## Teaching Algebra

http://www.teachers.tv/series/jonny-heeleys-masterclass

## grade 8 (DRAFT)

20 Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Algebra: Analyzing and representing linear functions and solving linear equations and systems

of linear equations

Students use linear functions, linear equations, and systems of linear equations to represent, analyze, and

solve a variety of problems. They recognize a proportion (y/x = k, or y = kx) as a special case of a linear

equation of the form y = mx + b, understanding that the constant of proportionality (k) is the slope and the

resulting graph is a line through the origin. Students understand that the slope (m) of a line is a constant rate

of change, so if the input, or x-coordinate, changes by a specific amount, a, the output, or y-coordinate,

changes by the amount ma. Students translate among verbal, tabular, graphical, and algebraic representations

of functions (recognizing that tabular and graphical representations are usually only partial representations),

and they describe how such aspects of a function as slope and y-intercept appear in different representations.

Students solve systems of two linear equations in two variables and relate the systems to pairs of lines that

intersect, are parallel, or are the same line, in the plane. Students use linear equations, systems of linear

equations, linear functions, and their understanding of the slope of a line to analyze situations and solve

problems.

Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using

distance and angle

Students use fundamental facts about distance and angles to describe and analyze figures and situations in

two- and three-dimensional space and to solve problems, including those with multiple steps. They prove

that particular configurations of lines give rise to similar triangles because of the congruent angles created

when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety

of problems, including those that ask them to find heights and distances. They use facts about the angles that

are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a

triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students

explain why the Pythagorean theorem is valid by using a variety of methods—for example, by decomposing a

square in two different ways. They apply the Pythagorean theorem to find distances between points in the

Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.

Data Analysis and Number and Operations and Algebra: Analyzing and summarizing data sets

Students use descriptive statistics, including mean, median, and range, to summarize and compare data sets,

and they organize and display data to pose and answer questions. They compare the information provided by

the mean and the median and investigate the different effects that changes in data values have on these

measures of center. They understand that a measure of center alone does not thoroughly describe a data set

because very different data sets can share the same measure of center. Students select the mean or the

median as the appropriate measure of center for a given purpose.

Algebra: Students encounter some nonlinear functions

(such as the inverse proportions that they studied in

grade 7 as well as basic quadratic and exponential

functions) whose rates of change contrast with the

constant rate of change of linear functions. They view

arithmetic sequences, including those arising from

patterns or problems, as linear functions whose inputs

are counting numbers. They apply ideas about linear

functions to solve problems involving rates such as

motion at a constant speed.

Geometry: Given a line in a coordinate plane, students

understand that all “slope triangles”—triangles created by

a vertical “rise” line segment (showing the change in y), a

horizontal “run” line segment (showing the change in x),

and a segment of the line itself—are similar. They also

understand the relationship of these similar triangles to

the constant slope of a line.

Data Analysis: Building on their work in previous

grades to organize and display data to pose and answer

questions, students now see numerical data as an

aggregate, which they can often summarize with one or

several numbers. In addition to the median, students

determine the 25th and 75th percentiles (1st and 3rd

quartiles) to obtain information about the spread of data.

They may use box-and-whisker plots to convey this

information. Students make scatterplots to display

bivariate data, and they informally estimate lines of best

fit to make and test conjectures.

Number and Operations: Students use exponents and

scientific notation to describe very large and very small

numbers. They use square roots when they apply the

Pythagorean theorem.