Transition to Algebra's 12 Units of Study

Unit 1: Language of Algebra

Language of Algebra presents algebra as a language for expressing patterns and relationships. Starting with a pictorial notation for the results of number tricks, students build intuition about algebraic solving processes. They develop algebraic language to describe these tricks as they move from pictures, to words, and finally to algebraic notation in order to track the transformations of an unknown starting number. Students also use the logic of mobile puzzles to learn about equivalence and to visualize the common sense behind algebraic solving steps.

Unit 2: Geography of the Number Line

Geography of the Number Line presents the number line as a tool for reasoning about integers and the relationships between integers, including order and distance. The number line is also used as a tool for making sense of the operations of addition and subtraction, first with numbers and then generalized to variables.

Unit 3: Micro-Geography of the Number Line

Micro-Geography of the Number Line extends the thinking students have used to make sense of integers in Unit 2: Geography of the Number Line to decimals and fractions. Students zoom in on the number line and observe that decimals and fractions continue to follow many of the conventions (in naming and ordering) and structures (that help to find distance and perform operations) that were previously explored with integers. Students repeatedly experience the idea that decimals and fractions follow the same logic as integers and that being able to locate a particular decimal or fraction on a number line is an important step in working with these numbers.

Unit 4: Area and Multiplication

Area and Multiplication builds a common-sense foundation for multiplying algebraic expressions by examining multiplication in the context of area. Students use area models to multiply integers and numerical expressions in order to support understanding the logic of the distributive property. They then extend their understanding of this model to multiplying algebraic expressions and identifying equivalent expressions. Students will revisit area models extensively in Unit 10: Area Model Factoring.

Unit 5: Logic of Algebra

Logic of Algebra builds facility with manipulating expressions and equations thoughtfully, logically, and accurately. Students use the common-sense logic they have developed in the Think of a Number tricks and mobile puzzles to make sense of the rules for solving equations. Solving for unknown weights in puzzles about balance builds the same logic as is required in solving equations and systems of equations: seeking every unknown value by performing only operations that preserve balance and substituting equivalent values or shapes as needed.  

Unit 6: Geography of the Coordinate Plane

Geography of the Coordinate Plane builds familiarity and facility with the conventions of the coordinate plane, graphs, and equation graphing. Students explore coordinated data to understand how a single point can represent (and connect) more than one piece of information. This understanding is furthered when students use transformations to affect a shape’s x-coordinates, y-coordinates, or both. By the end of the unit, students graph equations using the idea that every point on a graph represents a particular x and y pair whose values are related by the equation. As students use equations to test arbitrary points to see whether they are on the graph of an equation (and hence are solutions of the equation), they come to understand graphs as a collection of solution points, which will support future work in Unit 9: Points, Slopes, and Lines.

Unit 7: Thinking Things Through Thoroughly

Thinking Things Through Thoroughly is designed to stimulate mathematical thinking and to offer more practice solving problems, skills that take time and experience to develop. Problem solving requires a kind of thinking that cannot quite be captured in “steps.” Students must look for what is familiar and unfamiliar in the problem, look for entry points, poke around a bit, pay attention to hunches and intuitions, and assess whether their trials and experiments move them closer to an insight or a solution. Unit 7 invites students to consider what they can determine from a context and what possibilities there are before focusing on the answer requested. Students use tables and diagrams to organize information, and learn to repeat and generalize calculations to produce algebraic equations to describe problem contexts.

Unit 8: Logic of Fractions

Logic of Fractions builds on tools and strategies developed in earlier units, focusing on basic operations with rational numbers and rational expressions. Number lines and area models are used to help students make sense of additive and multiplicative operations with fractions. Mobile puzzles are used to build students’ intuition about proportional reasoning. In the last lesson, students see how graphs can be used to depict proportional relationships. Throughout the unit, students make connections between numerical quantities and algebraic expressions in ways that extend their understanding of the logic of fractions.

Unit 9: Points, Slopes, and Lines

Points, Slopes, and Lines aims to help students understand that graphs and equations help to visualize and describe a collection of points. Building on the ideas from Unit 6: Geography of the Coordinate Plane, students think about two relationships between any two points: distance and slope. They will compare points by describing the horizontal (x) and vertical (y) distances between them and will use these measurements to find the length of the straight-line path between any two points. They will also use the ratio of vertical to horizontal distance to quantify the slope of that straight-line path. They will use slope to determine collinearity, test whether points are on a line, generate new points along a line, and create an equation to describe the location of all points on a line.

Unit 10: Area Model Factoring 

Area Model Factoring presents factoring as one kind of “un-multiplying,” using the ideas of Unit 4: Area and Multiplication, which introduced the area model as a tool for organizing multiplication of numbers and polynomials. In this unit, students first divide with area models, working with a given area (the product of the multiplication that is being undone) and one dimension or “length” (a factor) of the area model rectangle to find the other length (the other factor). Students connect the models to multiplication and division equations that express the same relationship and explore area model puzzles which provide enough information about the factors and product to find the missing terms. This supports students in understanding the roles of and relationships between the pieces of the area model and their corresponding equations and in thinking flexibly with the model in preparation for factoring. The factoring problems in this unit focus primarily on expressing trinomial products as two binomial factors. Students use tables to organize possible sums for pairs of numbers with a given product and build logic about the zero product property and its role in solving equations by factoring.

Unit 11: Exponents

Exponents helps students make sense of exponential growth and leads students through extending the ideas of positive exponents to negative and fractional exponent. Students raise 2 to the 12th power by repeated multiplication. The exponent keeps track of the number of times 2 is used as a factor. Then students work backwards from 212, dividing by 2 until they reach the unfamiliar 21, 20, 2-1, and so on. Students see why, to keep notation consistent, it must be that 21 = 2, 20 = 1, and 2-1 = 1/2. They then generalize to a1 = a, a0 = 1, and a-1 = 1/a. Within the structure of multiplication and division, these make sense as a way to extend the existing pattern. Students practice using exponents in area models and in rational expressions and conclude the unit by examining rational exponents. Students consider why rational exponents would exist. For example in order to talk about numbers between 102 and 103, we might choose a notation like 102 1/2 to name them.

Unit 12: Algebraic Habits of Mind

Algebraic Habits of Mind reviews, consolidates, and extends several of the overarching themes of the year: building equations and expressions, distance and area modeling, and solving problems carefully with logic, persistence, and strategy. Lesson 1 reviews the use of variables and expressions as well as the kinds of manipulations that preserve balance. Lesson 2 revisits the use of the number line as a tool for reasoning about location, order, and basic operations. Lesson 3 reviews the area model and its use for multiplying, dividing, and factoring polynomial expressions. Lesson 4 distinguishes two types of measurement on the plane: distance and slope. Lesson 5 reviews the concept of solving and what a solution looks like in various contexts.