Download Pre-Algebra Seventh Grade v. 2016

Grade Seven Mathematics: Pre-Algebra
In 7th Grade Pre-Algebra, students will focus on five main areas: The Number System, Ratios and
Proportional Relationships, Expressions and Equations, Geometry, Statistics and Probability.
We will extend the use of addition, subtraction, multiplication and division of rational numbers to
solve real-world and mathematical problems. In addition, students will identify and order irrational numbers.
The concept of ratios and proportions will be used to solve various types of percent problems.
Students will be asked to calculate discounts, interest, taxes, tips, and percent of increase and/ or decrease.
Building on their prior knowledge of rational numbers, students will study one-variable equations and
inequalities extensively, including equations with variables on both-sides, compound inequalities and absolute
value. They will also use variables to represent quantities in real-world and mathematical problems.
The study of figures will include circles, triangles, quadrilaterals, polygons, cubes and right prisms.
Students will analyze two- and three-dimensional figures using area, volume and surface area.
The concept of random sampling will be studied as students will compare two population sets and
describe the differences between them.
Grade Level/ Subject: Grade Seven Pre-Algebra
Content: Rational numbers.
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Duration: August/Sept. (4 weeks)
Essential
How is computation with rational numbers similar and different to whole number computation?
Question:
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Assessment:
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Apply and extend previous understandings of addition and subtraction to add and
subtract rational numbers.
Understand subtraction of rational numbers as adding the additive inverse
o p-q = p + (-q)
Represent addition and subtraction on a horizontal or vertical number line.
Apply and extend previous understandings of multiplication and division and of
fractions to multiply and divide rational numbers.
Convert a rational number to a decimal using long division.
Demonstrate that the decimal form of a rational number terminates or eventually
repeats.
Solve real world and mathematical problems involving the four operations with rational
numbers.
Given rational numbers of different quantities, have students add, subtract, multiply
and divide the numbers.
Given rational numbers of different quantities, convert to decimals using long division.
Using a number line have students order rational numbers on that line. EX: 2/5, -6,
5/3, and -1.6.
Mathematics Course 2 (pages 96-105)
CC.2.1.7.E.1 Apply and extend previous understandings of operations with fractions to
Standards: operations with rational numbers.
Rational number – any number that can be expressed as the quotient or fraction, p/q, of two
integers, with the denominator q not equal to zero; Repeating decimal – a decimal fraction in
Vocabulary: which a figure or group of figures is repeated indefinitely, as in0.666… or as in 1.851851851….;
Terminating decimal – a decimal number that has digits that do not go on forever
Comments:
Grade Level/ Subject: Grade Seven Pre-Algebra
Content: Identifying Rational and Irrational Numbers
Essential
Question:
Duration: September (1 week)
How is mathematics used to quantify, compare, represent and model numbers?
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Skills:
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Determine whether a number is rational or irrational. For rational numbers, show that
the decimal expansion terminates or repeats (limit repeating decimals to thousandths).
Convert a terminating or repeating decimal to a rational number (limit repeating
decimals to thousandths).
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Assessment:
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4
2,0, −3, , − , 0.125,
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Are the following numbers rational or irrational? {
}
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−0.5,0. 3, √2, 𝜋
Resources: Mathematics Course 3, Prentice Hall, p. 106
Standards: CC.2.1.8.E.1 Distinguish between rational and irrational numbers using their properties.
Vocabulary: Irrational number – cannot be written as the quotient of two relatively prime whole numbers
Comments:
Grade Level/ Subject: Grade Seven Pre-Algebra
Content: Estimating Irrational Numbers
Essential
Question:
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Assessment:
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Duration: September (3 days)
What does it mean to estimate numerical quantities?
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Skills:
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Estimate the value of irrational numbers without a calculator (limit whole number
radicand to less than 144).
Use rational approximations of irrational numbers to compare and order irrational
numbers.
Locate/identify rational and irrational numbers at their approximate locations on a
number line.
√5 is between 2 and 3 but closer to 2
Mathematics Course 3, Prentice Hall, p. 107
Standards: CC.2.1.8.E.4 Estimate irrational numbers by comparing them to rational numbers.
Vocabulary:
Comments:
Grade Level/ Subject: Grade Seven Pre-Algebra
Content: Equivalent Expressions
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Duration: Oct (2 weeks)
Essential How can expressions, equations, and inequalities be used to quantify, solve, model and/or
Question: analyze mathematical situations?
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Assessment:
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Apply properties of operations to add, subtract, factor, and expand linear expressions
with rational coefficients.
Understand that rewriting an expression in different forms in a problem can shed light
on the problem and how the quantities in it are related.
Given an expression with different quantities can students write an equivalent
expression. EX: Is the expression 4w – 10 equivalent to 2(2w – 5)? EX: Does a + 0.05a
= 1.05a mean that increase by 5% is the same a multiply by 1.05a?
Mathematics Course 2 (pgs. 48-52)
Mathematics Course 3 (pages 266 – 269)
CC.2.2.7.B.1 Apply properties of operations to generate equivalent expressions.
Coefficient – a numerical or constant quantity placed before and multiplying the variable in an
algebraic expression (e.g., 4 in 4x y); Constant – is a number on its own, or sometimes a letter
Vocabulary: such as a, b or c to stand for a fixed number. Example: in "x + 5 = 9", 5 and 9 ...; Expression –
a mathematical phrase that can contain ordinary numbers, variables (like x or y) and operators
(like add, subtract, multiply, and divide)
Comments:
Grade Level/ Subject: Grade Seven Pre-Algebra
Content: Numerical and Algebraic Expressions, Equations,
and Inequalities.
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Duration: Oct/Nov (4 weeks)
Essential How can expressions, equations, and inequalities be used to quantify, solve, model and/or
Question: analyze mathematical situations?
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Skills:
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Assessment:
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Apply properties of operations to calculate with numbers in any form; convert between
forms as appropriate.
Use variables to represent quantities in mathematical problems
Construct simple equations and inequalities to solve problems by reasoning about the
quantities.
Solve multi-step problems with positive and negative rational numbers in any form.
Students will solve real world problems by writing simple equations. EX: If a woman is
making $25 an hour gets a 10% raise, what will her new hourly rate be?
Given equations with different values solve using inverse operations.
Resources: Mathematics Course 2(pages 169 – 223)
Standards:
CC.2.2.7.B.3 Model and solve real-world and mathematical problems by using and connecting
numerical, algebraic, and/or graphical representations.
Expression – a mathematical phrase that can contain ordinary numbers, variables (like x or y)
and operators (like add, subtract, multiply, and divide); Equation – a statement that the values
Vocabulary: of two mathematical expressions are equal (indicated by the sign =); Inequality – the relation
between two expressions that are not equal, employing a sign such as ≠ “not equal to,” >
“greater than,” or < “less than.”
Comments:
Grade Level/ Subject: Grade Seven Pre-Algebra
Content: Solving Equations
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Duration: November (3 weeks)
Essential
How do we set up and solve problems to find unknown pieces of information?
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Write, solve, and/or apply a linear equation.
Use and/or identify any algebraic property to justify any step in an equation-solving
process.
Combining like terms in algebraic expressions/equations
Interpret solutions to a problem in the context of the problem-solving situation.
Apply the distributive property to algebraic equations.
Set up a proportion with similar figures
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Solve equation such as:
2x + 3 = 17
2(x + 4) − 6 = 3(x − 1) + 2x
px − mx = h Solve for x.
Resources: Pearson Algebra I Common Core: Chapter 2: pgs. 81-114
CC.2.2.HS.D.8 - Apply inverse operations to solve equations or formulas for a given variable.
Standards: CC.2.2.HS.D.9 - Use reasoning to solve equations and justify the solution method.
Coefficient – the number multiplied by a variable; Constant – a number not multiplied by a
variable; Inverse operations – the opposite operation which solves an equation; Literal
Vocabulary: equation – an equation that relates multiple variables and has an expression as a solution;
Order of operations – an agreed upon order for performing arithmetic operations; Reciprocal
– the multiplicative inverse of a number i.e. flip the fraction
Comments: pgs.115 – 147 not Algebra I Keystone Standards
Grade Level/ Subject: Grade Seven Pre-Algebra
Content: Solving Inequalities
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Duration: December (3 weeks)
Essential
How do we represent a range of solutions to an inequality relationship?
Question:
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Solve one variable inequalities and represent solution set on a number line.
Identify or graph the solutions set to a linear inequality on the coordinate plane.
Solve multi-step inequalities
Solve compound inequalities
Solve equations and inequalities with absolute values
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Solve and graph solutions on a number line:
3x + 4 < 13
30 > −(5z + 15) + 10z
−3 < m − 4 < −1
|3x + 4| − 6 < 13
|3x + 4| − 6 = 13
Resources: Pearson Algebra I Common Core: Chapter 3: pgs. 165-220
CC.2.2.HS.D.7 - Create and graph equations or inequalities to describe numbers or
relationships.
Standards: CC.2.2.HS.D.10 - Represent, solve, and interpret equations/inequalities and systems of
equations/inequalities algebraically and graphically.
Absolute value – the distance away from zero; Inequality – a relationship where one thing is
greater than or less than another; Intersection – looks for values that are found in multiple sets;
Vocabulary: Solution set – the possible answers to an inequality; Unions – look for the total number of
unique value in multiple sets
Comments: Minimal emphasis on pgs. 194-199 and pgs. 214-220. If constrained by time, skip.
Grade Level/ Subject: Grade Seven Pre-Algebra
Content: Proportional Relationships
Essential
Question:
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Duration: January (3 weeks)
How can probability and proportional reasoning be used to make reasonable predictions?
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Skills:
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Compute unit rates associated with ratios of fractions, including ratios of lengths, areas,
and other quantities measured in like or different units.
Determine whether two quantities are proportionally related.
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams,
and verbal descriptions of proportional relationships.
Represent proportional relationships by equations.
Explain what a point (x, y) on the graph of a proportional relationship means in terms
of the situation, with special attention to the points (0, 0) and (1, r), where r is the unit
rate.
Use proportional relationships to solve multi-step ratio and percent problems.
Use proportional relationships to solve multi-step ratio and percent problems.
Examples: simple interest, tax, markups and markdowns, gratuities and commissions,
fees, percent increase and decrease.
Given ratios of different quantities, can students put them in simplest form. EX: Write
the ratio 10 s: 1 min. in simplest form.
Given ratios of different quantities, determine if they form a proportion.
Mathematics Course 2(pages 274 – 319)
CC.2.1.7.D.1 Analyze proportional relationships and use them to model and solve
real‐world and mathematical problems.
Commission – is earnings based on a percentage of the total amount of sales; Proportion –
two ratios (or fractions) are equal; Ratio – the quantitative relation between two amounts
Vocabulary: showing the number of times one value contains or is contained within the other ("the ratio of
men's jobs to women's is 8 to 1"); Unit rate – is a comparison of two different quantities when
they are combined together.
Comments:
Grade Level/ Subject: Grade Seven Pre-Algebra
Content: Angle Measure, Area, Surface Area,
Circumference, and Volume
Essential
Question:
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Assessment:
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Standards:
Duration: Jan/Feb (4 weeks)
How can the decomposition of 3-dimensional shapes aid in the understanding of surface areas
and volumes?
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Skills:
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Identify and use properties of supplementary, complementary, and adjacent angles in a
multi-step problem to write and solve simple equations for an unknown angle in a
figure.
Identify and use properties of angles formed when two parallel lines are cut by a
transversal (e.g., angles may include alternate interior, alternate exterior, vertical,
corresponding).
Know the formulas for the area and circumference of a circle and use them to solve
problems.
Solve mathematical problems involving area, volume, and surface area of two and three
dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right
prisms.
Given the measurement of one angle, find the measurement of its complement or
supplement.
Given multiple angles how are vertical angles and adjacent angles different?
Mathematics Course 2(pages 374 -397) (pages 410-425)
CC.2.3.7.A.1 Solve real‐world and mathematical problems involving angle measure, area, surface
area, circumference, and volume.
Adjacent – angles that have a common side and a common vertex; Area – the extent or
measurement of a surface or piece of land; Circumference – the enclosing boundary of a
curved geometric figure, especially a circle; Complementary – two angles that add to 90
Vocabulary: degrees; Parallel – lines in a plane which do not meet; Polygon – a plane figure with at least
three straight sides and angles, and typically five or more; Supplementary – two angles that add
to 180 degrees; Surface area – the total area that can be measured on the entire surface. This
can only be measured if the object is a three-dimensional object; Volume – the amount of space
that a substance or object occupies, or that is enclosed within a container
Comments:
Grade Level/ Subject: Grade Seven Pre-Algebra
Content: Properties of Geometric Figures.
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Duration: February (2 weeks)
Essential How can we use the relationship between surface area and volume to help us draw, construct,
Question: model, and represent real situations and/or solve problems of surface area and volume?
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Solve problems involving scale drawings of geometric figures, including finding length
and area from a scale drawing and reproducing a scale drawing at a different scale.
Draw (freehand, ruler and protractor, or with technology) geometric shapes with given
conditions.
Identify or describe the properties of all types of triangles based on angle and side
measures.
Use and apply the triangle inequality theorem.
Describe the two-dimensional figures that result from slicing three-dimensional figures,
as in plane sections of right rectangular prisms and pyramids.
Describe the quadrilaterals created by slicing plane sections of right rectangular prisms
and right rectangular pyramids.
Mathematics Course 3(page 122)
Mathematics Course 2(pages 336-339)(414-419)(258-263)
CC.2.3.7.A.2 Visualize and represent geometric figures and describe the relationships between
them.
Polyhedron – a solid figure with many plane faces, typically more than six; Prism – a solid
geometric figure whose two end faces are similar, equal, and parallel rectilinear figures, and
whose sides are parallelograms; Pyramid – a polyhedron that has a base and triangular faces
Vocabulary: meeting at a point; Scale – the ratio of the length in a drawing (or model) to the length of the
real thing; Triangle inequality theorem – any side of a triangle must be shorter than the other
two sides added together. If it was longer, the other two sides could not meet!
Comments:
Grade Level/ Subject: Grade Seven Pre-Algebra
Content: Statistical Measures
Essential
Question:
Duration: March (2 weeks)
How can we use the mean, median, mode, and range to describe a set of data?
Why do we need three different measures of central tendency?
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Skills:
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Assessment:
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Compare two numerical data distributions using measures of center and
variability.
Describe data using mean, median, mode and range.
Represent and interpret data using box and whisker plots.
Represent and interpret data using stem and leaf plots.
Use line plots, frequency tables, and histograms to represent data.
Given a set of data can students describe the mean, median, mode and
range. EX:Decide whether the words in a chapter of a seventh grade
science book are generally longer than the words in a chapter of a fourth
grade science book.
Using a line plot, can students note the difference between two sets of data.
Resources: Mathematics Course Three (chapter 9)
Standards:
CC.2.4.7.B.2 Draw informal comparative inferences about two populations.
Box and whisker plot – are uniform in their use of the box: the bottom and top
of the box are always the first and third quartiles, and the band inside the box is
always the second quartile (the median); Central tendency – the mean, median,
Vocabulary:
and mode; Measures of center – these statistics are commonly referred to
as measures of central tendency; Measures of variability – a mathematical
determination of how much the performance of the group as a whole deviates from
the mean or median (box and whisker)
Comments:
Grade Level/ Subject: Grade Seven Pre-Algebra
Content: Probability
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Duration: March (3 weeks)
How do we make predictions based on the outcomes of a probability experiment?
Essential
Question: How does the collection, analysis, organization, and interpretation of data help us to answer real
world questions
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Predict or determine whether some outcomes are certain, more likely, less likely, equally
likely, or impossible.
Find the experimental probability.
Find the probabilities of compound events using organized lists, tables, tree diagrams,
and simulation.
Given various events, can students find the probability of an event occurring? EX: Find
the approximate probability that a spinning penny will land heads up or that a tossed
paper cup will land open end up.
Given multiple events, find the probability of an event occurring.
Resources: Mathematics Course 2 (pages 580 – 605)
Standards: CC.2.4.7.B.3 Investigate chance processes and develop, use, and evaluate probability models.
Compound events –two or more events; Probability – is the chance that something will
Vocabulary: happen; Sample space – the collection of all possible outcomes in an experiment
Comments:
Grade Level/ Subject: Grade Seven Pre-Algebra
Content: Random Samples
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Duration: April (2 weeks)
How can we use proportionality represented through models of and models for ratio tables,
Essential
factor-of-change (scale factor), a unit rate, and cross-multiplication to solve real world
Question: problems?
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Assessment:
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Understand that statistics can be used to gain information about a population by
examining a sample of the population.
Determine whether a sample is a random sample given a real-world situation.
Identify a random sample and to write a survey question.
Use data from a random sample to draw inferences about a population with an
unknown characteristic of interest
Estimate population size using proportions
Using randomly sampled survey data, predict the outcome. EX: Predict the winner of a
school election based
Resources: Mathematics Course 2(pages 550-566)
Standards: CC.2.4.7.B.1 Draw inferences about populations based on random sampling concepts.
Biased question – a question that makes an unjustified assumption or makes some answers
Vocabulary: appear better than others; Sample – part of a population; Random sample – when each
member of a population has the same chance of being selected
Comments:
Grade Level/ Subject: Grade Seven Pre-Algebra
Content: Linear Equations and Slope
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Duration: May (2 weeks)
How does the representation support the linear relationship?
Essential
Question: Where in each representation can you find the rate of change, the y-intercept, etc.?)
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Graph proportional relationships, interpreting the unit rate as the slope of the graph.
Compare two different proportional relationships represented in different ways
Use similar right triangles to show and explain why the slope m is the same between
any two distinct points on a non-vertical line in the coordinate plane.
Given a linear equation, find several solutions. EX: Find three solutions of y = x + 5
Given graphs with different slopes, determine if the slope of the line positive, negative,
zero, or undefined.
Resources: Mathematics Course 2 (pages 486-508)
CC.2.2.8.B.2 Understand the connections between proportional relationships, lines, and linear e
Standards: quations.
Linear equation – the graph of its solutions lies on a line; Rise – the vertical change; Run –
Vocabulary: the horizontal change; Slope – ratio that describes the steepness of a line
Comments: This is an 8th grade standard to help students entering Algebra next year.