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Pre-Algebra Learning Progression (v2)

Page history last edited by Christine Koerner 2 years, 9 months ago

Welcome to the new progression for Pre-Algebra. This progression builds upon Math Framework Project Phase 1 work (see Progression v1 here), taking many of the best features and building in an Overarching Question, Essential Questions, and Big Ideas for each unit. This new model takes the work of bundling standards to the next level by grouping together grade level concepts under Big Ideas. The Big Ideas are designed to represent the critical mathematics of this grade level in a manner we believe to be more coherent and productive as a guide for planning instruction, assessment, and intervention. Big Ideas are not a replacement for the objectives.

 

Click on the unit numbers below to see essential understandings, student activities, and suggested sequencing.

 

Unit

Overarching Question

Essential Questions

Big Ideas

Full Objectives

Unit 0:

Growth Mindset

How does mindset affect learning mathematics?

 
  1. Math is about learning not performing

  2. Math is about making sense

  3. Math is filled with conjectures, creativity, and uncertainty

  4.  Mistakes are beautiful things  

  • Develop a deep and flexible conceptual understanding

  • Develop accurate and appropriate procedural fluency

  • Develop strategies for

    problem solving
  • Develop mathematical reasoning

  • Develop a productive mathematical disposition

  • Develop the ability to make conjectures, model, and generalize

  • Develop the ability to communicate mathematically 

Unit 1:

Numbers


Timing

 3-4 weeks


Objectives

PA.N.1.4

PA.N.1.5

How can you use Real Numbers to aid in the understanding of real-world situations?

  1. What strategies can you use to compare numbers not easily found on a number line?

  2. What characteristics do we use to classify real numbers?

  3. How can we use real numbers to identify real world situations?

  4. What strategies can be used to estimate the value of a non-perfect square root?

  5. Why does one need to distinguish between various types of real numbers? 

  1. Real Numbers are classified as rational or irrational.

  2. The estimation of a non-perfect square root is located between two whole numbers on a number line.

  3. Operations on Real Numbers results in rational or irrational values. 

PA.N.1.4 Classify real numbers as rational or irrational. Explain why the rational number system is closed under addition and multiplication and why the irrational system is not. Explain why the sum of a rational number and an irrational number is irrational; and the product of a non-zero rational number and an irrational number is irrational.

PA.N.1.5 Compare real numbers; locate real numbers on a number line. Identify the square root of a perfect square to 400 or, if it is not a perfect square root, locate it as an irrational number between two consecutive positive integers.

Unit 2: Expressions, Equations, and Inequalities


Timing

 4-5 weeks


Objectives

PA.A.3.1

PA.A.3.2 

PA.A.4.1

PA.A.4.2

PA.A.4.3

How can you use expressions, equations, and inequalities within real-world situations?

  1. In what real-world situations can substitution become a valuable tool?

  2. How can you determine if two expressions are equivalent?

  3. How can equivalent expressions be used in the real-world?

  4. How do linear equations give information for real-world situations?

  5. What are real-world situations that would yield one solution, no solution, or infinite solutions?

  6. How can we identify various situations in the world around us?

  1. Substitution is used to simplify and evaluate algebraic expressions.

  2. Properties of operations can be used to justify equivalent expressions.

  3. Real world problems that can be represented as linear equations will yield one, infinitely many, or no solution.

  4. Independent and dependent variables can be identified in the world around us. 

 

PA.A.3.1 Use substitution to simplify and evaluate algebraic expressions.

PA.A.3.2 Justify steps in generating equivalent expressions by identifying the properties used, including the properties of operations (associative, commutative, and distributive laws) and the order of operations, including grouping symbols.

PA.A.4.1 Illustrate, write, and solve mathematical and real-world problems using linear equations with one variable with one solution, infinitely many solutions, or no solutions. Interpret solutions in the original context.

PA.A.4.2 Represent, write, solve, and graph problems leading to linear inequalities with one variable in the form px + q > r  and px + q < r , where p, q and r are rational numbers.

PA.A.4.3 Represent real-world situations using equations and inequalities involving one variable.

Unit 3:

Linear Equations and Functions


Timing

5-6 weeks


Objectives

PA.A.1.1

PA.A.1.2

PA.A.1.3

PA.A.2.1

PA.A.2.2

PA.A.2.3

PA.A.2.4

PA.A.2.5

PA.D.1.3

How can we use graphs and other representations to gain knowledge of real-world situations?

  1. How can we identify relationships in the world around us?

  2. What information can the rate of change give you about a linear equation or real-world data?

  3. How can you represent linear relationships?

  4. How can you identify an equation or graph as linear or nonlinear?

  5. How can

    scatterplots

    be used to gain information about real-world events? 

  1. A function is a relationship between an independent and dependent variable.

  2. The rate of change describes how one quantity changes in respect to another.

  3. Multiple representations can be used to express and analyze linear relationships.

  4. Functions can be identified as linear if they can be expressed in slope intercept or graphed in a straight line.

  5.  Data can be displayed and interpreted using scatterplots. 

PA.A.1.1 Recognize that a function is a relationship between an independent variable and a dependent variable in which the value of the independent variable determines the value of the dependent variable.

PA.A.1.2 Use linear functions to represent and explain real-world and mathematical situations.

PA.A.1.3 Identify a function as linear if it can be expressed in the form y=mx + b or if its graph is a straight line.

PA.A.2.1 Represent linear functions with tables, verbal descriptions, symbols, and graphs; translate from one representation to another.

PA.A.2.2 Identify, describe, and analyze linear relationships between two variables.

PA.A.2.3 Identify graphical properties of linear functions including slope and intercepts. Know that the slope equals the rate of change, and that the y-intercept is zero when the function represents a proportional relationship.

PA.A.2.4 Predict the effect on the graph of a linear function when the slope or y-intercept changes. Use appropriate tools to examine these effects.

PA.A.2.5 Solve problems involving linear functions and interpret results in the original context.

PA.D.1.3 Collect, display and interpret data using scatterplots. Use the shape of the scatterplot to informally estimate a line of best fit. Use appropriate titles, labels and units.

Unit 4:

Exponents

 


Timing

 2-3 weeks


Objectives

PA.N.1.1

PA.N.1.2

 PA.N.1.3 

How can exponents be used to solve real-world situations?

  1. How can you apply strategies to simplify expressions?

  2. How can very large or very small numbers be written and applied in a way that is useful and meaningful? 

  1. Equivalent numerical and algebraic expressions can be generated by applying the properties of integer exponents.

  2.  Scientific notation creates a realistic way to utilize really small or really large numbers.

PA.N.1.1 Develop and apply the properties of integer exponents, including a0 = 1 (with a 0 ), to generate equivalent numerical and algebraic expressions.

PA.N.1.2 Express and compare approximations of very large and very small numbers using scientific notation.

PA.N.1.3 Multiply and divide numbers expressed in scientific notation, express the answer in scientific notation.

Unit 5: Pythagorean Theorem

 


Timing

 2-3 weeks


Objectives  PA.GM.1.1

PA.GM.1.2

How can the Pythagorean Theorem be used in real-world situations?

  1. What is the special relationship between the sides of a right triangle?

  2. How can the Pythagorean Theorem be used to solve real-world applications? 

  1. Pythagorean Theorem utilizes the special relationship between the three sides of a right triangle.

  2.  Pythagorean Theorem can be used to find the distance between two points on a coordinate plane. 

PA.GM.1.1 Informally justify the Pythagorean Theorem using measurements, diagrams, or dynamic software and use the Pythagorean Theorem to solve problems in two and three dimensions involving right triangles.

PA.GM.1.2 Use the Pythagorean Theorem to find the distance between any two points in a coordinate plane.

Unit 6:

Surface Area and Volume

 


Timing

 2-3 weeks


Objectives PA.GM.2.1

PA.GM.2.2

PA.GM.2.3

PA.GM.2.4

How are two dimensional and three-dimensional objects related to each other in real-world situations?

  1. How can formulas be used to better understand measurements of 3D objects?

  2. How can nets be used to analyze 3D figures and calculate surface area and volume?

  3. How are two dimensional and three-dimensional objects related to each other? 

  1. Surface area and volume can be calculated for a rectangular prism.

  2. Surface area and volume can be calculated for a cylinder. 

PA.GM.2.1 Calculate the surface area of a rectangular prism using decomposition or nets. Use appropriate measurements such as cm2.

PA.GM.2.2 Calculate the surface area of a cylinder, in terms of 𝜋 and using approximations for 𝜋, using decomposition or nets. Use appropriate measurements such as cm2.

PA.GM.2.3 Develop and use the formulas 𝑽=𝒍𝒘𝒉 and 𝑽=𝑩𝒉 to determine the volume of rectangular prisms. Justify why base area (𝑩) and height (𝒉) are multiplied to find the volume of a rectangular prism. Use appropriate measurements such as cm3.

PA.GM.2.4 Develop and use the formulas V = 𝜋 r2h and V = Bh to determine the volume of right cylinders, in terms of 𝜋 and using approximations for 𝜋. Justify why base area (B) and height (h) are multiplied to find the volume of a right cylinder. Use appropriate measurements such as cm3.

Unit 7:

 Measures of Central Tendency

 


Timing

 2-3 weeks


Objectives

PA.D.1.1

PA.D.1.2

How do data points affect mean and median in real-world situations?

  1. What effects can changing data points have?

  2. How do individual data points affect the entire data set? 

  1. The mean and median of a data set is impacted by inserting or deleting data points.

  2. Outliers can have an effect on measures of central tendency. 

PA.D.1.1 Describe the impact that inserting or deleting a data point has on the mean and the median of a data set. Know how to create data displays using a spreadsheet and use a calculator to examine this impact.

PA.D.1.2 Explain how outliers affect measures of central tendency.

Unit 8:

 Probability

 


Timing

 3-4 weeks


Objectives

PA.D.2.1

PA.D.2.2

PA.D.2.3

How can probability be used in real-world situations?

  1. How can probabilities be determined when actual probabilities are not known?

  2. How can predictions be made on unknown probabilities?

  3. How can you determine if the representation of a data population is fair?

  4. How are conclusions about a data set

    drawn

    using sampling? 

  1. Experimental probability can be calculated and the result can be expressed in multiple ways.

  2. Experimental probability can be used to make predictions.

  3. Samples are used to generalize a population. 

PA.D.2.1 Calculate experimental probabilities and represent them as percents, fractions and decimals between 0 and 1 inclusive. Use experimental probabilities to make predictions when actual probabilities are unknown.

PA.D.2.2 Determine how samples are chosen (random, limited, biased) to draw and support conclusions about generalizing a sample to a population.

PA.D.2.3 Compare and contrast dependent and independent events.

 

 

 

Culminating Unit  


Timing

 2-3 weeks


Objectives

PA.A.1.2

PA.A.2.1

PA.A.2.2

PA.A.2.3

PA.D.1.3

 

 

How can mathematical patterns be used to expand our understanding?
  1. What influences temperature?

  2. Which polygons can be tessellated?

  1. Patterns can be used to make predictions about data.
  2. Patterns and polygons can be used to create tessellations.

PA.A.1.2 Use linear functions to represent and explain real-world and mathematical situations.

PA.A.2.1 Represent linear functions with tables, verbal descriptions, symbols, and graphs; translate from one representation to another.

PA.A.2.2 Identify, describe, and analyze linear relationships between two variables.

PA.A.2.3 Identify graphical properties of linear functions including slope and intercepts. Know that the slope equals the rate of change, and that the y-intercept is zero when the function represents a proportional relationship.

PA.D.1.3 Collect, display and interpret data using scatterplots. Use the shape of the scatterplot to informally estimate a line of best fit. Use appropriate titles, labels and units.

Distance Learning Resources/
Supplemental Activities
How can students develop and show evidence of understanding?     Multiple objectives are covered in this material. These math tasks are designed to enhance current curriculum and support distance learning.

 


Introduction to the OKMath Framework

Pre-Algebra Introduction

 

 

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