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Waltham Public Schools

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Integrated I

WHS Math - Integrated 1

Integrated 1 aims to deepen and extend student understanding built in previous courses by focusing on developing fluency with solving linear equations, inequalities, and systems. These skills are extended to solving simple exponential equations, exploring linear and exponential functions graphically, numerically, symbolically, and as sequences, and by using regression techniques to analyze the fit of models to distributions of data.  On a daily basis, students in Integrated 1 use problem-solving strategies, questioning, investigating, analyzing critically, gathering and constructing evidence, and communicating rigorous arguments justifying their thinking. Under teacher guidance, students learn in collaboration with others while sharing information, expertise, and ideas. The course is well balanced among procedural fluency (algorithms and basic skills), deep conceptual understanding, strategic competence (problem solving), and adaptive reasoning (extension and application).

Unit Timeframe Big Ideas (Statements or Essential Questions) Major Learning Experiences from Unit
Unit 1
Function		 9/1-10/1 Can I identify important quantities in situations and describe their relationships using graphs?
  • investigate growth patterns
  • investigate characteristics of some graphs of nonlinear functions
  • describe graphs of functions
  • define what it means to be a function
  • use function notation
  • determine domain and range of a function
  • rewrite expressions with exponents
Unit 2
Linear Functions		 10/1-10/28 Can I create a representation of a problem, consider the units involved, and understand the meaning of the quantities using tables, graphs, and equations?
  • connect starting point and growth with the slope and y‑intercept on a graph.
  • measure the steepness of a line on a graph.
  • study the differences between lines that point upward, lines that point downward, and lines that are horizontal or vertical.
  • investigate how slope represents rate of change and speed
  • determine the slope and ‑intercept in various representations, and can convert readily between them.
  • develop an algebraic method for writing the equation of a line when given only two points on the line
Unit 3
Transformations and Solving		 10/29-12/8 Where will the new shape appear? How can I rewrite this expression? How can I solve this equation?
  • visualize the results of rigid transformations including rotations, translations, and reflections
  • use their knowledge of rigid transformations to find the slopes of parallel and perpendicular lines.
  • use definitions of rigid transformations to create new shapes (parallelograms, rectangles, rhombi, kites, and darts)
  • use area models to represent the products of binomials and other polynomials
  • Solve equations by
    • Looking inside (parentheses, square roots, absolute values, exponent bases, etc.)
    • Undoing the operation (including undoing fractions by multiplying through by the denominator)
    • Rewriting (distributing, simplifying, etc.)
Unit 4
Modeling Two-Variable Data		 12/8-1/8 Can I model relationships mathematically in order to describe, analyze, make predictions, and draw conclusions about a set of data?
  • draw a line of best fit by hand, and make a prediction based on their linear model what the slope and y-intercept will be.
  • graphically determine an upper and lower bound on the prediction they make from a linear best-fit model.
  • calculate the correlation coefficient and observe the scatter for various extremes of r.
Unit 5
Sequences		 5/15-5/23 When patterns are repeated, how can I use the patterns to write equations?
  • Describe two important types of sequences: arithmetic and geometric
  • Write the equations for the nth term of arithmetic and geometric sequences
  • Write recursive equations for sequences, and convert between explicit and recursive equations
  • Recognize the connections between arithmetic and geometric sequences and linear and exponential functions
Unit 6
Systems of Equations		 1/10-2/28 How do I solve systems of equations? Is there another approach I can take to help me solve this problem? How else can I look at it?
  • Students will write mathematical sentences (equations) for solving situational word problems.
  • Students will learn three algebraic methods for solving a system of equations, the Equal Values, Substitution, and Elimination Methods, and will know which solving method is most efficient.
  • Students will learn what it means for a system to have no solution or infinite solutions.
  • Students will make important connections among solving equations, multiple representations, and systems of equations.
Unit 7
Congruence and Coordinate Geometry		 4/10-5/2 How can I justify that these triangles are congruent? How do I prove different properties of shapes on a coordinate grid? How do I find midpoint and distance on a coordinate grid?
  • Students will prove that two triangles are congruent, justifying their reasoning using a flowchart.
  • Students will justify statements about shapes on coordinate grids.
  • Students will find the midpoint of a line segment on a coordinate grid.
Unit 8
Exponential Functions		 3/1-4/8 Am I making connections between the multiple representations making sense of the situation?
  • Enhance their understanding of exponential functions through multiple representations (tables, graphs, and equations) and applications
  • Distinguish between the growth in linear situations and exponential situations.
  • Model situations using step functions, especially simple and compound interest.
  • Learn how to graph exponential functions and use them to model everyday situations and solve problems
  • Learn how to find exponential equations when given two points
  • Fit an exponential function to scattered data and assess that fit residuals plots.
Unit 9
Inequalities		 5/2-5/26 Am I making connections between the multiple representations making sense of the situation?
  • Solve one-variable inequalities
  • Write mathematical sentences to for word problems that include inequalities
  • Represent one-variable inequality solutions on a number line
  • Solve one-variable absolute value inequalities
  • Solve two-variable inequalities
  • Represent two-variable inequalities on an xy-coordinate plane
  • Applying systems of equations to solve systems on inequalities
Unit 10
Functions and Data		 Am I considering all available tools as I approach this problem?
  • extend knowledge of statistical association between 2 variables
  • use probability to determine association between categorical variables
  • review the differences between graphical representations of single-variable data
  • distinguish between using a scatterplot or two histograms to compare data with two variables
  • compare the center, shape, spread, and outliers of two distributions
  • develop standard deviation as a method of reporting the variability, or spread, in a distribution
  • connect median to interquartile range (IQR), and mean to standard deviation, and decide which is appropriate considering the shape and outliers of the distribution
  • transform functions by using a vertical shift
Unit 11
Functions and Data		 Am I taking advantage of everything I have learned this year to understand the problems I am solving?
  • Construct a regular hexagon and an equilateral triangle.
  • Construct a perpendicular bisector and an angle bisector.
  • Construct a line parallel to a given line through a given point not on the line and how to construct a square.
  • Copy triangles.
  • Solve word problems involving work and mixture.
  • Sse graphs to approximate solutions.
  • Use knowledge of systems of equations and graphing to approximate solutions when no algebraic method is available.
  • Collect and analyze data.
  • Determine the equation of a least squares regression line, describe the association, verify the residual plot, create upper and lower boundary lines, and use the statistics to make a prediction.
  • Write and solve exponential functions and solve a system of inequalities.