Wednesday, April 17, 2024

# Transition To Algebra Unit 1 Answers

## Unit : Area Model Factoring

Common Core Algebra II.Unit 3.Lesson 6.Piecewise Linear Functions

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 and one dimension or length of the area model rectangle to find the other length . 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 : 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.

## Transition To Algebra Curriculum

Transition to Algebra is an initiative of the Learning and Teaching Division at Education Development Center, Inc. . Supported by;the;National Science Foundation, TTA seeks to quickly give students the mathematical knowledge, skills, and confidence to succeed in a standard first-year algebra class and to show;them that they can explore mathematics and actually enjoy it.

TTA is a full-year algebra support curriculum, published by;Heinemann;and distributed by their parent company Houghton Mifflin Harcourt,;designed to:

• run concurrently with first-year algebra ; and
• raise the competence and confidence of students who may benefit from supports for algebra success.

TTA is designed to build students’ algebraic habits of mind, key mathematical ways of thinking aligned with the Common Core Standards for Mathematical Practice. Students explore algebraic logic puzzles that connect to and extend algebra course topics and learn broadly-applicable tools and strategies to help them make sense of what they are learning in;algebra. Students discuss and refine their ideas as they work through mental mathematics activities, written puzzles, spoken dialogues, and hands-on explorations that engage them in cultivating mathematical knowledge, intuition, and skills.

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## Unit : 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 and vertical 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 1: 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.

The contents of this website are the results of work supported by the National Science Foundation under Grants No. 0917958 and 1621011. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. The text, images, and resources on this website, unless marked otherwise, are developed at EDC . EDC owns the copyright © 2011-2019.

## Unit : 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 more than one piece of information. This understanding is furthered when students use transformations to affect a shapes 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 , they come to understand graphs as a collection of solution points, which will support future work in Unit 9: Points, Slopes, and Lines.

## Unit : 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.;;

## Algebraic Habits Of Mind

One important purpose of learning mathematics is to develop useful, analytic, quantitative, logical ways of looking at the world and thinking about things. When mathematical ways of thinking begin to become automaticnot just ways one can use, but ways one is likely to useit is reasonable to call them habits: mathematical habits of mind.

The Standards for Mathematical Practice listed in the Common Core State Standards are one way of organizing and presenting a subset of mathematical habits of mind that are essential for mathematical proficiency. Transition to Algebra focuses on algebraic aspects of five of these: puzzling and persevering, seeking and using structure, using tools strategically, describing repeated reasoning, and communicating with precision. Transition to Algebra is designed to develop these algebraic habits by fostering and building on the common-sense logic that students bring.

## Unit : Thinking Things Through Thoroughly

Transition Math

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.

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## Teaching Guide Area And Multiplication Transition To Algebra Unit 4 Teaching Guide Unit

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1 4 Unit TEACHING GUIDE Area and Multiplication Transition To Algebra Unit 4 Teaching Guide June Mark E. Paul Goldenberg Mary Fries Jane M. Kang Tracy Cordner Unit Area and Multiplication 4 HEINEMANN Portsmouth, NH Research-based National Science Foundation-funded Learning transforms lives.

3 Unit 4 Area and Multiplication CONTENTS T4 Unit Introduction T7 Lesson 1: MysteryGrid Puzzles T11 Lesson 2: Multiplication Patterns T15 Lesson 3: Measuring Area T20 Lesson 4: Area Models T22 Student Reflections & Snapshot Check-in T23 Lesson 5: Area Models with Algebra T26 Lesson 6: Signs and Terms T28 Lesson 7: Squares and Like Terms T30 Lesson 8: Equivalent Expressions T32 Student Reflections & Unit Assessment T33 Exploration: Trains T35 Exploration: Building Squares T38 Game: Greatest Product Challenge RESOURCES T39 Latin Square Puzzles T40 MysteryGrid Puzzles T41 Multiplication Table T42 Areas to Calculate T43 Matching Cards Cutouts T44 Snapshot Check-in T45 Snapshot Check-in Answer Key T46 Unit Assessment T48 Unit Assessment Answer Key T50 MENTAL MATHEMATICS Base 10 structure: place value T51 Multiplying by 10 and 100 T52 Dividing by 10 and 100 T53 Dividing 2-digit numbers by 10 T54 Dividing by 10 repeatedly T55 Multiplying decimal numbers by 10 T56 Multiplication table with multiples of 10 T58 How many zeros? T60 Multiplying by 2 and 3 with area model thinking T61 Squaring T62 Squaring multiples of 10 T63 Square roots

## Geo1 Constructions And Rigid Transformations

In this unit, students first informally explore;geometric properties using straightedge and compass constructions. This allows them to build conjectures and observations before formally defining rotations, reflections, and translations. In middle school, students studied;transformations of figures in the coordinate plane. In this unit, they transition to more formal definitions;that don’t rely on the coordinate plane, and the focus shifts from transforming whole figures towards a more point-by-point analysis. Students then begin to use the rigorous definitions they developed to prove statements involving angles and distances, preparing them for congruence proofs in the next unit.

A blank reference chart is provided for students, and a completed reference chart for teachers.;The purpose of the reference chart is to be a resource for students to reference as they make formal arguments. Students will continue adding to it throughout the course. Refer to About These Materials;in the Geometry course for more information.

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## Unit : 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.

## Seeking And Using Structure:

Mathematically proficient students look closely to discern a pattern or structure… They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects…

Students in beginning algebra are often taught to solve equations like 2 + 4 = 24 by going through a particular set of steps written in a particular way. On successive lines, they write the original equation, -4 below each side, a new equation, ÷2 below each side, a new equation, and so on. That method works for all first-year algebra cases and helps students see the steps. But it doesnt encourage students to see beyond the trees and perceive the overall structure. Both are necessary. Seeing the structure helps students see the logic of algebra; it often also makes calculation much easier.

Consider the example to the right. Proceeding in the conventional way is much, much harder than seeing the equation as 11 minus something is 6 and concluding that the something equals 5. We can read that as fifty divided by something is 5. So, that something ;is 10. That leaves 3x 2 = 10, which is easier to solve.

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## Unit : 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 : 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.

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## Unit 5 Logic Of Algebra

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