algebra and geometry...lesson or activity ways to encourage dialog support for learning lesson solve...

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Algebra and Geometry

Video Workshop Answer Key

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Solving Linear Equations.......................................................................................................3

From Concrete to Symbolic ...............................................................................................4

Instructional Strategies ......................................................................................................6

Encouraging Mathematical Discourse ...............................................................................8

Standards........................................................................................................................11

Before and After ..............................................................................................................13

Group Activities ...............................................................................................................15

Wait Time ........................................................................................................................17

Extension Lesson ............................................................................................................19

Solving Open Sentences Involving Absolute Value .............................................................22

What’s the Big Idea? .......................................................................................................23

Make It Their Mathematics ..............................................................................................25

Lesson Elements.............................................................................................................27

What Comes Next? .........................................................................................................29

Graphing Systems of Linear Inequalities .............................................................................30

Concept Development .....................................................................................................31

Outline for a Lesson ........................................................................................................33

Recommendations ..........................................................................................................35

Warming Up . . . ..............................................................................................................37

Solving Quadratic Equations by Completing the Square .....................................................39

Prerequisite Knowledge...................................................................................................40

Next, please. ...................................................................................................................42

The Benefits Are . . .........................................................................................................44

Using Manipulatives ........................................................................................................46

Angle Relationships.............................................................................................................48

Learning Standards .........................................................................................................49

Goals for Improving Instruction........................................................................................51

Lesson Highlights ............................................................................................................53

Questioning Techniques ..................................................................................................55

Bisectors, Medians, and Altitudes........................................................................................57

Modeling Geometry Concepts .........................................................................................58

Words for the Wise..........................................................................................................60

Another Point of View......................................................................................................62

Communication Standards ..............................................................................................63

Tangents .............................................................................................................................65

Best Practices .................................................................................................................66

In the Flow.......................................................................................................................68

Exemplary Lesson Elements ...........................................................................................70

Cooperative Groups ........................................................................................................72

Analyzing Graphs of Quadratic Functions ...........................................................................73

Strategy-Based Outcomes...............................................................................................74

Lesson Strategies............................................................................................................77

Evaluation of Teaching Practices.....................................................................................79

Activities and Strategies ..................................................................................................81

Ellipses................................................................................................................................83

Lesson Time Management ..............................................................................................84

Instructional Strategies ....................................................................................................86

Lesson Process...............................................................................................................88

Piece of the Puzzle..........................................................................................................90

Solving Linear Equations


From Concrete to Symbolic 4

From Concrete to Symbolic

Sample content, topics, and answers will vary.

Concrete Representation

• Students act out the word problem. • Students use classroom currency sets (or create their own paper currency to use as

models). • Students can use the strategy of Guess and Check as they act out the problem. • Students next use plastic cups (to represent variables) and 2-color chips (to represent

integers to model the problem.

Using Language (vocabulary/narrative)

The cost of 1 children’s ticket to the zoo is $4.00 less than the cost of 1 adult ticket. Two adult tickets and 4 children’s tickets cost $38.00. How much does 1 adult ticket cost? How much does 1 children’s ticket cost?

Symbolic Representation<*>

2a + 4(a 4) = 38

Think and Discuss

Possible answers may include the following: What prerequisite vocabulary and skills must students have mastered before they

have a lesson on multi-step equations?

Participants may discuss the following prerequisites: • Vocabulary:

variable, equation, expression, evaluate, factor, exponent, base • Skills:

solving single-step equations using the order of operations understanding the concept of a variable and equality understanding the process of solving an equation


From Concrete to Symbolic 5

How would you determine your students’ readiness for this lesson?

• Review/assess student understanding of the order of operations by having students solve numeric expressions using the order of operations.

• Review/assess student ability to solve single-step equations • Review/assess student ability to represent single-step equations with concrete and/or

pictorial models.

Discuss other specific lessons that should come before a lesson on solving multistep

equations. • using order of operations to evaluate numeric expressions • solving single-step equations • writing algebraic expressions for verbal expressions Alternate Discussion Topic

Possible answers may include the following: Ask participants what new techniques they have learned in this or other professional

development that they will definitely try in their own classrooms. Participants may discuss the following techniques: • using vocabulary activities as warm-up activities and/or to informally assess students • using technology (e.g., such as graphing calculators or dynamic geometry software) to

demonstrate concepts and to test conjectures • using “Wait Time 2” (pausing after students give responses to questions before

indicating whether the response was correct to provide time for other students to think about the response and to give alternative responses)

• using algebra tiles for students to concretely model algebraic equations Have them discuss sources for new ideas other than formal professional development; for example, they might observe colleagues or use Internet courses.

Participants may discuss some of the following as ways to find new strategies and ideas for their lessons: • observing and collaborating with other teachers • reading professional journals • attending district workshops • using the County Office of Education resource library (where available) • attending conferences • using educator’s web sites • incorporating ideas from other subject areas, such as science, social studies, and language

arts


Instructional Strategies 6

Instructional Strategies Possible answers may include the following:

Instructional Strategies Used by Michael Cox Warm-Up Exercise Vocabulary Review Problem Solving Pair Work Real-Life Applications Using Manipulatives Informal Assessment Thoughtful Questioning Multiple Representations Writing in Mathematics

Lesson on Solving Multistep Equations

Most Effective Strategies:

• Making connections to prior learning • Using multiple representations (e.g., algebra tiles and equations) • Selecting preferred solution method (e.g., students selected either concrete or

symbolic representations) • Making connections to real-life applications • Explaining solution strategies • Modeling problem-solving strategies

My Lesson

Instructional Strategies • Warm-Up Exercise • Vocabulary Review • Pair Work • Multiple Representations • Informal Assessment

Implementation Notes As a warm-up activity, have students work in pairs to complete a vocabulary activity. • Students will create a representation of each

vocabulary term. • Students will be challenged to create multiple

representations for each term. • Students will have concrete materials available

(e.g., algebra tiles, cups and chips, geometry manipulatives).

• Students may also create drawings, write verbal representations, or write symbolic/algebraic representations.

Think and Discuss

Possible answers may include the following:

Michael Cox’s students were comfortable working in small groups and had pre-

assigned partners for pair work. What groupings have you used in your classroom? Participants may discuss the following groupings: • student pairs—mixed ability (i.e., peer tutoring or mentoring) or similar abilities

• small groups—heterogeneous or homogeneous (including student interests or language abilities)

• large groups—groups with 6–12 students



What have you done or would you do to prepare your students to work in small

groups or pairs? Participants may discuss the following strategies or techniques for preparing their students to work in groups or pairs: • creating task cards or handouts that list activity tasks, procedures, and evaluation criteria • creating a chart that lists and then discussing with the class, expectations for group

behavior (e.g., active listening, taking turns speaking, sharing tasks, ways to ask for help and ways to provide help).

• creating a chart that lists and then discussing with the class, roles for group members (e.g., Supplies Person, Spokesperson, Recorder)

• discussing with the class procedures for gathering, using, and returning materials, manipulatives, and tools.

• arranging the classroom (i.e., moving desks, clearing space for supplies, and making room for students to make presentations)

Alternate Discussion Topic Possible answers may include the following: Michael Cox’s homework assignment for his students included a parent participation

component. Have participants share ways they have successfully involved parents in

their students’ work. Participants may discuss the following suggested activities: • “Parent-Child: Graph-Equation: Team Work Activity”: The parent draws a line on a

coordinate grid and the student uses the slope–intercept form to determine the equation for the line. As an alternative, the parent can write an equation, and the student can create a table of values and then graph the equation.

• Students can bring home a sandwich bag of paper algebra tiles and show their parents how to use the algebra tiles to represent and solve algebraic equations. As an option, parents can record a more traditional symbolic approach to solving the equation as the student completes each step with the algebra tiles.

• Students can create a crossword puzzle that uses mathematical vocabulary and then have their parents complete the puzzle.

Have them also discuss any problems associated with parent involvement.

Participants may discuss some of the following as potential problems associated with parent involvement: • Parents may not have time, or may not feel comfortable taking the time, to complete

activities with students. • Parents may not have the required mathematical content knowledge required for the

activity. • Parents may not be available (due to unusual work schedules, or other factors). • Parents and students may not interact positively as they work on the activity.


Encouraging Mathematical Discourse 8

Encouraging Mathematical Discourse


Lesson or Activity Ways to Encourage

Dialog Support for Learning

Lesson Solve Multistep Equations.

• Lesson includes problem solving in small groups.

• Students share their work at the board.

• Students hear multiple-solution processes.

• Students get opportunities to use mathematical vocabulary.

• Students become more comfortable discussing mathematics.

Lesson Use coordinate grids to investigate transformations of plane figures.

• Students work in small-groups during the investigation.

• Students create oversized graphs that they share with the rest of the class.

• Students extend learning by applying knowledge of ordered pairs within the context of the activity (i.e., identifying vertices of shapes after transformations).

Activity Students complete vocabulary activities, such as word puzzles or graphic organizers.

• Students work in pairs for the activity.

• The nature of the activity (e.g., puzzles) encourages discussion as students complete the activity.

• Students discuss mathematical vocabulary.

• Students can help each other by sharing knowledge of terms.

Activity Students, working in pairs, recreate hidden designs drawn on coordinate grids by following partner’s verbal directions.

• Students work in pairs for the activity.

• Student giving directions needs to determine how to describe the hidden design so his or her partner can recreate it.

• Students discover the importance of communicating clearly and using precise language.



Think and Discuss Possible answers may include the following: Communication standards for mathematics exist at national, state, and local levels. What communication standards and objectives do you address during your lessons?

• If participants do not mention them, you may want to discuss the National Council of Teachers of Mathematics Communication Standards.

Instructional programs from pre-kindergarten through grade 12 should enable all students to: organize and consolidate their mathematical thinking through communication communicate their mathematical thinking coherently and clearly to peers, teachers,

and others analyze and evaluate the mathematical thinking and strategies of others use the language of mathematics to express mathematical ideas precisely

• Communication standards vary from state to state. Many state standards do not have

specific communication standards. Some states address communication skills within the larger strand of Problem Solving; for example, Texas standards for 8th grade include:

Underlying processes and mathematical tools. The student communicates about Grade 8 mathematics through informal and mathematical language, representations, and models. The student is expected to: (A) communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models; and (B) evaluate the effectiveness of different representations to communicate ideas.

How do you teach these standards? Do you address them specifically, or do you

expect that your lesson structure will cover them?

• Most participants will likely discuss that they do not teach communication standards specifically, but rather, plan instruction to include activities and assignments that incorporate communication.

• As an example, participants may discuss cooperative group activities they have implemented. This instructional strategy encourages communication because students are working in small groups where interaction, discussion, and sharing of ideas can take place in a nonthreatening setting. Many of these activities involve the use of models that can also encourage communication.

• Participants may also discuss activities that involve writing as a strategy to address communication standards.



Alternate Discussion Topic

Possible answers may include the following: Michael Cox had his students use algebra tiles to model solving multistep equations.

Have participants identify other lessons they have taught using algebra tiles. Have participants suggest other algebra topics supported by the use of algebra tiles.

Algebra tiles can be used to support other algebra topics, such as: • modeling polynomials • modeling and solving equations • modeling the distributive property • solving systems of equations • modeling operations with monomials, binomials, and polynomials • finding the square of a sum Discuss the importance of students having practice with the manipulatives before they are used to model concepts.

Students benefit from the opportunity to discover attributes of the manipulatives, explore how they can be used, and become familiar with the manipulatives prior to using them for specific tasks. This also helps alleviate some students’ interest in using the manipulatives to “play” or in otherwise inappropriate ways.


Standards 11

Standards Possible answers may include the following:

Using Standards Benefits and Concerns

Reasons Michael Cox gives Standards Documents to his students and discusses standards in class:

• He doesn’t want just to tell students the standards. He wants students to discover why they are doing an activity, and what they are doing or accomplishing with this activity.

• Gives students insight into where they are headed. • As they do things, they have a built-in reason why they are doing it. • Helps students and parents see why mathematics is so important, for their lives. Additional benefits from involving students in the application of standards

• helps students take ownership in their learning • helps prepare students for statewide assessments

Concerns regarding this approach:

• Students may feel too much pressure. • May place too much importance on mastery of standards instead of other important

outcomes such as reasoning abilities, interest, and motivation. • The standards may not have meaning for students. Think and Discuss


What other techniques do you use to involve your students in their own learning?

Participants may discuss some of the following strategies and activities as ways to involve students in their own learning: • cooperative-group activities • providing opportunities for students to make and test conjectures • providing opportunities for students to make presentations, share solution approaches,

and to explain their reasoning • having students research and bring in examples of real-life applications of mathematical

concepts • having students write personal goals for current units of study, and having students

revisit their goals at the end of the unit Think and Discuss

Discuss any group strategies that you have used for this purpose.

• Grouping strategies that may be discussed include: small groups (3-to-5 students) student pairs large groups (6-to-12 students)

• As an example, participants may describe creating small groups of students to prepare a whole-class presentation on a currently studied mathematical concept. Students in each group may be assigned specific roles or tasks to contribute to the presentation.


Standards 12

What was the outcome? Would you use the same strategy or strategies again?

• Participants may discuss successful experiences and the desire to use the same strategy again.

• If participants did not have a successful experience, they may discuss using different grouping arrangements the next time, such as student pairs instead of small groups. Participants may also discuss making modifications to the tasks, such as providing an outline for the final product to help students narrow their tasks.

Alternate Discussion Topic Possible answers may include the following: Michael Cox wants his students to understand that mathematics is a part of everyday

life. Have participants discuss real-world examples that could be applied to the study

of multistep equations.

The following example, from Glencoe Algebra 1, can be used to illustrate an interesting real-life example of using algebraic equations to solve problems: • The Washington Monument in Washington, DC, was built in two phases. During the first

phase, from 1848–54, the monument was built to a height of 152 feet. From 1854–78, no work was done. Then from 1878–88, the additional construction resulted in its final height of 555 feet. How much of the monument was added during the second construction phase? Write an equation to solve the problem.

• Students could be encouraged to research other famous monuments and write similar word problems. Students could research the Statue of Liberty, the Eiffel Tower, and the Great Archway in St. Louis, MO.

Have them suggest sources and resources for real-world problems for this or other

pre-algebra or algebra lessons.

Resources for real-world problems: • National Council of Teachers of Mathematics web resources, periodicals, and other

printed materials • colleagues and peers • textbook publishers’ resource materials • the television show Numb3rs • Web sites, such as:

Purple Math: http://www.purplemath.com/ USA Today Education: http://www.usatoday.com/educate/home.htm?Loc=vanity


Before and After 13

Before and After Possible answers may include the following.

Before a Lesson on

Graphing Linear Equations

After a Lesson on

Graphing Linear Equations

• Find and locate ordered pairs on a coordinate grid.

• Able to solve simple equations with 1 and 2 variables.

• Know how to perform operations with fractions and negative numbers.

• Know something about the standard form of a linear equation.

• Know which equations are going to be linear and which are not.

• Identify the x-intercept and y-intercept. • Determine whether a solution to a

graph of a linear equation is reasonable.

Think and Discuss

[Sample content, topics, and answers will vary.]

Bea Moore-Harris recognizes that experienced teachers are able to anticipate problems

that students may experience with new concepts. Choose another lesson topic. Discuss what students need to know when entering the lesson, identify the concepts with which

students will have difficulties, identify some common misconceptions, and discuss what

students should have mastered by the end of the lesson. • Topic: solving multistep equations • What students need to know entering the lesson: solving single-step equations, applying

the order of operations, completing operations with fractions and integers, properties of equality

• Concepts that may be difficult: consistently performing the opposite operation to simplify equations, particularly regarding signs of numbers (operations with negative numbers)

• Misconceptions students may have: using the order of operations in standard order instead of in reverse order (as the process of simplifying equations and isolating variables requires)

• What students should have mastered by the end of the lesson: solving multistep equations, using the order of operations to simplify equations, using more than one step to isolate a variable, checking solutions by replacing the variable in the original equation with the solution value, and checking for a true equation


Before and After 14


Possible answers may include the following: Bea Moore-Harris encourages teachers to continually try new and exciting things in

the mathematics classroom. Have participants discuss what she means by this. Bea Moore-Harris discusses the importance for both new teachers and veteran teachers to try new instructional strategies and to continually look for ways to improve their instruction and to keep their instruction fresh. Some suggestions she makes include: • incorporating technology as an effective and efficient tool • striving to have more effective communication in the classroom • having students become more responsible for articulating their ideas, either verbally or in

written statements • having students explain why they approached a problem the way they did • incorporating more group activities to encourage student interaction • considering appropriate ways to group students as well as appropriate tasks for students

to complete in the group setting • integrating the inquiry method from science in which students are more involved in

leading discussions and teachers act more as facilitators Ask how they find new strategies to try with their students. Have them suggest

resources for ideas to make their lessons different each year. Participants may discuss some of the following as ways to find new strategies and ideas for their lessons: • observing and collaborating with other teachers • reading professional journals • attending district workshops • using the County Office of Education resource library (where available) • attending conferences • using educator’s web sites • incorporating ideas from other subject areas, such as science, social studies, and language

arts


Group Activities 15

Group Activities


Cooperative Group Activity 1

Activity:

Students work in small groups at a computer station to design and create a spreadsheet to produce a table of values for a linear equation. Students will print their spreadsheets and discuss their procedures with the class. Informal Assessment: Writing Activity: Students are asked to write the steps to follow to create a spreadsheet for a given equation.


Activity: Students work in small groups writing word problems for linear equations. Each group has one linear equation that is shared by all groups and one unique linear equation. Groups create charts that show their word problems. Informal Assessment:

Observe groups: Look for student understanding of concepts and for misconceptions.


Activity: Students work in small groups on the following investigation: How can linear equations in slope–intercept form be used to classify figures graphed on coordinate grids as parallelograms? Students draw parallelograms on oversized coordinate grids, use ordered pairs from lines that form the sides of the figures to write corresponding linear equations in slope–intercept form. Students investigate the slopes of parallel and perpendicular lines. Informal Assessment:

Extension activity: When given four linear equations in slope–intercept form, can students determine if the resulting figure (when drawn on a coordinate grid) would be a parallelogram?

Think and Discuss

Possible answers may include the following: Cooperative groups require careful classroom management. Identify several cooperative group activities that have worked well in your classrooms.

• Participants will reference their activities above.


Group Activities 16

What strategies did you implement before, during, and after the lesson to ensure the

success of these activities? • Before: Identify tasks for students, consider seating arrangements, gather and prepare

materials, prepare student task handouts and recording sheets, and identify activity criteria • During: Communicate expectations for group behavior, including roles of group

members; discuss and distribute task activities, procedures, and criteria; monitor groups; clarify concepts for students and address misconceptions; and when needed, encourage student interaction.

• After: Review effectiveness of instruction: Were students able to complete the activity tasks in the allotted time? Did students stay on-task with a minimum of off-task or disruptive behavior? Did students have the right tools and materials to complete the activity? Were the grouping assignments appropriate? Did students understand what was expected of them? Review student work, check for evidence of concept understanding, and identify students who may need remediation.

Alternate Discussion Topic Possible answers may include the following: Tashana Howse used a warm-up activity in her lesson. Have participants describe any

effective warm-up activities they have used with their students.

Participants may discuss the following activities: • Vocabulary Activities: There are a variety of vocabulary activities students can complete

as warm-up activities, including writing definitions and providing examples; completing vocabulary grids in which some definitions, examples, and terms are listed and students must fill in the missing items; and completing crossword puzzles or word search puzzles.

• Error Analysis: Present to students a sample of “student work” in which an error has been made. Students write an error analysis in which they describe the error and explain why they student may have made that error.

• Journal Writing: As an example, students can be asked to write everything they know about a topic (within a time limit). Students can share statements that can be recorded on a chart, the board, or an overhead transparency. Students can discuss the validity of each statement.

• “Problem of the Day”: Use a Problem of the Day–type problem-solving activity as a warm-up. This problem can be presented to students as they first enter the classroom (e.g., displaying it on the overhead projector, written on the board, or on handouts at students’ desks).

Ask if they use warm-ups in every lesson and, if so, how the warm-ups vary from day

to day. Participants may discuss using warm-up activities everyday as a way to help students focus and prepare for the day’s instruction. Participants who use warm-up activities everyday may likely use a pool of activities that may or may not have a direct relation to the day’s lesson. Have them discuss the purpose of a warm-up and what it should contribute to the

lesson.

Participants may discuss the following goals for warm-up activities: • Make connections to prior learning. • Motivate students. • Encourage discussion. • Teach or reinforce vocabulary. • Informal assessment. • Reinforce or assess prerequisites.


Wait Time 17

Wait Time Possible answers may include the following.

Think and Discuss

Possible answers may include the following: Bea Moore-Harris suggested that Tashana Howse could have used Wait Time 2 in her

lesson on graphing linear equations. Where in the lesson would it have been

appropriate to use that strategy? Participants may discuss the following opportunities within Tashana Howse’s lesson to use Wait Time 2:

• When she asks about the equation =3

4x y , and she asks: “How can you make x a

fraction? [Student replies: Put it over 1.] • When she asks Sierra: “Is [example] D a linear equation? [Sierra replies: No, because

the x and the y are not separated by addition or subtraction.] • When solving the equation 2x + y = 6 for the x-intercept and y-intercept, she asks

Xavier: “If that is the x-intercept, what is that point? [Xavier replies: The point would be (3, 0).]


Wait Time 18

Are there any other suggestions that Mrs. Howse could have used to support or

expand her lesson? Participants may make the following suggestions: • Incorporate technology into the lesson, such as dynamic geometry software or graphing

calculators • Incorporate real-life examples of linear equations.

Alternate Discussion Topic Possible answers may include the following: Bea Moore-Harris commented on the modeling of notes in the classroom lesson.

Have participants discuss the various ways that they can help students learn to take

good notes. If participants do not mention them, bring up guided notes, model notes,

and verbal instructions. Have participants discuss how they help students develop note-taking skills in their classrooms. • Graphic organizers are helpful tools that students can use to assist with note taking. • Use self-sticking notes to restate, summarize, or otherwise highlight critical areas of text

is another helpful strategy. • Additional strategies include:

Students working in pairs read, discuss, and take notes Students writing pre-reading organizing questions

• Participants may discuss the importance of having students rewrite their notes to further strengthen their understanding of the material and to aid in remembering the content.


Extension Lesson 19

Extension Lesson Possible answers may include the following:

Discuss with participants the instructional strategies that Tashana Howse mentions in her Reflections video.

Tashana Howse mentions the following instructional strategies: • addressing student misconceptions • using small-group activities • using warm-up activities (“Motivators”) • guiding students to take notes • asking students to explain answers


Lesson Topic

Graphing linear equations using slope–intercept form

Objectives:

1. Identify the slope and y-intercept of a graph. 2. Write and graph linear equations in slope–intercept form.

Manipulatives, Media

• student handouts for warm-up activity; matching graphs for each linear equation • graph paper for students to graph linear equations

Lesson Plan • Warm-Up/Introduction: Review graphing linear equations by finding a table of values

and plotting the ordered pairs on a coordinate grid. Post graphs of linear equations around the classroom. Distribute to student pairs a handout that lists four equations. Student pairs will create a table of values for each linear equation and then use the table of values to identify the graph (displayed somewhere in the room) that corresponds to each of their linear equations.

• Teach: Students learn how to graph linear equations using slope–intercept form. Review x-intercept and y-intercept with students and how they can be found (by solving the equation with x = 0 or y = 0). Review how to graph lines using this approach. Discuss with students that there is another approach: using the slope and the y-intercept. Discuss slope with students. Discuss the process of writing linear equations in slope–intercept form. Demonstrate how to graph a line using slope–intercept form.

• Practice/Apply: Students compare and analyze graphs, discussing variations in the corresponding linear equations and how those variations affect the characteristics of each line.

• Assess: Students complete a writing activity individually at the end of the lesson.


Extension Lesson 20

Individual Activities Pair/Group Activities

Informal Assessment:

• Students, at the end of the activity, complete a writing activity.

• Students are asked to respond to the following prompt:

Describe the lines for the following equations.

Discuss the similarities and differences between the lines:

+=1

14

xy

= 14xy

=1

14

xy

+= 24xy

+=1

24

xy

Warm-Up Activity:

• Student pairs complete warm-up activity.

• Students, after creating a table of ordered pairs, find matching graph for each of their assigned linear equations

Practice/Apply Activity:

• Students work in small groups comparing and analyzing graphs and their linear equations written in slope-intercept form.

• Students make observations regarding how the slope and y-intercept affect the characteristics of each line.

• Informal Assessment: Complete individual writing activity, described above. • Remediation: Review graphing linear equations by creating a table of values. • Enhancement: Use graphing calculators to investigate lines of linear equations. Think and Discuss

Possible answers may include the following: Tashana Howse asks students to explain their thinking so that she can determine if

they understand concepts or if they are simply copying what she has done. What

informal assessment strategies do you use with your students? What information do these strategies give you about students’ understanding of concepts or their mastery

of skills? Which informal assessments are most effective?

Some of the informal assessment strategies that participants use may include: • Questioning: Including asking “why” questions, which encourage students to think

critically and reason effectively about the content (especially when used along with wait time), requires students to justify their answers, helps students build logical reasoning, and provides teachers information about students’ thinking.

• Writing Activities: Writing activities used as informal assessment can be very effective. Students may feel more comfortable expressing ideas in writing (than verbally), thus providing more insight into student understanding of concepts and misconceptions they may have.

• Modeling Concepts: Asking students to model concepts is another effective informal assessment strategy. Students can be asked to use concrete or pictorial models to represent concepts or procedures. This strategy can provide opportunities for students to demonstrate understanding in alternative ways (from symbolic, abstract, verbal or written forms). Asking students to model concepts or procedures may reveal gaps in student understanding as some students may be able to complete traditional exercises involving the concept without a deep conceptual understanding.


Extension Lesson 21


Possible answers may include the following: Part of lesson planning is gathering and creating materials in advance so that the

lesson runs smoothly. Ask participants what evidence of planning was shown in the video lesson.

Evidence of planning that participants may mention include: • writing outline notes on the whiteboard • preparing the classroom for the warm-up activity: using tape to mark grid on the floor,

determining ordered pairs for each student desk, and writing ordered pairs on cards for each student

• creating charts with sample linear equations • preparing small-group activity: creating student work sheets, distributing chart paper for

students to use to create their graphs Have them discuss the preparations they commonly make for a lesson, including

research materials preparation, etc.

Participants may discuss the following preparation tasks: • identifying the goals for the lesson • clearly identifying the mathematical content for student activities • identifying criteria for student success • identifying any assessment that will take place during or after the lesson • identifying real-life examples of the concepts • gathering models to use to represent or demonstrate concepts or procedures • writing outline of notes • writing and solving sample problems • determining grouping arrangements for the lesson • writing activity tasks, procedures, and student expectations on charts, overhead

transparencies, or student handouts • preparing materials for activities (e.g., placing manipulatives or other tools in one central

location, placing items in bins or sandwich bags, distributing items to student desks • arrangement of the classroom • previewing software for any demonstrations

22

Solving Open Sentences Involving Absolute Value


What’s the Big Idea? 23

What’s the Big Idea? Possible answers may include the following:


Prerequisite Skills

• Perform operations with integers. • Solve inequalities. • Graph inequalities on a number line. • Solve 1- and 2-step equations and inequalities. Core Concepts

• Solve absolute value equations. • Solve absolute value inequalities.

Mastery Concepts

• Understand magnitude and that distance is always positive. • Solve equations by removing absolute value symbols and representing their

magnitude in a positive and negative way (solving for the two cases). • Graph solution sets. • Understand concept of equality and of inequality and the order being held.

Think and Discuss

Possible answers may include the following: Discuss the “big idea” concept of equality and identify topics, in addition to absolute value, that rely on an understanding of this “big idea.”

Some of the topics that involve equality include: • working with algebraic expressions and equations • solving equations, including multistep equations • understanding and applying a wide variety of mathematical properties (i.e.,

properties of numbers, properties of operations, and properties of equality) • investigating linear relationships and working with linear equations • investigating relationships between and among rational numbers • working with ratios and proportions • investigating functions (i.e., linear, quadratic, and exponential) Discuss how these concepts and skills are connected and how they develop

across lessons.

Some of the ways these concepts and skills are connected and develop across lessons include: • Students simplify and solve expressions and equations and then move to linear

equations. • Students examine linear functions and then expand that to quadratic and exponential

functions. • Students solve 1-step equations, move to multistep equations, move to quadratic

equations, and eventually move to systems of equations. • Students works with monomials, binomials, and polynomials. • Students work with rational expressions and then radical expressions.


What’s the Big Idea? 24

Alternate Discussion Topic Possible answers may include the following: Carol Molloy talks about the need for a cohesive curriculum build from various resources and guidelines, such as textbooks and state standards. Have

participants discuss the resources and guidelines that they use to plan lessons

and how they incorporate information and requirements from multiple sources. • Resources and guidelines that participants discuss may include:

state standards district curriculum frameworks statewide assessments graduation requirements textbooks and textbook resources and materials

• Strategies for incorporating resources that participants may discuss include: creating time lines for the yearly instructional calendar and then incorporating

resources into the time line creating and using assessments to determine where gaps in student

understanding are and then tailoring lessons to meet student needs, including identifying appropriate resources

planning collaboratively with colleagues using statewide assessments as an overarching framework, incorporating state

standards into this framework, and then incorporating resources


Make It Their Mathematics 25

Make It Their Mathematics


Helping Students Make the Mathematics Their Mathematics

Students should be encouraged to think about the mathematics and how it (the mathematics) can be extended to new situations.

Ways I Can Encourage My Students to Do This: • Make connections to prior learning. • Provide activities that encourage students

to construct relationships, apply concepts and procedures to problem-solving situations, reflect about their learning, and articulate what they know.

• Encourage students to notice relationships among concepts.

• Ask students to describe connections between multiple representations.

Students should be encouraged to reflect on the mathematics, articulate it, and talk about it with their classmates.

Ways I Can Encourage My Students to Do This: • Ask the right kinds of questions (e.g., Why

did your solution work? How is that solution like the other solution? How did you decide to solve the problem that way?).

• Have students make and test conjectures. • Have students discuss alternative

strategies. • Have students discuss the efficiency of

various strategies and approaches.

Students should be encouraged to challenge each other’s mathematical conclusions.

Ways I Can Encourage My Students to Do This: • Have students share their solutions and

explain their procedures. • Have students question students about

their solution strategies. • Use Wait Time 2 (waiting after students

give a response).


Make It Their Mathematics 26

Think and Discuss Possible answers may include the following:

Describe an example from your teaching in which you provided an opportunity for students to think about, extend, reflect, articulate, and discuss mathematics in a

meaningful way. What was the lesson topic? Describe the activity and any

grouping arrangement that helped foster this experience. May want to discuss the example Jason Willcoxon used in his Reflections video segment. • Lesson Topic: Determining Interest • Students worked in small groups to solve interest problems. Students used the

context of determining interest that they could earn from their own savings.

Alternate Discussion Topic Possible answers may include the following: In Carol Molloy’s discussion of best practices, she discusses the importance of

developing effective instructional transitions within a lesson. Have participants

discuss and describe strategies for ensuring smooth instructional transitions within lessons.

Participants may discuss strategies such as: • planning transitions ahead of time • letting students know what is expected of them during transitions • letting students know the time constraints they will be working under • making sure students bring one activity to closure before moving on to the next

activity


Lesson Elements 27

Lesson Elements


Instructional Practice

Example from Lesson Student Benefits

Using Physical Models • Created a large number line on the floor

• Had two students stand back to back at 0 and then walk in opposite directions to model the absolute value of 3

• Helped students visualize the meaning of absolute value and reinforced the mathematical principal that absolute value is always positive

Using Application Problems

• Used the example of someone (e.g., at a carnival) trying to guess someone’s weight within 3 pounds

• Used the example to model identifying the constant and the parameters and setting up Case 1 and Case 2 and how to solve for each case

• Engaged student interest • Connected the application

problem to the symbolic representation

Small-Group Instruction

• Students worked together to solve open sentences involving absolute value.

• Students used chart paper to show their solutions.

• Enabled the teacher to informally assess students

• Enabled peer-to-peer teaching

• Promoted student discussion

Students Sharing Results

• Students presented their solutions (and charts) to the whole class.

• Students discussed the two cases, solution sets, and graphs.

• Enabled teacher to provide guidance and to help students identify errors

• Reinforced students’ understanding of concepts and procedures

• Saw varied solution approaches

• Promoted student discussion


Lesson Elements 28

Think and Discuss Possible answers may include the following: Discuss the effectiveness of instructional strategies used by Jason Willcoxon in

his lesson. Do you believe this was an effective lesson? What would you have done differently?

• Some of the effective strategies that participants may discuss include: making connections to prior learning encouraging student discussion using physical models using a real-life application using small-group activities informally assessing students having students share results

• Ways participants may have approached the lesson differently may include: moving from one activity to another a little more slowly (Jason moved from one

activity to another pretty quickly) using more wait time, both after teacher questions and after student responses having students model more problems on the physical model (i.e., the oversized

number line) having students solve application problems in their small-group activity


Possible answers may include the following: One of the strategies used by Jason Willcoxon was to provide an application

problem as an example of the lesson concepts. Providing real-life examples can engage student interest in lesson concepts. Have participants describe other real-

life examples for absolute value concepts.

Other real-life examples of absolute value concepts include: • the margin of error in polls (e.g., voter polls) • keeping your heart rate in its target range during exercise • the temperature in a refrigerator staying within a recommended range • precision in measurements • tire pressure within the manufacturer’s recommended range • a contestant on a game show guessing the value of a car, within $1,500, without

going over • having students guess how long 1 minute is and then determining the average range

of guesses (e.g., if guesses are within 6 seconds, what is the range of guesses?)


What Comes Next? 29

What Comes Next? [Sample content, topics, and answers will vary.]

Extending Lesson

Topic:

• Graphing Inequalities on the Coordinate Plane Use of Multiple Representations • Create an oversized coordinate grid on the floor. • Students use graphs on handouts. • Students create oversized coordinate grids on chart paper. • Use technology (i.e., graphing calculators). Making Connections Between Models and Symbols

• As pairs of students model the lines on the oversized coordinate grid on the floor, the class records the same solutions on their handouts.

• When students present their graphs to the class, one student discusses the graph and another student writes the symbolic representation on the board.

Use of Application Problems

• Students solve application problems (e.g., as target heart range, tire pressure). recommendations, temperature settings for a refrigerator, and game show contestants guessing prices of items

Think and Discuss Possible answers may include the following: Are there certain elements that you try to incorporate in your extension lessons?

Do you assign written activities? Application problems? Discuss the strategies,

techniques, and student activities that are appropriate for extension lessons. Extension lesson strategies that participants discuss may include the following: • using technology to further investigate concepts (e.g., graphing calculators) • solving real-life applications • completing written assignments • completing project-based activities or investigations Alternate Discussion Topic Possible answers may include the following: Point out to participants that prior to starting an extension lesson or to assigning students an extension activity, it is important to assess student mastery of lesson

concepts. Have them discuss and describe assessments that would be

appropriate to gauging student mastery of absolute value concepts. Assessments that could be used to gauge student mastery of absolute value concepts may include: • presenting graphs and having students write the corresponding absolute value

equation or inequality • having students solve absolute value application problems • presenting an absolute value equation or inequality and asking students to describe

what the graph of its solution would look like

30

Graphing Systems of Linear Inequalities


Concept Development 31

Concept Development Possible answers may include the following:

Progression of Understanding

Concept: Graphing Systems of Linear Inequalities Prerequisite Concept

• understanding equality using single-variable equations and inequalities Build On • solving systems of linear equations Apply To

• solving systems of linear inequalities Concept: Open Sentences Involving Absolute Value Prerequisite Concept • understanding integers and graphing integers on number lines Build On

• understanding absolute value and using number lines to represent absolute value Apply To

• solving open sentences involving absolute value Think and Discuss

Possible answers may include the following: Describe some inequality concepts that you teach in which you help your

students build their understanding. How do you extend your students’ knowledge

of prerequisites to new concepts? How do you help your students apply their new learning to more advanced concepts?

Some of the continuums of skills that participants may discuss may include the following: • Prerequisite Concept: Solve equations involving absolute value. • Concept: Solve inequalities involving absolute value. • Apply to New Concepts: Solve real-life applications involving inequalities involving

absolute value.

• Prerequisite Concept: Simplify expressions before solving linear equations. • Concept: Simplify expressions before solving linear inequalities. • Apply to New Concept: Solve problems involving linear inequalities.

• Prerequisite Concept: Graph linear equations. • Concept: Graph linear inequalities. • Apply to New Concept: Solve systems of linear inequalities and sketch the solution

sets.


Concept Development 32

Alternate Discussion Topic Possible answers may include the following: Jack Price discusses the importance of teachers knowing the major algebra concepts that are expected of students in their district and state. Discuss with

participants the major algebra concepts that their students are expected to master

according to their district and state guidelines. • Participants’ state standards and district guidelines will vary. • You may want to consider discussing the Algebra standards from the National Council

of Teachers of Mathematics:

Instructional programs from pre-kindergarten through grade 12 should enable all students to: understand patterns, relations, and functions represent and analyze mathematical situations and structures using algebraic

symbols use mathematical models to represent and understand quantitative relationships analyze change in various contexts


Outline for a Lesson 33

Outline for a Lesson


Steve Alfi’s Lesson My Lesson

Warm-Up

Activity

• Activated prior knowledge • Worked one-on-one with

individual students • Conducted informal assessment • Provided individual remediation

• Students reviewed graphing inequalities and the characteristics of these graphs.

• Students worked in pairs matching inequalities to their corresponding graphs.

Direct Instruction

• Reviewed systems of linear equations

• Used whole-group instruction to demonstrate the process for graphing linear inequalities

• Had students read problems out loud to reinforce mathematical terminology

• Used whole-class demonstration to introduce graphing systems of inequalities

• Worked at overhead projector with assistance from student volunteers to combine pairs of graphs from Warm-Up activity to model solution sets for systems of inequalities

Peer Work • Students discussed their thinking and shared ideas.

• Students worked cooperatively to solve problems.

• Students communicated about and in the language of mathematics.

• Students worked in small-groups on an application problem.

• Students created overhead transparencies of their graphs and solution sets to present to the class.

Shared Results

• Students checked their work against the work that others had done.

• Students heard concepts and procedures explained in multiple ways.

• Students communicated about mathematics and used mathematical language.

• Students presented solutions from small-group activity (i.e., solving application problem) to the whole class.

• Students discussed how they approached the problem, how they found a solution, and how they verified that their solutions were reasonable.

Think and Discuss

Possible answers may include the following: Are there any additional instructional strategies that you could use to teach a

similar lesson to students? Describe any alternative activities you could use with

students. • Use real-life application problems for students to solve that require graphing systems

of inequalities to determine a solution. • Use technology (e.g., graphing calculators) for students to investigate graphing

systems of inequalities. • Use written activities for students to describe verbally the process they undertake to

solve systems of inequalities by graphing.


Outline for a Lesson 34

Discuss any unique needs that your students have that would affect your

approach to a similar lesson. • Participants may discuss that their students’ need to have real-life connections to

provide a meaningful context for the concepts they are learning. • Participants may also discuss that their students’ need to better understand how to

ask for and provide help during peer and small-group activities. This may include the use of role modeling to demonstrate for students how to do this.

Alternate Discussion Topic Possible answers may include the following: Effective use of wait time is an important element of instruction. Discuss with

participants their observations of wait time as used by Steve Alfi in his lesson.

Participants may discuss seeing a lack of wait time used by Steve Alfi during his lesson. You may want to consider asking participants to suggest places within his lesson where Mr. Alfi could have incorporated wait time, for example: • During the Warm-Up activity, when the student finishes showing how she solved a

warm-up problem, Mr. Alfi could have used wait time to see if any students had any comments about the problem, or alternative solutions, rather than immediately providing feedback to the student.

Have participants discuss their own experiences with wait time. Have them share

any effective techniques they have used.

• Participants may discuss that wait time is important because students need time to think about the question, to process the question, and to think about how to answer the question. Even if a student does not answer the question, he or she needs time to process the question to be ready to listen to the other students’ responses.

• Participants may also discuss how wait time can help students become critical thinkers because they have time to reflect on the question, to apply their reasoning skills to the question and to learn not to be reliant on the teacher to supply answers.

• Participants may discuss the use of Wait Time 1 and Wait Time 2: Wait Time 1: Teacher asks a question and allows time for the student to think

about the question and to then give a response. Wait Time 2: After a student has given a response, teacher waits to see if anyone

else will elaborate on the response, or can ask other students if they agree or disagree to see if there are different answers, or alternative thinking or approaches, rather than immediately giving students feedback about their response.

• Participants may discuss the difficulties they may have in fostering patient and positive behavior in students, which is important in creating a classroom environment that is supportive of students taking time to think about questions without feeling pressure to answer quickly.

• Participants may also discuss the importance of distinguishing the difference between a student who just needs a little bit more time to think about the question before giving an answer and a student who is not capable of answering the question.


Recommendations 35

Recommendations Possible answers may include the following: Example: Make connections to students’ prior learning Implementation Ideas: Use a technology demonstration for a Warm-Up activity that connects to students’ prior learning. Demonstrate a variety of graphs and have students share information about each graph; for example, students may discuss the slope and intercepts of linear graphs or students may discuss attributes of graphs of inequalities (e.g., open dot, closed dot).

Example: Monitor student understanding and progress to facilitate remediation. Implementation Ideas: Have students graph a solution (to a system of inequalities) and display it on their desks, circulate from desk to desk to scan the graphs, and determine if students are ready to move on. Use student work to provide information about student understanding of concepts to use when forming groups for group work.

Example: Vary the lesson to keep all students engaged. Implementation Ideas: Plan a variety of activities for the lesson: Begin the lesson with either a technology demonstration or bring in a few real-life examples. Next, use direct instruction that involves the use of multiple models. For practice, have students work in small groups. After group work, have students share their work. End the lesson with an independent “Quick Write” writing activity in which students process and summarize the key content from the lesson. Think and Discuss

Possible answers may include the following: Jack Price discusses the importance of variety in instructional approaches to

keep students focused on a lesson. What instructional approaches work best with

your current students? Which instructional approaches are not suited for your current students?

• Instructional approaches that work best with current students may include: small-group or pairs activities because students like opportunities to share ideas,

work collaboratively, and communicate with each other the use of models (i.e., concrete, pictorial, and via technology) because students

are visual and kinesthetic learners, or because students need concrete experiences to develop understanding of concepts before moving to abstract representations

real-life connections because they increase student motivation and interest in the content

• Instructional approaches that are not suited for current students may include: small-group activities because students are too talkative and the social interaction

leads to off-task and disruptive behavior, or because students prefer to work individually

use of technology (e.g., geometry software or graphing software) because it is too abstract for students; students need hands-on, concrete experiences to visualize and to understand concepts


Recommendations 36


Possible answers may include the following: Jack Price discusses the use of real-life examples as an extension to Steve Alfi’s

lesson. Have participants describe real-life examples of linear inequalities that they could use with their students.

• Real-life examples of linear inequalities include situations in which a quantity must fall within a range of possible values; for example, scores on a quiz to qualify for certain grades, a range of “healthy” cholesterol values, or scores that a diver must receive to qualify for a medal in a competition.

• Real-life examples of linear inequalities in two variables include planning a party that meets the requirements of a budget; for example, Xavier can spend $70 on food for the party; pizzas cost $12 and submarine sandwiches cost $8. Write an inequality that represents the situation and graph the solution set.

• Real-life examples of systems of linear inequalities include planning a healthy diet: Roberta wants to eat between 1,800 and 2,000 calories per day and limit her total fat intake between 55 and 65 grams. Create a graph to illustrate the appropriate amounts of calories and fat for Roberta.


Warming Up . . . 37

Warming Up . . . Possible answers may include the following:

Benefits of Warm-Up Activities

• Steve Alfi’s comments: provide consistent review (i.e., review concepts everyday) enable him to “touch the table”—to sit with a student one-on-one provide additional information about student understanding besides test results help determine pacing for the lesson

• Additional benefits: activates prior knowledge reinforces vocabulary

provides an opportunity to use peer interaction

helps students focus and prepare for the day’s instruction

My Warm Up-Activity

• Activity: Students work in pairs to complete a vocabulary activity: Students unscramble vocabulary terms and then write a definition and an example

for each term. • Goals:

reinforce vocabulary assess students informally review prerequisites

Think and Discuss Possible answers may include the following: Steve Alfi mentions that warm-ups permit him to make informal assessment of his

students’ understanding of the current topic and recently learned concepts.

Describe any informal assessment strategies that you have used for similar

lessons with your students. Participants may discuss informal assessment strategies such as: • using questioning • observing group work, peer work, or peer tutoring • having students make presentations • using student portfolios • using mathematics journals (or other writing activities) • having students create posters or other displays • having students use models to demonstrate concepts


Warming Up . . . 38

How have the results of informal assessment impacted your instruction?

• Questioning: Participants may discuss how using questioning can provide a window into students’ thinking. The questioning can reveal what students understand and what they do not understand. Questioning can also reveal student misconceptions or gaps in understanding, and may alert teachers to areas in which students need more reinforcement or extra practice.

• Small-Group Activities: Participants may discuss how they can use small-group activities to assess students informally. Collaborative activities and the use of models, such as oversized materials (e.g., large grids), can provide a way for students to talk about the content and to reveal their level of understanding or misconceptions. Observing the group work can let teachers know if students are ready to move on or if they need more practice and/or discussion about the concepts.


Possible answers may include the following: Steve Alfi talks briefly about keeping his students motivated during a lesson. Have participants discuss the issue of student motivation as it relates to their students.

Have them describe any techniques they have found to be effective in keeping

their students motivated. Some techniques that participants may discuss as a means to increase student motivation include: • social interaction—such as with peer and small-group activities • real-life connections—to provide a meaningful context for students and to increase

interest in the topic • use of technology—to enable students to explore and investigate concepts in an

open-ended manner that may yield interesting discoveries

39

Solving Quadratic Equations by Completing the Square


Prerequisite Knowledge 40

Prerequisite Knowledge Possible answers may include the following:

Completing the Square Prerequisite Knowledge

My Students’ Understanding <Participants’ ratings will vary.>

Students are able to perform algebraic computation, including squares and square roots.

low high

Students are familiar with algebra tiles and their values.

low high

Students know the definition of a quadratic equation.

low high

Students understand graphical and symbolic representations of quadratic equations and functions.

low high

Prerequisite Concept or Skill Students are familiar with algebra tiles and their values.

Ideas to Help Increase Student Understanding • Students will create their own set of algebra tiles: Students will cut out algebra tiles

that have been duplicated on colored paper and will store their “tiles” in plastic sandwich bags. Students will review the value of each tile: x2, x, and unit tiles.

• Students will use the algebra tiles to review modeling and solving linear equations, including multistep equations. Included in this review will be a review of zero pairs and how to model zero pairs with algebra tiles.

• Students will record multiple representations to accompany the concrete models: Students will draw the models (pictorial) and record the algebraic equations (symbolic).

• Students will then move on to using the algebra tiles to represent quadratic equations.

Think and Discuss

Possible answers may include the following: What are your instructional goals during a lesson on completing the square? How does the prerequisite knowledge you described above help students reach these

goals? What are potential problems for students who do not understand these

prerequisite concepts and skills? How can you teach the content so that students

overcome these problems? • Instructional Goals: Students learn two methods to solve quadratic equations by

completing the square: Use algebra tiles to build a perfect square trinomial, and algebraically by factoring the trinomial.

• Prerequisite Knowledge: Understand squares and square roots, understand the definition of a square, and understand how to manipulate equations by keeping them balanced.


Prerequisite Knowledge 41

• Potential Problems: If students do not have a facility with manipulating equations, they may have difficulties transforming equations to perfect squares.

• Instruction to Help Students Overcome Problems: Using models, such as algebra tiles, can be used to help students manipulate the equations concretely. Additionally, having students work in pairs can help students overcome problems by creating a nonthreatening setting in which students can ask for and receive help from their peers.

Alternate Discussion Topic Possible answers may include the following: Roger Day explains that students need to have both procedural and conceptual

understandings of ways to solve quadratic equations. Have participants discuss

why both are important. Ask them how they can teach, so students leave a lesson

with both. • This is important because students need to see a connection between the geometry

of the square (conceptual) and the symbolic representation of the quadratic equation (procedural).

• Both representations are important to help students see how completing the square fits into the bigger picture of what a quadratic equation is and the methods for solving them.

• This understanding helps students understand and successfully use the methods for solving quadratic equations: by graphing, by factoring, by completing the square, and by using the quadratic formula.

• Instruction that can help students achieve this includes having students use both concrete and symbolic representations and discussing how the two representations are related.


Next, please. 42

Next, please. Possible answers may include the following:

Extending Lesson

Topic: Using the Quadratic Formula to Solve Quadratic Equations Introduce/Warm Up: Review solving quadratic equations by factoring and by completing the square. Teach: Use direct instruction in the whole-class setting to introduce the Quadratic Formula: Present this method to solve quadratic equations to students and demonstrate using it. Show a second example, and this time, have a student volunteer use the overhead projector to show how to solve the same equation via another method (i.e., completing the square with algebra tiles or factoring). Lead a discussion that compares the methods. Practice/Apply: In pairs, students complete practice activities: Students apply two methods (at their discretion: factoring, completing the square, formula) and discuss the methods. Students share their solution methods with the rest of the class. Assess: Students write about the three methods, comparing and contrasting the methods, discussing the advantages and disadvantages of each method. Students select the one they prefer and explain why. Think and Discuss

Possible answers may include the following: Discuss the instructional sequence Jennifer Lamanski used in her lesson. Was this sequence effective? How did this sequence of instruction contribute to her

students’ learning?

• Ms. Lamanski began by discussing the objectives for the day’s lesson: Students next completed a warm-up activity that reviewed prerequisite skills

(activated prior knowledge). Students practiced getting perfect trinomials. Students were given sandwich bags of algebra tiles and worked in pairs to

complete several problems. Students had recording sheets to record their work. Ms. Lamanski circulated

among the student pairs to provide instruction and encourage student interaction. • This was an effective instructional sequence because students were able to:

review prerequisites before starting on the content for the day use concrete models to help develop their conceptual understanding of the

concepts work together to provide support and assistance to each other.

Is there anything you would have done differently? In the whole-class setting, Ms Lamanski could have led several more demonstrations with the algebra tiles and included student volunteers to record the algebraic representations as she completed each step.


Next, please. 43


Possible answers may include the following: Jennifer Lamanski closes her lesson by suggesting to her students that they

should look for a pattern in what they are doing. She says that they will look at this pattern in a subsequent lesson. Have participants discuss what is

accomplished by informing students about the content of future lessons.

• This helps create interest and anticipation for the next lesson. • This alerts students that what they are doing will be applied to something else; they

are not learning concepts in isolation. • It helps students see that what they are learning has application, and they can use it

to generalize situations. Ask them if there are any drawbacks.

• This may create anxiety in students if they are still a little confused about the current concepts.

• If circumstances arise that require lesson content to change, this may create confusion or disappointment in students.


The Benefits Are . . . 44

The Benefits Are . . . Possible answers may include the following:

Instructional Practice Mathematics Specialist’s Comments

My Thoughts

Tell students the objectives of the lesson.

• Students are aware of what they are learning and where they are going.

• Helps students focus on lesson goals

Use manipulatives to create concrete models.

• Builds conceptual understanding and moves students in the right direction

• Helps students “touch it” and “see it,” which strengthens learning by addressing kinesthetic and visual learning modalities

• Enables students to demonstrate understanding in a variety of ways

• Increases student discourse

• Increases student motivation

Have students work in pairs or cooperative groups.

• Enables students to share results, compare results, and raise questions

• May want to incorporate more “Think–Pair–Share” activities

• Enables opportunities for informal assessment and individual instruction

• As students talk about concepts, they strengthen their understanding of the concepts.

Help students connect symbolic representations to models.

• May want to consider having each student do both: use algebra tiles (concrete) and record algebraic equations (symbolic), and want to make sure students can do both

• Students consider the underlying concepts and processes as they transfer from one model to another.

Think and Discuss Possible answers may include the following: Roger Day would like to see each student illustrate both concrete and symbolic

representations of the process for completing the square. Do you agree that this

would be a beneficial practice?

Yes, this would be a beneficial practice. Student understanding is strengthened as they translate from one representation to another. By translating from the concrete to the symbolic, students consider the underlying concepts and procedures represented by the concrete models. This also helps students think about concepts in a variety of ways.


The Benefits Are . . . 45

What are the management issues around implementing this suggestion? What

could you do to make this happen in your classroom? • Management issues include having ample space to use the algebra tiles and to

record the algebraic equations, which may entail larger recording sheets. Also, this means that there needs to be enough algebra tiles for all students. Additionally, ample instructional time needs to be set aside to enable students to complete the activities.

• To make this happen in a classroom, teachers could have students create their own sets of algebra tiles by cutting them out of colored paper and storing them in plastic sandwich bags. Teachers can also prepare recording sheets that provide space for students to draw a representation of the tiles and to record the corresponding algebraic (symbolic) representations. Teachers will need to model this process several times to ensure that students understand the process. As students present their solutions to the class, they can use an overhead projector and overhead algebra tiles and overhead markers to model the concrete and symbolic representations.


Possible answers may include the following: Roger Day mentions that the strategies used in the Completing the Square lesson have application in many other lessons and for many other topics. Have

participants suggest other lessons in which they might use one or more of these

instructional practices. For one of these lessons, have them describe how these

strategies would contribute to a successful lesson. • Lesson: Solving 2-Step Equations • Strategies from the lesson on Completing the Square:

Tell students the objectives of the lesson: Begin the lesson by discussing the state standards that are covered in the day’s lesson. Have students use handouts to read the standards and suggest the specific standards that they may be addressing. This gives students insight into where they are headed and helps students

understand what they are trying to accomplish with the lesson. Use manipulatives to create concrete models: Students use algebra tiles to model

and solve 2-step equations. This helps students develop conceptual understanding of the concepts and

procedures. Have students work in pairs or cooperative groups: Students work in pairs to solve

2-step equations. This encourages students to help each other and to talk about the concepts

and procedures. This can also expose students to alternative approaches. Help students connect symbolic representations to models: As students work in

pairs, one student uses the algebra tiles to model the problems concretely and one student uses algebraic equations to model the problems symbolically. This strengthens student understanding of the concepts and procedures.


Using Manipulatives 46

Using Manipulatives Possible answers may include the following: Using Manipulatives Benefits:

• helps kinesthetic learners understand concepts • enables the use of multiple representations to strengthen student understanding • enables students to touch, to feel, and to put their hands on the objects, which helps

them understand what they are doing • encourages communication about mathematical concepts and procedures Management Issues: • Prepare manipulatives ahead of time, including preparing them for distribution to

individual students or to groups. • Determine the tasks students will complete with the manipulatives. • Prepare any applicable recording sheets for students to use with the manipulatives. Think and Discuss Possible answers may include the following: Jennifer Lamanski mentions several manipulatives that she uses in her classroom. Which manipulatives do you use in your classroom? For which topics

are these manipulatives applicable? How do they help you teach various Algebra

1 skills and concepts? • algebra tiles (including plastic tiles, paper sets, and sets for the overhead projector) • cups and chips (models for variables and integers)

used to model and solve equations helps students visualize the equations helps students develop a conceptual understanding of the procedures used to

simplify and solve equations • 2-color chips

used to model integers, including zero pairs helps students develop conceptual understanding of zero pairs helps students develop conceptual understanding of operations with integers

• graphing calculators • demonstration graphs (such as oversized coordinate grids)

used to create a wide variety of graphs helps students explore and investigate graphing and graphing concepts helps students make interesting discoveries regarding graphs, their characteristics,

and their corresponding equations or inequalities


Using Manipulatives 47


Possible answers may include the following: Jennifer Lamanski discusses a lesson sequence that she uses when she teaches

a new concept. This sequence includes the use of pictorial representations, manipulatives or hands-on learning, patterns, and algebraic equations, rules, and

methods. Have participants select an Algebra 1 topic and discuss how this

progression would apply to that lesson. • Sequence:

use of representations (pictorial, concrete) discussion of patterns use of algebraic equations, rules, and methods

• Suggested implementation: Topic: Solving Systems of Linear Equations Representations: Create an oversized coordinate grid on the classroom floor. Use

an overhead transparency of a coordinate grid. Students use chart paper to create oversized grids. Students record solutions on handouts.

Patterns: Students make observations about and discuss the slopes and intercepts of parallel and perpendicular lines and how this impacts the solution sets.

Algebraic Equations, Rules, and Methods: Students record algebraic representations to accompany each graph of the solution set.

Ask if they need every item in the sequence and if there are some things they

might need to add.

• Discussion of patterns may not be applicable to all lessons and all concepts. • In some lessons it may be appropriate to add the use of technology.

Have them consider if some elements of the sequence need repeating. • Instruction may be more effective if use of representations was repeated again after

discussion of algebraic equations, rules, and methods. • Additionally, for some topics, it may be beneficial to discuss patterns again after

using algebraic representations.

Ask them how they could change or generalize this sequence so that it could be

applied to a variety of lessons. • Suggested generalized sequence:

representations (e.g., broadened to include concrete, pictorial, symbolic, and real-life examples and the use of technology)

patterns algebraic representations (e.g., equations, rules, and methods) patterns

48

Angle Relationships

Angle Relationships

Learning Standards 49

Learning Standards Possible answers may include the following:

Angle Relationships Prerequisite Skills

• Explain what an angle is and use proper vocabulary to describe angles. • Understand how to name angles: when to use one letter or number and when to use

three letters or numbers. • Understand how to measure angles and that degrees are the units of measure that

used. • Classify angles: acute, right, obtuse, and reflex.

Angle Relationships

Lesson Concepts or Skills • Understand adjacent angles; identify numerically and algebraically. • Understand vertical angles; identify numerically and algebraically. Angle Relationships

Application Skills

• Use properties of adjacent angles and vertical angles to solve problems. • Expand understanding of adjacent and vertical angles to triangles, quadrilaterals,

and all other polygons.

Think and Discuss Possible answers may include the following: Describe the continuum of content and skills for angle relationships. Discuss the

prerequisite skills that your students should bring with them to angle relationship

lessons.

• Participants will reference their notes from Learning Standards, above. • Additional prerequisite skills include:

Understand and identify lines, line segments, and rays. Understand and identify parallel, intersecting, and perpendicular lines. Understand and identify supplemental angles and complementary angles. Understand and use vocabulary of angles vertex, endpoint, interior points and

exterior points, and congruent angles. Have the ability to use protractors. Have the to set up and solve equations.

How prepared are your students for this content? Are there any particular prerequisite skills that many of your students lack? Describe any intervention

strategies you have used to help address any gaps in your students’

understanding of angle relationship prerequisites. • Prerequisite Skill: Participants may discuss students lacking facility in setting up and

solving equations. • Intervention Strategy: Can provide extra practice for students in setting up equations;

for example, can present illustrations of angles. Ask students to write algebraic equations that could be used to determine missing angle measures.

Angle Relationships

Learning Standards 50


Possible answers may include the following: Kathy Dawson discusses the importance of students being able to algebraically

describe, when appropriate, adjacent angles and vertical angles. Discuss this and other examples of ways that geometry and algebra can be connected during a

lesson on angle relationships.

• Geometry and Algebra Connections: setting up and solving equations, including solving for an unknown to determine

angle measures applying triangle congruency theorems to help analyze angles and angle

relationships applying the Pythagorean theorem to help prove angle relationships and to solve

problems

Angle Relationships

Goals for Improving Instruction 51

Goals for Improving Instruction Possible answers may include the following:

Goals for Improving Instruction • Ask students questions continually, engaging them in the learning process. • Have students defend and explain their answers. • Have students determine how concepts can be applied in different situations. • Use essay questions that require students to articulate and explain their reasoning. • Find real-life examples for geometry concepts. • Use a wide variety of instructional strategies. Goal: Find real-life examples for geometry concepts. Ideas for Improving My Instruction • Assign students, as a homework activity, to look for and bring in examples of

adjacent and vertical angles in their environment. • Have students bring in photographs, sketches, or clippings from magazines and

newspapers. • Use the gathered examples to discuss “Always, Sometimes, and Never” statements

about adjacent and vertical angles; for example, “Vertical angles are always congruent.”

Goal: Have students defend and explain their answers. Ideas for Improving My Instruction • Have students, in a small-group activity, complete formal proofs. • Have each group create two charts: one that lists the steps in the proof and one that

lists the reasons. • Have each group present and explain its proofs. • Have the rest of the class ask questions of the groups as they make their

presentations. Think and Discuss Possible answers may include the following: Discuss which of Kathy Dawson’s instructional goals are easier to implement and

which may be more challenging.

• Some goals that participants may discuss as easier to implement include: asking students questions continually, engaging them in the learning process having students defend and explain their answers finding real-life examples for geometry concepts

• Some goals that participants may discuss as more challenging to implement include:

having students determine how concepts can be applied in different situations using essay questions that require students to articulate and explain their

reasoning using a wide variety of instructional strategies

Angle Relationships

Goals for Improving Instruction 52

Describe why some goals are more difficult to achieve than others.

• Participants may discuss the following reasons why some goals are more difficult to achieve than others: Having students determine how concepts can be applied in different situations:

requires a strong understanding of the concepts and higher-order thinking skills to make the applications to different situations

Using essay questions that require students to articulate and explain their

reasoning: requires additional planning on the part of the teacher to find or write the essay questions, requires additional time on the part of the teacher to evaluate the student essays, and requires a strong understanding of the geometry concepts for students to answer the essay questions

Using a wide variety of instructional strategies: requires familiarity on the part of the teacher of varied instructional strategies and requires additional planning and preparation time

What can you do to ensure that you will use the ideas that you listed for improving instruction?

• Participants may discuss incorporating the goals into their instructional calendars or planning books, discussing their improvement ideas with colleagues, and setting deadlines by which time they will have tried at least one new improvement idea.

Alternate Discussion Topic Possible answers may include the following:

Have participants describe successful writing activities that they have implemented in their geometry classes. Have them describe the assignments,

skills, or concepts that students needed to have mastered to complete the

assignment, and how the assignment was evaluated. Discuss with participants the following writing activity that can be used to determine student understanding of angle relationships: • Present a drawing to students. • Ask students to write about the drawing, identifying as much as they can about the

angles in the drawing. • Ask students to use precise vocabulary and to support their statements with

mathematical reasoning; for example, students could be presented with the following drawing:

XX

W

V

ST

U Y

• Evaluate the assignment by using a criteria checklist. The checklist could include the

number of statements students made about the angles, specific terms used, known angle measures, and the reasoning used to defend each statement.

Angle Relationships

Lesson Highlights 53

Lesson Highlights Possible answers may include the following:

Lesson Highlights • using warm-up activities • using spiral review to review prior learning • emphasizing mathematical vocabulary • having students explain answers • having students solve application problems • permitting alternate approaches • using peer tutoring • using technology to explain and model concepts Think and Discuss Possible answers may include the following: Which lesson elements engaged students best?

Elements of the lesson that engaged students included: • the use of technology • working in pairs/peer tutoring and peer interaction • the use of tools (protractors)

Were there any special classroom management or lesson planning elements that Mr. Skelly incorporated into this lesson?

Mr. Skelly: • prepared the technology demo ahead of time • prepared the warm up spiral review

Angle Relationships

Lesson Highlights 54


Possible answers may include the following: Art Skelly used technology throughout his lesson on angle relationships. Have

participants discuss the kinds of technology available to them and how they can best use that technology in the geometry classroom. Encourage them to give

examples of successful lessons involving technology.

• Examples of technology that participants may have available include: interactive whiteboard dynamic geometry software graphing calculators video/DVD players overhead projector

• Examples of using technology in geometry lessons may include: Students use graphing calculators to make discoveries about vertical angles and

adjacent angles. Students can use the graphing calculators to explore 2 lines as they move and

watch the angles and how they change as the lines move.

Teacher uses overhead transparencies on which various angles and lines have been drawn to illustrate angle properties; for example, place two transparencies on top of each other to show an overlapping angle and then rotate the top transparency to show an adjacent angle.

Angle Relationships

Questioning Techniques 55

Questioning Techniques Possible answers may include the following:

Questioning Techniques • Art Skelly:

asks students to apply new information in context asks students to explain how they arrived at their answers asks “if, then” questions: “If this is [statement], then what is [statement]? asks “why” questions: “Why can’t we do that? [name the angle with a single letter] asks “how” questions: How did you know that? [the measure of the angle is 70 ] restates student answers to reinforce concepts

Think and Discuss


Appropriate questions can be used to assess students’ prior knowledge and to identify misconceptions. What questioning techniques do you use to help you

assess your students’ prior knowledge?

Examples of questioning techniques that can be used to assess student prior knowledge include: • questions that focus on recall of facts; for example, What is the angle measure for

supplementary angles? • asking students to make statements about a given topic or concept; for example,

students can be asked to provide statements about triangles (This can also be done as a writing activity.)

• making statements and then asking students if they agree or disagree with the statement, and to explain why; for example, “All adjacent angles form straight lines.”

What questioning techniques do you use to check for misconceptions?

• Identify common student misconceptions about concepts ahead of time and form questions around the misconceptions; for example, Roger said that all supplementary angles are congruent. Is Roger correct? Why or why not?

Angle Relationships

Questioning Techniques 56


Possible answers may include the following: Art Skelly used peer tutoring during his geometry lesson. Have participants

discuss the potential benefits of this practice plus any content or management considerations they must keep in mind when organizing peer tutors. Have them

discuss any experiences they have had with peer tutoring.

• Peer Tutoring: Benefits: Enables students to:

help each other work cooperatively to solve problems see alternative approaches to problems communicate about mathematics practice using mathematical vocabulary

Enables the teacher to: assess students informally provide individual, or targeted, instruction, remediation, or enrichment

• Peer Tutoring: Content or Management Considerations: Determine how students will be paired together: Similar abilities? Mixed abilities? Consider room arrangement: Do desks need to be moved? Determine tasks for pair work: What are the tasks each pair will need to complete?

Are the tasks appropriate for pair work, can they be completed individually, or would the tasks be better suited for larger groups?

Consider materials and tools students may need: How will the materials be distributed? Do both of the students need their own supplies or can the supplies be shared?

• Peer Tutoring: Experiences:

Participants may discuss using mixed-ability peer tutoring, in which one student with strong skills is paired with one student with weaker skills; for example, this strategy can be used when students are assigned application problems to solve.

Participants may discuss that often the “stronger” student may be most comfortable using abstract thinking and algebraic representations, and the “weaker” student may be most comfortable using visual or concrete representations. This combination can work successfully for some application problems due to the broad approaches the students may use and the varied problem-solving strategies they may apply to the problem.

57

Bisectors, Medians, and Altitudes


Modeling Geometry Concepts 58

Modeling Geometry Concepts Possible answers may include the following.

Geometry Concept • Angle Relationships

Example for Each Model

Paper Folding Manipulatives Technology

AAG

C

H

DIE

F B

• Instruct students to create

a folded paper square, as shown above.

• Ask students to identify all the angle relationships that they can within the picture.

• Have them explain how they know each fact.

• Students use poster board to create models that illustrate various angle relationships and properties.

• Students can present their models to the class, and/or the models can be used to create a display in the classroom (e.g., bulletin board display).

• Use dynamic geometry software to draw a series of lines, forming a variety of angles.

• Ask students, after each line is drawn, to identify any angles they notice and to describe the attributes of the angles.

Think and Discuss

Possible answers may include the following: Many geometry concepts can be modeled in a variety of ways. Discuss the

management and planning that are required for using models or hands-on

activities in your classroom. What are some pre-lesson tasks that you need to complete to have a successful lesson involving hands-on activities for students?

Some of the management, planning, and preparation tasks for hands-on activities may include: • Gather supplies and materials. • Prepare student recording sheets. • Determine grouping arrangements. • Assign students to groups or pairs. • Review rules for group behavior if necessary. • Prepare classroom for activity: • Move desks together, if necessary, and clear a space for materials. • Prepare task cards, overhead transparency, or chart that lists the tasks and

procedures.


Modeling Geometry Concepts 59


Possible answers may include the following: Ask participants if they feel that their students are well prepared when they enter

the geometry classroom. Ask them to identify any particular areas of weakness for students and the strategies that they use to address any gaps in prerequisite skills

or knowledge.

• Participants may discuss students’ weaknesses in some of the following prerequisite skills for geometry lessons, such as those involving bisectors, medians, and altitudes: familiarity with triangles and their properties familiarity with lines and properties of lines familiarity with angles and properties of angles familiarity with the Pythagorean theorem ability to complete simple constructions ability to set up and solve equations ability to reason and to support statements, conjectures, and conclusions with

reasoning • Strategies to address gaps:

pretesting students to identify areas of weakness and mastery of prerequisite skills assigning extra practice for students who need to strengthen prerequisite skills conducting spiral review of prerequisite skills, recently studied concepts, and new

concepts using peer tutoring to enable students to provide help and support to each other


Words for the Wise 60

Words for the Wise Possible answers may include the following:

Term: perpendicular

Example

• Matthew Roberson: discussed the term with students used technology (interactive whiteboards) to display a definition and an example

• Students: underlined terms they thought were important in the definition drew an example (using small whiteboards) and shared the example with the rest

of the class Term: angle bisector

Example: Matthew Roberson: • connected the term to prior learning • discussed the term within the context of a drawing of a triangle Term: median

Example: Students were positioned with their backs to each other; one student provided a verbal description of the term, and the other student attempted to draw an illustration of the term.

Term: altitude

Example: • Matthew Roberson used a large cardboard cutout of a triangle, along with additional

illustrations on the interactive whiteboard, to discuss altitudes in various types of triangles.

• Students completed a vocabulary activity in which they related everyday meanings of the term to the mathematical meaning.


Words for the Wise 61

Think and Discuss

Possible answers may include the following: What techniques and activities have you used successfully to help students

master vocabulary terms? Describe your activities and techniques. • Playing a Jeopardy-style game: Answers are supplied and students need to

determine the question. Vocabulary definitions are provided and students must determine the term. This can be played as a whole-class activity or with small groups competing against each other.

• Having students create their own set of vocabulary flash cards: The cards are created by students writing their own definition of the term on one side and drawing an illustration of the term on the other side.

• Developing an introductory lesson that involves learning centers: Students may complete a variety of vocabulary-based activities. At one center, students can use online dictionaries to look up and record definitions and examples. At another center, they can complete a creative writing exercise in which they use a list of vocabulary words in a paragraph. At a third center, you can have students act out vocabulary terms that they draw out of a hat in a charades-type game. At a final center, students can use concrete materials to build or draw examples of each term.

Alternate Discussion Topic Possible answers may include the following: Matthew Roberson gives students several study suggestions that will help them prepare for tests. Have participants discuss and describe some techniques they

use to help students prepare for geometry tests.

Discuss the strategies Kathy Dawson discussed in the Angle Relationships lesson that can be used by students to prepare for tests (or to review previously learned content): • Students write questions from review topics. • Students solve the problems as a small-group activity. • Peer tutoring, in which a stronger student is paired with a weaker student, is used to

help students review the material.


Another Point of View 62

Another Point of View

Lesson Elements

Benefits

Using Interactive and

Student

Whiteboards

• This is an innovative instructional technique. • It keeps students involved in the lesson. • Students used small whiteboards, too.

Having Students Share

Results

• Students recorded solutions on large chart paper. • Students explained the steps they used to solve the problems. • Students could see a variety of methods for solving the problem.

Applying Vocabulary in

Student Pair

Activity

• This illustrated the importance of using precise vocabulary. • If students weren’t accurate in their vocabulary, the drawings

weren’t accurate. • This result is okay. • Not every time we try something will it work perfectly.

Additional Notes

Using technology, the interactive whiteboard, can save time, make it easier to discuss examples, and can help teachers stay on track and keep things moving in a lesson. Students can take notes along with the technology demonstrations.

Think and Discuss Possible answers may include the following: Matthew Roberson used several instructional strategies that effectively involved students in the lesson. Discuss the various types of student involvement that you

saw during the lesson. Which of Mr. Roberson’s strategies was most effective in

engaging students? In contributing to student learning? Strategies that engaged students and contributed to learning included: • using technology (i.e., interactive whiteboard) to demonstrate and illustrate terms • completing the pair activity in which students either described a figure or drew the figure • using small whiteboards to illustrate terms and share the illustrations with the class • solving application problems in small groups • sharing problem solutions with the rest of the class Alternate Discussion Topic

Possible answers may include the following: Matthew Roberson used a variation of a pair–share activity in his lesson. Have

participants discuss any experiences they have had with pair–share activities. Have them describe those activities that were most effective in their classrooms.

• Think–Pair–Share activities can be effective for students to discuss “Always, Sometimes, and Never” statements. Students can be given a statement or an illustration of a property and then complete a Think-Pair-Share activity to discuss if they think the statement or property is always, sometimes, or never true. Students “think” about the question or problem, discuss their ideas with their partner (“pair”), and then “share” their ideas with the rest of the class.


Communication Standards 63

Communication Standards Possible answers may include the following:

Organize Thinking Instructional programs should enable all students to organize and consolidate their mathematical thinking through communication.

• Instructional Strategies: Use writing activities. Use peer interaction (i.e., pairs, small groups, presentations).

• Activity idea: Have students work in pairs to complete formal proofs. Have pairs create two charts to present to the class: one that

lists the steps in the proof and one that lists the supporting reasons for each step.

Have pairs present their proofs to the class.

Communicate

Thinking to Others Instructional programs should enable all students to communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

• Instructional Strategies:

Use writing activities. Use peer interaction (i.e., pairs, small groups, presentations). Have students share results. Have students use mathematical vocabulary.

• Activity idea: Complete the following small-group activity:

Divide the class into groups. Each group creates a drawing that contains a variety of polygons, lines, and angles. The groups exchange drawings with one another. The groups use the drawing they received and create a chart that lists as much information as they can come up with about the drawing in a set time limit (e.g., 15 minutes). Each group presents its charts to the rest of the class.

Analyze Thinking

of Others Instructional programs should enable all students to analyze and evaluate the mathematical thinking and strategies of others.

• Instructional Strategies

Have students justify their thinking. Use technology. Use peer interaction (i.e., pairs, small groups, presentations). Have students share results. Have students use mathematical vocabulary.

• Activity idea: Have students use technology (i.e., interactive whiteboard,

graphing calculator, dynamic geometry software) to demonstrate a geometry concept to the rest of the class.

Have students ask questions about the concept to the presenters.

Use Mathematical

Language

Instructional programs should enable all students to use the language of mathematics to express mathematical ideas precisely.

• Instructional Strategies

Have students justify their answers. Use writing activities. Have students share results. Have students use mathematical vocabulary. Use peer interaction (i.e., pairs, small groups, presentations).

• Activity idea: Have students work in pairs to complete drawings that are

described to them by their partner.


Communication Standards 64

Think and Discuss

Possible answers may include the following: Some students like working with peers. Some students prefer to work alone. Other

students may be better at written communication than oral communication. Discuss how activities and strategies may be adapted to address these

differences. Give specific examples.

Participants may discuss some of the following strategies for addressing student communication preferences: • When planning some activities, allow students to select how they want to complete

the activity. Permit students to work alone, with a partner, or with a small group. • When creating assignments, allow for students to choose between written activities

or oral presentations. • Assign students to pairs or small groups with a mix of students who prefer written

and oral communication. Ask the group to make both a written and an oral presentation.

• Students who do not prefer writing may be more inclined to do so when given a choice of topics or prompts from which to choose.

• Students who do not prefer making oral presentations may feel more comfortable when part of a group presentation or when using models as part of their presentation, such as technology demonstrations or overhead manipulatives.


Possible answers may include the following: Communication-based activities can be engaging for students as well as

providing them with opportunities to strengthen their understanding of concepts. However, these activities may not be as easy to assess as more traditional

practices. Have participants discuss strategies for assessing communication-

based activities. Have them share any successful assessments they have used for

these kinds of activities. • Rubrics, criteria charts, or checklists can be used to assess communication-based

activities. • Categories on an oral presentation (or written assignment) rubric may include

content, organization, creativity, materials, use of vocabulary, support for conclusions, and ability to answer audience questions.

• Student evaluation can also be used. Students can use short criteria lists to record their evaluations of student presentations. Items on the criteria lists may include presentation was clear and easy to follow, ideas were illustrated and supported with examples, questions were answered, and presentation helped me understand the concept.

65

Tangents

Tangents

Best Practices 66

Best Practices Possible answers may include the following:

Geometry Best Practices My Rating

<Participants’ ratings will vary.>

Make connections between algebra and geometry.

low high

Address varied depth of understanding in students.

low high

Use a variety of instructional approaches.

low high

Make teaching a life-long learning process.

low high

Best Practices Goal Ideas to Increase Implementation

Use a variety of instructional approaches.

At least once a week, lead a lesson that utilizes at least three different instructional approaches. At least once a week, incorporate a new instructional approach suggested to you by a colleague.

Make connections between algebra and geometry.

Plan one lesson a week that incorporates algebra and geometry. Include one algebra problem on daily warm-up review activities.

Think and Discuss

Possible answers may include the following: For the best practices listed, discuss how you accomplish each in your

classroom.

• Make connections between algebra and geometry: Students are asked to set up and solve algebraic equations to solve tangents problems.

• Address varied depth of understanding in students: Students are assigned to mixed-ability groups to work on problem-solving applications cooperatively.

• Use a variety of instructional approaches: Lessons are planned so that they incorporate a variety of instructional approaches; for example, lessons begin with a warm-up activity, real-life examples or technology demonstrations are used for direct instruction, students work in cooperative groups, and lessons end with a writing activity.

• Make teaching a life-long learning process: Collaborative planning is held with colleagues to share ideas and experiences.

Share ideas for ways to improve [your implementation of] each strategy with your

colleagues.

• Participants will reference their notes, above.

Tangents

Best Practices 67


Possible answers may include the following: Jerry Cummins discusses some prerequisites for studying tangents. Discuss with

participants the prerequisites that he mentions. Have participants share their ideas for any additional prerequisites they believe students may need prior to studying

tangents.

• Tangents Prerequisites: Understand perpendiculars and definitions of perpendicular lines. Understand congruent triangles and their primary theorems and postulates.

• Additional Prerequisites: Understand and apply the Pythagorean theorem. Understand how to set up and solve equations. Perform basic constructions. Understand properties of lines and angles. Understand properties of circles.

Tangents

In the Flow 68

In the Flow Possible answers may include the following:

Tangents Lesson

Lesson Activities Instructional Approaches

Warm-Up Activity/Anticipatory Set

• Mr. Elizalde discusses lesson goals with students.

• Mr. Elizalde introduces the concept (tangents) with multiple models/real-life examples.

Application Problems/Investigations/Use of Technology

• Students solve problems that apply tangents concepts.

• Students discuss the problems together, help each other, and correct mistakes

• Mr. Elizalde uses dynamic geometry software to test student claims and statements.

Small-Group Activities/Use of Manipulatives

• Students work in small groups to complete a hands-on activity in which they use linguini to model tangent lines.

Make and Test Conjectures/Use of Technology/Formal Proofs

• Mr. Elizalde uses technology (dynamic geometry software) to test conjectures students made about tangents.

• Students work in small groups to completeformal proofs on tangents concepts.

• Students make and present charts of their proofs and reasoning.

Constructions

• Mr. Elizalde uses the overhead projector to demonstrate a construction.

• Students use compass, protractor, and handouts to complete constructions.

Think and Discuss

Possible answers may include the following: Are there any additional instructional strategies that you could use to teach a

similar lesson to your students? Describe any alternative activities you might include.

• Small-group activity: Students create interior design or landscape design plans that require construction and application of tangents and triangle properties to determine measurements. Students present their plans to the class.

Tangents

In the Flow 69

Discuss any unique needs of your students that would affect your approach to a

similar lesson. • This lesson was conducted with an honors geometry class, so students were capable

of moving through multiple activities in one lesson. Other classes may need to move more slowly from one activity to the next to allow more time for students to understand the concepts from each activity. Some participants may discuss spreading out this lesson over several instructional days.

Alternate Discussion Topic Possible answers may include the following: Andrew Elizalde demonstrated natural and appropriate use of mathematical

language throughout his tangents lesson. Have participants point out some

examples of mathematical terms and vocabulary that Mr. Elizalde used during his

lesson. Some examples of mathematical terms and vocabulary used by Mr. Elizalde during his lesson include: • When students were working in small groups with the linguini, “What we are forming

is segments, not lines.” “Break the linguini off at the point of tangency.” • When discussing the student’s statement, “She made a statement earlier regarding

the relationship between the radius and the tangent line.” • During the technology demonstration, “Regardless of where that external point is, we

know that the two segments have to be congruent to one another.”

Tangents

Exemplary Lesson Elements 70

Exemplary Lesson Elements Possible answers may include the following:

Jerry Cummins’

Exemplary Elements

Example from Andrew

Elizalde’s Tangents Lesson

Ideas for a Similar Lesson

Using Handheld Calculators Students use handheld calculators when solving problems (applying the Pythagorean theorem and tangent properties) to determine the unknown length of a line.

Students use handheld calculators to assist in findingmeasures of unknown anglesduring a lesson on angle relationships.

Using Models Mr. Elizalde used a variety of models in his tangents lesson: a roll of toilet paper, a paintbrush roller, a Frisbee, a hula-hoop, pointer, and linguini.

Students are asked to find an example of a tangent line in their lives outside of school. Students bring in examples (or write descriptions or create drawings of the examples).

Making Algebra and Geometry Connections

Students set up and used equations to solve problems involving the Pythagorean theorem.

Students use equations to solve problems involving measures of unknown angles in a lesson on angle relationships.

Using Dynamic Geometry (software)

Mr. Elizalde used dynamic geometry software to demonstrate tangent concepts and properties and to test conjectures about tangents.

Create a demonstration that uses dynamic geometry software to investigate properties of bisectors, medians, and altitudes.

Using Cooperative Groups Students worked in small groups to solve application problems, to use hands-on materials to model tangents, and to write formal proofs.

Students work in cooperative groups to create architectural plans for a house. Students apply concepts of tangents, angle relationships, and the Pythagorean theorem to create the drawings and determine measurements.

Tangents

Exemplary Lesson Elements 71

Think and Discuss

Possible answers may include the following: Andrew Elizalde used a variety of tangent models to help his students understand

tangents concepts. Discuss his use of models and the technology that he used in his lesson to illustrate tangents concepts.

• Andrew Elizalde used a variety of concrete/real-life models to represent tangents, a roll of toilet paper, a paint roller, a Frisbee, and a hula-hoop, and pointer.

• Andrew also used technology to demonstrate tangents (dynamic geometry software) and tangents properties.

• Andrew had his students use linguini to model tangents in a small-group activity. Describe your ideas for modeling tangents concepts in a similar lesson.

• Have students bring in examples of tangents from their lives outside of school. • Discuss tangents in the context of races on circular or curved courses, such as

bicycle races, car races, or running races, in which competitors “cut the tangents” to complete the turns in the most direct route around the curve. If available, show video of examples from car or bicycle races.


Possible answers may include the following: Jerry Cummins discusses the importance of integrating geometry and algebra in

secondary mathematics classes. Have participants discuss any algebra connections they observed in Mr. Elizalde’s tangents lesson.

• Students applied the Pythagorean theorem to solve problems. • Students set up and solved equations.

Have participants share any activities in which they have integrated algebra and

geometry in their lessons.

• Using the slopes of lines on coordinate grids to determine if pairs of lines are parallel (the slopes of parallel lines are equal)

• Using the slopes of lines on coordinate grids to determine if triangles are right triangles (perpendicular lines have opposite slopes)

Tangents

Cooperative Groups 72

Cooperative Groups Possible answers may include the following: Cooperative Groups Advantages

• Students see different approaches and perspectives. • Students can learn from three peers versus one teacher. • Teachers can have dialog with a student and can conduct reteaching in a setting that

is not intimidating. • Students can glance around the group and check to see if they are on the right track,

can see the other students on task, and are motivated to stay on task too. Management Strategies

• Need to prepare activity tasks and procedures ahead of time. • Caution: Some students can become over-dependent on peers, and some students

may only follow or may only lead with no collaboration. Solution: Need to make sure you are supervising the cooperative groups by walking around the room and keeping each student’s participation accountable.

• Can sometimes have inappropriate behavior, but good classroom rules can help address behavior problems.

Think and Discuss Possible answers may include the following: Describe your experiences using cooperative groups as an instructional strategy.

Discuss the benefits that you have observed and the ways you have addressed

related classroom management issues. • Benefits: Students communicate about mathematics, students are engaged in their

learning, students help each other, and students practice using new vocabulary in a non-threatening environment.

• Classroom Management Issues: Clearly identify activity tasks and procedures ahead of time and communicate these to students (via chart, overhead transparency, or handouts), establish and reinforce rules for group behavior, carefully assign students to groups, assign roles to group members (i.e., Recorder, Supplies Person, and Spokesperson), arrange classroom as needed (move desks), and circulate and monitor groups and facilitate student interaction.

Alternate Discussion Topic Possible answers may include the following: Andrew Elizalde discusses how he strives to show his students that there are

multiple approaches to solving problems. He accepts that sometimes a student may

present a better approach to solving a problem than he does. Have participants share any experiences they have had where their students have come up with a

better approach to solving a problem than the one that they have proposed.

• Participants may discuss students using alternative approaches to a proof, using different steps, or reasoning, where the resulting proof may illustrate a valid approach that may make more sense to other students than the traditional one presented by the teacher.

73

Analyzing Graphs of Quadratic Functions


Strategy-Based Outcomes 74

Strategy-Based Outcomes


Strategy: Have students solve open-ended problems.

Activity: Have students work in small groups to create a “Quadratic Functions Game Show.” Each group will decide the format of the game and will write questions or tasks that “contestants” must complete to win the game; for example, students can create a Jeopardy-style game in which answers are presented and students need to provide the corresponding questions. Students can write answers about characteristics of parabolas. Strategy: Have students complete project-based activities.

Activity: Have students complete the following project-based activity: • Have students toss a ball in the air, estimate its maximum height, and measure how

far it goes before landing. • Have students find a quadratic function that models the flight of the ball. The height

of the ball can be estimated by tossing it next to a tree, building, or some other object of known height.

• Videotape the experiment with a meter stick or yardstick in the view. • Repeat the experiment for different heights and distances to develop a set of

findings. • Ask students to make generalizations and allow them to present their results to the

rest of the class.

Strategy: Have students communicate about mathematics. Activity: Have students write a summary that explains how the constants affect graphs of quadratic functions.

Strategy: Have students use multiple representations.

Activity: Divide the class into three groups to graph quadratic functions; one group will create graphs by hand, one group will use graphing calculators, and one group will use graphing software. Have students, as a whole class, compare each type of graph and the process completed to create the graph. Then, have students discuss their preferred method and explain why it is their favorite.



Think and Discuss:

Possible answers may include the following: How do the prerequisite skills mentioned by Ms. Moore-Harris connect to the

concepts that Algebra 2 students learn as they study quadratic functions? • Participants may discuss the following prerequisites: Students know:

the parent form of the function for a family of graphs how the equation is going to be written if it is a quadratic function how to factor quadratic equations vocabulary involved when working with quadratic functions: symmetry, axis of

symmetry, parabola how to use graphing calculators and how to graph by hand

• Skills students learn as they study quadratic functions: Have the ability to give a reasonable summation of what the graph depicts. Understand how the constants will affect graphs. Understand what a graph should look like if the constant is changing, when the

graph opens upward, opens downward, gets wider, or gets narrower. Students should have an intuitive sense of when each would happen.

Understand the vertex form of the function. • Examples of how the skills connect include:

Students need to be able to factor quadratic equations to set up these equations in proper forms for graphing and analyzing the functions.

Students need to have a familiarity with attributes of graphs and their corresponding equations to help develop their intuitive sense of characteristics of quadratic graphs.

How would a lack of these prerequisite skills hinder student success?

• Students who do not have strong graphing skills will have a difficult time creating and analyzing the more complex graphs of quadratic functions. Additionally, these students may work too slowly (to create the graphs) to keep up with the rest of the class.

• Students who have not mastered factoring quadratic equations may set up the equations improperly, thus resulting in graphs that are not accurate.

What can you do before you introduce new concepts to ensure students have mastered prerequisite skills?

• Assess students for mastery of prerequisites prior to starting a unit on graphing quadratic functions.

• Use the information gathered from the assessment to plan remediation and group activities.

• Provide extra practice for students who may need to strengthen prerequisites. • Group students heterogeneously to enable peer tutoring, or use homogeneous groups to

enable the teacher to provide extra instruction or remediation to those students who need it.




Possible answers may include the following: Bea Moore-Harris suggests that teachers in an Algebra 2 classroom give students

the opportunity to solve open-ended problems, as well as problems for which there are more than one right answer. Have participants identify content from their

curricula that lends itself to open-ended problems. Have them discuss their own

comfort level and the comfort level of their students with such problems. Participants may discuss the following regarding open-ended problems: • Their own comfort level: Participants may describe being uncomfortable with open-

ended problems because of the very “open” nature of the problems. Some teachers may be more comfortable knowing clearly ahead of time the answers students will come up with for assigned problems. Additionally, some participants may discuss difficulties evaluating open-ended problems. These types of problems often require the use of a rubric, often created by the teacher specifically for the problem.

• Comfort level of students: Participants may describe some students who are uncomfortable with open-ended problems because it can be difficult to determine exactly how to start working on the problem, what the goal of the problem is, and appropriate strategies to use to reach the goal. It is for the same reasons that some students enjoy open-ended problems. These students like the flexibility involved and the process of trial and error that may be used along the way.

• Content that lends itself to open-ended problems:

Content for which students can make predictions; for example, having students make predictions about parabolas due to characteristics of their corresponding quadratic functions, and how the parabola will change when changes are made to the functions, and then testing these predictions

Content for which there are multiple answers; for example, challenging students to write multiple equations that will satisfy given requirements for parabolas (e.g., open downward and have a vertex in quadrant II)


Lesson Strategies 77

Lesson Strategies Possible answers may include the following:

Using Technology • Mr. Roberson used an interactive whiteboard to review prior learning. • The students used graphing calculators to graph and investigate quadratic functions. Making Connections

• Mr. Roberson: used the metaphor of a tree to make connections to parent functions and families

of graphs made connections to prior learning made connections between forms of quadratic functions guided students to make connections between the vertex of the parabola and the

form of the equation had his students think of real-life examples of quadratic functions

Cooperative Groups

• Students worked in cooperative groups to: graph by hand quadratic functions complete a graphing activity with graphing calculators and hand-created graphs make large charts to present their work to the rest of the class analyze constants and write a rule about the constant and their graphs discuss real-life examples of quadratic functions

Think and Discuss


What classroom management techniques contributed to the effectiveness of Mr.

Roberson’s strategies? Classroom management techniques include: • assigning students to groups • arranging desks for cooperative groups • preparing the activity for groups to complete • gathering supplies (e.g., chart paper and markers) for the group activity • preparing technology ahead of time (e.g., notes displayed on interactive whiteboard) • establishing an environment in which students feel comfortable articulating their

thoughts and ideas

What instructional strategies, besides the ones listed, did you notice in the lesson?

Additional strategies include: • emphasizing mathematical vocabulary • writing in mathematics class • use of multiple representations


Lesson Strategies 78

Would you have changed any part of this lesson or used different strategies? If

so, explain what you would have done and why. Alternative ideas for the lesson include: • Introducing the lesson content with a real-life example to create student interest at

the beginning of the lesson.


Possible answers may include the following: Matthew Roberson has his students write out the rules they discover as they

analyze the constants in quadratic functions. He also uses writing as part of his homework. Have participants describe the ways they incorporate writing into their

classes. Ask them to discuss the importance of writing in a mathematics class.

Participants may discuss the following regarding writing activities in mathematics classes: • Writing activities:

provide a way for students to clarify their understanding of concepts and procedures

enable students to express confusion in a nonthreatening manner enable students to strengthen their reasoning abilities as they write explanations enable students to demonstrate mastery in an alternative manner provide valuable insight to teachers regarding their students’ levels of

understanding or misconceptions • Writing activities can be incorporated:

at the beginning of a lesson as a way to review prior learning and check for student understanding of prerequisites

during a lesson as a way for students to apply newly learned concepts to new situations

at the end of a lesson as a way for students to summarize learning from the day


Evaluation of Teaching Practices 79

Evaluation of Teaching Practices Possible answers may include the following:

Mathematics Specialist’s Evaluation

Your Evaluation

1 Students graphed using multiple

formats, including paper and pencil and graphing calculators.

• Evaluation: This was an effective way to strengthen student understanding of graphing quadratic functions. It also helped students strengthen their understanding of the characteristics of these graphs.

• Concerns: This approach may take too long to complete all tasks in one lesson.

2 The teacher asked questions to activate prior knowledge.

• Evaluation: This was an effective strategy to inform the teacher about his students’ readiness for the lesson content and to identify any areas that may require extra focus.

• Concerns: May lose some students if their prior knowledge is lacking.

3 Students shared their results, presenting summaries and explanations of their reasoning.

• Evaluation: This was an effective strategy to help students clarify their understanding. It also enabled the teacher to assess his students informally.

• Concerns: Need to ensure appropriate classroom environment so that students feel comfortable.

4 Students used mathematical

vocabulary appropriately and communicated using verbal and written language.

• Evaluation: This was an effective strategy to strengthen student understanding of concepts and procedures.

• Concerns: Do not have enough instructional time to include writing activities in all lessons; writing activities require extra time to read and evaluate.

5 Students identified real-life

applications of lesson content. • Evaluation: This was an effective

strategy to engage student interest. It also helped students understand that what they are studying has application in the world outside of the mathematics classroom.

• Concerns: Not all lesson content presents easy-to-identify real-life applications.


Evaluation of Teaching Practices 80

Think and Discuss Possible answers may include the following: Students in the lesson were asked to identify real-world uses for quadratic

functions and parabolas. What real-world examples can you add to what they said?

Real-world examples of quadratic functions and parabolas include: • the flight of golf balls or baseballs after being hit • the timing and path of fireworks • architectural designs for arches • the analysis of “hang time” for the flight of a football that has been kicked

How effective are real-world examples in engaging student interest? Are some

lessons and examples more effective than others?

• When it is appropriate to the lesson content, real-world examples that are meaningful to students are effective in engaging student interest and increasing motivation. Real-world examples can provide a context for students; that is much more meaningful than abstract mathematical contexts. Real-world examples help provide reasons for students to study and learn concepts and help students see that mathematics has application and utility outside of the mathematics classroom.


Possible answers may include the following: Have participants discuss the best way to have groups present their findings to

the whole class. Talk about the limitations of having selected students report to

the class rather than the whole group. Ask participants to propose ways to ensure that all students in a cooperative group participate equally and ways to

appropriately assess group work.

Participants may discuss the following regarding group work: • Strategies to encourage participation by all members:

Assign specific tasks to each group member. Assign specific roles to each group member (e.g., Recorder, Reporter, Checker,

and Facilitator). Give assignments that have both individual and group accountability.

• Strategies to help all students feel prepared and comfortable presenting to the class: Use a strategy similar to the one discussed by Matthew Roberson: have students

present their ideas and findings in the small group first. This helps ensure that all students understand the findings and are better prepared to present to the class.

Assign numbers to students, randomly draw a number, and the selected student makes the presentation.

Have students get in pairs and present to each other first to practice. • Strategies for assessing group work include:

All students are assigned a final product that will be evaluated. Criteria checklists are used to evaluate the group’s project.


Activities and Strategies 81

Activities and Strategies


Lesson Topic Write Equations to Go with Graphs of Quadratic Functions

Cooperative Groups • Give several graphs of quadratic functions to each group. (Can use all unique

graphs, all similar graphs, or one similar graph and the rest unique.) • Ask students to determine the equations for each graph. • Have students use graphing calculators to verify their solutions. Using the proposed

equation, test it on the graphing calculator to see if the same graph results

Writing in Mathematics • At the end of the activity, students write a summary of how to determine the equation of

a graph of a quadratic function. Real-World Connections

• Students solve the following problem: The Bayside Bombers football team analyzes the hang time of kickoffs made by their two kickers. This is done by analyzing the graphs of the football’s trajectory on the kickoffs. Unfortunately, due to a technology glitch, the labels for the graphs were deleted. The students will need to determine the equations for each graph and then use the data to determine the hang time for each kick.

Think and Discuss


Matthew Roberson wants students to understand that their current lessons are

just the beginning of higher-level mathematics. How do you convey this idea to

your students? What activities could you use that would lead students to this conclusion?

• One strategy to accomplish this is to present a “sneak peek” of an advanced mathematics topic that has connections to the currently studied topic. Teachers can present an application problem that uses advanced levels of the current concept to students and discuss how what they are learning now will enable them to solve such a challenging-looking problem in the near future.


Activities and Strategies 82


Possible answers may include the following: Matthew Roberson wants students to be able to graph quadratic functions by

hand, as well as to be able to use a graphing calculator. Have participants discuss the reasons that students need to be able to graph by hand when they could rely

on technology.

• Graphing by hand helps students better understand the nature of quadratic functions and the characteristics of their parabolas. Graphing by hand also forces students to carefully follow each step in the process, and when errors occur, students can trace backwards through each step to identify where an error occurred.

Have them suggest appropriate and best uses of graphing calculator technology

in the Algebra 2 classroom.

• Graphing calculator technology is appropriate for investigations, activities that involve predictions, or activities that involve analysis of patterns. Graphing calculators enable students to create multiple graphs in less time than if doing them by hand. Students can also easily manipulate aspects of the graphs or their functions.

Ask how they balance teaching students to use a graphing calculator effectively

without relying completely on it. • One strategy to accomplish this is to have students begin any type of graphing lesson

by creating a graph by hand first before moving on to the use of graphing calculators. This can help students clarify the graphing process for the particular type of graph and can help teachers know whether students are ready to move on to the use of technology.

83

Ellipses

Ellipses

Lesson Time Management 84

Lesson Time Management

Activity or Grouping Lesson Length ____ Minutes

Time Allotted

<Participant’s responses will vary.>

Warm-Up Activity 5 minutes 20 minutes 35 minutes

Whole-Group Instruction 5 minutes 20 minutes 35 minutes

Whole-Group Practice 5 minutes 20 minutes 35 minutes

Cooperative Group Activity 5 minutes 20 minutes 35 minutes

Individual Practice 5 minutes 20 minutes 35 minutes

Closing Activity 5 minutes 20 minutes 35 minutes

Assessment 5 minutes 20 minutes 35 minutes

Small-Group Intervention 5 minutes 20 minutes 35 minutes

Think and Discuss Possible answers may include the following: Consider the time management decisions Gilbert Cuevas mentions. How do you

make decisions about the time needed to develop skills, to develop an

understanding of the concept, or to apply the concept? How do state or local

standards contribute to your decisions? Participants may discuss some of the following regarding time management decisions: • State standards can be used as a guide to help focus instructional goals and

prioritize content to cover. • Analyzing standards, “unpacking” standards, and identifying “big ideas” are

strategies that can be used to help clarify standards, cluster standards, and to plan larger units of study.

• Assessment data (including that from statewide or other high-stakes tests) provides valuable information that can also be used to help make instructional decisions based on the performance of students and identify any areas that may need more attention.

• Formative assessment, such as assessments designed to measure student mastery of prerequisites, can be used to help structure an upcoming unit of study. Information from this assessment can identify areas that require reteaching as well as areas that may require just a brief review.

• Informal assessment also plays an important role in time management decisions. Feedback from informal assessment can help a teacher make instructional decisions, such as whether students are ready to move on to the next topic, whether students need more practice in specific skills, or whether students need more time to develop conceptual understanding of the material.

Ellipses

Lesson Time Management 85


Possible answers may include the following: Gilbert Cuevas talks about the prerequisite skills students need coming to a

lesson on ellipses. Have participants discuss any additional prerequisites they think students need.

• Prerequisite skills discussed by Gilbert Cuevas: Students should: be comfortable graphing and understand the coordinate grid system have had opportunities to graph quadratic functions and to analyze the graphs and

their equations know the Pythagorean theorem and be able to apply in it problem-solving

situations • Participants may discuss the following additional prerequisite skills for a unit on

ellipses: Students should: be comfortable working with radical expressions understand the relationships between equations and their graphs understand how to use the coordinate graphing system to find the distance

between 2 points understand how the slope of lines on a coordinate grid can be used to analyze

lines and polygons be comfortable setting up and solving a wide variety of equations

Have them discuss which prerequisites cause the most difficulty for students and

how they ensure that students have the necessary prerequisites before beginning

the lesson. • Prerequisites that may cause difficulty for students include:

comfort working with a wide variety of equations, including radical expressions and factoring quadratic equations

remembering to simplify and solve equations carefully and to check each step in the solution for errors

• Strategies to ensure that students have the necessary prerequisites include: assessing students for mastery of prerequisites, including student understanding of

quadratic functions and their graphs using information gathered from the assessment to plan remediation and group

activities grouping students heterogeneously to enable peer tutoring, or using homogeneous

groupings to enable the teacher to provide extra instruction or remediation to those students who need it

assigning extra practice to students who may need to strengthen specific prerequisites

Ellipses


Instructional Strategies Possible answers may include the following:

Instructional Strategies from Doug Roberts’ Lesson on Ellipses

Strategy 1: Emphasizing Vocabulary

• Doug Roberts: defined and clarified important terms related to the study of ellipses: focus, foci,

major axis, minor axis used real-life contexts to help develop meaning and to connect to prior learning used words and pictures (via the use of technology) to clarify definitions

Strategy 2: Informal Assessment • Doug Roberts:

used networked calculators to determine students’ mastery of prerequisites used the feedback from the calculators to identify remediation needs and tailor the

pace and scope of the lesson • Students were able to look at student answers and conduct an error analysis for

incorrect responses.

Strategy 3: Real-Life Applications • The context of planetary orbits was used as a real-life connection. This:

engaged student interest provided an example of how ellipses are used in science connected the abstract mathematical concepts to a visual model

Strategy 4: Modeling Thinking Processes • Doug Roberts modeled analytic thinking throughout his lesson. He:

used an ordered process to uncover information about ellipses explained each step in the process used diagrams to support his thinking asked guiding questions to help students follow along

Think and Discuss Possible answers may include the following: Doug Roberts emphasizes vocabulary with his students. In what ways do you help your students understand and retain mathematical vocabulary? How do you

encourage students to use this vocabulary during your lessons?

Participants may discuss the following strategies for strengthening student understanding of mathematical vocabulary: • Model and reinforce correct usage of terms throughout lessons. • Lead activities in which students need to use precise language (e.g., having students

work in pairs to create drawings that are hidden from the view of the student drawing and described verbally by the student viewing the drawing).

• Have students complete warm-up activities that involve mathematical vocabulary.

Ellipses



Possible answers may include the following: Doug Roberts uses technology that allows students to communicate with him

individually while he teaches the class as a whole. Have participants discuss the benefits and drawbacks of this technology.

Participants may discuss the following benefits and drawbacks: • provides immediate feedback to the teacher regarding students’ understanding of

concepts and procedures • enables teachers to use that information to make adjustments to the pacing and

content of the lesson • lets students see how their classmates are answering questions • lets students conduct error analysis of incorrect responses • although it is anonymous, some students may nevertheless be intimidated by the

process • may discourage teachers to use other forms of informal assessment, such as

questioning, asking students to explain solutions, or having students use models to represent concepts or procedures

Have them suggest ways to accomplish similar tasks without the use of this

technology. • Students can use individual whiteboards on which they write answers and then hold

up the whiteboards for the teacher to view answers. • Students can use calculators (i.e., traditional or graphing) and place the calculator

prominently on their desks, so the teacher can walk around the room and look at the student answers on their calculators.

• Students can work in pairs to solve problems and then use a thumbs-up signal when their answer choice is given by the teacher (i.e., Did any pairs get 5.25 units as their answer? 215 units? 2.15 units?).

Ellipses

Lesson Process 88

Lesson Process


Outline for a Lesson

Lesson Element and Content Strategies and Implementation Requirements

Warm-Up • Review prerequisites required for the

study of ellipses, including graphing on the coordinate plane and the Pythagorean theorem.

• Students work in pairs to solve several review problems.

• Students share solutions.

Teach • characteristics of ellipses, and

terminology • how to find the coordinates of the center

and foci of an ellipse

• how to find the lengths of the major and minor axis

• Begin with a real-life example (e.g., physics, other science applications, or within the context of engineering or architecture).

• Have students make constructions of ellipses, including major and minor axis to illustrate concepts.

Practice/Apply • Apply ellipses concepts to solve

problems. • Use Pythagorean theorem to find

missing measures.

• Students work in small groups to solve problems.

• Students create large charts to show their solutions.

• Students present their charts to the class.

Assess

• Determine student ability to apply concepts and procedures to problem-solving application.

• Present the following problem for students to solve independently:

A landscaper is planning to build an elliptical walkway around a flagpole in a park. The major axis will be 30 feet long, and the minor axis will be 20 feet long. The landscaper will lay out the ellipse using a rope tied to stakes driven in the ground at the foci. Describe how to find the location of the foci.

Ellipses

Lesson Process 89

Think and Discuss

Possible answers may include the following: Gilbert Cuevas suggested that Doug Roberts needs to bring the lesson together at

its end. How would you suggest that he do this? What closing activity might you suggest that Mr. Roberts use?

Participants may discuss the use of a writing activity as a way to bring a lesson to an end. Students could be given the following application problem and asked to write an explanation for how they would solve it.

Planets travel in elliptical orbits with the Sun at one focus. Describe how to find an equation for the orbit of Mercury, given that the length of the major axis is about 1.1582 108 km and the distance from Mercury to the center of the Sun is

about 4.600 107 km at its closest point.

Alternate Discussion Topic Possible answers may include the following: Gilbert Cuevas says that mathematics teachers need to teach students how to

“think like a mathematician.” Have participants identify what that means in their

classrooms. Participants may discuss the following attributes of “thinking like a mathematician”: • using a logical, sequential development of ideas and procedures (as in formal proofs) • using visual models to support and clarify statements or conclusions • making and testing conjectures • noticing patterns and using patterns to make generalizations • using metacognitive strategies, “what operation does that term signify? Is that result

reasonable? Have I completed all necessary steps?

Ask them how they might encourage this type of thinking and how they would

know when it occurs. Participants may discuss the following ways to encourage this thinking and to recognize it when it occurs: • When presenting a problem, before asking students to solve the problem, ask

students: “How would you go about solving this problem? What strategies would you use?

• Ask students if a particular problem-solving approach could be applied to another problem. Ask students to explain why or why not.

• Have students write about concepts, procedures, or problem-solving strategies. • Ask students if they can think of an alternative way to solve a problem. Note: Students who are able to make connections between concepts and procedures may be demonstrating “mathematical thinking.” Additionally, students who can explain why a certain conjecture is true in a logical manner may be demonstrating “mathematical thinking.”

Ellipses

Piece of the Puzzle 90

Piece of the Puzzle


Lesson Piece Completing the Square to Write Equations for Ellipses

How You Would Teach It • Use algebra tiles to review completing the square to solve quadratic functions. • Apply a similar process to completing the square with equations for ellipses. • Model using Algeblocks® to represent the process of completing the square with

equations for ellipses. • Have students record the algebraic steps as they complete each step concretely

(with the Algeblocks®). Formative Assessment

• Present an illustration that shows Algeblocks® used to represent the process of completing the square for an equation for an ellipse. Have students write the corresponding algebraic equations for each step.

• Present several equations for ellipses that are not in standard form. Have students complete the square algebraically to transform the equation to standard form.

Think and Discuss

Possible answers may include the following: Consider the overall organization of the lesson Doug Roberts used and the way

his “jigsaw” approach to new content fit within the overall lesson structure. What were the discrete parts of the lesson? How did lesson organization support his

approach to new content?

Participants may discuss the following lesson structure: • Review prerequisites that students will need to use during current lesson. • Use networked-calculators to get feedback regarding student understanding of

prerequisites. • Teach vocabulary and provide context for ellipses. • Apply new vocabulary terms within the context of whole-class demonstration of

ellipses concepts and properties. • Have students work in cooperative groups to solve problems, using new skills and

concepts. • Use guided questioning to encourage mathematical discourse about new concepts and

for students to practice using new vocabulary.

Ellipses

Piece of the Puzzle 91


Possible answers may include the following: Doug Roberts likes to use multiple representations for mathematical concepts. He

feels that students often need the connection between graphical or pictorial and abstract representations. Have participants identify other topics supported by

multiple representations. Have them suggest ways to represent those concepts.

• You may want to suggest additional ways that ellipses concepts can be represented. Discuss the following representation and activity ideas with participants: To help kinesthetic learners or for a hands-on activity for a class, consider having

students go out to a field with a measuring tape and a rope and having them create ellipses, including marking the foci, and the major and minor axis.

The relationships between parts of an ellipse may be more meaningful to kinesthetic and visual-spatial learners if they create their own ellipses using a loop of string. Draw axes on a piece of paper and label the foci. Place the paper on a piece of thick cardboard and place a thumbtack in each focus. Place the loop of string around the thumbtacks. (The string should be loose.) With the tip of a pencil, stretch the string loop tight, then move the pencil in an arc, keeping the tip tight against the string to trace an ellipse. Students can use a ruler to measure the axes and the distance from the center to each focus. They can also measure the string they used and use the Pythagorean theorem to relate these measures.

• Other topics supported by multiple representations include: Quadratic Functions: Students can use algebra tiles to concretely represent

completing the square to solve the equation. Students can also draw pictures of the tiles (pictorial representations) and can write the algebraic equations (symbolic representations).

Using visual (grid paper) and concrete (e.g., paper cutouts, rulers, and protractors) models to investigate properties of similar triangles and to represent triangle postulates.

Using centimeter paper and base-10 blocks to interpret powers of a binomial geometrically.