportion based on a verbal
description.
4OO Find the percent of a number.
4.0 Students solve simple linear equations and inequalities over the rational numbers.
2
Learning Targets
4PP Use cross multiplication to solve proportion.
Mathematical Reasoning:
1.0 Students make decisions about how
to approach problems
2.0 Students use strategies, skills, and
concepts in finding solutions.
Learning Targets
4P Find percent change (increase/decrease), or use percent to
change a number, and explain the difference.
4Q Solve real world problems involving distance, discounts, markups, and commission, and the construction of scale
models.
2.1 Understand negative wholenumber exponents. Multiply
and divide expressions involving exponents with a
common base.
CAHSEE Prep Instructional Guide 2011-12
Supporting Medium/
Low Priority Standards
& Learning Targets
able, equation, term, coefficient, inequality, expression, constant) correctly.
4QQ Students determine a solution is complete and
move beyond a particular problem by generalizing to other situations.
2
Learning Targets
4R Evaluate a power with a negative exponent.
4S Explain the meaning of a negative exponent.
2.2 Add and subtract fractions by
using factoring to find common denominators.
2
Learning Targets
4T Add and subtract fractions with like denominators.
4U Add and subtract fractions with unlike
denominators.
2.4 Use the inverse relationship
2
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
7
Unit
High Priority Standards
And
Learning Targets*
between raising to a power
and extracting the root of a
perfect square integer; for an
integer that is not square, determine without a calculator
the two integers between
which its square root lies and
explain why.
# Q2
Items
CAHSEE Prep Instructional Guide 2011-12
Supporting Medium/
Low Priority Standards
& Learning Targets
Learning Targets
4V Explain the meaning of the square root of
a number, and why two such numbers
exist.
4W Find the two square roots of a positive
integer.
4X Describe the principal square root.
4Y Find the principal square root of a positive integer.
4Z Explain the method for estimating the
square root of an integer that is not a
perfect square.
4AA Find the cube root of a perfect cube,
and solve equations of the form .
2.5 Understand the meaning of
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.
2
Learning Targets
4BB Explain the meaning of absolute value.
4CC Find the absolute value of rational
numbers.
4DD Solve absolute value equations (|x|=b).
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
8
CAHSEE Prep Instructional Guide 2011-12
Unit
Unit Five: Statistics, Data Analysis and Probability
This unit begins with a discussion of the concept of a
data set. Different statistical measures (range, mean,
median, mode) are used to analyze different data sets.
Tables and graphs are used to represent data that has
been collected through a variety of sampling methods.
The focus then shifts to probability, with a discussion of
theoretical and experimental probability. Using real
world examples, the experimental probability of an event
is found. Given a compound event, all possible outcomes are determined and the theoretical probability is
calculated for each outcome. This concept is framed in a
real world context and used to solve problems involving
proportion and probability, continuing the study of the
unknown.
High Priority Standards
And
Learning Targets*
Statistics, Data Analysis, and
Probability:
1.1 Compute the range, mean,
median, and mode of data
sets.
# Q2
Items
2
Learning Targets
5A Find the range, mean, median, and
mode of a set of different data sets
and explain the results
1.2 Understand how additional
data added to data sets
may affect these computations of measures of central tendency.
Learning Targets
5H Find the range, mean, median, and mode of a set
of different data sets and explain the results
2
Learning Targets
5C Solve real world problems involving
experimental probability.
5D Find all possible outcomes of a compound event.
5E Find the theoretical probability of the
outcome of a compound event.
5F Explain the difference between theoretical and experimental probability.
5G Solve real world problems involving
proportion and probability.
3.5 Understand the difference
between independent and
dependent events
1.4 Know why a specific measure of central tendency (mean, median, mode)
provides the most useful information
in a given context.
Learning Targets
5I Use tables and graphs to represent data sets and
how representation of the data sets can influence
conclusions reached.
Learning Targets
5B Find the range, mean, median, and
mode of a set of different data sets
and explain the results
2.5 Identify claims based on
statistical data and, in
simple cases, evaluate the
validity of the claims.
3.0 Students determine theoretical and experimental
probabilities and use
these to make predictions
about events: SDP 3.1,
SDP 3.2 and SDP 3.3
Supporting Medium/
Low Priority Standards
& Learning Targets
Statistics, Data Analysis, and
Probability:
1.3 Understand how the inclusion or exclusion of outliers affect measures
of central tendency.
2
2.1 Compare different samples of population with the data from the entire
population and identify a situation in
which it makes sense to use a sample.
Learning Targets
5J Collect data using different sampling methods.
2
2.4 Identify data that represent sampling
errors and explain why the sample
(and the display) might be biased.
Learning Targets
5K Use tables and graphs to represent data sets and
how representation of the data sets can influence
conclusions reached.
5L Collect data using different sampling methods.
3.2 Use data to estimate the probability
of future events (e.g., batting averages or number of accidents per
mile driven).
Learning Targets
5M Find the theoretical probability of the outcome of a
compound event.
2
3.4 Understand that the probably of either of two disjoint events occurring
is the sum of the two individual
probabilities and that the probability
of one event following another, in
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
9
Unit
High Priority Standards
And
Learning Targets*
# Q2
Items
CAHSEE Prep Instructional Guide 2011-12 10
Supporting Medium/
Low Priority Standards
& Learning Targets
independent trials, is the product of
the two probabilities.
Learning Targets
5N Explain the difference between theoretical and
experimental probability.
Unit
Unit Six: Algebra and Functions
An expression is a collection of variables linked together
with operations. The value of an expression can be
found for many numbers. An equation, on the other
hand, is simply a true or false statement about the
equality of two expressions. If a value is used for a variable in an equation and each side is evaluated, a statement is made about the validity of the equation. The
meaning of the process of solving an equation is to find
the set of numbers that make the equation true.
The comparison between expressions and equations
begins by combining variables and operations to write
expressions and equations that represent verbal descriptions. The relationship between expression and
equation is thoroughly explored, and a wide variety of
examples are given. Equations of the form
are
introduced. Elementary transformations (e.g. adding to
both sides) are introduced as a way of solving simple
equations.
An inequality is then introduced. An inequality is also
true or false statement, just like an equation. However,
while an equation tests the equivalence of two expressions, an inequality tests the relative size of the two
quantities. The solution set of an inequality is discussed, and examples are given. Elementary transformations are also used to solve an inequality. An illustration of the reason for switching the direction of the inequality sign when multiplying or dividing by a negative
High Priority Standards
And
Learning Targets*
Algebra and Functions:
1.1 Use variables and appropriate operations to write
an expression, an equation, an inequality, or a
system of equations or inequalities that represents
a verbal description (e.g.,
three less than a number,
half as large as area A).
# Q3
Items
2
Learning Targets
6U Explain the difference between a function and an
equation.
6W Explain the definition of a function.
6X Write a function as an input-output table and as a
graph; explain the connection between the table
and the graph.
6Y Explain how the input, output and function are related.
Learning Targets
6A Write a function as an equation in two
variables.
6B Solve word problems involving area
and perimeter.
6C Find the surface area of cylinders,
prisms, and cones.
1.2 Use the correct order of
operations to evaluate algebraic expressions such
as 3(2x + 5)2.
1.5 Represent quantitative relationships graphically
and interpret the meaning
of a specific part of a
graph in the situation represented by the graph.
Learning Targets
6D Write a function as a mapping or set
of ordered pairs.
2.1 Interpret positive wholenumber powers as repeat-
Supporting Medium/
Low Priority Standards
& Learning Targets
Algebra and Functions:
1.0 Students express quantitative relationships by using algebraic terminology, expressions, equations, inequalities, and graphs.
1.4 Use algebraic terminology (e.g., variable, equation, term, coefficient, inequality, expression, constant) correctly.
2
Learning Targets
6Z Explain the difference between a function and an
equation.
6AA Graph linear equations in the coordinate plane.
2
3.0 Students graph and interpret linear
and some nonlinear functions
Learning Targets
6BB Write a function as an input-output table and as a
graph; explain the connection between the table
and the graph.
Number Sense:
1.2 Add, subtract, multiply, and divide
rational numbers (integers, fractions, and terminating decimals) and
take positive rational numbers to
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
Unit
number is explored. The solution sets of equations and
inequalities are compared. The number of solutions of
an inequality (i.e. infinitely-many solutions) illustrates the
need to draw the graph of an inequality. This leads to a
discussion of boundary points, and the conditions for
when a boundary point is included in the solution set,
and when it is not. Inequalities are written based on information from a number line graph. Finally, equations
and inequalities will be viewed in a real world context,
solving problems involving distance, rate, and time.
High Priority Standards
And
Learning Targets*
ed multiplication and negative whole-number powers as repeated division or
multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents.
# Q3
Items
2
CAHSEE Prep Instructional Guide 2011-12 11
Supporting Medium/
Low Priority Standards
& Learning Targets
whole-number powers.
Learning Targets
6CC Simplify both sides of an equation before solving,
including using the distributive property and combining like terms.
6DD Use the multiplicative inverse property to solve
more complex proportions and explain how cross
multiplication is a process used to solve these
proportions.
Learning Targets
6E Solve equations of the form and
explain why the solution is +/- a.
6FSolve equations of the form .
2.2 Multiply and divide monomials; extend the process
of taking powers and extracting roots to monomials when the latter results
in a monomial with an integer exponent.
2
Learning Targets
6G Solve equations of the form and
explain why the solution is +/- a.
6H Solve equations of the form .
3.1 Graph the functions of the
form y = nx2 and y = nx3
and use in solving problems.
Learning Targets
6I Graph functions of the form y=nx2.
6J Graph functions of the form y=nx3.
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 quantities.
2
2
Learning Targets
6K Solve problems where the slope is
the average rate of change.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
Unit
High Priority Standards
And
Learning Targets*
# Q3
Items
CAHSEE Prep Instructional Guide 2011-12 12
Supporting Medium/
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& Learning Targets
6L Solve word problems involving linear
functions.
3.3 Graph linear functions,
noting that the vertical
change (change in yvalue) per unit of horizontal change (change in xvalue) is always the same
and know that the ratio
(“rise over run”) is called
the slope of a graph.
2
Learning Targets
6M Explain the meaning of the slope of a
line, and give examples of an equation whose graph has positive slope
and negative slope.
6N Solve word problems involving linear
functions.
6O Graph linear equations in the coordinate plane.
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.
2
Learning Targets
6P Solve multi-step equations using
transformations.
6Q Solve an equation with variables on
both sides, and explain the situations where there may be no solution or infinitely many solutions.
6R Simplify both sides of an equation
before solving, including using the
distributive property and combining
like terms.
6S Use the multiplicative inverse property to solve more complex proportions and explain how cross multiplication is a process used to solve
these proportions.
6T Read a completed solution to an
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
Unit
High Priority Standards
And
Learning Targets*
# Q3
Items
CAHSEE Prep Instructional Guide 2011-12 13
Supporting Medium/
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& Learning Targets
equation or inequality, find any errors, and rewrite the solution, justifying each step in the process.
6U Solve multi-step inequalities using
transformations, and graph the solution set.
4.2 Solve multistep problems
involving rate, average
speed, distance, and time
or a direct variation.
Learning Targets
6V Write the formulas for volume as a
function of height.
Unit
Unit Seven: Algebra
The introduction to Algebra I unit begins with a discussion of the conceptual understanding of a variable and
how variables are used with operations to build expressions representing a range of different situations. These
expressions are then evaluated using a variety of values
for the variable, further representing the concept of the
variable. Order of operations is reviewed as expressions are evaluated. It is natural to study the individual
terms within an expression, in order to determine if multiple terms can be combined as one. Two terms are defined as like terms if they have the same variable and
the same exponent. Like terms may have different coefficients. Addition and subtraction is used to combine like
terms into one term. Algebraic expressions involving
more than one set of like terms are simplified. The multiplication and division properties of exponents are then
introduced and used to simplify expressions where the
exponents of the factors are integers. It is emphasized
that while multiplying or dividing changes the exponent
and the coefficient, addition and subtraction of like terms
only changes the coefficient. Also, it is emphasized that
multiplication and division can always be used to com-
High Priority Standards
And
Learning Targets*
Algebra and Functions:
2.0: Students understand and
use such operations as
taking the opposite, finding the reciprocal, taking a
root and raising to a fractional power. They understand and use the rules of
exponents
2
# Q3
Items
2
Learning Targets
7A Use the multiplication property of
exponents while using the distributive property to simplify expressions.
3.0: Students solve equations
and inequalities involving
absolute values
Learning Targets
7LL Explain how the process of factoring out the GCF
of an expression is related to the distributive
property
2
24.0: Students use and know simple aspects of a logical argument
Learning Targets
7MM Simplify an expression by combining like terms
and explain how the expressions are affected by
the algebraic properties.
7NN Use the algebraic properties to justify solving
multi-step single variable equations.
Learning Targets
7B Differentiate between and explain the
process of solving single and twostep equations (in one variable).
7C Explain the meaning of an absolute
value and solve multi-step equations containing absolute values.
4.0: Students simplify expressions before solving linear
Supporting Medium/
Low Priority Standards
& Learning Targets
Algebra and Functions:
11.0: Students apply basic factoring
techniques to second- and simple
third-degree polynomials. These
techniques include finding a common factor for all terms in a polynomial, recognizing the difference or
two squares, and recognizing perfect
squares of binomials.
2
25.0: Students use properties of the
number system to judge the validity
of results, to justify each step of a
procedure, and to prove or disprove
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
Unit
bine expressions, while addition and subtraction can
only be used to combine like terms. Multiplication of
expressions, and combining like terms are both used to
simplify expressions where a single term is being multiplied by an expression with several like terms (e.g. ).
These types of expressions are can be simplified by
combining like terms and then multiplying the resulting
two terms or first by multiplying term by term and then
combining like terms. This naturally leads to the question of how to multiply a single term by an expression
that does not contain like terms (e.g. ). The distributive
property is then introduced and used to simplify a wide
variety of such expressions. Other properties of real
numbers are then discussed including the commutative,
associative, identity, inverse, and closure properties.
The concept of the greatest common factor (GCF) of a
set of numbers from previous courses is generalized to
an understanding of the GCF of an algebraic expression, which is used to factor an expression using reverse
distribution. An expression is simply a collection of variables linked by operations. However an equation is a
statement that two expressions are equal. The process
of solving an equation is really the process of finding a
set of numbers (called the solution set) that, when substituted for the variable in the expressions, make the
equation a true statement. The difference between a
single solution, and a solution set is highlighted. In this
unit, properties of equality are used to develop methods
of solving equations. The algebraic properties of equality
for the four operations (i.e. the elementary transformations) are introduced and used to solve single step
and multi-step linear equations in one variable. Methods
of simplification (combining like terms, distributive property) are used to simplify both sides of an equation before solving. Equations in one variable with no solution
(e.g. ), one solution (e.g. ), and infinitely-many solutions
(e.g. ) are compared and contrasted. The concept of
absolute value is then reviewed in terms of distance
from 0 on a number line, which leads to an exploration
of equations with an absolute value term. The different
High Priority Standards
And
Learning Targets*
equations and inequalities
in one variable, such as
3(2x-5) + 4(x-2) = 12
# Q3
Items
Learning Targets
7OO Evaluate expressions using the order of operations correctly and justify each step
7PP Use variables and operations to construct expressions that represent mathematical situations
and explain your reasoning
7QQ Simplify an expression by combining like terms
and explain how the expressions are affected by
the algebraic properties.
7RR Simplify equations before solving multi-step
equations and explain the properties used to simplify
7SS Solve equations with variables on both sides of
the equation and explain how the algebraic properties are used in the process.
Learning Targets
7D Simplify an expression by combining
like terms and explain how the expressions are affected by the algebraic properties.
7E Explain how the process of factoring
out the GCF of an expression is related to the distributive property.
7F Explain how the concept of equivalence is used in simplifying algebraic expressions and connect to the
property of equality.
7G Simplify equations before solving
multi-step equations and explain
the properties used to simplify
7H Solve equations with variables on
both sides of the equation and explain how the algebraic properties
are used in the process.
5.0: Students solve multi-step
problems, including word
problems, involving linear
equations and linear inequalities in one variable
and provide justification
for each step.
16.0: Students understand the concepts
of a relation and a function, determine whether a given relation defines a function, and give pertinent
information about given relations
and functions.
2
Learning Targets
7TT Find the domain and range of a set of data when
given the ordered pairs
7UU Graph a line by building a function table and use
the function table to describe how the line represents a function.
18.0: Students determine whether a relation defined by a graph, a set of ordered pairs, or symbolic expression
is a function and justify the conclusion
Learning Targets
7I Use variables and operations to construct expressions that represent
mathematical situations and explain
your reasoning
7J Differentiate between and explain the
process of solving single and twostep equations (in one variable).
7K Use relationships within word problems to solve complex applied
problems
6.0: Students graph a linear
equation and compute the
x- and y- intercepts (e.g.,
graph 2x + 6y = 4). They
are also able to sketch the
region defined by linear
CAHSEE Prep Instructional Guide 2011-12 14
Supporting Medium/
Low Priority Standards
& Learning Targets
statements.
Learning Targets
7VV Graph a line by building a function table and use
the function table to describe how the line represents a function.
21.0: Students graph quadratic functions and know that their roots are xintercepts
2
Learning Targets
7WW Find the x and y intercept of a linear equation
and explain the difference between the two.
7XX Define the root/zero/solution of a linear function
and find the roots/zeros/solutions of a linear function.
24.0: Students use and know simple as-
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
Unit
conditions for the number of solutions of an absolute
value equation are also explored. Finally, equations in
one variable are studied from a real world context, including rate problems, work problems, and percent mixture problems.
High Priority Standards
And
Learning Targets*
inequality (e.g., they
sketch the region defined
by 2x + 6y < 4).
# Q3
Items
Learning Targets
7YY Graph a line by building a function table and use
the function table to describe how the line represents a function.
Learning Targets
7L Solve an equation for one variable in
terms of another and justify each
step
7M Derive the formula for slope and use
it to find the slope of a given line.
7N Graph a line using slope intercept
form and explain the process of
transforming the equation into a
graph.
7O Find the x and y intercept of a linear
equation and explain the difference
between the two.
7P Define the root/zero/solution of a
linear function and find the
roots/zeros/solutions of a linear
function.
Learning Targets
7Q Graph a system of linear equations
and estimate the solution based on
the graph.
7R Solve a system of equations by substituting one equation into the other
and solving for x and y coordinate
of the solution and explain the process
7.0: Students verify that a
point lies on a line, given
an equation on the line.
Students are able to derive linear equations by
using the point-slope formula.
Learning Targets
7S Determine graphically whether a
point is a solution to a linear equation.
7T Determine algebraically whether a
point lies on the line (whether or not
the ordered pair is a solution to the
equation).
7U Write the equation for a line when
given the slope and a point on the
line and explain which form (slopeintercept/point-slope) I prefer and
CAHSEE Prep Instructional Guide 2011-12 15
Supporting Medium/
Low Priority Standards
& Learning Targets
pects of a logical argument.
25.0: Students use properties of the
number system to judge the validity
of results, to justify each step of a
procedure, and to prove or disprove
statements.
Learning Targets
7ZZ Solve an equation for one variable in terms of
another and justify each step
7AAA Write the equation for a line when given two
points that lie on the line and explain the process
one goes through to do this.
24.0: Students use and know simple aspects of a logical argument.
Learning Targets
7BBB Explain the meaning of a system of linear equations and a solution to a system of linear equations
7CCC Explain the reason why not all systems of equations have the same number of solutions.
2
25.0: Students use properties of the
number system to judge the validity
of results, to justify each step of a
procedure, and to prove or disprove
statements.
Learning Targets
7DDD Verify that an ordered pair is a solution to a
system by substituting the x and y coordinate into
each equation of the system and explain each
step of the process
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
Unit
High Priority Standards
And
Learning Targets*
# Q3
Items
CAHSEE Prep Instructional Guide 2011-12 16
Supporting Medium/
Low Priority Standards
& Learning Targets
why.
7V Write the equation for a line when
given two points that lie on the line
and explain the process one goes
through to do this.
7W Verify that an ordered pair is a solution to a system by substituting the
x and y coordinate into each equation of the system and explain each
step of the process
8.0: Students understand the
concepts of parallel lines
and perpendicular lines
and how those slopes are
related. Students are able
to find the equation of a
line perpendicular to a
given line that passes
through a given point.
2
Learning Targets
7X Determine graphically whether a
point is a solution to a linear equation.
7Y Derive the formula for slope and use
it to find the slope of a given line.
7Z Write the equation for a line when
given the slope and a point on the
line and explain which form (slopeintercept/point-slope) I prefer and
7AA Use knowledge of slopes to identify
systems of parallel and perpendicular lines as well as write equations
for lines that are parallel or perpendicular and explain
9.0: Students solve a system
of two linear equations in
two variables algebraically
and are able to interpret
the answer graphically.
Students are able to solve
a system of two linear inequalities in two variables
and to sketch the solution
sets.
2
Learning Targets
7BB Explain the meaning of a system of
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
Unit
High Priority Standards
And
Learning Targets*
# Q3
Items
CAHSEE Prep Instructional Guide 2011-12 17
Supporting Medium/
Low Priority Standards
& Learning Targets
linear equations and a solution to a
system of linear equations
7CC Use knowledge of slopes to identify
systems of parallel and perpendicular lines as well as write equations
for lines that are parallel or perpendicular and explain
7DD Explain the reason why not all systems of equations have the same
number of solutions.
7EE Graph a system of linear equations
and estimate the solution based on
the graph.
7FF Verify that an ordered pair is a solution to a system by substituting the
x and y coordinate into each equation of the system and explain each
step of the process
7GG Solve a system of equations by
substituting one equation into the
other and solving for x and y coordinate of the solution and explain
the process
7HH Solve a system of linear equations
using linear combinations and explain the process of doing so.
7II I can justify my reasoning for choosing one method of solving a system
of equations over another method,
while still explaining how all methods could lead to the same solution.
10.0: Students add, subtract,
multiply, and divide monomials and polynomials.
Students solve multi-step
problems, including word
problems, by using these
techniques.
2
Learning Targets
7JJ Multiply two polynomials to determine the relationship between factored form and standard form of the
quadratic.
15.0: Students apply algebraic
techniques to solve rate
problems, work problems,
and percent mixture prob-
2
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
Unit
High Priority Standards
And
Learning Targets*
lems
# Q3
Items
CAHSEE Prep Instructional Guide 2011-12 18
Supporting Medium/
Low Priority Standards
& Learning Targets
Learning Targets
7KK Use relationships within word problems to solve complex applied
problems
Unit
High Priority Standards
And
Learning Targets*
# Q3
Items
Supporting Medium/
Low Priority Standards
& Learning Targets
Unit Eight: Geometry Review
This unit is a focused Geometry review based on Quarter Two Benchmark results.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.