kids see the math. teachers see results. · 2018-09-05 · kids see the math. teachers see results....

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Program Overview

Kids See the Math. Teachers See Results.

ALGEBRA 1|GEOMETRY|ALGEBRA 2

2 3

Kids See the Math. Teachers See Results.

BUILT FOR FLORIDA

It Works in Every Classroom. Made for Blended, Full Digital Classrooms, or 1:1 Learning

1 2Assessment Formative and summative assessments align to the Mathematics Florida Standards and EOC Test Item Specifications.

3Instructional Support Meaningful, accessible teaching support provides flexibility for planning and instruction.

Understanding Problem-based Learning and Visual Learning deepen conceptual understanding of mathematics.


You’re going to love what you see. New enVision Florida A|G|A helps you teach Mathematics Florida Standards (MAFS), assess in the format of End-of-Course (EOC) Assessments, and manage student data.

Video

ACT 2 Develop a Model

5. Use the math that you have learned in the topic to refine your conjecture.

ACT 3 Interpret the Results

6. Did your refined conjecture match the actual answer exactly? If not, what might explain the difference?

354 TOPIC 8 Quadratic Functions Go Online | PearsonRealize.com

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MATHEMATICAL MODELING IN 3 ACTS

Video

The Long ShotHave you ever been to a basketball game where they hold contests at halftime? A popular contest is one where the contestant needs to make a basket from half court to win a prize. Contestants often shoot the ball in different ways. They might take a regular basketball shot, a hook shot, or an underhand toss.

What is the best way to shoot the basketball to make a basket? Think about this during this Mathematical Modeling in 3 Acts lesson.

ACT 1 Identify the Problem

1. What is the first question that comes to mind after watching the video?

2. Write down the Main Question you will answer.

3. Make an initial conjecture that answers this Main Question.

4. Explain how you arrived at your conjecture.

ACT 2 Develop a Model

5. Use the math that you have learned in the topic to refine your conjecture.

ACT 3 Interpret the Results

6. Did your refined conjecture match the actual answer exactly? If not, what might explain the difference?

MAFS.912.A-REI.4.10–Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Also F-IF.2.4

MAFS.K12.MP.4.1

TOPIC 8 Mathematical Modeling in 3 Acts 345

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MATHEMATICAL MODELING IN 3 ACTS

Video

The Long ShotHave you ever been to a basketball game where they hold contests at halftime? A popular contest is one where the contestant needs to make a basket from half court to win a prize. Contestants often shoot the ball in different ways. They might take a regular basketball shot, a hook shot, or an underhand toss.

What’s the best way to shoot the basketball to make a basket? Think about this during this Mathematical Modeling in 3 Acts lesson.

ACT 1 Identify the Problem

1. What is the first question that comes to mind after watching the video?

2. Write down the Main Question you will answer.

3. Make an initial conjecture that answers this Main Question.

4. Explain how you arrived at your conjecture.

MATHEMATICAL MODELING IN 3 ACTS The Long Shot 353

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Engage and motivate students with reality-based mathematical modeling that makes math inviting and interesting for all students.

See What They Can Do

UNDERSTAND ING

Focus on Mathematical Modeling

• Mathematical Modeling in 3 Acts lessons are available for every topic and engage students in the complete modeling cycle.

• Student Companion worktext organizes students’ thinking to actively develop a model.

enVision STEM Project

Science, Technology, Engineering, and Math(STEM)Projects provide opportunities for students to explore situations that address real social, economic, and environmental issues that foster mathematical connections across topics.

Anytime Interactive Learning with DesmosComprehensive integration of Desmos into Pearson Realize™ offers a groundbreaking interactive experience.

• Develop conceptual understanding through ready-to-go examples that bring mathematical concepts to life, available online and offline.

• Extend learning with Anytime Tools powered by Desmos.

• Save time with prebuilt interactives that help students focus on the math not the tool.

• Exclusive to enVision, switches, sliders, and buttons enable more focused student exploration.

Groundbreaking digital experiences let students see the math anytime, anywhere.

Mathematical Modeling Mathematical Modeling in 3 Acts is a collaborative task where students develop a mathematical model to explain a real-world problem. The high-interest, low-entry task develops students’ conceptual understanding, procedural fluency, and adaptive reasoning as they test out different models and conjectures to answer the question posed.

An engaging video introduces the question and gets

students talking.

Students determine what resources they need

and develop a solution to answer the question.

The final video reveals the answer and students

analyze the results.

ACT 1: THE HOOK ACT 2: MODEL WITH MATH ACT 3: THE SOLUTION

Learn More! Teacher’s Edition Program Overview Algebra 1: p. 62 Geometry: p. 60 Algebra 2: p. 56

Visual Glossary

EnglishA

Acute triangle

Adjacent angles

Alternate interior (exterior) angles

Altitude

Altitude of a triangle

17°

60° 45°

75°

Example

1 34 2

∠1 ∠2∠3 ∠4

Example Altitude

Spanish

Ángulo agudo

Triángulo acutángulo

Ángulos adyacentes

Ángulos alternos internos (externos)

Altura

Altura de un triángulo

G2

Step 3Practice & Problem Solving

Step 4Assess & Differentiate

Step 2Understand & Apply

Step 1Explore

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I Can See Clearly Now! Starting on a firm foundation of conceptual understanding, students can connect and apply math ideas in amazing ways.

UNDERSTANDING

A simple lesson design provides a clear, intentional pathway.

Step 1: Explore

Step 2: Understand & Apply

Lesson-opening explorations foster the development of conceptual understanding through a problem-solving experience.

MAFS are cited right on the

student page for easy reference.

Explore & Reason Students explore a mathematical concept and use reasoning to draw conclusions.

Model & Discuss Students develop proficiency with the full modeling cycle by focusing deeply on aspects of the modeling cycle.

Critique & Explain Students are required to construct mathematical arguments. They may also be asked to evaluate examples of mathematical reasoning and correct the reasoning if necessary.

Habits of Mind questions develop proficiency with

the Mathematical Practices.

enVision Florida A|G|A helps you teach mathematics through problem solving. Multiple examples support a balanced pedagogy: Conceptual Understanding, Proof, Skill, and Application.

Conceptual Understanding examples are designed to help students focus deeply on mathematical understanding of lesson content.

Proof examples require students to build, justify and analyze formal and informal proofs in enVision Florida Geometry.

Skill examples help students build fluency with the lesson content.

Application examples show students how the lesson’s mathematical content can be applied to solve real-world problems.

The Concept Summary provides

multiple representations to consolidate

student understanding.

Embedded digital interactives powered

by Desmos make graphical, numerical,

and algebraic connections.

English Language Learners (ELL) A complete library of resources supports English Learners:

• ELL instruction for various proficiency levels in every lesson

• Spanish closed captioning for video tutorials

• Spanish text and audio for Algebra 1 problem statements

• English/Spanish Glossary

• Multilingual Handbook

Learn More! Teacher’s Edition Program Overview

Algebra 1: pp. 48, 72 Geometry: pp. 46, 70 Algebra 2: pp. 42, 66

UNDERSTAND PRACTICE

Additional Exercises Available Online

Practice Tutorial

PRACTICE & PROBLEM SOLVING

For Exercises 17 and 18, does each transformation appear to be a rigid motion? Explain. SEE EXAMPLE 1

17.

preimage image

18.

preimage image

For Exercises 19–24, suppose m is the line with equation x = −5 , line n is the line with equation y = 1, line g is the line with equation y = x, and line h is the line with equation y = −2 . Given A(9, −3), B(6, 4), and C(−1, −5), what are the coordinates of the vertices of △AʹBʹCʹ for each reflection? SEE EXAMPLES 2 AND 3

19. R x-axis 20. R y -axis

21. R m 22. R n

23. R g 24. R h

For Exercises 25–28, what is a reflection rule that maps each triangle and its image? SEE EXAMPLE 4

25. D(3, 6), E(−4, −3), F(6, 1) and Dʹ(1, 6), Eʹ(8, −3), Fʹ(−2, 1)

26. G(9, 12), H(−2, −15), J(3, 8) and Gʹ(9, −2), Hʹ(−2, 25), Jʹ(3, 2)

27. K(7, −6), L(9, −3), M(−4, 6) and Kʹ(7, −4), Lʹ(9, −7), Mʹ(−4, −16)

28.

x

y

−2

2−2

P′

R′Q′

P

R

Q

O

29. Trace the diagram below. Where does the shopper in a dressing room see her image in each mirror? SEE EXAMPLE 5

shoppermirror 3mirror 1

mirror 2

11. Look for Relationships Becky draws a triangle with vertices A(6,7), B(9,3), and C(4, −2) on a coordinate grid. She reflects the triangle across the line y = 4 to get △AʹBʹCʹ . She then reflects the image across the line x = 3 to get △AʺBʺCʺ .

a. What are the coordinates of △AʹBʹCʹ and △A″B″C″ ?

b. Write a rule for each reflection.

12. Use Structure Under a transformation, a preimage and its image are both squares with side length 3. The image, however, is rotated with respect to the preimage. Is the transformation a rigid motion? Explain.

13. Error Analysis Jacob is playing miniature golf. He states that he cannot hit the ball from the start, bounce it off the back wall once, and reach the hole in one shot. Is Jacob correct? Trace and label a diagram to support your answer.

14. Higher Order Thinking For the miniature golf hole in Exercise 13, Jacob wants to bounce the ball off the back wall and then the right wall. Draw a diagram to show how Jacob can hit the ball so that it reaches the hole after two bounces.

15. Mathematical Connection Dana reflects point A(2, 5) across line ℓ to get image point A (6, 1). What is an equation for line ℓ?

16. Look for Structure Can a figure be reflected across three lines of reflection so the image is the original figure? Explain.

LESSON 3-1 Reflections 111

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PRACTICE & PROBLEM SOLVING

APPLY ASSESSMENT PRACTICE

Mixed Review Available Online

Practice Tutorial

30. Look for Relationships Which of the numbered stones shown cannot be mapped to another with a rigid motion?

1 2 3

4

5

910

67

8

11

12

13 14

31. Use Structure Reese is inside a shop and sees the sign on the window from the back. Draw the letters as they would appear from the outside of the shop. Is the transformation a rigid motion?

COFFEESHOP

32. Make Sense and Persevere Look at the floor plan below. Abdul sees the image of a clock in the mirror on the door.

Abdul Imageof ClockDoor

a. Trace the diagram. Where is the line of reflection? Explain.

b. Where is the clock located? Explain.

c. Find where Abdul’s image is located relative to the line of reflection. Can Abdul see himself in the mirror? Explain.

33. Classify whether each pair of figures appears to be a rigid motion or not a rigid motion.

preimage image preimage image

preimage image preimage image

34. SAT/ACT Consider the following reflection. Preimage: A(3, 9), B(2, –7), C(6, 14)

Image: Aʹ(–25, 9), Bʹ(–24, –7), Cʹ(–28, 14)

Suppose p is the line with equation x = 11, q is the line with equation x = 22, r is the line with equation x = –11, and s is the line with equation x = –22. What is the rule for the reflection?

Ⓐ R p (x, y) Ⓒ R r (x, y)

Ⓑ R q (x, y) Ⓓ R s (x, y)

35. Performance Task Sound echoes from a solid object in the same way that light reflects from a mirror. A hiker at point A shouts the word hello. The hiker at point B first hears the shout directly and later hears the echo.

AB

1,600 ft

Cliff

2,400 ft

Part A Trace the diagram. Show the path taken by the sound the hiker at point B hears echoing from the cliff.

Part B Sound travels at about 1,000 feet per second. After how long does the hiker at point B hear the shout directly? After how long does he hear the echo? Show your work.

G-CO.1.2

112 TOPIC 3 Transformations Go Online | PearsonRealize.com

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Practice with a Purpose Personalized and adaptive learning encourages students to build their mathematical understanding and demonstrate proficiency.

UNDERSTANDING

Step 3: Practice & Problem Solving

enVision Florida A|G|A features a uniquely balanced exercise set. Meet the rigor of the FSA with assessment practice in every lesson.

Step 4: Assess & Differentiate

Understand Develops conceptual understanding of lesson content by explaining reasoning, constructing arguments, and analyzing errors.

Apply Requires students to apply math to solve real-world problems.

Practice Builds procedural fluency with lesson content.

Assessment Practice Every lesson includes:

• EOC Practice

• ACT/SAT Practice

• Performance Task

Virtual Nerd Tutorial Videos• Tutorial videos are provided for every

lesson in the program.

• Three different viewing windows let students review math concepts in the visual way that best helps them learn.

• Students can easily drill down to another video to review prerequisite content.

• Available with Spanish closed captioning!

BouncePages Launch an interactive video from the print student page with the FREE BouncePages app.

Robust Practice Powered by MathXL® for School Embedded MathXL® for School in Pearson Realize provides a seamless experience for students and teachers with instant feedback, powerful interactive learning aids, and auto-graded assignments on ONE platform.

• Daily Homework and Practice

• Mixed Review

• Differentiated Learning for remediation, additional practice, and enrichment

Ensure MAFS mastery through multiple daily formative assessments. The differentiation library has print and digital resources to meet the needs of a wide range of learners.

Adaptive Practice Powered by Knewton• Easy to assign, instant access!• Assign from the enVision Florida Lesson Table of

Contents on Pearson Realize. No need to go to an outside web site or additional resources for adaptive learning.

• Remediation occurs for the skills taught that day. Delivers both instruction and practice automatically in real time.

• Pinpoints the right course-level and prerequisite skill.


EOC Practice in Every Lesson!

10 11

ASSESSMENT

Show Me What You Know enVision Florida uses a strategic blend of assessments aligned to the Florida Mathematics Standards and EOC Test Item Specifications to help determine what students know and are able to do.

Focus on the MAFS• Effectively assess the MAFS• Aligned to the EOC Test Item Specifications• Adheres to the Florida Content Complexity

expectations (DOKs)

Ongoing Florida Standards Assessment Practice format prepares students for the FSA.

• Assessment Practice items in every lesson in the Student Edition

• Topic Assessments• Benchmark Assessments• End-of-Course Practice Tests• Progress-Monitoring Assessments

(Forms A, B, and C) are item-for-item parallel and each covers the full year.

Florida Standards Assessment Practice Workbook provides additional weekly targeted MAFS practice and three, full-year EOC Practice Tests in print and online formats.

SUMMATIVE Assessment • Topic Assessments (Print/Online)• Topic Performance Assessments (Print/Online)• ExamView® Test Bank DVD-ROM• Benchmark Assessments (Print/Online)• End-of-Course Practice Tests (Print/Online)• Build Your Own Custom Assessment (Online)

FORMATIVE Assessment • Common Errors (Print/Online)• Habits of Mind Questions (Print/Online)• Try Its! (Print/Online)• Do You Understand? (Print/Online)• Do You Know How? (Print/Online)• Lesson Quiz (Print/Online)

DIAGNOSTIC Assessment • Course Readiness Assessment (Print/Online)• Topic Readiness Assessments (Print/Online)


Many assessments also available as editable

word documents

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ASSESSMENT

A variety of auto-generated reports show MAFS mastery on assessments, overall progress, and usage data. It’s all on PearsonRealize.com.

Gain Meaningful Insight

Data reports help drive differentiation.Mastery AnalysisIn-depth information is provided about MAFS coverage and mastery for an assignment.

Item AnalysisView individual test items across the class to gauge difficulty and make informed decisions.

Real-Time Data ReportsAddress individual learning needs quickly with real-time data reports. Adaptive practice powered by Knewton drives daily instructional decisions.

MAFS Aligned ResourcesIndividual student’s mastery or classwide mastery for each standard are linked to resources that can be immediately assigned.

NamePearsonRealize.com

GEOMETRY

4-5 Reteach to Build UnderstandingCongruence in Right Triangles

1. Which theorem shows each pair of right triangles is congruent? Explain.

A

C B D F

EA

C B D F

E

Example ASA; because two angles and the included side are congruent

a. ; because two angles and a nonincluded side are congruent

A

C B D F

EA

C B D F

E

b. ; because two sides and the included angle are congruent

c. ; because the hypotenuse and one leg of a right triangle are congruent

2. Carmen wrote a proof to show the triangles are congruent. What was her error?

∠C and ∠F are rt. ∠s. Given

AB = DE, AC = DF Given

AB ≅ DE , AC ≅ DF Congruent segments have equal measures.

△ABC ≅ △DEF SSA

3. Match each item on the left to only one item on the right.

Congruent hypotenuses HL

Right angles AB , DE

Congruent sides AC , DF

Theorem showing △ABC ≅ △DEF ∠C, ∠F

A D F

E

C4

5

4

5

B

A C

B

D F

E

enVision® Florida Geometry • Teaching Resources

NamePearsonRealize.com

GEOMETRY

4-5 Mathematical Literacy and VocabularyCongruence in Right Triangles

The first column shows the steps used to prove that AB ≅ ED . Use the first column to answer each question in the second column.

Problem ≅ in Right TrianglesGiven: C is the midpoint of AE and BD .Prove: AB ≅ ED

1. What is the definition of the midpoint of a line segment?

C is the midpoint of AE and BD ; AB ∥ DE ; m∠B = 90°.Given

2. How do you know that AB ∥ DE and m∠B = 90°?

AC ≅ EC ; BC ≅ DC Definition of midpoint

3. What does the symbol ≅ between two segments mean?

∠B ≅ ∠DAlternate Interior Angles Theorem

4. What does the word interior mean?

∠B and ∠D are right angles.Definition of a right angle

5. What is the measure of ∠D?

△ABC ≅ △EDCHypotenuse-Leg (HL) Theorem

6. What information is necessary to apply the HL Theorem?

AB ≅ ED Corresponding parts of congruent triangles are congruent.

7. What are the corresponding angles and sides for △ABC and △EDC?

A B

C

D E

enVision® Florida Geometry • Teaching Resources

14 15

ASSESSMENT

Differentiation options for each lesson and every Mathematics Florida Standard encourage and challenge students of all learning levels.

Focus on Each Learner enVision Florida A|G|A provides both a fully adaptive system for Response to Intervention and a library of resources to support a wide range of students.

Give all students what they need for success. Each lesson includes:Reteach to Build Understanding: Guided reteaching offers a fresh approach. Stepped-out, scaffolded support solidifies understanding.

Enrichment: Higher-order thinking activities help students develop deeper understandings. Additional Practice: More practice for each lesson. Mathematical Literacy & Vocabulary: Scaffolded support helps students build vocabulary. Virtual Nerd Videos: Tutorials are available for every lesson.

Adaptive Practice Powered by Knewton: Adaptive practice is a daily option to support students on prerequisite skills not yet mastered or to move advanced students through the skills more efficiently.

Interactive digital intervention instruction

Based on results from the Topic Readiness Assessment, individual study plans are automatically created to fill in gaps on prerequisite knowledge and help students focus on specific areas to experience success.

Interactive digital intervention practice

Individual Study Plans• Available for every Topic

• Automatically prescribed digital intervention instruction and practice help students master prerequisite skills.

• Interactive instruction with explicit examples

• Powerful learning aids in multiple modalities

Additional Examples• Additional explicit instruction assists teachers in

meeting their classroom needs.

• The “Try Another” feature, which algorithmically generates new problem statements, allows for endless classroom instruction and practice opportunities.

Enrichment Examples Extend the learning to enhance Algebra 2 students’ understanding and application of lesson concepts.

Individualized Learning Pathways

Available as an editable worksheet.

Available as a MathXL for School digital assignment.


INSTRUCTIONAL SUPPORT

MATH BACKGROUND FOCUS

TOPIC 7

Polynomials and Factoring

Operations of PolynomialsAdd and Subtract Polynomials In Lesson 7-1, students learn that a monomial is a real number, a variable, or the product of a real number and one or more variables, and a polynomial is a monomial or the sum or difference of two or more monomials (terms). Students learn that polynomials are closed under the operations of addition and subtraction. They add and subtract polynomials by combining like terms.

4x 3x2+ x2+2x+ 5+

xxx

x

x

x

1 1 1 1 1x2

x2

x2 x2

Multiply Polynomials In Lesson 7-2, students understand that polynomials form a system similar to integers when they recognize that polynomials are also closed under multiplication. Students multiply polynomials by applying the Distributive Property or by using tables and area models.

2x

x

1

3

{

Product of Binomials In Lesson 7-3, students see that the product of the square of a binomial (a + b) 2 always follows the same pattern: the square of the first term, plus twice the product of the first and last terms, plus the square of the last term. Students recognize that the product of two binomials in the form (a + b)(a − b) results in the difference of the two squares.

Topic 7 focuses on extending polynomials. Students identify the parts and factors of polynomials. Students understand how to factor trinomials using the greatest common factor, binomial factors, and special patterns. Students learn methods to add, subtract, and multiply polynomials.

Factoring PolynomialsGreatest Common Factor In Lesson 7-4, students learn that the greatest common factor of a polynomial is the greatest common factor of the coefficients combined with the variables that are common factors of each term.

Quadratic Trinomials (when a = 1) In Lesson 7-5, students recognize and understand that when a trinomial is in the form x 2 + bx + c , the factors are found by identifying a pair of integer factors of c that have a sum of b. Students then use the factors to write binomials that have a product equal to the trinomial.

x xx

xx

1 1 11 1 1

x2

(x + 3)

(x + 2)

Quadratic Trinomials (when a ≠ 1) In Lesson 7-6, students recognize and understand that when a trinomial is in the form ax 2 + bx + c , the factors are found by identifying a factor pair of ac that have a sum of b. When ac and b are positive, the second terms in the binomial factors are also positive. When ac is negative, the second terms in the binomial factors have opposite signs.

Special Cases In Lesson 7-7, students identify patterns that are used to factor a perfect-square trinomial and a difference of two squares.

Factoring a Perfect-Square Trinomial a 2 + 2ab + b 2 = (a + b) 2 a 2 − 2ab + b 2 = (a − b) 2

Factoring a Difference of Two Squares a 2 − b 2 = (a + b)(a − b)

TOPIC 7 258A TOPIC OVERVIEW

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MATH BACKGROUND COHERENCE

TOPIC 7

Polynomials and Factoring

MAKING MATHEMATICAL CONNECTIONS

Looking BackHow does Topic 7 connect to what students learned earlier?

GRADE 8• Factoring Students factored expressions

by identifying the greatest common factor and using the Distributive Property. Students apply this knowledge in Topic 7 when factoring polynomials, including trinomials with binomial factors.

• Operations and Properties Students learned that operations with integers and exponents were similar to the concept of adding and subtracting digits with the same place value. In Topic 7, students recognize that polynomials form a system that is similar to integers and use the same properties of equality to add, subtract, and multiply polynomials.

4x3x2

6x4x2

x2 5

5

2x

xxx

x

x

x

1 1 1 1 1x2

x2

x2 x2

• Polynomials Students were previously introduced to polynomials and identified different types of polynomials. In Topic 7, students extend this understanding of polynomials by adding, subtracting, multiplying, and factoring polynomials.

ALGEBRA 1• Multiply Exponents In Topic 6, students

learned about multiplication properties of exponents and solved exponential equations. In Topic 7, students apply these properties of exponents when multiplying polynomials and finding products of binomials.

Students learn best when concepts are connected through the curriculum. This coherence isachieved within topics, across topics, across domains, and across grade levels.

IN THIS TOPICHow is content connected within Topic 7?

• Addition, Subtraction, and Multiplication Students recognize that the operations of addition, subtraction, and multiplication are closed for polynomials (Closure Property). In Lesson 7-1, students add and subtract polynomials by combining like terms. In Lesson 7-2, students multiply polynomials using different methods.

• Products In Lesson 7-3, students identify patterns in the square of a binomial, (a + b) 2 or (a − b) 2 , and in the products of two binomials in the form (a + b)(a – b). Students apply these patterns to simplify expressions and solve problems.

• Factors of Polynomials In Lesson 7-4, students factor polynomials by finding the greatest common factor of the terms. In Lesson 7-5, students factor a trinomial in the form x 2 + bx + c by finding a pair of integer factors of c that have a sum of b and then using the factors of c to write the binomial factors. In Lesson 7-6, students factor a trinomial in the form ax 2 + bx + c by finding a factor pair of ac that has a sum of b. In Lesson 7-7, students factor special cases of polynomials such as a perfect-square trinomial and a difference of two squares.

1 and −6

−1 and 6

−5

5

Sum of FactorsFactors of −6

MAKING MATHEMATICAL CONNECTIONS

Looking AheadHow does Topic 7 connect to what students will learn later?

ALGEBRA 1• Quadratics In Topic 7, students

recognize that a quadratic trinomial in the form ax 2 + bx + c is factored by finding a factor pair of ac that has a sum of b. In Topic 8, students will identify the key features of a quadratic function and use quadratic functions in various forms to model real-world problems. Then, in Topic 9, students will solve quadratic equations using tables, graphs, and factoring.

ALGEBRA 2• Operations on Polynomials In Topic 7,

students learn how to add, subtract, and multiply polynomials. In Algebra 2, students will expand their knowledge of operations on polynomials to include division of polynomials.

8

x2

5x

8x

40

(x + 5)(x + 8)

x

x

5

CALCULUS • Procedures with Exponents In Topic 7,

students apply their understanding of multiplying exponents to multiplying and factoring polynomials. In Calculus, students will extend their knowledge of multiplying and factoring polynomials and expressions containing exponents to find derivatives of powers of x.

TOPIC 7 258B TOPIC OVERVIEW

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AVAILABLE ONLINE

Factoring Polynomials

7-4Activity Assess

I CAN… factor a polynomial.

MODEL & DISCUSS

A catering company has been asked to design meal boxes for entrees and side dishes.

A. Design a meal box that meets each of these requirements:

a. Equal numbers of sections for entrees and side dishes

b. More sections for entrees than for side dishes

c. More sections for side dishes than for entrees

B. Use Structure For each meal box from Part A, write an algebraic expression to model the area of the meal boxes.

STUDY TIPRecall that finding the prime factorization of a number is expressing the number as a product of only prime numbers.

EXAMPLE 1 Find the Greatest Common Factor

What is the greatest common factor (GCF) of the terms of 12 x 5 + 8 x 4 − 6 x 3 ?

Step 1 Write the prime factorization of the coefficient for each term to determine if there is a greatest common factor other than 1.

12 8 6

2 ∙ 2 ∙ 3 2 ∙ 2 ∙ 2 2 ∙ 3

Step 2 Determine the greatest common factor for the variables of each term.

x 5 x 4 x 3

x ∙ x ∙ x ∙ x ∙ x x ∙ x ∙ x ∙ x x ∙ x ∙ x

The greatest common factor of the terms 12 x 5 + 8 x 4 − 6 x 3 is 2 x 3 .

One instance of 2 is the only common factor of the numbers, so the GCF of the coefficients of this trinomial is 2.

The sections for theentrees must be square.

The sections for the side dishes are half the length and width of the entree sections.

Three instances of x are the only common factors of the terms, so the GCF of the variables is x 3 .

Try It! 1. Find the GCF of the terms of each polynomial.

a. 15 x 2 + 18 b. −18 y 4 + 6 y 3 + 24 y 2

ESSENTIAL QUESTION How is factoring a polynomial similar to factoring integers?

MAFS.912.A-SSE.1.2–Use the structure of an expression to identify ways to rewrite it. For example, see x4 − y4 as (x2)2 − (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 − y2)(x2 + y2).

Also A-APR.1.1

MAFS.K12.MP.1.1, MP.4.1, MP.7.1

LESSON 7-4 Factoring Polynomials 283

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a. b.

c.

A.

B. a. 5x2; b. 6x2; c. 4x2

Where x is the length of a side of an entree section.

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STEP 1 ExplorePearsonRealize.com

Activity

MODEL & DISCUSSINSTRUCTIONAL FOCUS Students experiment with possible configurations of different size squares having side lengths that are multiples of a single factor. Students write equations for the area of these configurations. The activity prepares students to learn how to factor polynomials and factor a polynomial model.

STUDENT COMPANION Students can complete the Model & Discuss activity on page 287 of their Student Companion.

Before WHOLE CLASS

Implement Tasks that Promote Reasoning and Problem Solving Q: How does the size of the side dish sections compare to the size

of the entrée sections? [The side dish sections are half the size of the entrée sections on each side, so a quarter of the area.]

During SMALL GROUP

Support Productive Struggle in Learning Mathematics Q: What do you know about the relationship between the length

and width of each section? [Since the entrée sections are square, the length and width are equal. The length and width of the side dish sections are also equal since they are half the length and width of the entrée sections.]

For Early FinishersQ: Explain how to determine the maximum value for the side

lengths of entrée and side dish sections if the overall size of the meal box is 8 in. × 6 in. [Since the side length of an entrée plus side dish is 3x, and the smallest dimension of the meal box is 6 in., let 3x = 6 or x = 2 . So side dishes are 2 in. and entrée sections are 4 in.]

After WHOLE CLASS

Facilitate Meaningful Mathematical Discourse Q: What do all the terms of your equations for area have in

common? [The variable for the side length of the sections.]

Q: How is the formula for area affected when there is more than one of a particular section? [The area is multiplied by the number of sections.]

SAMPLE STUDENT WORK

STUDENT EDITION, PAGE 283

HABITS OF MINDConstruct Arguments Can you meet more than one of the three requirements with the same-sized meal box? Use a mathematical argument to support your answer.

[Yes, a meal with more sides than entrées and a meal with more entrées than sides could fit in the same box. A box with 8 sides and 4 entrées has the same area as a box with 5 entrées and 4 sides.]

Use with MODEL & DISCUSS

TOPIC 7 283B LESSON 4

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MANAGING CLASSES

Ideas, inspiration, and teaching methods. Math background for every Topic and Lesson serves as an easy-to-access math methods course.

Classroom Videos show a classroom in action. Interviews with the teacher cover planning and reflection.

Professional Development videos on PearsonRealize.com give important perspectives on math concepts and show the program in action.

myPearsonTraining.com features many online tutorials and quick-start guides for enVision Florida. Available 24/7!

Make every lesson perfect for you. Access all digital content, assessments, and management tools at PearsonRealize.com.• Search by keyword or MAFS• Customize lessons• Reorder lessons and Topics• Align to your district framework• Access Google Classroom™• Add OpenEd resources• Upload your own content• Use online discussion boards

Gain a new perspective on your teaching with embedded strategies, methods, and a wide range of professional development opportunities.

See the Big Picture

Every math teacher is a master teacher.

Easy drag-and-drop customization

Research-Based PracticesThe enVision Florida A|G|A Teacher’s Edition features embedded professional development. Effective Teaching Practices (ETP) are probing questions based on NCTM’s Principles to Action.

Learn More! Teacher’s Edition Program Overview

Algebra 1: pp. 66, 80 Geometry: pp. 64, 78 Algebra 2: pp. 60, 74

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INSTRUCTIONAL SUPPORT

Teach using multiple modalities and tiers. All components support the MAFS and prepare students for FSA success. You don’t have to look anywhere else!

Florida Resources

Student Companion two-color, consumable student worktext increases in-class engagement as students explain their thinking, solve problems, and make it their own.

Florida Standards Assessment Practice Workbook provides weekly targeted MAFS practice and three, full-year EOC Practice Tests.

Teacher’s Edition two volumes include all support for teaching the program in print or digitally.

Teacher’s Edition Program Overview is a user guide and PD resource in one! Explore pacing, lessons, differentiated instruction, components, and correlations to the MAFS.

Teacher Assessment Sourcebook includes an Assessment Guide, Readiness Test Masters, Topic Assessment Masters, Topic Performance Task Masters, Lesson Quizzes, Benchmark Assessment Masters, and Progress-Monitoring Assessment Masters.

Student Edition includes all instructional content. Available digitally with the Student Companion at point of use through the Interactive Student Edition: Realize Reader.

Robust math tools powered by Desmos include a graphing calculator, scientific calculator, and geometry tools available online and offline.

Author Professional Development Videos provide practical tips on implementing the program in a high school math classroom.

Classroom Videos show a classroom in action. Interviews with the teacher cover planning and reflection.

Pearson Realize™ Learning Management System offers a full suite of personalized teaching and learning tools to help students master the MAFS.

Mathematical Modeling in 3 Acts videos accompany Mathematical Modeling in 3 Acts lessons.

enVision® STEM Projects include videos and blackline masters.

Interactive Digital Lessons are easily projected to bring concepts to life, available online and offline.

Additional Examples help students in need of more instruction.

Enrichment Examples extend the learning and enhance student understanding in Algebra 2.

Ready-to-go, easily customizable and auto-scored MathXL® for School assignments support daily practice, mixed review, remediation, additional practice, and enrichment.

Adaptive Practice powered by Knewton automatically adjusts to student performance and intervenes with instructional support as needed.

Virtual NerdTM Tutorial Videos act as a 24/7 personal tutor with closed captioning in Spanish.

Editable Teacher Resource Masters for vocabulary support, remediation, additional practice, enrichment, graphing calculator activities, assessments, and more!

Ready-made, auto-scored assessments provide auto-assigned remediation.

Technology-enhanced items throughout the program prepare students for the EOC.

Wealth of reporting options include Mastery, Progress, and Usage.

Answers and Solutions software application provides answers and solutions to textbook problems.

ExamView® desktop test generator software includes test banks with thousands of additional questions.


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kids see the math. teachers see results. · 2018-09-05 · kids see the math. teachers see results....

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