Pre-Algebra - McGraw-Hill Professional DevelopmentPre-Algebra Video Workshop Answer Key . 2 ... poor understanding of integer concepts and operations as they apply to integers. •

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Pre-Algebra

Video Workshop Answer Key

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Integers ...........................................................................................................................3

Goal for Instruction ......................................................................................................4

Lesson Elements.........................................................................................................6

Instructional Benefits ...................................................................................................8

Learning Styles..........................................................................................................10

Linear Equations ...........................................................................................................12

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

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

Wait Time ..................................................................................................................18

Extension Lesson ......................................................................................................20

Solving Multistep Equations ..........................................................................................23

From Concrete to Symbolic .......................................................................................24

Instructional Strategies ..............................................................................................26

Encouraging Mathematical Discourse .......................................................................28

Standards..................................................................................................................31

The Pythagorean Theorem............................................................................................33

Conceptual Development ..........................................................................................34

An Investigation-Based Lesson .................................................................................35

Lesson Highlights ......................................................................................................38

Goals for Instruction ..................................................................................................40

Surface Area and Volume .............................................................................................43

Best Practices ...........................................................................................................44

An Exemplary Lesson................................................................................................46

Another Look .............................................................................................................48

Ranking of Goals.......................................................................................................50

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Integers

Integers

Goal for Instruction 4

Goal for Instruction

Goals for Integer Instruction My Instruction

<Participant responses will vary.> Build student understanding of integers from prior knowledge of

whole numbers. beginning average high

Have students compare integers. beginning average high

Have students perform operations

with integers. beginning average high

Have students apply integers in

problem-solving situations. beginning average high

Teach students the mathematical

vocabulary of integers. beginning average high

Allow students to explore the rules

of integer operations. beginning average high

Help students avoid

misconceptions. beginning average high

Allow students to move from the

concrete to the pictorial to the rules

of integer operations. beginning average high

Think and Discuss

Possible answers may include the following: Discuss any additional instructional goals that you may have for integers and

their operations in the middle grades.

• Students develop understanding for the concept of zero pairs (and the additive identity) through concrete and pictorial representations.

• Students use concrete and pictorial models to represent integer operations. • Students use integers in equations.

Are you able to achieve all the goals?

• Participants may discuss being able to meet most of the goals, but not all. • Participants may discuss time constraints and the need to address all grade-level

standards as potential barriers to achieving all the goals. Discuss any goals you would like to give more focus. • Participants may discuss giving more focus to:

helping students build understanding of integers in context developing understanding for operations with integers, especially multiplication

and division, so students understand more than just the easily-memorized rules: “positive times a positive is positive,” “negative times a negative is positive,” and “positive times a negative is negative.”

solving equations that involve integers

Integers

Goal for Instruction 5

Alternate Discussion Topic

Possible answers may include the following: Adelina Alaniz discusses misconceptions that students may have with integer

operations. Have participants describe any misconceptions that they have observed in student understanding of integer concepts. Ask how they can clarify

these misconceptions for students.

• Misconception: Students make errors in performing operations with integers due to a poor understanding of integer concepts and operations as they apply to integers.

• How to Clarify: Ensure that student understanding of the meaning of operations with whole numbers; develop student understanding of the meaning of integer operations through hands-on activities and the use of multiple representations.

• Misconception: Students are unable to provide multiple examples of integers in real-

life context. Students have a limited experience in providing multiple examples for integers and may just be able to supply an example of negative temperatures.

• How to Clarify: Incorporate multiple real-life examples of integers; for example, debt, elevations and elevation maps, depth below sea level, ocean tides (minus low tides), lost yardage in football games, and golf scores under par.

• Misconception: Students have difficulty solving equations involving integers,

especially remembering to check for using the right number signs throughout the solution steps.

• How to Clarify: Have students use a highlighter (marker) to highlight the number signs in equations; reinforce the process of going back through a solution and checking each step for mistakes.

Integers

Lesson Elements 6

Lesson Elements Possible answers may include the following:

Adding Integers Instructional strategies and techniques observed in Darla Schwolert’s lesson:

• emphasis on mathematics vocabulary • connections to prior learning • use of small-group activities (i.e., to encourage communication and helping each

other) • use of multiple representations (e.g., 2-color chips, number lines, drawings, and

rules) • monitoring small-groups to provide guidance and support and to clarify concepts and

procedures • having students explain thinking and solution strategies • having students move from the concrete (chips) to the pictorial (number lines and

drawings) to the abstract (rules and equations) Think and Discuss

Possible answers may include the following: Which instructional approaches were especially effective in this lesson?

• Using multiple representations to model integer concepts and operations (e.g., chips, number lines, drawings, rules, and equations)

• Emphasizing communication (e.g., small groups interacting, students explaining thinking, and using mathematics vocabulary)

• Enabling students to develop conceptual understanding by moving from concrete to pictorial to abstract representations of integers and integer addition

• Having students share, explain, and compare answers (e.g., two students at front of room, one at overhead using chips, one at chart using number line)

• Providing clear instructions for procedures to follow during the small-group activity Discuss any strategies that you observed in this lesson that you will try in your classroom.

• Participants may discuss trying most of the same strategies they listed as those that were effective, above.

Integers

Lesson Elements 7


Possible answers may include the following: Darla Schwolert begins her lesson with a vocabulary activity. Have participants

describe any interesting vocabulary activities that they have used in their mathematics classrooms.

• Playing a variation of the Jeopardy game in which 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.

• Having students play a Concentration-style game with vocabulary cards: Students can match cards with a definition to cards with an illustration of the term.

• Providing students 6-to-10 vocabulary terms and challenging them to write a paragraph, or a story, that uses all of the vocabulary terms.

Integers

Instructional Benefits 8

Instructional Benefits Possible answers may include the following:

Instructional Practice

Mathematics Specialist’s Comments

My Thoughts

Students engage in

vocabulary activities.

• Students write vocabulary definitions and examples—this is a form of informal assessment that can drive your instruction.

• Knowing vocabulary is important for standardized assessment, so students can read directions and problems with accurate comprehension

Students use

manipulatives and models.

• Your success rate will be greater when you use manipulatives and models. Students need to be able to “see” concepts. These experiences help students remember concepts and apply what they remember on assessments. Manipulatives are important to use at any grade level.

• Important for students to transfer concrete models to pictorial and symbolic representations as they complete activities

Students explain their answers.

• When students are given the opportunity to explain their thinking, this enables the teacher to check for understanding, to ensure that students do not get confused.

• Students can learn from each other, from listening to other students’ explanations.

Classroom activities address the affective

domain of learning.

• For many pre-algebra students, their affective domain becomes increasingly important. In mathematics classes the teacher can set the tone, can turn students on to mathematics or off. It is important to ensure that all students can experience success in mathematics and that all students are interested in mathematics.

• Remember to be aware of the subtle attitudes you as a teacher convey to your students about mathematics.

Integers

Instructional Benefits 9

Think and Discuss Possible answers may include the following: How did Adelina Alaniz’s comments affect your opinion of each instructional

practice? Did her comments give you any new insights about the benefits of each instructional practice?

• Participants may mention: an increased awareness of the importance of vocabulary activities, especially as

an informal assessment technique the impact manipulatives can have on students’ remembering concepts and their

ability to apply these remembered concepts during assessments having students explain their thinking as an informal assessment technique that

enables teachers to check for student understanding and to clarify misconceptions

the importance of considering students’ affective domain and the critical role this can play in influencing students’ attitudes toward mathematics

Alternate Discussion Topic Possible answers may include the following: For this lesson, students use multiple representations for integer addition

problems. Have participants discuss this instructional practice and ways it can be

implemented in similar or related lessons.

• Students can use 2-color chips (to model zero pairs), number lines, drawings, rules and equations to model integer subtraction.

• Students can use 2-color chips and number lines to model the concept of absolute value as well as additive inverse.

• Students can use 2-color chips and number lines to model integer multiplication (to represent multiple groups of equal sets of integers; for example, to model 3 x –5, students can show 3 groups of 5 negative (red) chips for a total of 15 negative (red) chips, or –15.

Integers

Learning Styles 10

Learning Styles

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

Integer Concept: Integer Multiplication

Instructional Strategies for Mastering Concepts

Hands-On Visual Representation Abstract Rule

Two-color chips (red and black) are used to model integer multiplication. Students can use the chips to show groups of positive chips or negative chips to represent integer multiplication problems; for example, to model 3 –5, show 3 sets of –5 (red) chips, for a product of –15.

Students create drawings to represent the groups of chips used to model integer multiplication. Students can use number lines to represent integer multiplication by drawing lines to represent repeated addition; for example, 3 –5 can be thought of

as adding –5 three times, or taking 3 sets of –5 and adding them to 0: 0 + –5 + –5 + –5 = –15 or on a number line, starting at 0, draw 3 lines of –5 (to the left of 0), ending at –15.

Students develop a concrete understanding of integer multiplication using the concrete and pictorial representations to help lead them to the rules of integer multiplication: positive positive

= positive negative negative

= positive positive negative = negative or If the multiplier is positive and the multiplicand is positive, add positive sets to 0. If the multiplier is positive and the multiplicand is negative, add negative sets to 0. If the multiplier is negative and the multiplicand is positive, subtract positive sets from 0. If the multiplier is negative and the multiplicand is negative, subtract negative sets from 0.

Integers

Learning Styles 11

Think and Discuss

Possible answers may include the following:

How important is it for teachers to address their students’ various learning styles?

How does this impact instruction? Participants may discuss the importance of designing instruction so that it addresses students’ various learning styles as much as possible. However, some participants may discuss the balance of trying to address students’ varied needs by differentiating instruction (as well as other strategies) and proving a variety of activity types with the reality of how much extra time these instructional practices require to plan, prepare, deliver, and assess.

Discuss any successful strategies or techniques that you have used to help

address your students’ various learning styles.

• Participants may discuss the utilization of differentiated instruction to address students’ varied learning styles.

• Participants may discuss students’ learning styles: visual, kinesthetic, and auditory, and how varied activities should be planned to address each learning style. (Providing activities that appeal to visual learners—seeing, kinesthetic learners—manipulating, and auditory learners-hearing)

• Participants may discuss Howard Gardner’s theories on multiple intelligences, how they have observed students who possess different types of intelligences (i.e., linguistic, logical-mathematical, musical, bodily-kinesthetic, spatial, interpersonal, and intrapersonal), and how activities can be planned to address the varied types of intelligences. (Students write and perform a song to teach the rules of integer operations, students write a newspaper-style article reporting on examples of integers in daily living, students work cooperatively to solve integer problems, or students use concrete models to explore integers.)


Possible answers may include the following: Darla Schwolert discusses the importance of students taking ownership of their

learning. Ask participants how they can help their students take ownership of their learning. Have them discuss their ideas for types of experiences that may

encourage this to occur.

• Having students complete small-group activities and pair activities in which the students are responsible for researching a topic and creating an instructional activity for the rest of the class.

• Providing opportunities for students to share their problem solutions and to explain the procedures and strategies they used to solve the problems.

• Asking students to provide examples of real-life applications of concepts (e.g., students can bring in newspaper or magazine articles, graphs, or advertisements that show examples of integers).

• Having students create displays (e.g., bulletin boards) that show representations of concepts (e.g., drawings of number lines and zero pairs to represent integer operations).

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Linear Equations

Linear Equations

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


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, and 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

Linear Equations

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 (e.g., science, social studies, and

language arts

Linear Equations

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 for linear equations in slope–intercept form, can students determine if the resulting figure (when draw 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.

Linear Equations

Group Activities 16

What strategies did you implemented 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, when needed; and 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.


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; for example, 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.

Linear Equations

Group Activities 17

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. • Conduct informal assessment. • Reinforce or assess prerequisites.

Linear Equations

Wait Time 18

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 1 Wait Time 2

Similarities

–Provides time for students to think and reflect

–Provides time for students to consider responses

–Enables teachers to informally assess students

–Teacher asks a question and allows time for the student to think about the question and to give a response.

–Used as an informal assessment technique, monitors student understanding.

–Students have an opportunity to organize their thoughts and articulate more complete answers.

–After a student has given a response, wait to see if anyone else will “piggyback” on the response, or can ask other students if they agreeor disagree to see if there are different answers, or alternative thinking or approaches, rather than immediately letting a student knowif he or she is correct.

–Encourages students to take ownership of their learning.

Linear Equations

Wait Time 19

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.


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. • Using self-sticking notes to restate, summarize, or otherwise highlight critical areas

of text is another helpful strategy. • Additional strategies include:

students working in pairs to read, discuss, and take notes writing prereading 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.

Linear Equations

Extension Lesson 20

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 to 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 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.

Linear Equations

Extension Lesson 21

Individual Activities Pair/Group Activities

Informal Assessment

At end of activity, students 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 of the lines:

y =

1

4x + 1

y = 4x – 1

y =

1

4x – 1

y = 4x + 2

y =

1

4x + 2

• Warm-Up Activity

Student pairs complete warm-up activity: After creating 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: Assess 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.

Linear Equations

Extension Lesson 22


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.

Participants may discuss the following preparation tasks: • identifying the goals for the lesson • identifying clearly 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 plastic baggies, or distributing items to student desks)

• determining the arrangement of the classroom • previewing software for any demonstrations

23

Solving Multistep Equations


From Concrete to Symbolic 24

From Concrete to Symbolic


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 multistep equations?

Participants may discuss the following prerequisites: • Vocabulary:

variable, equation, expression, evaluate, factor, exponent, base


From Concrete to Symbolic 25

• 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

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., 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 26

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-to-12 students


Instructional Strategies 27

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, and 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)


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 28

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 grade 8 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.




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 31

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 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 32

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

33

The Pythagorean Theorem


Conceptual Development 34

Conceptual Development Possible answers may include the following:

Layers of Understanding Concept: Triangles Informal, Elementary Study » Level of Understanding » Abstract, Formalized Study

• angles: measuring angles and identifying types of angles • finding the sum of the measure of the interior angles in polygons • identifying supplementary and complementary angles and applying

these concepts to geometric proofs

Think and Discuss Possible answers may include the following: Describe some geometry concepts that you teach in which students also use early

algebra concepts. Discuss how increased understanding of the geometry

contributes to increased understanding of the algebra.

• Spatial visualization skills: Providing hands-on activities involving transformational geometry helps students develop spatial visualization skills that are important for more abstract activities such as using coordinate grids to plot transformations.

• Properties of figures: Developing understanding of similarity, congruence, properties of lines, and properties of figures helps students apply these properties to formal geometric proofs.

• Geometric Formulas: Developing understanding of formulas and using formulas to determine measurements of plane figures helps students feel comfortable using variables and solving for unknowns.

Alternate Discussion Topic Possible answers may include the following: The study of geometry often figures prominently in a pre-algebra curriculum. Have

participants discuss why this is so.

Many geometry concepts can be used to build understanding, first concretely and pictorially, and then apply this understanding to abstract applications.

If they do not mention it, point out the way geometry lends itself to a concrete–

pictorial–abstract progression both for geometric and algebra topics. Participants may discuss some of the following as examples of this progression: • similarity and congruence of figures (i.e., informal understanding develops into

applications of concepts in formal proofs) • angles (i.e., classifying angles and identifying angle relationships, leads to using

concepts in proofs) • formulas (i.e., concrete and pictorial representations to build meaning for the

formulas, leads to application of formula to solve problems) • the Pythagorean theorem (i.e., hands-on investigations, leads to application to

problem-solving activities)


An Investigation-Based Lesson 35

An Investigation-Based Lesson Possible answers may include the following:

Investigation: The Pythagorean Theorem

Lesson Elements Implementation Notes

Students engage in hands-on investigation that builds meaning for the theorem.

• Amy begins by presenting an open-ended investigation for her students to explore the concepts behind the Pythagorean theorem.

• She presents a visual representation showing that the area of the square formed by the hypotenuse of a triangle is equal to the area of the squares formed by the other two sides.

• Students use centimeter paper and triangle patterns to investigate if this works for all triangles.

• Students discover that this property is true only for right triangles.

Students apply what they have learned to a second task that extends their understanding of the theorem.

• After their first investigation, students connect their discoveries to the Pythagorean theorem.

• The investigation helped provide concrete meaning for the theorem.

• Students next investigate Pythagorean triplets, and use the Pythagorean theorem to check results.

Students connect concepts to real-life applications.

• Amy leads a discussion about ways the theorem might be used outside of the mathematics classroom.

• Students discuss how architects might use the theorem. • This helps students also see that what they are studying in the

mathematics classroom has application outside of class.

The class applies the formula to problem-solving situations.

• The investigation and their discoveries lead students to understand the Pythagorean theorem.

• Amy then helps students see how the theorem can be used to help solve problems, such as finding the unknown length of a side or hypotenuse of a triangle.

Discuss the importance of the summary discussion that Amy leads at the end of

the lesson.

• It is important that teachers guide the discussion at the end of an activity to help ensure that students are making the appropriate mathematical discoveries.

• It is important for students to share their ideas. Some students may have thought about the mathematics in one way, and another student may have thought about the mathematics in another way. It is important that the teacher is clear on the underlying mathematics ideas and guides students’ comments toward the targeted concepts.

• Students may think that what they just did was an entertaining activity and that they got to do some neat things, but it is important that the teacher help them see the mathematics behind what they just did.



Think and Discuss

Possible answers may include the following: Describe a lesson that you can lead on this or a related concept.

• Students can investigate the sum of the measure of the interior angles of polygons. • Can begin the investigation by having students investigate the sum of the measure of

the interior angles of a triangle. Students can draw triangles, cut them apart into the three vertices, and then rotate and arrange the angles, so they form a straight line, or 180 degrees.

• Students can apply their discoveries to problem-solving situations in which they need

to find the missing measure of an angle in a triangle. • Students can extend the investigation to quadrilaterals. • Students can discuss how architects, landscape designers, craftsmen, and interior

designers can use these principles in their design activities. Describe your preparation for the lesson, the teaching strategies you would use,

and any classroom management strategies that you would find helpful.

• Preparation: Identify clearly the mathematical goals for the investigation. Gather materials (e.g., construction paper, scissors, or tape if desired) for the

investigation. Prepare several problem-solving situations in which triangles and quadrilaterals

have missing angle measures. • Teaching Strategies:

small-group activity hands-on materials problem-solving applications



• Classroom Management: Determine how students will gather and distribute materials (Will one student

from each group get the materials from a designated place in the classroom?). Determine how to group students and assign students to groups or pairs (if

using). Discuss rules for group behavior. Determine how investigation tasks and procedures will be communicated to

students (via chart, via overhead transparency, via handout). Alternate Discussion Topic

Possible answers may include the following: Ask participants to identify several plane-geometry concepts that are well suited

for investigation-based lessons.

• transformational geometry • measurement activities: including finding the area and perimeter of polygons,

investigating pi, and circumference of circles • using coordinate geometry to examine properties of plane figures Have them identify concepts better supported by other instructional approaches,

such as direct instruction. • applying formulas to problem-solving situations • using angle types to classify triangles • using relationships of corresponding sides to determine similarity of triangles


Lesson Highlights 38

Lesson Highlights Possible answers may include the following: Exemplary Lesson Elements • Amy creates a mathematical environment that is perfect for middle grades’ students:

They work together, conduct investigations, question each other, and explain their thinking.

• Amy is very clear on the underlying mathematics concepts that she wants to get across.

• Amy uses questioning effectively to help students reason and prove or disprove theories.

• Amy helps students form a strong, internalized understanding of the mathematics concepts they are investigating, so they can apply them to new experiences.

• Amy helps students make connections to real-life examples, which is especially important for middle grades’ students.

• Amy helps students realize that they are surrounded by mathematics, that it has utility, application, and beauty. Geometry and Measurement are strands that lend themselves well to this goal.

• Amy understands that students learn in a variety of ways and that it is important to provide activities that enable students to learn in the manner in which they are most comfortable.

• Amy uses multiple representations. Students could use concrete or pictorial models or could use symbolic representations.

Think and Discuss

Possible answers may include the following: Heidi Janzen discusses the mathematics “environment” created by Amy Doherty in

her classroom. Describe your observations of the environment that Ms. Doherty has

created for her mathematics lesson. Describe the classroom-management strategies and techniques required to create and maintain this kind of environment.

• Students in Amy’s class are comfortable conducting investigation-type activities and discussing the results.

• Having students frequently conduct investigation-type activities in which they make discoveries and then share and discuss their results: The more students share ideas and listen to each other’s ideas, the more comfortable students are in doing this. It is important to guide the discussions to encourage students to explain their thinking and to describe their reasoning.

• Students in Amy’s class work together cooperatively, help each other, and explain concepts and procedures to each other.

• Having clearly established rules for group behavior, as well as expectations for individual student behavior: It is important to frequently review expectations for group behavior, and reinforce desired individual behavior. You may even consider having your students role-model desired behavior.

• Amy understands that her students learn in a variety of ways. • Planning and preparing activities that use a variety of models provides a variety of

ways for students to feel comfortable and confident: This lends to the overall positive environment in a classroom because students are able to experience success.


Lesson Highlights 39


Possible answers may include the following: Amy Doherty uses guiding questions to help her students reason and explain

their thinking. Have participants describe some questioning techniques that they use to encourage their students to reason and explain their thinking?

• Asking “What if” questions: What if we were missing the measures for 2 angles, what could we do to find the measures?

• Asking questions that encourage students to explain step-by-step their reasoning processes: After you found the measure of the 2 sides, what did you do next?

• Asking questions that encourage students to explain how they could use their discoveries in new application: How do you know if this process would work for hexagons?

• Asking questions that do not have a simple yes or no answer, but instead are open-ended and require students to provide an explanation and expand on their results: How can you use what you know about triangles to apply this to rectangles?


Goals for Instruction 40

Goals for Instruction Possible answers may include the following:

Goals for Instruction • Begin concept study with a hands-on activity. • Have manipulatives available for students. • Lead discussion to focus on key concepts. • Help students see relationships between concepts. • Make connections to real-world examples. • Represent algebraic equations concretely, pictorially, and symbolically. Goal: Begin concept study with hands-on activity. Ideas for Applying This Goal in My Instruction: • Use grid paper and cut out shapes to build understanding of formulas for finding the

area of parallelograms and trapezoids. • Use string and a variety of circular objects (e.g., jar lids) to investigate circumference. • Use grid paper and paper cutouts to investigate the golden ratio and golden

rectangles. Goal: Make connections to real-world examples. Ideas for Applying This Goal in My Instruction: • Provide real-life examples of where there is a real-life need to find the area and

perimeter of figures, such as in landscape design or interior design. • Discuss how the golden ratio and golden rectangles are used in the design of

commercial products, such as boxes for perfume and cereal. Think and Discuss Possible answers may include the following:

Discuss which of Amy Doherty’s instruction goals are easy to implement and which are more difficult to implement.

• Some goals that may be easier to implement include the following: Begin concept study with hands-on activity. Have manipulatives available for students. Represent algebraic equations concretely, pictorially, and symbolically.

• Some goals that may be more difficult to implement include the following: Make connections to real-world examples. Lead discussion to focus on key concepts. Help students see relationships between concepts.

Discuss why some goals are easier to achieve than others.

• Some goals may be easier to achieve because they require less: content knowledge on the part of teachers preparation time instructional time prerequisite understanding on the part of students



What can you do to ensure that you will apply the ideas that you listed for your

instruction? • Include relevant lesson plans in your instructional calendar. • Gather real-life examples of geometry and measurement concepts (e.g., start a

binder that contains newspaper or magazine clippings). • Gather necessary tools and materials for the activities (e.g., start a collection of jar

lids). • Share ideas with grade-level teachers. Alternate Discussion Topic


Have participants identify a geometric formula that can be represented concretely,

pictorially, and algebraically. Have them describe how they can have their

students represent this formula in these three ways. • the formula for the area of a parallelogram • concretely:

Students use colored-construction paper to cut out and arrange triangles, squares, rectangles, and parallelograms to represent the area of these figures and to investigate their equivalencies



• Pictorially

Students use grid paper to draw parallelograms and rectangles. Students can use drawings to show that parallelograms can be changed to rectangles by moving a triangular-sized piece to the opposite side of the parallelogram and flipping it.

• Algebraically

A = bh

43

Surface Area and Volume


Best Practices 44

Best Practices Possible answers may include the following:

Instructional Goals for Geometry and Measurement of Solid Figures in the Middle Grades

• Provide opportunities for students to look for and analyze patterns. • Provide opportunities for students to make connections among geometry concepts. • Provide opportunities for students to make connections to algebra concepts. • Provide opportunities for students to explore concepts and build meaning for

formulas.

Ideas for My Instruction

• Use function tables and graphs for students to display data gathered during investigations of measurements of solid figures. This can enable students to notice patterns and to make connections to algebra concepts.

• Provide hands-on investigations into surface area and volume of solid figures to help students develop meaning for the formulas.

• Connect prior learning about area and perimeter of solid figures to surface area and volume of solid figures to help students make connections among geometry concepts.

Think and Discuss Possible answers may include the following: Discuss the kinds of mathematical experiences that students must previously have had to successfully participate in a lesson that includes the elements listed above.

Participants may discuss prior experiences in which students: • used concrete and pictorial models to investigate geometry concepts, such as area

and perimeter of plane figures • analyzed patterns in units of measure in the metric system • practiced writing equations to represent formulas for area and perimeter of plane

figures • compared and contrasted attributes of plane and solid geometry figures

How could you address the lack of such experience?

To address a lack of such experience, participants may discuss: • providing remedial exercises for students to strengthen understanding of the

attributes of plane and solid geometric figures • planning hands-on activities for students to investigate how area and surface area

are related • encouraging students to look for and analyze patterns that can be found in the

geometry and measurement of both plane and solid figures (e.g., patterns that can be found between the number of faces of a solid figure and the number of edges and vertices)


Best Practices 45


Possible answers may include the following: Ask participants what instructional goals they may have for their lessons on solid

geometry, in addition to the ones listed on their handouts. Ask what they do to meet those goals.

• Goal: helping students apply the appropriate formula in problem-solving situations; for example, when working on word problems, knowing which formula to apply to solve the problem

• How to Meet Goal: provide hands-on and investigation-based activities for students to build conceptual meaning for the formulas; have students complete exercises in which they do not solve word problems but instead create a pictorial representation of the situation described in the problem

• Goal: helping students recognize when results are reasonable; for example, determining if the results of calculations to find volume or surface area of solid figures are reasonable

• How to Meet Goal: model and reinforce the problem-solving strategy of checking results for reasonableness after finding a solution; have students conduct hands-on and investigation-based activities to determine volume and surface area of solid figures and when results are discussed, emphasize the reasonableness of the results

• Goal: providing opportunities for students to strengthen their spatial visualization skills

• How to Meet Goal: providing multiple opportunities for students to use concrete models of geometric solids; have students create 2-dimensional representations of 3-dimensional objects; have students identify views of solid figures from different perspectives


An Exemplary Lesson 46

An Exemplary Lesson Possible answers may include the following:

Meeting Instructional Goals Provide opportunities for students to explore concepts and build meaning for

formulas. Strategies:

• Brad’s activity used concrete models (i.e., linking cubes) to explore and apply formulas.

• Students used a variety of strategies to find surface area and volume of the solid figures.

• Students found and used short cuts to find volume and surface area and then used variables to generalize to unknown scenarios (e.g., what it would be for the 10th step, what it would be for the xth step).

Provide opportunities for students to look for and analyze patterns. Strategies:

• Brad used linking cubes to create a variety of trios of solid figures that formed patterns in surface area and volume.

• Students recorded data (i.e., surface area and volume) in function tables and used the function tables to help recognize and analyze the patterns.

• Students used the patterns to help generate a rule for the function (i.e., an equation or a formula).

• The use of calculators provided a greater opportunity and ease for students to explore the patterns and to generalize the patterns to equations, instead of being bogged down by cumbersome calculations.

Provide opportunities for students to make connections among geometry

concepts.

Strategies: • Students related area and surface area. • Students related attributes of cubes and prisms to the linking cube figures. Provide opportunities for students to make connections to algebra concepts.

Strategies:

• Students used function tables to display and analyze data. • Students related the change from one figure to the next to that of slope. • Students made connections to the previously studied algebra concept of linear

functions.


An Exemplary Lesson 47

Think and Discuss


Throughout this lesson, students exhibited several behaviors that are desirable in

a mathematics class. They worked together on their tasks, made discoveries, and explained their thinking. What were some strategies or techniques that Brad

Fulton used to encourage this behavior?

• Brad provided an interesting investigation that led to interesting discoveries. The nature of the activity engaged students and motivated them to make discoveries.

• Brad established a collaborative classroom environment. Brad modeled a positive and an enthusiastic attitude toward mathematical investigations, encouraged students to contribute their thoughts and ideas to discussions, and established clear procedures for completing the activity tasks.

• Brad guided students to explain their thinking, through the use of probing questions, and strengthened their observations and conclusions by reinforcing and restating their comments.

Alternate Discussion Topic Possible answers may include the following: Brad Fulton made use of cooperative groups. This instructional strategy requires

careful management before and during a lesson. Have participants discuss the

strategies used by Mr. Fulton to manage the cooperative groups.

Brad Fulton: • had materials ready in bins and station cards prepared to accompany the materials;

instructed one member from each group to get the materials and bring them back to their group

• prepared recording sheets for students to use during the activity • monitored the groups and intervened when students needed clarification or when a

“teachable moment” arose • communicated procedures clearly for the groups


Another Look 48

Another Look


Outline for a Lesson Lesson Topic: How Changes in Dimension Affects Volume of Rectangular Prisms

Instructional Practice

Implementation Notes for My Lesson

Use questioning techniques to help students make discoveries and to reinforce correct answers.

Questioning Techniques: Use probing questions and open-ended questions; for example, present a rectangular solid and ask: What if I changed 1 dimension? 2? All 3? How would that change the volume? What if I changed the dimension by doubling the length of 1 side; would the volume double?

Model activity task and guide students through the process.

Activity to Model, Steps in Process:

• Use interlocking centimeter cubes to build a 3 4 5 rectangular prism. • Guide students through steps to find volume (60 cubic centimeters). • Permit students to share a variety of strategies to find the volume. • Ask students what they think would happen to the volume if you doubled

the length of 1 side (1 dimension) of the prism; for example, made it a 6

4 5 prism, would this double the volume? Would the same results occur

for all rectangular prisms? What if you doubled all 3 dimensions?

Help students see multiple approaches to a problem by having students share their solutions.

Ways for Students to Share Solutions:

• Students may count the cubes individually to find volume. • Students may apply the formula. • Students may find the volume of 1 layer of the prism and then multiply by

the height to find the total volume.

Use manipulatives to help students move from a concrete understanding of a concept to a more abstract understanding.

Concept, Hands-On Activity:

Concept: Patterns in the changes in volume when the dimensions of prisms change Activity: • Students use interlocking centimeter cubes to build and analyze a variety

of rectangular prisms. • Students have recording sheets on which they record the dimensions

and volumes of each prism. • Students use the data they collect to look for and analyze patterns.

Think and Discuss Possible answers may include the following: Describe the preparation tasks for this lesson and strategies you selected. Describe any classroom management strategies that will be involved in this

lesson.

• Gather supplies and materials and prepare student recording sheets. • Determine grouping arrangements and assign students to groups. If necessary,

review rules for group behavior.


Another Look 49

• Prepare classroom for activity. If necessary, move desks together, clear a space for materials.

• Prepare task cards, overhead transparency, or chart that lists the tasks and procedures.

Alternate Discussion Topic Possible answers may include the following: Adelina Alaniz discussed the questioning techniques Brad Fulton used in his

lesson. She discussed the importance of “drawing out” the information rather

than giving students an answer. Have participants discuss questioning techniques that can be used to help students “discover” concepts as they

complete lesson activities.

• Asking questions that encourage students to think about the underlying mathematics and to apply it to generalities or to new situations: How can you use this information to find the volume when 1 side is unknown?

What if we did the same thing with a triangular prism? What about a cylinder?

What if you were told how the volume changed, but not how the dimensions were changed, could you figure out how the dimensions were changed?

• Asking questions that encourage students to look for and analyze patterns. What do you notice about the numbers in your tables? If we added another row of values to the table, what would they be? How do you

know that?


Ranking of Goals 50

Ranking of Goals Possible answers may include the following:

Ranking of Instructional Goals

Instructional Goal

Rank (1 to 5)

<Participant responses

will vary.>

Teach concepts in context and make connections to

other branches of mathematics.

Help students look at problems in more than one way

and help them see that there is more than one way to solve a problem.

Use manipulatives as a tool—available when

appropriate and not available when not appropriate.

Use manipulatives to help middle grades’ students learn advanced topics.

Ensure that students do not simply perform the

mathematics but that they also understand the mathematics.

Think and Discuss


Brad Fulton strives to help his students look at the structure of problems, not just

the answers. Describe some activities that could help students develop this faculty.

Problems that present unknown values and/or that ask students to make generalizations to broader applications are designed to help students look at the structure of a problem more than the answer. Likewise, problems that involve patterns and relationships are also designed to help students look more at the structure of the problem than the answer: • Have students explore the patterns that can be found in the number of edges, faces,

and vertices of solid figures. • Ask students to identify the relationship between the number of faces of a solid figure

and the number of edges and vertices. • Have students create tables to record the information and to look for and analyze

patterns. • Ask students if they could tell you the number of faces of a solid figure if they knew

the number of edges.


Ranking of Goals 51


Possible answers may include the following: Have participants describe some concepts from the Geometry [strand] of solid

figures in which students can benefit from hands-on activities. • analyzing the faces of solid figures and recognizing solid figures from different

perspectives • connecting 2- and 3-dimensional representations of solid figures (nets and solids) • discovering the formula for the volume of prisms and cylinders