1
Overview of CAHSEE Blueprint
Algebra 1 Rescue! Correlations Document
First, it’s important to note that Algebra 1 Rescue! was written to answer the question, “How do I
meet the needs of all my students so they experience success in algebra?” With the increased
mathematics requirements for graduation, teachers are faced with a different population of
students taking higher-level mathematics. Algebra 1 Rescue! provides the materials for small
group intervention, as a supplement to your basal series program, or as a standalone program for
students who need another chance to succeed at algebra.
Algebra 1 Rescue! is an intervention program based specifically on algebra concepts and skills.
The objectives, or outcomes, are specified for each chapter; with approximately five objectives
for each chapter. There are 8 to 10 activities per lesson covering a range of mathematical content
strands with the main focus of supporting the algebra objective in the lesson. Students use
problem-solving and critical thinking methods to gain conceptual understanding of the content
area.
As you look at the correlations document, you will see that there are different types of
connections between the materials and the exit exam blueprint. For each of the main content
areas (highlighted in gray with the total number of items indicated), we provide a quick-glance
correlation of lessons to the strand. Then in the sub-headings under the main strand, we provide
one or more examples of problems used in the materials to support the particular standard in
question. Sometimes, Algebra 1 Rescue! is a direct match with the item in the blueprint
document (e.g. in the 7th grade and Algebra 1 level standards), and sometimes it provides a
preskill or definition necessary for understanding the item. We list the main objective of the
lesson and then provide an example of how the activity within the lesson supports the objective.
Because the materials are based on the algebra standards, there are some content areas that are not
covered in as much depth. We have noted these areas in the document. However, many of the
other mathematical strands, such as geometry, measurement, data, probability and statistics, are
introduced in the lessons as supporting activities. This provides students with rich opportunities
to use the algebra skills in contextual and application activities.
In the area of Mathematical Reasoning and Proof, we provide a paragraph describing the way in
which the program uses the process strands in virtually every lesson – communication, problem
solving, representation, connections and reasoning. These process strands are an important part
of teaching for conceptual understanding. Students do a lot with informal proof and derivation of
formulas and mathematical ideas and procedures, as well as checking for validity and
appropriateness within the context in which the problem is framed.
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2
Correlations Document for Algebra 1 RESCUE!
and the California High School Exit Exam (CAHSEE)
Mathematics Blueprint – Revised July 2003
California Content Strand
Algebra 1 RESCUE! Coverage
Grade 6 – Statistics, Data Analysis, and The following Algebra 1 Rescue! Lessons cover
Probability (TOTAL of 8 Items)
Statistics, Data Analysis, and Probability topics:
2-1, 2-3, 3-5, 4-2, 4-3, 5-3, 9-1
1.0 Students compute and analyze statistical
measurements for data sets.
1.1 Compute the range, mean, median, and mode
of data sets (3 items)
NOTE: One or more examples are given for each of
the specific standards below, where applicable.
2-1 Objective: Graph rational numbers on the
number line.
Sample Supporting Activity: Students collect
information from classmates regarding random
rolling of dice. Students compute the average
number of rolls to reach a certain benchmark.
1.2 Understand how additional data added to data
5-3 Objective: Draw a best-fit line and find the
sets may affect these computations of measures equation of the best-fit line for a scatter plot.
of central tendency. (0 items)
Sample Supporting Activity: Students collect data
about boys’ heights in the class and compare the data.
Then they collect data from other classes. They look
at averages of these different data sets.
1.3 Understand how the inclusion or exclusion of
outliers affects measures of central tendency
(0 items)
5-3 Objective: Draw a best-fit line and find the
equation of the best-fit line for a scatter plot.
Sample Supporting Activity: Students discuss the
best positioning of the line for a scatter plot. Some of
the discussion points include why it’s not drawn
through the point at the top or the point at the bottom.
1.4 Know why a specific measure of central
tendency (mean, median, mode) provides the
most useful information in a given context.
(0 items)
Algebra 1 Rescue! is an intervention program based
specifically on algebra standards. Therefore, some
of the other math strands that are not directly
connected to algebra and/or do not provide support
of specific algebra topics are not covered in these
materials.
3
2.0 Students use data samples of a population and
describe the characteristics and limitations of the
samples:
NOTE: One or more examples are given for each of
the specific standards below, where applicable.
2.1 Compare different samples of a population
with the data from the entire population and
identify a situation in which it makes sense to
use a sample. (0 items)
5-3 Objective: Draw a best-fit line and find the
equation of the best-fit line for a scatter plot.
Sample Supporting Activity: Students compare data
they collect themselves re: boys’ heights from their
class and from other classes. Students compare the
collection of this data with projections about how
data is collected for growth charts used in a doctor’s
office.
2.2 Identify different ways of selecting a sample
(e.g., convenience sampling, responses to a
survey, random sampling) and which method
makes a sample more representative for a
population. (0 items)
5-3 Objective: Draw a best-fit line and find the
equation of the best-fit line for a scatter plot.
Sample Supporting Activity: Students compare data
they collect themselves re: boys’ heights from their
class and from other classes; and then make
predictions about the data. Then the discussion is
extended to how growth charts are created for
doctors’ offices dealing with a larger population.
2.3 Analyze data displays and explain why the way
in which the question was asked might have
influenced the results obtained and why the
way in which the results were displayed might
have influenced the conclusions reached.
(0 items)
Algebra 1 Rescue! is an intervention program based
specifically on algebra standards. Therefore, some
of the other math strands that are not directly
connected to algebra and/or do not provide support
of specific algebra topics are not covered in these
materials.
2.4 Identify data that represent sampling errors
and explain why the sample (and the display)
might be biased. (0 items)
Algebra 1 Rescue! is an intervention program based
specifically on algebra standards. Therefore, some
of the other math strands that are not directly
connected to algebra and/or do not provide support
of specific algebra topics are not covered in these
materials.
4
2.5 Identify claims based on statistical data and,
in simple cases, evaluate the validity of the
claims. (1 item)
3.0 Students determine theoretical and
experimental probabilities and use these to make
predictions about events:
9-1 Objective: Find the greatest common factor
through prime factorization for integers and sets
of monomials.
Sample Supporting Activity: collect data and make
predictions in class experiment and then analyze the
results.
NOTE: One or more examples are given for each of
the specific standards below, where applicable.
3.1 Represent all possible outcomes for compound
events in an organized way (e.g., tables, grids,
tree diagrams) and express the theoretical
probability of each outcome. (1 item)
2-3 Objective: Compare/Order Rational Numbers.
Sample Supporting Activity: Find the average
monthly temperatures for many different cities (using
an almanac or other reference book on climate) and
make a chart that shows average high and low
temperature. Graph, rank, and compare the average
temperatures.
3.2 Use data to estimate the probability of future
events (e.g., batting averages or number of
accidents per mile driven). (0 items)
5-3 Objective: Draw a best-fit line and find the
equation of the best-fit line for a scatter plot.
Sample Supporting Activity: Make predictions about
data.
3.3 Represent probabilities as ratios, proportions,
decimals between 0 and 1, and percentages
between 0 and 100 and verify that the
probabilities computed are reasonable; know
that if P is the probability of an event, 1 –P is
the probability of an event not occurring.
(2 items)
3-5 Objective: Solve a proportion with
a missing part.
Sample supporting Activity: Students look at
outcomes of events and collect, record, and graph
data. Students use a proportion to solve for missing
parts of probabilities.
3.4 Understand that the probability 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 independent
trials, is the product of the two probabilities.
(0 items)
Algebra 1 Rescue! is an intervention program based
specifically on algebra standards. Therefore, some
of the other math strands that are not directly
connected to algebra and/or do not provide support
of specific algebra topics are not covered in these
materials.
5
3.5 Understand the difference between
independent and dependent events. (1 item)
4-3 Objective: Determine the range for a given
domain of a relation.
Sample Supporting Activity: Students look at
independent vs. dependent variables.
Grade 7 – Number Sense
(TOTAL of 14 items)
The following Algebra 1 Rescue! Lessons cover
Number Sense Topics: 1-1, 1-2, 1-3, 1-4, 2-1, 2-2,
2-3, 2-4, 2-5, 3-1, 3-2, 3-3, 3-4, 3-5, 3-6, 4-1, 4-2, 43, 4-4, 4-5, 5-1, 5-2, 5-3, 5-4, 5-5, 6-1, 6-2, 6-3, 6-4,
6-5, 7-1, 7-2, 7-3, 7-4, 7-5, 8-1, 8-2, 8-3, 8-4, 8-5, 91, 9-2, 9-3, 9-4, 9-5, 10-1, 10-2, 10-3, 10-4, 11-1, 112, 11-3, 11-4, 11-5, 12-1, 12-2, 12-3, 12-4, 12-5.
1.0 Students know the properties of, and compute
with, rational numbers expressed in a variety of
forms:
NOTE: One or more examples are given for each of
the specific standards below, where applicable.
1.1 Read, write, and compare rational numbers in
scientific notation (positive and negative
powers of 10) with approximate numbers using
scientific notation. (1 item)
8-2 Objective: Write numbers in scientific
notation and find products and quotients of these
numbers.
Sample Supporting Activity: Students review place
value with large numbers and with small numbers
(the places to the right of the decimal). Students
review the terms scientific and standard notation.
Students are involved in problem-solving activities
that support the topic – “magic chessboard” and
“bacteria growth”
1.2 Add, subtract, multiply, and divide rational
numbers (integers, fractions, and terminating
decimals) and take positive rational numbers to
whole-number powers. (3 items)
2-2 Objective: Add and subtract rational
numbers.
Sample Supporting Activity: Students work with the
rules for rules for adding and subtracting rational
numbers and graph the results on a number line.
2-4 Objective: Multiply and divide rational
numbers.
Sample Supporting Activity: Students work with the
rules for multiplying and dividing rational numbers
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by proving or disproving through the use of
examples.
8-1 Objective: Multiply and divide monomials
and simplify expressions.
Sample Supporting Activity: Students look at powers
as repeated multiplication; rules about performing
operations on powers.
1.3 Convert fractions to decimals and percents and
use these representations in estimations,
computations, and applications. (2 items)
3-6 Objective: Use proportions to solve percent
problems
Sample Supporting Activity: Students convert fraction
to decimal to percent; problem solving real-life
percent problems; use newspapers and magazines to
collect data to make up percent problems.
1.4 Differentiate between rational and irrational
numbers. (0 items)
11-5 Objective: Solve rational equations.
Sample Supporting Activity: Students review the
definition of rational numbers.
1.5 Know that every rational number is either a
terminating or repeating decimal and be able to
convert terminating decimals into reduced
fractions (0 items)
11-1 Objective: Simplify rational expressions.
Sample Supporting Activity: Students review the
definition of rational expressions and review
reducing fractions.
1.6 Calculate the percentage of increases and
decreases of a quantity. (1 item)
10-4 Objective: Graph exponential functions and
solve problems using the graphs.
Sample Supporting Activity: Students discuss
exponential growth and decay; as well as, investment
problems.
1.7 Solve problems that involve discounts,
markups, commissions, and profit, and
compute simple and compound interest.
3-6 Objective: Use proportions to solve percent
problems.
Sample Supporting Activity: Students are given the
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(2 items)
opportunity to do some problem solving with real-life
percent problems. Students use newspapers and
magazines to collect data to make up percent
problems of their own.
2.0 Students use exponents, powers, and roots, and
use exponents in working with fractions:
NOTE: One or more examples are given for each of
the specific standards below, where applicable.
2.1 Understand negative whole-number exponents.
Multiply and divide expressions involving
exponents with a common base. (1 item)
8-1 Objective: Multiply and divide monomials
and simplify expressions.
Sample Supporting Activity: Students review rules
about performing operations on powers.
10-4 Objective: Graph exponential functions and
solve problems using the graphs.
Sample Supporting Activity: Students review zero
and negative powers.
2.2 Add and subtract fractions by using factoring
to find common denominators. (1 item)
9-1 Objective: Find the greatest common factor
through prime factorization for integers and sets
of monomials.
Sample Supporting Activity: Students review the
terms factor and greatest common factor (GCF).
11-4 Objective: Add and subtract rational
expressions.
Sample Supporting Activity: Students review
common denominators and least common
denominators.
11-5 Objective: Solve rational equations.
Sample Supporting Activity: Students continue to
review least common denominators.
2.3 Multiply, divide, and simplify rational numbers 8-1 Objective: Multiply and divide monomials
by using exponent rules. (1 item)
and simplify expressions.
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Sample Supporting Activity: Students review rules
about performing operations on powers.
2.4 Use the inverse relationship 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. (1 item)
2-5 Objective: Find the principal square root of a
number.
Sample Supporting Activity: Students look at the
inverse relationship between raising a power and
extracting the root of a perfect square by examining
the inputs and outputs of operations in an imaginary
“computing machine”.
Another Sample Supporting Activity (within the same
lesson): Students practice estimating square roots by
finding benchmarks on each side of an integer that is
not a perfect square. Students use a number line to
help find the benchmarks.
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. (1 item)
Grade 7 – Algebra & Functions
(TOTAL of 17 items)
1.0 Students express quantitative relationships by
using algebraic terminology, expressions,
equations, inequalities, and graphs
1.1 Use variables and appropriate operations to
6-4 Objective: Solve and graph the solution set of
compound inequalities and inequalities involving
absolute value.
Sample Supporting Activity: Students look at a
problem such as |x| = 4 on a number line. Which
numbers on the number line are 4 units from 0?
Students discuss absolute value and distance.
The following Algebra 1 Rescue! Lessons cover
Algebra & Functions Topics: : 1-1, 1-2, 1-3, 1-4,
2-1, 2-2, 2-3, 2-4, 2-5, 3-1, 3-2, 3-3, 3-4, 3-5, 3-6, 41, 4-2, 4-3, 4-4, 4-5, 5-1, 5-2, 5-3, 5-4, 5-5, 6-1, 6-2,
6-3, 6-4, 6-5, 7-1, 7-2, 7-3, 7-4, 7-5, 8-1, 8-2, 8-3, 84, 8-5, 9-1, 9-2, 9-3, 9-4, 9-5, 10-1, 10-2, 10-3, 10-4,
11-1, 11-2, 11-3, 11-4, 11-5, 12-1, 12-2, 12-3, 12-4,
12-5.
NOTE: One or more examples are given for each of
the specific standards below, where applicable.
1-1 Objective: Translate verbal expressions into
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write an expression, an equation, an inequality,
or a system of inequalities that represents a
verbal description (e.g., three less than a
number, half as large as area A). (2 items)
mathematical expressions and vice versa.
Sample Supporting Activity: Students translate
verbal statements to numeric expressions and vice
versa.
3-4 Objective: Solve problems that can be
represented as equations.
Sample Supporting Activity: Students translate word
problems to equations and vice versa.
6-3 Objective: Solve and graph the solution set of
inequalities using more than one operation.
Sample Supporting Activity: Students translate word
problems to inequalities and vice versa.
7-5 Objective: Solve systems of inequalities by
graphing.
Sample Supporting Activity: Students translate
word problems to systems of inequalities, and vice
versa.
1.2 Use the correct order of operations to evaluate
algebraic expressions such as 3 (2x + 5)².
(1 item)
1-2 Objective: Evaluate expressions by using the
order of operations.
Sample Supporting Activity: Students begin solving
numeric expressions using the order of operations.
10-3 Objective: Solve quadratic equations by …
using the quadratic formula.
Sample Supporting Activity: Students work with
order of operations and algebraic expressions
throughout the materials. Applying order of
operations to solve quadratics is the high-end of this
concept.
1.3 Simplify numerical expressions by applying
properties of rational numbers (e.g., identity,
inverse, distributive, associative, commutative)
and justify the process used. (0 items)
1-4 Objective: Use mathematical properties to
evaluate expressions.
Sample Supporting Activity: Most of the
properties are introduced in this lesson.
The properties are discussed in more depth in the
following lessons :
-
additive identity – Lessons 1-4 and 2-2
associative – Lessons 1-4 and 8-4
10
-
1.4 Use algebraic terminology (e.g., variable,
equation, term, coefficient, inequality,
expression, constant) correctly. (0 items)
1.5 Represent quantitative relationships
graphically and interpret the meaning of a
specific part of a graph in the situation
represented by the graph.
commutative – Lessons 1-4, 2-4, and 8-4
density – Lesson 2-1
distributive – Lessons 1-4, 8-5, 9-1, 9-2, 10-4
and 12-1
multiplicative identity – Lessons 1-4 and 10-3
multiplicative inverse – Lessons 1-3 and 1-4
properties of equality – Lessons 3-1, 3-2, 3-3,
3-5, and 5-4
zero product property - Lessons 9-2, 9-3, 9-4,
10-2, and 10-3
Algebraic terminology is defined and used
throughout the materials. For example, the
following words are defined in the indicated lessons:
- “variable” and “expression” – Lessons 1-1
- “equation” – Lesson 3-1
- “coefficient”, “constant”, “term” – Lesson 3-1
- “inequality” – first defined in Lesson 2-3 and
then used extensively in 6-1 through 6-5.
5-1 Objective: Determine the slope of a line from a
graph or given two points on the line.
Sample Supporting Activity: Students find linear
graphs in a newspaper or magazine and interpret
them.
5-5 Objective: Use the slope of lines to determine
if two lines are parallel or perpendicular.
Sample Supporting Activity: Students determine
whether lines are parallel or perpendicular by
associating parts of the equation with parts of the
graph.
2.0 Students interpret and evaluate expressions
involving integer powers and simple roots.
NOTE: One or more examples are given for each of
the specific standards below, where applicable.
2.1 Interpret positive whole-number powers as
repeated multiplication and negative wholenumber powers as repeated division or
multiplication by the multiplicative inverse.
Simplify and evaluate expressions that include
1-1 Objective: Translate verbal expressions into
mathematical expressions and vice versa.
Sample Supporting Activity: Students discuss the
mathematical terms “powers”, “squared”, and
“cubed”.
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exponents. (1 item)
8-1 Objective: Multiply and divide monomials
and simplify expressions.
Sample Supporting Activity: Students look at the
rules for operating with powers. They simplify and
evaluate expressions with exponents, such as:
• 5³ · 5² = (5 · 5 · 5) · (5 · 5); and
• x³/x³ = x³ ¯ ³ = xº = 1 (x ≠ 0)
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. (1 item)
8-1 Objective: Multiply and divide monomials
and simplify expressions.
Sample Supporting Activity: Students extend the
process of taking powers and extracting roots by
writing and exchanging problems to solve. Students
are given directions to write algebraic expressions
for the multiplication and/or division of two
monomials, with the following criteria:
• each must have a coefficient
• each may have up to three variables
• and must have exponents from 1 to 10
Students work in groups and each student should
work all the problems his/her group created.
Multiplication Example: (7x³ y²)(-2xy)
Division Example: 6xy²
2x³y
3.0 Students graph and interpret linear and some
nonlinear functions.
NOTE: One or more examples are given for each of
the specific standards below, where applicable.
3.1 Graph functions of the form y = nx² and y =
nx³ and use in solving problems. (1 item)
10-4 Objective: Graph exponential functions and
solve problems using the graphs.
Sample Supporting Activity: Students discuss the
shape of the graph for a problem like y = 2ⁿ.
Students solve problems involving exponential
growth and decay.
12
3.2 Plot the values from the volumes of threedimensional shapes for various values of the
edge lengths (e.g., cubes with varying edge
lengths or a triangle prism with a fixed height
and an equilateral triangle base of varying
lengths). (0 items)
12-4 Objective: Find the distance between two
points in the coordinate plane.
Sample Supporting Activity: Students draw triangles
on a coordinate grid and show the distance formula
generalized from the Pythagorean theorem as a
preskill activity [students do not look at 3-D shapes
plotted on a graph].
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. (2 items)
5-1 Objective: Determine the slope of a line from a
graph or given two points on the line.
Sample Supporting Activity: Students explore the
concept of slope using 2 rulers. They discuss slope
as being the ratio represented by “rise over run”.
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 a line equals the
quantities. (1 item)
5-3 Objective: Draw a best-fit line and find the
equation of the best-fit line for a scatter plot.
Sample Supporting Activity: Students look at average
costs for phone rates and compare one long-distance
company to another. Student plot the results on a
graph and get a scatter plot. Students are asked to
draw the “best fit line. Students determine the slope
from the line they draw.
4.0 Students solve simple linear equation sand
inequalities over the rational numbers.
NOTE: One or more examples are given for each of
the specific standards below, where applicable.
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. (3 items)
3-3 Objective: Solve linear equations using one or
more operations.
Sample Supporting Activity: Students solve linear
equations using the properties of equality. Students
identify the principal operator in a problem. The
problems include a variety of numbers – fractions,
decimals, signed numbers, whole numbers.
6-3 Objective: Solve and graph the solution set of
inequalities using more than one operation.
Sample Supporting Activity: Students solve two-step
inequalities and discuss changes in the properties
from working with linear equations, particularly
when to reverse the relation. Students substitute the
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values for the variable back into the problem to check
it. Is the statement of inequality true? Students
solve inequality word problems.
4.2 Solve multistep problems involving rate,
average speed, distance, and time or a direct
variation. (2 items)
5-4 Objective: Write linear equations in the slopeintercept form to find the slope, x-intercept, and
y-intercept, and sketch the graph.
Sample Supporting Activity: Students are asked a
series of questions given the following data about a
long-distance phone service: the phone rates for
company D are computed by using the formula R =
.07t +.26, where R is the rate and t is the time in
minutes.
Grade 7 – Measurement and Geometry
(TOTAL of 17 items)
The following Algebra 1 Rescue! Lessons cover
measurement and geometry topics: 1-2, 2-5, 3-4,
3-5, 4-1, 4-3, 4-4, 5-1, 5-2, 5-3, 5-4, 5-5, 6-1, 6-2, 63, 6-4, 6-5, 7-1, 7-2, 7-3, 7-4, 7-5, 8-1, 8-4, 9-2, 9-3,
9-4, 10-1, 10-2, 11-1, 11-5, 12-1, 12-2, 12-3, 12-4,
12-5.
1.0 Students choose appropriate units of measure
and use ratios to convert within and between
measurement systems to solve problems:
NOTE: One or more examples are given for each of
the specific standards below, where applicable.
1.1 Compare weights, capacities, geometric
measures, times, and temperature within and
between measurement systems (e.g., miles per
hour and feet per second, cubic inches to cubic
centimeters). (2 items)
Algebra 1 Rescue! is an intervention program based
specifically on algebra standards. Therefore, some
of the other math strands that are not directly
connected to algebra and/or do not provide support
of specific algebra topics are not covered in these
materials.
1.2 Construct and read drawings and models made
to scale. (1 item)
12-5 Objective: Find the unknown measures of the
sides of similar triangles.
Sample Supporting Activity: Students create models
to scale that connect similarity with proportions and
ratios. We also define the “constant of
proportionality” in this lesson.
1.3 Use measures expressed as rates (e.g., speed,
density) and measures expressed as products
5-3 Objective: Draw a best-fit line and find the
equation of the best-fit line for a scatter plot.
14
(e.g., person-days) to solve problems; check
the units of the solutions; and use dimensional
analysis to check the reasonableness of the
answer. (2 items)
Sample Supporting Activity: Students look at average
costs for phone rates and compare one long-distance
company to another. Students measure using
products and dimensional analysis to check the
reasonableness of the answer.
2.0 Students compute the perimeter, area, and
volume of common geometric objects and use the
results to find measures of less common objects.
They know how perimeter, area and volume are
affected by changes of scale:
NOTE: One or more examples are given for each of
the specific standards below, where applicable.
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
rectangles, parallelograms, trapezoids, squares,
triangles, circles, prisms, and cylinders. (3
items)
1-2 Objective: Evaluate expressions by using the
order of operation.
Sample Supporting Activity: Students review solving
formulas for perimeter and area.
8-1 Objective: Multiply and divide monomials
and simplify expressions.
Sample Supporting Activity: Students look at how
changes in dimension of a square affect perimeter
and area.
11-5 Objective: Solve rational equations.
Sample Supporting Activity: Students determine a
missing dimension given the perimeter of a
parallelogram and triangle.
12-3 Objective: Use the Pythagorean theorem to
solve problems.
Sample Supporting Activity: Students calculate the
measurements for a 3-dimensional shape.
2.2 Estimate and compute the area of more
complex or irregular two- and threedimensional figures by breaking the figures
down into more basic geometric objects.
(2 items)
12-3 Objective: Use the Pythagorean theorem to
solve problems.
Sample Supporting Activity: A quadrilateral is drawn
inside a square with dimensions a, b, and c. Students
are asked to show that the area of the larger square is
equal to the sum of the areas of the four triangles plus
the area of the smaller square.
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
Algebra 1 Rescue! is an intervention program based
specifically on algebra standards. Therefore, some
of the other math strands that are not directly
15
solids. Understand that when the lengths of all connected to algebra and/or do not provide support
dimensions are multiplied by a scale factor, the of specific algebra topics are not covered in these
surface area is multiplied by the square of the
materials.
scale factor and the volume is multiplied by the
cube of the scale factor. (1 item)
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]). (1 item)
Algebra 1 Rescue! is an intervention program based
specifically on algebra standards. Therefore, some
of the other math strands that are not directly
connected to algebra and/or do not provide support
of specific algebra topics are not covered in these
materials.
NOTE: One or more examples are given for each of
3.0 Students know the Pythagorean theorem and
the specific standards below, where applicable.
deepen their understanding of plane and solid
geometric shapes by constructing figures that
meet given conditions and by identifying attributes
of figures:
3.1 Identify and construct basic elements of
geometric figures (e.g., altitudes, mid-points,
diagonals, angle bisectors, and perpendicular
bisectors; central angles, radii, diameters, and
chords of circles) by using a compass and
straightedge. (0 items)
12-3 Objective: Use the Pythagorean theorem to
solve problems.
Sample Supporting Activity: Students look at the
concept of a diagonal in a 3-dimensional figure.
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. (2 items)
Algebra 1 Rescue! is an intervention program based
specifically on algebra standards. Therefore, some
of the other math strands that are not directly
connected to algebra and/or do not provide support
of specific algebra topics are not covered in these
materials.
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
12-3 Objective: Use the Pythagorean theorem to
solve problems.
Sample Supporting Activity: Students look at the
Pythagorean theorem geometrically. Next, students
8-1 Objective: Multiply and divide monomials
and simplify expressions.
Sample Supporting Activity: Students review
definitions surrounding circles; and find the area of a
circle.
16
situations, empirically verify the Pythagorean
theorem by direct measurement. (2 items)
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. (1 item)
3.5 Construct two-dimensional patterns for threedimensional models, such as cylinders, prisms,
and cones. (0 items)
3.6 Identify elements of three-dimensional
geometric objects (e.g., diagonals of
rectangular solids) and describe how two or
more objects are related in space (e.g., skew
lines, the possible ways three planes might
intersect). (0 items)
draw triangles and squares on graph paper and look at
the concept there, deriving the formula. Students
then solve problems involving the Pythagorean
theorem.
Algebra 1 Rescue! is an intervention program based
specifically on algebra standards. Therefore, some
of the other math strands that are not directly
connected to algebra and/or do not provide support
of specific algebra topics are not covered in these
materials.
Algebra 1 Rescue! is an intervention program based
specifically on algebra standards. Therefore, some
of the other math strands that are not directly
connected to algebra and/or do not provide support
of specific algebra topics are not covered in these
materials.
Algebra 1 Rescue! is an intervention program based
specifically on algebra standards. Therefore, some
of the other math strands that are not directly
connected to algebra and/or do not provide support
of specific algebra topics are not covered in these
materials.
Grade 7 – Statistics, Data Analysis, and The following Algebra 1 Rescue! Lessons cover
Probability (TOTAL of 4 Items)
Statistics, Data Analysis, and Probability topics:
2-1, 2-3, 3-5, 4-2, 4-3, 5-3, and 9-1.
1.0 Students collect, organize, and represent data
sets that have one or more variables and identify
relationships among variables within a data set by
NOTE: One or more examples are given for each of
the specific standards below, where applicable.
17
hand and through the use of an electronic
spreadsheet software program:
1.1 Know various forms of display for data sets,
including a stem and leaf plot or box and
whisker plot; use the forms to display a single
set of data or to compare two sets of data.
(2 items)
9-1 Objective: Find the greatest common factor
through prime factorization for integers and sets
of monomials.
Sample Supporting Activity: Students collect data
and make predictions in class experiment and then
analyze the results.
1.2 Represent two numerical variables on a
scatterplot and informally describe how the
data points are distributed and any apparent
relationship that exists between the two
variables (e.g., between time spent on
homework and grade level). (2 items)
5-3 Objective: Draw a best-fit line and find the
equation of the best-fit line for a scatter plot.
Sample Supporting Activity: Students discuss the
best positioning of the line for a scatter plot. Some of
the discussion points include why it’s not drawn
through the point at the top or the point at the bottom.
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. (0 items)
Algebra 1 Rescue! is an intervention program based
specifically on algebra standards. Therefore, some
of the other math strands that are not directly
connected to algebra and/or do not provide support
of specific algebra topics are not covered in these
materials.
Grade 7 – Mathematical Reasoning (TOTAL of The following Algebra 1 Rescue! Lessons cover
8 Items Plus Integrated Into Other Strands)
Mathematical Reasoning: 1-1, 1-2, 1-3, 2-4, 2-5,
3-1, 3-2, 3-6, 4-1, 4-2, 4-3, 4-4, 4-5, 5-1, 5-2, 5-3, 54, 5-5, 6-1, 6-5, 7-2, 7-3, 7-4, 7-5, 8-5, 9-1, 9-2, 9-3,
9-4, 9-5, 10-2, 11-1, 11-2, 11-3, 1-4, 11-5, 12-1, 12-2,
12-3, 12-4, 12-5.
1.0 Students make decisions about how to
approach problems:
NOTE: One or more examples are given for each of
the specific standards below, where applicable.
1.1 Analyze problems by identifying relationships,
distinguishing relevant from irrelevant
information, identifying missing information,
sequencing and prioritizing information, and
Every lesson has Problem-Solving Activities where
the problem solving process is directly or
indirectly applied and/or reinforced.
18
observing patterns. (2 items)
1.2 Formulate and justify mathematical conjectures
based on a general description of the
mathematical question or problem posed.
(1 item)
Every lesson has Problem-Solving Activities where
the problem solving process is directly or
indirectly applied and/or reinforced.
1.3 Determine when and how to break a problem
into simpler parts. (0 items)
Every lesson has Problem-Solving Activities where
the problem solving process is directly or
indirectly applied and/or reinforced.
2.0 Students use strategies, skills, and concepts in
finding solutions:
NOTE: One or more examples are given for each of
the specific standards below, where applicable.
2.1 Use estimation to verify the reasonableness of
calculated results. (2 items)
2-5 Objective: Find the principal square root of a
number.
Sample Supporting Activity: Students estimate
square roots and check them with perfect square
benchmarks above and below.
2.2 Apply strategies and results from simpler
problems to more complex problems. (0 items)
1-2 Objective: Evaluate expressions by using the
order of operation.
Sample Supporting Activity: Students solve numeric
expressions using order of operations and then in the
next lesson (1-3) they extend these strategies and
results to expressions with variables.
2.3 Estimate unknown quantities graphically and
solve for them by using logical reasoning and
arithmetic and algebraic techniques. (1 item)
5-2 Objective: Write the equation of a line in
standard form given two points on the graph of the
line.
Sample Supporting Activity: Students look at a graph
representing the miles flown by an airplane in one
day. Students answer questions about how far the
plane will fly in certain amounts of time by analyzing
the slope and what it represents in the graph.
Students then find the equation of the line and
19
explain what the equation means. Students do a
similar activity with average heights of girls from
ages 2 -14 years.
2.4 Make and test conjectures by using both
inductive and deductive reasoning. (1 item)
8-5 Objective: Multiply two binomials and
simplify the expressions, including special
products of (a + b)(a + b) and (a + b)(a – b)
Sample Supporting Activity: Students first use
algebra tiles to find the product (x + 1) (x – 1). Then
they move to (x + 2) (x – 2) and (x + 3) (x – 3) and so
on. Once they have made these observations,
students are asked to generalize what happens when
(a + b) and (a – b) are multiplied. During this
exploration the following vocabulary is developed “a
sum times a difference equals a difference of two
squares”.
2.5 Use a variety of methods, such as words,
numbers, symbols, charts, graphs, tables,
diagrams, and models, to explain mathematical
reasoning. (0 items)
6-5 Objective: Graph inequalities in the
coordinate plane.
Sample Supporting Activity: Students make and test
conjectures about a line on a plane (e.g. the solution
to an inequality with 2 variables is represented by all
the points on one side or the other side of a line). And
other kinds of point and line observations on a
number line and on a plane. They do this by drawing
the point, the line, the inequality, finding coordinates
of the points in an x/y table, and drawing the line on
the plane and looking on the line, below the line, and
above the line.
2.6 Express the solution clearly and logically by
using the appropriate mathematical notation
and terms and clear language; support
solutions with evidence in both verbal and
symbolic work. (0 items)
1-3 Objective: Solve open sentences by performing
arithmetic operations.
Sample Supporting Activity: Students determine if a
mathematical statement is true, false, or open. They
work in a group and come to a consensus. One
member of the group is selected randomly to share
the answer with the entire class. The entire group
earns points for the correct answer.
3-1 Objective: Solve linear equations with addition
and subtraction.
Sample Supporting Activity: Students solve linear
equations and then check their answers through
20
substitution and/or estimation.
2.7 Indicate the relative advantages of exact and
approximate solutions to problems and give
answers to a specified degree of accuracy.
(0 items)
10-2 Objective: Estimate the roots of a quadratic
equation by graphing the associated quadratic
function.
Sample Supporting Activity: Students find the roots
of quadratic equations by graphing the associated
functions. If the roots are not exact, students are
asked to name the consecutive integers between
which the roots lie. Students see the advantage of
finding these benchmarks prior to solving for the
exact solution on a calculator.
2.8 Make precise calculations and check the
validity of the results from the context of the
problem. (0 items)
10-4 Objective: Graph exponential functions and
solve problems using the graphs.
Sample Supporting Activity: Students estimate which
problem would amount to more money in each pair
of compounded interest problems given and then
students work the problems to determine the accuracy
of the estimate.
3.0 Students determine a solution is complete and
move beyond a particular problem by generalizing
to other situations:
NOTE: One or more examples are given for each of
the specific standards below, where applicable.
3.1 Evaluate the reasonableness of the solution in
the context of the original situation. (0 items)
5-3 Objective: Draw a best-fit line and find the
equation of the best-fit line for a scatter plot.
Sample Supporting Activity: Students look at the
concept of “best-fit line” in the context of comparing
average costs of phone rates. Students check several
rates to be sure that they have found the “average”.
3.2 Note the method of deriving the solution and
demonstrate a conceptual understanding of the
derivation by solving similar problems. (0
7-5 Objective: Solve systems of inequalities by
graphing. Sample Supporting Activity: Students are
shown a series of graphs of inequalities. Students
21
items)
determine which graphs contain certain points in the
solution region. Then students do the reverse to
show conceptual understanding. They write two
inequalities of the form y __ mx + b, where one blank
is replaced with > and the other with <. Students
then form groups and graph the systems of inequalities created by all of the students in the group.
3.3 Develop generalizations of the results obtained
and the strategies used and apply them to new
problem situations. (1 item)
12-3 Objective: Use the Pythagorean theorem to
solve problems.
Sample Supporting Activity: Students derive the
Pythagorean Theorem geometrically placing three
squares to form a right triangle in the middle.
Students then solve problems using the derived
formula.
Algebra 1
(Total 12 Items)
The following Algebra 1 Rescue! Lessons cover
Algebra: 1-1, 1-2, 1-3, 1-4, 2-1, 2-2, 2-3, 2-4, 2-5,
3-1, 3-2, 3-3, 3-4, 3-5, 3-6, 4-1, 4-2, 4-3, 4-4, 4-5, 51, 5-2, 5-3, 5-4, 5-5, 6-1, 6-2, 6-3, 6-4, 6-5, 7-1, 7-2,
7-3, 7-4, 7-5, 8-1, 8-2, 8-3, 8-4, 8-5, 9-1, 9-2, 9-3, 94, 9-5, 10-1, 10-2, 10-3, 10-4, 11-1, 11-2, 11-3, 11-4,
11-5, 12-1, 12-2, 12-3, 12-4, 12-5.
1.0 Students identify and use the arithmetic
properties of subsets of integers and rational,
irrational, and real numbers, including closure
properties for the four basic arithmetic operations
where applicable:
1.1 Students use properties of numbers to
demonstrate whether assertions are true or
false. (0 items)
2.0 Students understand and use such operations as
taking the opposite, finding the reciprocal, and
1-4 Objective: Use mathematical properties to
evaluate expressions.
Sample Supporting Activity: Most of the
properties are introduced in this lesson. The
properties are discussed in more depth in the
following lessons :
-
additive identity – Lessons 1-4 and 2-2
associative – Lessons 1-4 and 8-4
commutative – Lessons 1-4, 2-4, and 8-4
density – Lesson 2-1
distributive – Lessons 1-4, 8-5, 9-1, 9-2, 10-4 and 12-1
multiplicative identity – Lessons 1-4 and 10-3
multiplicative inverse – Lessons 1-3 and 1-4
properties of equality – Lessons 3-1, 3-2, 3-3, 3-5, 5-4
zero product property - Lessons 9-2 – 9-4, 10-2, 10-3
8-1 Objective: Multiply and divide monomials
and simplify expressions.
22
taking a root, and raising to a fractional power.
They understand and use the rules of exponents (1
item).
Sample Supporting Activity: Students look at powers
as repeated multiplication; rules about performing
operations on powers.
3.0 Students solve equations and inequalities
involving absolute values. (1 item)
6-4 Objective: Solve and graph the solution set of
4.0 Students simplify expressions before solving
linear equations and inequalities in one variable,
such as 3 (2x – 5) + 4 (x -2) = 12. (2 items)
8-5 Objective: Multiply two binomials and
simplify the expressions, including special
products of (a + b)(a + b) and (a + b)(a – b)
Sample Supporting Activity: Students review the
distributive property and simplify expressions.
5.0 Students solve multistep problems, including
word problems, involving linear equations and
linear inequalities in one variable and provide
justification for each step.
3-4 Objective: Solve problems that can be
represented as equations.
Sample Supporting Activity: Students solve multistep
word problems involving linear equations.
compound inequalities and inequalities involving
absolute value.
Sample Supporting Activity: Students discuss the
definition of the absolute value of a number as the
distance of the number from 0 and solve inequalities
involving absolute values.
6-3 Objective: Solve and graph the solution set of
inequalities using more than one operation.
Sample Supporting Activity: Students solve
inequality word 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 inequality (e.g., they sketch the region
defined by 2x + 6y < 4). (2 items – 1 graphing
item; 1 computing item)
4-4 Objective: Graph linear equations.
Sample Supporting Activity: Students graph linear
equations and learn how to find x and y intercepts.
7.0 Students verify that a point lies on a line, given
an equation of the line. Students are able to derive
linear equations. by using the point-slope formula.
(1 item)
5-1 Objective: Determine the slope of a line from a
graph or given two points on the line.
Sample Supporting Activity: Students look at an
equation of a line and graph it and then are asked to
name some points on the line.
23
8.0 Students understand the concepts of parallel
lines and perpendicular lines and how their slopes
are related. Students are able to find the equation
o f a line perpendicular to a given line that passes
through a given point. (1 item)
5-5 Objective: Use the slope of lines to determine
if two lines are parallel or perpendicular.
Sample Supporting Activity: Students are given three
equations for lines and they are asked to graph all
three lines. Then they are asked “which two graphs
are parallel lines?” Next, students are given
equations and they are asked to give a parallel graph
to each of these equations.
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. (1 item)
7-1 Objective: Solve systems of equations by
graphing. Sample Supporting Activity: Students
solve 2 equations with 2 unknowns by graphing
them.
7-2 Objective: Determine whether a system of
equations has one solution, no solutions, or
infinitely many solutions. Sample Supporting
Activity: Students look at the graph of 2 equations
with 2 unknowns and ask “how many times do the
lines intersect?”
7-3 Objective: Solve systems of equations by using
the substitution method. Sample Supporting
Activity: Students solve 2 equations with 2
unknowns by solving one equation for one of the
variables and substituting that value into the other
equation.
7-4 Objective: Solve systems of equations by
eliminating one variable. Sample Supporting
Activity: Given 2 equations with 2 unknowns,
students add the 2 equations together to eliminate one
of the variables. For example, 2x – y = 7 and x + y =
2, we can add the 2 equations together and the –y + y
eliminates the variable y.
7-5 Objective: Solve systems of inequalities by
graphing. Sample Supporting Activity: Students
solve a system of 2 linear inequalities by sketching
the solution on a graph.
24
10.0 Students add, subtract, multiply, and divide
monomials and polynomials. Students solve
multistep problems, including word problems, by
using these techniques. (1 item)
8-3 Objective: Add and subtract polynomials and
express the answer so the powers of the terms are
in descending order.
Sample Supporting Activity: Students solve problems
adding and subtracting polynomials.
8-4 Objective: Multiply a polynomial by a
monomial and arrange the power of the terms in
descending order.
Sample Supporting Activity: Students solve word
problems involving operations with polynomials.
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 of two squares, and recognizing perfect
squares of binomials. (0 items)
9-3 Objective: Factor quadratic trinomials of the
form ax² + bx + c and solve equations by
factoring. Sample Supporting Activity: Students
factor polynomials in an activity Making Area Rugs.
12.0 Students simplify fractions with polynomials in
the numerator and denominator by factoring both
and reducing them to the lowest terms. (0 items)
11-1 Objective: Simplify rational expressions.
Sample Supporting Activity: Students simplify
rational expressions.
13.0 Students add, subtract, multiply and divide
rational expressions and functions. Students solve
both computationally and conceptually challenging
problems by using these techniques. (0 items)
11-2 Objective: Multiply and divide rational
expressions. Sample Supporting Activity: Students
perform operations on complex expressions and
indicate any excluded values (e.g., x ≠ some value).
14.0 Students solve a quadratic equation by
factoring or completing the square. (0 items)
9-3 Objective: Factor quadratic trinomials of the
form ax² + bx + c and solve equations by
factoring. Sample Supporting Activity: Students
solve quadratics by factoring.
9-4 Objective: Factor quadratic polynomials that
are perfect squares or differences of squares and
solve equations by factoring. Sample Supporting
Activity: Students factor polynomials in an activity
Perfect Square Trinomial Area Rugs. Then students
move to the activity Difference of Squares Area
Rugs. Students also play a game called Perfect
Squares Jeopardy.
9-4 Objective: Factor quadratic polynomials that
25
are perfect squares or differences of squares and
solve equations by factoring. Sample Supporting
Activity: Students solve quadratics by factoring.
9-5 Objective: Solve quadratic equations by
completing the square. Sample Supporting Activity:
Students solve quadratics by completing the square.
15.0 Students apply algebraic techniques to solve
rate problems, work problems, and percent mixture
problems. (1 item)
3-6 Objective: Use proportions to solve percent
problems Sample Supporting Activity: Students
solve rate problems and percent problems using
algebraic and proportional thinking.
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. (0 items)
4-5 Objective: Determine whether a relation is a
function and find a value for a given function.
Sample Supporting Activity: In small groups and in
large class discussions, students discuss “what is a
function?” and “what is not a function?”
17.0 Students determine the domain of independent
variables and the range of dependent variables
defined by a graph, a set of ordered pairs, or a
symbolic expression. (0 items)
4-3 Objective: Determine the range for a given
domain of a relation.
Sample Supporting Activity: Students discuss
independent and dependent variables.
18.0 Students determine whether a relation defined
by a graph, a set of ordered pairs, or a symbolic
expression is a function and justify the conclusion.
(0 items)
4-5 Objective: Determine whether a relation is a
function and find a value for a given function.
Sample Supporting Activity: In small groups and in
large class discussions, students discuss “what is a
function?” and “what is not a function?”
19.0 Students know the quadratic formula and are
familiar with its proof by completing the square.
10-3 Objective: Solve quadratic equations by
factoring or using the quadratic formula.
26
(0 items)
Sample Supporting Activity: Students complete the
square of a quadratic of the form ax² + by + c = 0.
Students look at the derivation of the quadratic
formula.
20.0 Students use the quadratic formula to find the
roots of a second-degree polynomial and to solve
quadratic equations. (0 items)
10-2 Objective: Estimate the roots of a quadratic
equation by graphing the associated quadratic
function.
Sample Supporting Activity: Students look at the
graph of a quadratic function and find the zeros.
Students then look at the relationship of roots to an
equation and zeros of a function.
21.0 Students graph quadratic functions and know
that their roots are the x-intercepts. (0 items)
10-2 Objective: Estimate the roots of a quadratic
equation by graphing the associated quadratic
function.
Sample Supporting Activity: In this lesson, we
reinforce the concept that another name for the xintercepts is the zeros of the function.
22.0 Students use the quadratic formula or
factoring techniques or both to determine whether
the graph of a quadratic function will intersect the
x-axis in zero, one, or two points. (0 items)
10-3 Objective: Solve quadratic equations by
factoring or using the quadratic formula.
Sample Supporting Activity: Students look at a
variety of solutions to quadratics and determine
where and how many places the graph will cross the
x-axis.
23.0 Students apply quadratic equations to physical 10-3 Objective: Solve quadratic equations by
problems, such as the motion of an object under the factoring or using the quadratic formula.
Sample Supporting Activity: Solve problems
force of gravity. (0 items)
involving quadratic equations.
24.0 Students use and know simple aspects of a
*Students informally use logical argument
throughout the materials. Formal statements of
27
logical argument.
24.1 Students explain the difference between
inductive and deductive reasoning and
identify and provide examples of each.
(0 items)
24.2 Students identify the hypothesis and
conclusion in logical deduction.
(0 items)
24.3 Students use counterexamples to show
that an assertion is false and recognize
that a single counterexample is
sufficient to refute an assertion.
(0 items)
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.
25.1 Students use properties of numbers to
construct simple, valid arguments
(direct and indirect) for, or formulate
counterexamples to, claimed assertions.
(0 items)
25.2 Students judge the validity of an
argument according to whether the
properties of the real number system and
the order of operations have been
applied correctly at each step. (0 items)
25.3 Given a specific algebraic statement
involving linear, quadratic, or absolute
value expressions or equations or
inequalities, students determine whether
the statement is true sometimes, always,
or never. (0 items)
hypothesis, conclusion, inductive/deductive
reasoning and counterexamples, however, are not
part of the program.
*Students informally use proof throughout the
materials. Formal statements of proof, however, are
not part of the program.