Unit: Creating a Community of Mathematical Thinkers

Curriculum Map: Grade 6 Pre-Algebra Course: PreAlgebra Sub-topic: Algebra

Grade(s): None specified Course Description This course is designed to complete the 6th grade standards of Ratio and Proportional reasoning, Geometry and Statistics and Probability. The course will also complete 7th grade standards of developing understanding of and applying proportional relationships; developing understanding of operations with rational numbers and working with expressions and linear equations. It will solve problems including scale, area, surface area, and volume. Statistics and Probability will include drawing comparative inferences from random sampling, and investigating chance.

Unit: Creating a Community of Mathematical Thinkers Subject: Mathematics Timeline: Week 1 Purpose: The purpose of the Establishing a Community of Mathematical Thinkers Module is to launch the important mathematical processes and practices that will support the growth and independence of our students as mathematical thinkers in a learning community. Stage One - Desired Results Enduring Understandings:What will students understand (about Essential Questions:What arguable, recurring, and thought- what big ideas) as a result of the unit? "Students will understand provoking questions will guide inquiry and point toward the big ideas that..." of the unit?

Mathematical thinkers effectively use their understanding, Why problem solve (or why think mathematically)? knowledge and skills. How does participation in a community of learners support Mathematical thinkers learn through collaboration and problem thinking and learning? solving. How do structures, strategies, and routines in problem solving support thinking and learning? How does productive mathematical thinking support critical thinking, growth and independence?

Learning Targets:

I can attack a math problem and solve it (MP1)

I can make sense of a problem (MP2)

I can prove my answer (MP3)

I can assist others when I do not agree with their answer (MP3)

I can use the appropriate vocabulary and symbols to represent a problem or solution (MP4)

I can use math tools to help me solve/show how I solved a problem (MP5)

I can explain the steps I used to solve a problem, and I am careful in my calculations (MP6)

I can use what I already know to help me solve a new problem (MP7)

I can use multiple ways to find a solution to a problem (MP8) STANDARDS

Unit: Unit 1 - Compacted 5/6 Review Subject: Mathematics Timeline: Week 2 to 3 Purpose: The purpose of this learning unit is for students to develop a learning path of previously taught 6th grade skills and concepts in a differentiated workshop model. Stage One - Desired Results Enduring Understandings:What will students understand (about Essential Questions:What arguable, recurring, and thought- what big ideas) as a result of the unit? "Students will understand provoking questions will guide inquiry and point toward the big ideas that..." of the unit?

Computational fluency includes understanding not only the What makes a strategy both effective and efficient? meaning, but also the appropriate use of numerical operations. How do you know if your solution is reasonable? Mathematical rules and expressions depict mathematical How do you determine what algorithm( including relationships operations,GCF, LCM, and distributive property) is There are multiple representations for any number. appropriate when solving a word problem. How can models and representations depict mathematical relationships?

Learning Targets:

I can use the distributive property to find the sum of two numbers that share a common factor. (example:36 + 8= 4(9 + 2). I can solve problems involving operations (+, -, x, ÷) with whole numbers, decimals, straight computation, or word problems. I can write, graph, interpret and or explain relationships between rational numbers in real-world contexts.

STANDARDS STATE: PA Core Standards (2014) CC.2.1.6.E.1 (Practiced) Apply and extend previous understandings of multiplication and division to divide fractions by fractions. CC.2.1.6.E.2 (Practiced) Identify and choose appropriate processes to compute fluently with multi-digit numbers. CC.2.1.6.E.3 (Practiced) Develop and/or apply number theory concepts to find common factors and multiples. CC.2.1.6.E.4 (Practiced) Apply and extend previous understandings of numbers to the system of rational numbers. CC.2.2.6.B.1 (Practiced) Apply and extend previous understandings of arithmetic to algebraic expressions. CC.2.2.6.B.2 (Practiced) Understand the process of solving a one-variable equation or inequality and apply it to real-world and mathematical problems. CC.2.2.6.B.3 (Practiced) Represent and analyze quantitative relationships between dependent and independent variables.

Unit: Unit 3 - Computation with Rational Numbers Subject: Mathematics Timeline: Week 7 to 10 Purpose: The purpose of this learning unit is for students to solve problems with rational numbers including fractions, decimals, and integers and to develop an understanding of how to use representations to model solutions. Stage One - Desired Results Enduring Understandings:What will students understand (about Essential Questions:What arguable, recurring, and thought- what big ideas) as a result of the unit? "Students will understand provoking questions will guide inquiry and point toward the big ideas that..." of the unit? Rational numbers can be represented in multiple forms How can the properties of whole numbers help us to perform (fraction, decimal, percent). operations of fractions and decimals? The same properties that were used to add, subtract, multiply, divide whole numbers can be applied to operations with all How can representations be used to model the solutions when rational numbers. adding and subtracting rational numbers?

Learning Targets: I can apply properties of operations to add, subtract, multiply & divide rational numbers, including real-world contexts. I can represent addition & subtraction on a horizontal or vertical number line. STANDARDS STATE: PA Core Standards (2014) CC.2.1.7.E.1 (Practiced) Apply and extend previous understandings of operations with fractions to operations with rational numbers.

Unit: Unit 4 - Expressions and Equations Subject: Mathematics Timeline: Week 11 to 15 Purpose: The purpose of this learning unit is for students to write and solve equations and inequalities, with an emphasis on real-world applications. Stage One - Desired Results Enduring Understandings:What will students understand (about Essential Questions:What arguable, recurring, and thought- what big ideas) as a result of the unit? "Students will understand provoking questions will guide inquiry and point toward the big ideas that..." of the unit? How do algebraic representations relate to one another? The properties of numbers can be applied to expressions, How can variables, expressions, equations and inequalities help equations and inequalities, to expand, simplify or solve. us solve problems? Learning Targets: I can apply properties of operations to add, subtract, factor and expand linear expressions with rational coefficients. I can apply properties of operations to calculate with numbers in any form and convert between forms as appropriate. I can use the distributive property to solve word problems with rational numbers. I can translate a word problem or real-world situation into an algebraic equation or inequality, solve that equation/inequality, and graph the solution set. I can reason about my answer, and determine if it is reasonable in the context of the problem. STANDARDS STATE: PA Core Standards (2014) CC.2.2.7.B.1 (Focus) Apply properties of operations to generate equivalent expressions. CC.2.2.7.B.3 (Focus) Model and solve real-world and mathematical problems by using and connecting numerical, algebraic, and/or graphical representations.

Unit: Unit 5 - Ratios and Proportional Reasoning Subject: Mathematics Timeline: Week 12 to 15 Purpose: The purpose of this unit is for students to develop an understanding of ratios and proportional relationships, with an emphasis on solving problems including unit rates, percents, and scale and reasoning through them. Stage One - Desired Results Enduring Understandings:What will students understand (about Essential Questions:What arguable, recurring, and thought- what big ideas) as a result of the unit? "Students will understand provoking questions will guide inquiry and point toward the big ideas that..." of the unit?

Ratios can be represented in numerous ways. How can ratios be represented? An understanding of proportional relationships provides the How are proportional relationships used in the real world? basis for solving many types of problems.

Learning Targets: I can find the unit rate a/b associated with a ratio a:b (with b ≠0) and use rate language in the context of a ratio relationship. I can make tables of equivalent ratios showing quantities with whole-number measurements and use them to find missing values, plot coordinates on a graph, and compare ratios. I can find a percent of a quantity as a rate per 100; solve problems involving find the whole, given a part and the percentage. I can use ratios to convert measurements from one unit to another and solve problems that involve multiplying and dividing quantities in more than one unit of measurement. I can solve unit rate problems, including those involving unit price and constant speed. I can determine whether 2 ratios are proportionally related. I can identify and interpret the constant of proportionality (unit rate) in tables, graphs, equations, diagrams & word problems. I can write equations to represent proportional relationships. I can solve multi-step ratio and percent problems (tax, discount, mark up / down, tips, commission, % change), using proportional relationships. I can solve problems involving scale & scale drawings. STANDARDS STATE: PA Core Standards (2014) CC.2.1.6.D.1 (Focus) Understand ratio concepts and use ratio reasoning to solve problems. CC.2.1.7.D.1 (Focus) Analyze proportional relationships and use them to model and solve real-world and mathematical problems.

Unit: Unit 2 - Statistics 6 Subject: Mathematics Timeline: Week 2 to 5 Purpose: The purpose of this learning unit is for students to collect, display and analyze data and communicate their statistical reasoning. Stage One - Desired Results Enduring Understandings:What will students understand (about Essential Questions:What arguable, recurring, and thought- what big ideas) as a result of the unit? "Students will understand provoking questions will guide inquiry and point toward the big ideas that..." of the unit? In what ways can data sets be evaluated and described? The question to be answered determines the data to be collected and how best to collect and display it.

Learning Targets:

I can understand that a statistical question will have a variety of answers.

I can display numerical data in plots on a number line, including line plots, histograms, and box-and-whisker plots. I can determine quantitative measures of center (e.g., median, mean, mode) and variability (e.g., range, interquartile range, mean absolute deviation). I can describe any overall pattern and any deviations from the overall pattern with reference to the context in which the data were gathered.

STANDARDS STATE: PA Core Standards (2014) CC.2.4.6.B.1 (Focus) Demonstrate an understanding of statistical variability by displaying, analyzing, and summarizing distributions.

Unit: Unit 6 - Geometry 6 Subject: Mathematics Timeline: Week 19 to 22 Purpose: The purpose of this learning unit is for students to use geometric concepts, properties, and relationships in two and three dimensions to model and solve problems, with an emphasis on communicating their reasoning. Stage One - Desired Results Enduring Understandings:What will students understand (about Essential Questions:What arguable, recurring, and thought- what big ideas) as a result of the unit? "Students will understand provoking questions will guide inquiry and point toward the big ideas that..." of the unit? Geometry is all around us in 2-D and 3-D figures. How does what we measure affect how we measure? Linear measure, area and volume are fundamentally different How can space be defined through numbers/measurement? but may be related to one another in ways that permit How do decomposing and recomposing shapes help us build calculation of one given the other. our understanding of mathematics? How can geometry be used to solve problems about real-world situations, spatial relationships, and logical reasoning? How can you solve problems by drawing polygons in the coordinate plane? Learning Targets:

I can determine the area of triangles and special quadrilaterals (i,e square, rectangle, parallelogram, rhombus, and trapezoid). I can determine the volume of right rectangular prisms with fractional edge lengths. I can use the coordinates to find side lengths and area of a polygon (limited to triangles and special quadrilaterals) when given coordinates for the vertices of a polygon in the plane. I can represent three-dimensional figures using nets made of rectangles and triangles. I can determine the surface area of triangular and rectangular prisms (including cubes.)

STANDARDS STATE: PA Core Standards (2014) CC.2.3.6.A.1 (Focus) Apply appropriate tools to solve real-world and mathematical problems involving area, surface area, and volume.

Unit: Unit 7 - Geometry 7 Subject: Mathematics Timeline: Week 23 to 27 Purpose: The purpose of this learning unit is for students to understand how to apply their knowledge of volume, surface area, circumference and area of circles, and scale to solve real-world problems. Students will extend their understanding of spatial relationships on the coordinate plane using transformations. Stage One - Desired Results Enduring Understandings:What will students understand (about Essential Questions:What arguable, recurring, and thought- what big ideas) as a result of the unit? "Students will understand provoking questions will guide inquiry and point toward the big ideas that..." of the unit? Two-dimensional and three-dimensional shapes take up space. How are spatial relationships, including shape and dimension, Geometric relationships can be described, analyzed, and used to draw, construct, model and represent real situations or classified based on spatial reasoning and/or visualization. solve problems?

Patterns exhibit relationships that can be extended, described, How can the application of attributes of geometric shapes and generalized. support mathematical reasoning and problem solving?

Measurements can be calculated by choosing and applying a How can geometric properties and theorems be used to formula and thinking logically and spatially to solve problems describe, model, and analyze situations? related to real-world situations.

Learning Targets: I can identify or describe the properties of all types of triangles based on angle and side measure. I can use and apply the triangle inequality theorem. I can describe the 2-dimensional figures that result from slicing 3-dimensional figures. I can use the properties of complementary, supplementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. I can identify and use properties of angles formed when two parallel lines are cut by a transversal. I can solve problems involving the area and circumference of a circle. I can solve problems involving scale drawings of geometric figures. I can solve real-world & mathematical problems involving the area, volume, and surface area of two and three dimensional figures, composed of triangles, quadrilaterals, polygons, cubes, right prisms.

I can identify corresponding, alternate interior, and alternate exterior angles.

I can determine if two polygons are similar by using a proportion. I can find unknown lengths of similar figures using a proportion.

I can determine an unknown angle measure in a triangle. I can graph a translation, reflection, rotation, and dilation. I can identify lines of symmetry and determine if a figure has rotational symmetry. I can describe the effect of dilation on two dimensional figures using coordinates and determine the scale factor of dilation. I can determine the sequence of transformations that shows two figures are similar. I can use transformations to determine if two figures are similar.

STANDARDS STATE: PA Core Standards (2014) CC.2.3.7.A.1 (Focus) Solve real-world and mathematical problems involving angle measure, area, surface area, circumference, and volume. CC.2.3.7.A.2 (Focus) Visualize and represent geometric figures and describe the relationships between them. CC.2.3.8.A.1 (Focus) Apply the concepts of volume of cylinders, cones, and spheres to solve real-world and mathematical problems. CC.2.3.8.A.2 (Focus) Understand and apply congruence, similarity, and geometric transformations using various tools. CC.2.3.8.A.3 (Focus) Understand and apply the Pythagorean Theorem to solve problems.

Unit: Unit 8 - Statistics and Probability 7 Subject: Mathematics Timeline: Week 28 to 33 Purpose: The purpose of this unit is for students to examine populations, with an emphasis on random sampling and measures of center and variability. Students will draw conclusions about a population or populations, using experimental and theoretical probability. Stage One - Desired Results Enduring Understandings:What will students understand (about Essential Questions:What arguable, recurring, and thought- what big ideas) as a result of the unit? "Students will understand provoking questions will guide inquiry and point toward the big ideas that..." of the unit? Making predictions and determining the likelihood of an event helps you plan future events. How do you know if a prediction is reasonable? In what ways can data sets be evaluated and described?

Learning Targets: I can determine and predict whether some outcomes are certain, more likely, less likely, equally likely, or impossible. I can determine the probability of a chance event given relative frequency. Predict the approximate relative frequency, given the probability. I can find the probability of a simple event, including the probability of a simple event not occurring. I can find probabilities of independent compound events using organized lists, tables, tree diagrams & simulation. I can determine whether a sample is random given a real-world situation. I can use data from a random sample to draw inferences about a population with an unknown characteristic of interest. I can compare numerical data distributions using measures of center and variability. STANDARDS STATE: PA Core Standards (2014) CC.2.4.7.B.1 (Focus) Draw inferences about populations based on random sampling concepts. CC.2.4.7.B.2 (Focus) Draw informal comparative inferences about two populations. CC.2.4.7.B.3 (Focus) Investigate chance processes and develop, use, and evaluate probability models.

Unit: Unit 9 - Number System and Functions Subject: Mathematics Timeline: Week 34 to 37 Purpose: The purpose of this unit is for students to learn how to represent numbers using a variety of structures with an emphasis on rational and irrational numbers. Students will create an understanding of functions. Stage One - Desired Results Enduring Understandings:What will students understand (about Essential Questions:What arguable, recurring, and thought- what big ideas) as a result of the unit? "Students will understand provoking questions will guide inquiry and point toward the big ideas that..." of the unit? How do we use estimation and determine its reasonableness in There are many ways to represent a number and an exact the real-world? answer is not always needed to solve a problem. Functions can be used to model relationships between How can the relationship between two quantities be shown? quantities. In what ways can functions be used to model real-world situations ?

Learning Targets: I can determine whether a number is rational or irrational. I can convert a rational number to show that the decimal expansion terminates or repeats. I can convert a terminating decimal to a rational number. I can compare rational numbers. I can find the square root of a perfect square. I can find the cubed root of a perfect cube. I can order rational numbers. I can locate a rational number’s location on a number line. I can estimate the value of an irrational number. I can use approximations to compare, order and graph irrational numbers. I can classify parts of a graph as linear or nonlinear. I can determine whether a relation is a function I can identify the domain and range of a function. I can identify possible solutions of an equation containing two variables. I can graph a linear equation by using an x/y table. I can write a function rule to describe the relationship between two variables STANDARDS STATE: PA Core Standards (2014) CC.2.1.8.E.1 (Practiced) Distinguish between rational and irrational numbers using their properties. CC.2.1.8.E.4 (Practiced) Estimate irrational numbers by comparing them to rational numbers. CC.2.2.8.C.1 (Practiced) Define, evaluate, and compare functions.