Focus Areas for Integrated Math I
- Extend understanding of numerical manipulation to algebraic manipulation
- Synthesize understanding of functions
- Deepen and extend understanding of linear relationships
- Apply linear models to data that exhibit a linear trend
- Establish criteria for congruence based on rigid motions
- Apply the Pythagorean Theorem to the coordinate plane
Math Standards by Domain
Number and Quantity
What Students Learn:
Students reason quantitatively and use units as a tool for problem-solving. They learn to interpret units in formulas, choose appropriate scales for graphs, and select accuracy levels appropriate to the context.
Key Skills:
- Use units to understand and guide the solution of multi-step problems
- Choose and interpret units consistently in formulas
- Choose and interpret scales and origins in graphs and data displays
- Define appropriate quantities for descriptive modeling
- Choose accuracy levels appropriate to measurement limitations
When calculating distance using d = rt, if rate is 55 mph and time is 3 hours, the units guide us: d = 55 miles/hour × 3 hours = 165 miles
Algebra
What Students Learn:
Students interpret the structure of linear and exponential expressions to understand the quantities they represent. They learn to see expressions as composed of parts that can be interpreted individually.
Key Skills:
- Interpret expressions in context, identifying terms, factors, and coefficients
- Interpret P(1+r)ⁿ as P times a growth factor
- View complicated expressions by treating parts as single entities
In 500(1.08)ᵗ: 500 is the initial amount, 1.08 is the growth factor (8% increase), and t is the number of time periods.
What Students Learn:
Students create equations and inequalities to represent relationships, including absolute value equations, multi-variable equations, and systems. They learn to rearrange formulas to highlight specific quantities.
Key Skills:
- Create equations with absolute value
- Create equations in two or more variables and graph them with labels and scales
- Represent constraints using equations, inequalities, or systems
- Interpret solutions as viable or non-viable in modeling contexts
- Rearrange formulas (e.g., solve V = IR for R)
A company's profit P depends on items sold x: P = 25x - 500. The 25 is profit per item, -500 represents fixed costs.
What Students Learn:
Students solve linear equations and inequalities, systems of equations, and understand solving as a reasoning process. They represent solutions graphically and understand intersections as solutions.
Key Skills:
- Explain each step in solving equations as following from equality
- Solve linear equations and inequalities, including literal equations
- Solve absolute value equations and inequalities
- Prove that replacing one equation with a sum produces equivalent systems
- Solve systems of linear equations exactly and approximately
- Understand graphs as solution sets and intersections as solutions
- Graph linear inequalities as half-planes
- Graph systems of linear inequalities
Solve: 2x + 5 = 13
2x + 5 - 5 = 13 - 5 (subtraction property)
2x = 8
x = 4 (division property)
Functions
What Students Learn:
Students understand functions as relationships where each input has exactly one output. They learn function notation, interpret key features of graphs, and analyze functions in multiple representations.
Key Skills:
- Understand domain, range, and function notation
- Evaluate functions: if f(x) = 3x - 2, find f(5) = 13
- Recognize sequences as functions with integer domains
- Interpret key features: intercepts, increase/decrease, maxima/minima
- Relate domain to context (e.g., number of engines must be positive integers)
- Calculate average rate of change
- Graph linear and exponential functions
- Compare functions in different representations
For f(x) = 2ˣ, the average rate of change from x = 1 to x = 3 is:
[f(3) - f(1)] / (3-1) = (8 - 2) / 2 = 3
What Students Learn:
Students build functions from contexts and combine them arithmetically. They write arithmetic and geometric sequences recursively and explicitly, and understand function transformations.
Key Skills:
- Write functions from context or patterns
- Determine explicit and recursive formulas
- Combine functions using arithmetic operations
- Write arithmetic sequences: aₙ = a₁ + (n-1)d
- Write geometric sequences: aₙ = a₁ · rⁿ⁻¹
- Translate between recursive and explicit forms
- Identify effects of transformations: f(x) + k, kf(x), f(kx), f(x + k)
Sequence: 2, 6, 18, 54,... This is geometric with r = 3
Explicit: aₙ = 2(3)ⁿ⁻¹
Recursive: a₁ = 2, aₙ = 3aₙ₋₁
What Students Learn:
Students distinguish between linear and exponential growth patterns. They construct models from data, prove growth properties, and understand that exponential growth eventually exceeds linear growth.
Key Skills:
- Distinguish linear (equal differences) from exponential (equal factors)
- Prove linear functions grow by equal differences over equal intervals
- Prove exponential functions grow by equal factors over equal intervals
- Recognize constant rate vs. constant percent change
- Construct functions from graphs, descriptions, or pairs of values
- Understand exponential growth eventually exceeds linear growth
- Interpret parameters in context
Linear: A car's value decreases by $2000 per year
Exponential: A car's value decreases by 15% per year
The exponential decay will eventually result in lower values than linear.
Geometry
What Students Learn:
Students work with transformations in the plane and understand congruence through rigid motions. They make formal geometric constructions and establish congruence criteria for triangles.
Key Skills:
- Know precise definitions of angles, circles, perpendicular and parallel lines
- Represent transformations as functions that take points as inputs
- Compare transformations that preserve distance vs. those that don't
- Describe rotations and reflections that carry regular figures onto themselves
- Develop definitions of rotations, reflections, and translations
- Draw transformed figures using various tools
- Use rigid motion definition of congruence
- Show triangles are congruent using side and angle correspondence
- Explain triangle congruence criteria (ASA, SAS, SSS)
- Make geometric constructions: copy segments/angles, bisect, construct perpendiculars and parallels
- Construct equilateral triangles, squares, and hexagons inscribed in circles
To show △ABC ≅ △DEF, demonstrate a sequence of rigid motions (translation, rotation, reflection) that maps △ABC onto △DEF.
What Students Learn:
Students use coordinates to prove geometric theorems algebraically, building on the Pythagorean Theorem. They verify properties of special triangles and quadrilaterals and analyze slopes.
Key Skills:
- Use coordinates to prove geometric theorems
- Prove slope criteria for parallel and perpendicular lines
- Find equations of lines parallel or perpendicular to given lines
- Compute perimeters and areas using the distance formula
Prove that a quadrilateral with vertices A(0,0), B(4,0), C(4,3), D(0,3) is a rectangle by showing opposite sides are parallel (equal slopes) and adjacent sides are perpendicular (slopes are negative reciprocals).
Statistics and Probability
What Students Learn:
Students summarize, represent, and interpret data on single and two-variable datasets. They create various displays, fit models to data, and interpret linear models in context.
Key Skills:
- Create dot plots, histograms, and box plots
- Use appropriate statistics (mean/median, standard deviation/IQR)
- Compare center and spread of multiple data sets
- Interpret differences in shape, center, spread, and outliers
- Create and interpret two-way frequency tables
- Recognize associations and trends in categorical data
- Create scatter plots and describe relationships
- Fit functions to data and use them to solve problems
- Assess fit by analyzing residuals
- Fit linear functions to scatter plots
- Interpret slope and intercept of linear models
- Compute and interpret correlation coefficients
- Distinguish between correlation and causation
A scatter plot of height vs. weight has line of best fit: w = 4h - 100
Slope 4: Each inch of height is associated with 4 pounds of weight
Intercept -100: A mathematical artifact (doesn't make physical sense)
Strong positive correlation doesn't prove height causes weight gain.