California Professional Development for McGraw-Hill ...€¦ · Classroom Management and Differentiated Instruction Whole-Class Differentiation Workshop Activity 12 Whole-Class Differentiation

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California Professional Development for McGraw-Hill Mathematics, Grades K–3

Video Workshop Answer Key

2

Academic Language........................................................................................................3

Overview Workshop Activity ........................................................................................4

The Language of Mathematics Workshop Activity .......................................................5

Meaningful Connections Workshop Activity .................................................................6

Academic Language in Context Workshop Activity ......................................................7

Classroom Management and Differentiated Instruction ...................................................8

Differentiating Instruction Workshop Activity ................................................................9

Classroom Lesson Workshop Activity........................................................................10

Differentiation with Groups Workshop Activity ...........................................................11

Whole-Class Differentiation Workshop Activity ..........................................................12

Classroom Management Workshop Activity ..............................................................13

Managing Space Workshop Activity ..........................................................................14

Managing Process Workshop Activity........................................................................15

Data-Driven Decision Making ........................................................................................16

Overview Workshop Activity ......................................................................................17

First Steps Workshop Activity ....................................................................................18

Next Steps Workshop Activity....................................................................................19

Continuing the Cycle Workshop Activity ....................................................................21

Mathematical Reasoning ...............................................................................................22


Classroom Lesson Workshop Activity........................................................................24

Representation Workshop Activity .............................................................................25

Communication Workshop Activity ............................................................................26

Connections Workshop Activity .................................................................................27

Problem-Solving Workshop Activity ...........................................................................28

Motivation......................................................................................................................29


Concrete Experiences Workshop Activity ..................................................................31

Sharing Workshop Activity.........................................................................................32

Student View Activity .................................................................................................33

Teacher View Activity ................................................................................................34

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Academic Language

Academic Language

Overview Workshop Activity 4

Overview Workshop Activity

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

Mathematics Topic: Fractions

Content-Specific:

• numerator • denominator

• fraction

• unit fraction

• equivalent

Symbolic:

• fraction bar (—) • equals sign (=)

• greater than (>)

• less than (<)

Academic:

• part

• whole • fractional piece

• greater

• less • compare

• equal parts, sets

Discussion

Possible answers may include the following:

• Students have difficulties remembering new terms and using them in appropriate contexts.

• Students may interchange terms; for example, they may not remember which term is

the numerator and which is the denominator. • Possible strategies:

using a word wall

writing definitions or drawing pictures in journals

making connections to familiar references using mnemonic devices

Academic Language

The Language of Mathematics Workshop Activity 5

The Language of Mathematics Workshop Activity

[Participants may find additional terms.]

Content-Specific: • subtraction

• digit

• tens place • ones place

• odd number

• even number

• addition • sum

• addend

• minuend

Academic: • before

• between

• even • strategy

• odd

• even • count

• split apart

• same

• matched pair • answer

Discussion


• Students can confuse mathematical terms with similar terms used in their social

language. They may also use the wrong term to describe a mathematical process or skill, or they may confuse terms from a previous lesson with those in the current

lesson.

• Students generally need at least three exposures to a term in context to begin to assimilate it into their own language. This demonstrated use is one way that teachers

may know that proficiency is achieved.

• Listening to a student use the term properly and seeing its use in journal writing are some informal ways that teachers can assess proficiency.

Academic Language

Meaningful Connections Workshop Activity 6

Meaningful Connections Workshop Activity


Mathematical Term: numerator Content Strategies:

• Using a mnemonic. “N” is for north, which is at the top of most maps, and numerator

is at the top of a fraction. • Making connections to related terms. Numerator has the same root as “number,” and

it is the number of pieces that you have in a fraction.

Mathematical Term: denominator Content Strategies:

• Using a mnemonic. “D” is for down, and the denominator is the lower number in a

fraction, or the one that is down on the bottom. • Making connections to related terms. In money, the denomination of a bill is its

value, and the denominator identifies the value, or size, of a fractional piece.

Mathematical Term: equivalent

Content Strategies:

• Use counter-examples. Have students identify quantities that are not equivalent, such as one-half a pizza is not equivalent to one-fourth of a pizza.

• Make connections to related terms. Equivalent is similar to the word equal, and they

have similar meanings.

Discussion


• Second language learners are already working on language acquisition, so they may

struggle with the content vocabulary and the words used to define this vocabulary.

This adds a level of difficulty to work through that other students may not have. • These are some strategies to help second language learners develop fluency with

academic language:

using a word wall making real-life connections to vocabulary terms

rephrasing answers to model academic language

allowing peer teaching in collaborative group activities

Academic Language

Academic Language in Context Workshop Activity 7

Academic Language in Context Workshop Activity


Your Goal: Have students identify the fractional portion of a circle and order unit fraction pieces from smallest to largest.

Content Terms: • fraction

• numerator

• denominator

• fraction name • number name

Academic Terms:

• part of a whole

• greater than • less than

• larger

• smaller

Activity:

• Using circle fraction transparency pieces, show pieces and have students identify their names.

• Write names using symbols and words.

• Place the pieces in order from smallest to largest.

• Have students write a rule for ordering unit fractions. • Rule: If the numerator is 1, then the greater the denominator, the smaller the fraction

piece.

Discussion


• Academic language is the language of problem solving. It provides the clearest and most precise way to describe problem-solving processes.

• Students need language that explains their thinking and mathematical processes,

and academic language is used to convey these in a way that is universally understandable.

• Students who understand mathematical language are able to read problems to find

key information, and they are able to determine the mathematics needed to solve these problems.

• Academic language provides students with key information that is necessary to both

solve problems and communicate solutions clearly.

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Classroom Management and Differentiated Instruction


Differentiating Instruction Workshop Activity 9

Differentiating Instruction Workshop Activity

[Participant answers may vary.]

My Strategies: [Answers will be strategies that participants use in their classrooms already. If participants can not think of any strategies that they currently use, they may

leave this blank. Any answers that match Vicki Gibson’s strategies will be placed in the

middle of the Venn diagram.]

Vicki Gibson’s Strategies:

• using leveled materials

• using small-group and whole-group instruction • varying practice activities

• allowing student collaboration

• adjusting instruction based on informal assessment • designing instruction based on data

• grouping homogenous/same-ability students for teaching and assessment

• grouping heterogeneous/mixed-ability students for peer teaching

Discussion:

One possible strategy that participants may find difficult is to vary the materials and assignments used for various small groups.

• This can cause difficulty because of the following:

It requires a great deal of time to prepare multiple sets of materials and assignments.

It requires extra time to assess these different work products.

Varied assignments can take different amounts of time, causing some groups to complete their assignments before others.

• These may be addressed by using some of the following strategies and techniques:

Prepare bins of materials for each group with prewritten instructions that may be

reused in subsequent years. Create a rubric for assessing multiple assignments that allows each to be graded

and recorded in a similar way.

Provide an additional journal activity that can be available for groups who finish early. This activity can provide another means of informal assessment to provide

data for further differentiation.


Classroom Lesson Workshop Activity 10

Classroom Lesson Workshop Activity

[Participants may find additional terms.]

Behavioral: • placing student in row and letter groups

• providing clear instructions, assigning table monitors

• dismissing one group at a time, monitoring small groups • establishing a rotation schedule

Environmental:

• using a Math Station Board • using visual and verbal cues for activities

• placing table materials in tubs

• positioning small groups around a central teacher location • using posted vocabulary cards/word wall

• using premade manipulative sets

Instructional:

• providing different assignments for small groups

• having students write about their activities

• giving small-group instruction • using multiple representations

• using questions for individual informal assessment

Discussion


• The three types of strategies are actually interrelated.

Environmental strategies set up an easy-to-use classroom with well-managed

materials. This contributes to fewer behavioral problems during instruction and class

transitions.

Students are always aware of expectations and do not wonder where they need to find materials or instructions.

Greater instructional differentiation is possible because there is less time spent

correcting behavior and moving materials and students from one place to

another.


Differentiation with Groups Workshop Activity 11

Differentiation with Groups Workshop Activity


Group Topic: addition

Main Lesson Concept: adding one-digit numbers with sums to 18

Learning Station 1:

• Students roll a number cube and add one-digit numbers using linking cubes.

Learning Station 2: • Students are given one-digit addition problems on paper that they solve using base-

10 manipulatives and a place-value mat.

Learning Station 3:

• Students respond to a journal prompt that says, “How would you describe what it

means when you add two numbers together? Draw a picture and explain your answer.”

Learning Station 4:

• Students draw arrows on a number line to show addition of one-digit numbers.

Discussion

• Possible answers for small group and transition strategies may include the following:

having assignments on a board

using table monitors or materials managers using an audible signal to identify when it is time to change stations

having expectations and rules clearly marked in the classroom

• Possible answers for materials management may include the following:

keeping manipulatives in marked bins storing materials in a central location

premaking tubs or bags with small-group materials

color-coding materials by group


Whole-Class Differentiation Workshop Activity 12

Whole-Class Differentiation Workshop Activity


Lesson Concept: adding one-digit numbers with sums to 18

Ways to differentiate instruction and materials for the whole class or large group:

• Activate prior knowledge by reviewing counting. Show sets of objects and count them as a group.

• Begin with a journal activity, asking students to explain or draw a picture showing

what it means to “add” something to something else.

• Introduce the concept using concrete materials. Have students use linking cubes at their desks to model sample addition problems.

• Involve kinesthetic learners using a classroom floor number line. Have a student help

model a problem. Start the student at 0 and have him or her move the first addend’s number of spaces. Then, have the student count on and move the second addend’s

number of spaces to find the sum. Have other students follow along using counters

and a number line at their desks. • Place key words on a class word wall, including addition, sum, and addend.

Discussion

One possible answer is that large-group differentiation is more difficult than

small-group differentiation.

Reasons may include the following: • It requires that all students meet together, so activities must engage all students.

• All students need to start at relatively the same point and come with the same

prerequisite skills. • It allows for less individual attention and less informal assessment during the lesson.

• It does not usually allow for students to learn in multiple modalities at the same time.

Only one modality can be the focus at any point during the lesson.

• Less vocal students may not provide feedback during the lesson. • Success of the lesson may not be evident until students have complete independent

practice.


Classroom Management Workshop Activity 13

Classroom Management Workshop Activity

[Individual rankings will vary.]

Low ratings may occur for many reasons, including the following: • There are limited classroom materials.

• Financial limitations do not permit the purchase of extra resources, such as enough

manipulatives for all students. • There is not enough time to meet standards and tailor all lessons for individual

students.

• Multiple levels of learners may be in the same classroom.

• Students may have many different learning modalities, and each lesson cannot cover them all.

• Classrooms may have limited space or may be shared with other teachers or

resource specialists.

Discussion

Many participants will agree with Vicki Gibson’s statement that they spend “more

time on behavioral and environmental management” than on instructional

management.

Reasons may include the following: • Student behavior is a crucial need and must be addressed immediately.

• Disruptive behavior is a hindrance to good instruction.

• It takes a lot of practice to establish good routines. • Students do not begin the school year knowing classroom rules, which then need to

be taught and practiced.

• Students move a great deal throughout the day (e.g., whole-group instruction, small-group instruction, learning centers, and to other locations on the school grounds).

• Classroom layout needs to be constantly changing to allow for good instruction.

• There are many materials and manipulatives to manage, and each needs a storage

space and a way to get the materials to students in an efficient manner.


Managing Space Workshop Activity 14

Managing Space Workshop Activity

[Sample content; participant answers will vary.]

Instructional, Behavioral, and Environmental Goals:

Instructional

• Promote peer teaching and interaction in small groups.

Behavioral

• Students will transition quietly between whole-group and small-group activities.

Environmental

• Materials for small-group work will be easily located and put away.

Classroom Layout:

[Individual answers will reflect the space and furniture available in participants’ classrooms.]

Possible layouts can include the following:

• a centralized location for direct instruction • tables organized in groups so that students’ seats may also be used for small-group

work

• learning stations around the periphery of the classroom • materials bins located in a set of wall compartments

• texts and other resources located near the teacher’s desk

• clearly-posted behavioral rules • a bell or other sound indicating when it is time to direct attention to the teacher or to

rotate stations

Discussion


• Organized space minimizes disruptions during transitions. • The least amount of student moment provides the least commotion.

• Students can be in close proximity to small-group and learning-station areas.

• Managed space allows for informal assessment and monitoring.

• Materials located near where students need to use them minimizes noise and disruptions.

• Students who have their materials are less likely to interrupt other students’ work.

• Students who know where items are do not need to ask questions when getting them out or putting them away.


Managing Process Workshop Activity 15

Managing Process Workshop Activity

[Sample content; individual answers will vary.]

Managing Group Behavior: • Post expectations clearly at the beginning of the year.

• Write out clear instructions for group work.

• Assign jobs to group members so that they are aware of their own responsibilities. • Monitor group work and provide feedback.

Managing Transitions:

• Use an auditory signal to stop work. • Give students warnings 1–2 minutes before the completion of an activity so that they

have time to clean up.

• Do not allow students to leave the group area until they have prepared it for the next group.

• Establish a rotation schedule.

Managing Materials:

• Use labeled bins or bags for each type of manipulative.

• Pre-make tubs or bins for each activity that includes instructions and expectations.

• Have a materials manager in each group who is responsible for getting and replacing materials.

• Make a list of materials used for each activity that you can use again the following

year.

Discussion


• Assign tasks to each group member.

• Require individual work from each student.

• Monitor groups and provide feedback. • Include a written component to group work.

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Data-Driven Decision Making





How would you define data-driven decision making and how does it relate to your performance expectations?

Data-driven decision making is using students’ results from various assessments to plan mathematics instruction. After instruction the data are used to gauge student progress

and to identify whether expectations have been met.

What sources of data are available to you to help you identify students’ past

performance levels in mathematics?

Participant answers may include: • standardized tests

• prior-years’ grades • assessments

• diagnostic tests

What sources of data are available to you to help you identify students’ current

performance levels in mathematics?

Participant answers may include: • chapter tests

• performance assessments • homework assignments

• informal assessment

Discussion

Possible merits of using data to make instructional plans may include the following:

• Decisions can be made based on what students already know. • Instruction can be planned based on what students still need to learn.

• Targeted instruction makes the most efficient use of already limited instructional

time.

A possible goal for using data in instructional planning for the coming year may be the

following:

• We will identify the most critical area in the state standardized test and tailor instruction to meet our student needs within that area.

• We will use data to identify topics of difficulty for many students and use this

information to create instruction that addresses the multiple learning styles of our students.


First Steps Workshop Activity 18

First Steps Workshop Activity

[Sample data; participant strands and standards will vary.]

Kindergarten Number Sense Strength • K NS1.0 Students understand the relationship between numbers and quantities (i.e.,

that a set of objects has the same number of objects in different situations regardless

of its arrangement).

Grade 1 Number Sense Challenge

• 1 NS1.2 Compare and order whole numbers to 100 by using the symbols for less

than, equal to, or greater than (<, =, >).

Grade 2 Number Sense Critical Need

• 2 NS 2.3 Use mental arithmetic to find the sum or difference of 2 two-digit numbers.

Discussion


• Annual data provide a big picture while classroom data can be more targeted to

specific lessons.

• Annual data may provide guidance for the school’s whole grade level. • Classroom data can give specific information about a student’s learning style or

preferences.

• Annual data may be a number or score while classroom data can provide more narrative information.

• Classroom data are more readily available for formative assessment.

Participants may find classroom data more useful because it addresses lessons that

the teacher is already teaching. It shows where needs are and where additional

instruction is necessary. It can provide immediate diagnostic and formative feedback to

allow instantaneous changes to be made to instruction.

Other participants may find annual data more useful because it focuses on standards

that are of state- and district-wide importance. It may also show how students score in relation to other students of the same level. It provides baseline data for where students

are and identifies where they need to be. Because it is used in multiple years, it provides

a good indicator of where students have improved and where there still exists a critical

need.


Next Steps Workshop Activity 19

Next Steps Workshop Activity

[Sample data; participant topics will vary.]

Focus Standard: • 2 NS 2.3 Use mental arithmetic to find the sum or difference of 2 two-digit numbers

Instructional Plan: • Activate prerequisite knowledge.

Play flash card game to reinforce addition and subtraction facts.

Use addition chart to identify easy facts or those with mnemonics (e.g., any

number plus 1 is like counting, any number plus 9 has a 1 in the tens place, and the ones place is 1 less than the number).

Design journal writing prompt to have students explain how to think-through a

two-digit addition problem. • Provide direct instruction.

Use base-10 manipulatives to model ways to group tens and ones when adding

two-digit numbers. Have students work with manipulatives at their desks while another student

models a problem using the overhead projector.

Reinforce strategies for mental math, including solving a related problem, group

ones to make tens, or solving known facts. • Provide small-group work.

Play mental addition game, such as Race to 100, or mental subtraction game,

such as Race to 0. Have students roll four number cubes and create a two-digit addition problem for

another student to solve.

Encourage peer teaching and sharing strategies. Pull out small groups or provide one-on-one reteaching, as necessary.

Assessment:

• Have students write about strategies in mathematics journals. • Use a timed drill where students show answers to problems written on the board on

their own small slates or whiteboards.

• Have students take a timed quiz where answers and the strategy used are provided for each answer.


Next Steps Workshop Activity 20

Discussion


• It provides an indicator where additional lesson plans and activities will be needed.

• Whole-class challenges and critical needs can direct instructional goals that were not

fully met. • Assessment results show which state standards need additional focus in the

classroom.

• Critical needs may indicate where classroom assignments and assessment did not align with what was on state tests.

• Topics covered in previous-years’ assessment can be assumed to be on this year’s

assessment. • More classroom instruction, small-group activities, and learning centers can be

devoted to the standards that were a critical need in previous years.


Continuing the Cycle Workshop Activity 21

Continuing the Cycle Workshop Activity


Identify the Mathematical Concepts: • place value

• subtraction of two- and three-digit numbers

• borrowing, in subtraction • reasonableness of answers

Identify the Likely Reasons for Errors:

A. The student subtracted the smaller number from the larger number, even when this meant switching the subtrahend and the minuend.

B. The student did not bring down the 0 placeholder for the tens place in the difference,

so the answer included the 1 in the tens place, not the hundreds place. C. The student made the correct answer.

D. The student either did not borrow correctly from the tens place or did not subtract

13 – 7 correctly.

State the Standard(s):

• 2 NS 2.0 Students estimate, calculate, and solve problems involving addition and

subtraction of two- and three-digit numbers. • 2 NS 2.2 Find the sum or difference of two whole numbers up to three-digits long.

Discussion


• Participants may feel that they can plan effective instruction because they have many sources of data available to them, including state tests, previous-years’

assessments, and classroom assessment.

• Participants may feel that they can not plan effective instruction because test data

provide a number and not specific feedback on student errors or because they do not yet have enough classroom data on individual students.

• Additional potential data sources may include state assessments, progress reports,

resource specialist reports, informal assessments, portfolios, or diagnostic tests.

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Mathematical Reasoning




[Sample content, topics, and answers will vary based on group choices.]

Mathematics Topic or Strand: Grade 2 Number Sense—recognize and name fractions

Representation: • fraction circle manipulatives

• fractional portions of sets, such as the number of linking cubes in a group that are

the same color

• fractional parts on a number line • numerical representations

Communication: • Have students describe fractional parts of a whole in journal entries or small groups.

• Have students name fractions using words.

Connections:

• Identify real-world contexts that use fractions, including time (1 half-hour).

• Have students explain fractional parts of a whole, such as a pizza or a fractional part

of a dozen cookies.

Problem Solving:

• Describe a fractional quantity and have students show that fraction with manipulatives.

• Have students determine the fraction of a pie each student will receive if a group of

them shares it equally.

Discussion

• Possible answers may include the following: Participants may feel that communication is a difficult aspect of mathematical

reasoning.

• Reasons may include the following: Mathematics and language arts are generally seen as separate.

Students may learn algorithms without being able to explain the mathematics

behind the processes.

English language learners struggle with language skills necessary to communicate mathematical concepts.

Mathematical vocabulary may be difficult for many students.

• Ways to overcome these difficulties may include the following: Encourage students to communicate using academic language in small groups.

Ask students to write answers in mathematics journals.

Have students keep vocabulary lists and post a list on a word wall or similar classroom display.

Model using academic language and writing in mathematics.


Classroom Lesson Workshop Activity 24

Classroom Lesson Workshop Activity


Representation: • using a hundreds chart

• counting linking cubes

• counting on a number line • writing explanations in mathematics journals

• using symbols to write mathematical problems

Communication: • emphasizing key vocabulary terms

• modeling academic language

• having students explain their thinking • asking students to work in peer groups

Connections: • explaining the ways that science and mathematics are related

• discussing mathematics in real-world activities

Problem Solving: • asking students to explain how they solved problems

• having students solve problems in multiple ways

• finding key information in a word problem

Discussion


• Being able to repeat what is taught (such as a definition) shows that students have

committed it to memory.

• Increased mathematical reasoning becomes evident when students understand and apply the skill or concept to their own problem solving.

• Higher levels of mathematical reasoning are shown as students analyze and

evaluate their work and determine errors or show multiple ways to solve problems. • Students at the highest levels of mathematical reasoning are able to apply what they

have learned to similar situations or create new ways to use this information or skill.


Representation Workshop Activity 25

Representation Workshop Activity


Lesson Topic 1: Identify unit fractions. Representations:

• Concrete

Fold a piece of paper in half.

Show

1

4 of a set of 4 linking cubes.

• Pictorial

Identify where

1

2 falls on a number line.

Draw a circle and shade

1

4 of it.

• Symbolic

Write unit fractions.

Name unit fractions written in symbolic form.

Lesson Topic 2: Recognize fractional parts of a whole or group.

Representations: • Concrete

Create a set of 2-color counters that is

3

4 red.

Make 1-whole fraction strip using various-sized pieces (i.e.,

1

2 and

2

4).

• Pictorial

Draw a picture to solve a word problem involving fractional parts.

Show a pie that is

3

4 eaten and ask students to identify the missing part.

• Symbolic

Write fractions.

Name fractions written in symbolic form.

Discussion

Possible answers may include the following: • Student gives correct representation.

Encourage the student to explain the representation.

Ask the student to find connections to known representations. Discuss the benefits of alternate representations.

Reinforce that the answer is correct and can be derived in several ways.

• Student gives incorrect representation.

Affirm the student for trying. Provide corrective feedback.

Ask leading questions to see if the student can find his or her own error.

Have the student attempt to find a correct way to solve the problem that makes sense to him or her.


Communication Workshop Activity 26

Communication Workshop Activity


Lesson Topic 1: Identify unit fractions.

Leading Questions—Questions that lead students to conceptual understanding:

• What is a fraction? • How many fractional parts of the same size make a whole?

• What does the bottom number in a fraction tell you?

• What does the top number in a fraction tell you?

Evaluative Questions—Questions that check for student understanding:

• How did you determine the name of the fraction?

• Which fraction is greater?

• Which picture shows a shaded area of

1

4?

• What is a numerator?

Question Stems—Question starters that help students check for understanding

with each other:

• How can you make . . . ?

• Which fraction is greater, . . . or . . . ? • What is the definition of . . . ?

Discussion


• Encourage and model the use of mathematical vocabulary. • Monitor groups and provide immediate feedback.

• Create heterogeneous groups with students of various ability levels to provide peer

tutoring.

• Provide question stems to encourage mathematical communication. • Require groups to produce a written product for assessment.


Connections Workshop Activity 27

Connections Workshop Activity


Curriculum Standard:

• 2 NS 4.1 Recognize, name, and compare unit fractions from

1

12 to

1

2.

Prerequisite Skills: • Count, read, and write whole numbers.

• Compare and order whole numbers.

• Compare and order areas. • Count objects in a group.

• Separate objects and areas into equal parts.

Cross-curricular and Real-Life Connections: • Share an apple by dividing it into equal parts.

• Slice a pizza or pie.

• Identify money, such as a quarter or half-dollar. • Tell time, such as a quarter hour or half hour.

• Make literature connection: The Doorbell Rang by Pat Hutchins.

Discussion


• Do a warm-up activity that focuses on prerequisite skills. • Model problems step-by-step during direct instruction to emphasize skills necessary

to find the solution.

• Pair a struggling student with one who understands the concept to take advantage of peer teaching.

• Make adjustments to lesson pacing based on student feedback.

• Use small-group work and learning centers to engage other students while you pull

out a small group for reteaching. • Provide informal and formal assessment that includes problems focusing on

prerequisite skills to identify which need additional instructional time.


Problem-Solving Workshop Activity 28

Problem-Solving Workshop Activity


Story Problem: Mr. Taylor brought a pizza for his class to share. There are 12 students in the class.

What fraction of the pizza does each student get?

Problem-Solving Steps:

• Direct students to explain how pizza is shared: that it is cut into fractional slices

(connections).

• Have students read the problem and identify the key information and key words in the text (communication).

• Ask students to build the pizza using twelfths fraction pieces (representation).

• Have students explain how much pizza each student receives (communication). • Ask a volunteer to explain his or her answer to the class, using overhead fraction

pieces (problem solving, communication).

• Have a second volunteer write the solution using symbols (representation).

Discussion

Participants should see that the areas of mathematical reasoning are interconnected.


• All mathematical reasoning is conveyed to others by being spoken or written, which ties them all to communication.

• Representations are a form of communication, and they can be used as part of the

problem-solving process. • Using multiple representations often ties one content area to another and helps

relate problems to real-world contexts, providing connections.

• Problem solving requires an understanding of mathematical vocabulary, which is a

part of good communication.

29

Motivation

Motivation



[Individual rankings and participant responses will vary.]

Strategy that I use least often: using small-group work

Reasons that I do not use the strategy:

I have students at various levels, and preparing activities to engage all students in small groups effectively is time consuming.

Ways to incorporate the strategy:

• Provide learning centers with small-group activities for students on a rotation schedule.

• Use activities and assignments that are normally for whole groups in small-group

settings. • Allow students to work in pairs on some assignments.

• Make small-group activities a regular part of lesson planning.

Discussion


• Measurement activities have application in cooking. • Fraction skills have application in equal sharing of toys, food, or other common

items.

• Subtraction skills have application in determining how many pages are left to read in a book.

• Time and calendar skills have application during the school day or determining how

long an activity lasts. • Solid geometric shapes are found in common objects such as ice cubes, balls, and

cans.

Motivation

Concrete Experiences Workshop Activity 31

Concrete Experiences Workshop Activity


Mathematics Topic 1: counting groups of coins

Manipulatives: real or play money

Conceptual and Motivational Purposes:

Coins are engaging to students who enjoy pretend play and like to feel that the

mathematics that they learn is the same that grown-ups use.

Mathematics Topic 2: adding one-digit numbers

Manipulatives: teddy bear counters

Conceptual and Motivational Purposes: Counters help students solidify one-to-one correspondence between the objects and

numbers. Bright colors and fun shapes engage students who like the toy quality of this

manipulative.

Mathematics Topic 3: identifying fractional quantities

Manipulatives: fraction circles

Conceptual and Motivational Purposes:

• Kinesthetic and visual learners are both engaged by this manipulative because it provides a concrete representation of fractions and helps students see fractions’

relative sizes.

• Fraction circles also help students make real-life connections, such as fractional parts of a pie or pizza.

Discussion


• Base-10 manipulatives are used to play games such as Race to 100. They can also

be used to model early experiences in borrowing for subtraction problems. • Linking cubes are used to play games such as Number Trains. They can also be

used for nonstandard measurement, such as measuring a book that is 4 cubes long.

• Attribute blocks are used to make patterns or to play a Pattern Description game.

They can also be used to represent fractional quantities, such as a trapezoid is

1

2 of

a hexagon.

Motivation

Sharing Workshop Activity 32

Sharing Workshop Activity


Small-Group Collaboration Topic or Assignment: Students work together to solve two-digit subtraction word problems and present

answers to the class.

Drawings or Artwork Topic or Assignment:

Students use pictures to represent plane geometry vocabulary words in their

mathematics journals.

Classroom Presentation Topic or Assignment:

Students create a poster board display showing several different forms of AABB

patterns, including those with shapes, colors, numbers, and manipulatives.

Discussion


• Assign jobs to each student in the group, so they are all engaged during group work.

• Have all students turn in individual written or drawn work related to the group

assignment. • Ask students who are not speaking during the presentation to be note-takers or

script-writers for the group.

• Direct questions to nonpresenting group members after the presentation. • Allow English language learners to draw pictures or use manipulatives as part of

their presentations.

• Post vocabulary terms in a prominent place in the classroom, so all students, including English learners, can see them during the presentation.

Motivation

Student View Activity 33

Student View Activity


Mathematics Topic: Unit fractions

[Insert into Venn diagram. Sample content, topics, and answers will vary based on group

choices.]

Activities that interest students:

• discussing television characters

• going to movies • acquiring collections (e.g., trading cards, stickers, stuffed toys)

Content and student interest: • eating favorite foods (e.g., dividing items to share with a group)

• playing sports (e.g., games that are divided into halves)

• purchasing toys at the store (i.e., using money, such as half dollars or quarters) • playing video games (i.e., playing for a half hour at a time)

Activities that support content:

• modeling with manipulatives (e.g., fraction strips or fraction circles) • creating linear models

• writing fractions with symbols

• focusing on vocabulary terms

Discussion


• Subscribe to popular culture magazines.

• Have students bring in favorite items for sharing.

• Spend time in stores and restaurants that children enjoy. • Watch movies or television shows that children enjoy.

Motivation

Teacher View Activity 34

Teacher View Activity


Brad Fulton “Ah-ha” Moment: • Linear equations are made fun by creating a connection to the x- and y-axes

between eXperts and Yo-yos.

• Mr. Fulton acts as though he is discovering the linear pattern along with students. • Mr. Fulton points out that students are able to predict the pattern better than they

were at the beginning of the lesson. (What seemed difficult has become do-able.)

• Students see that they can predict using the graph when Mr. Fulton stands in front of it.

• Students enjoy discovering that they can now solve the equation for a larger number, such as 20.

Juanita Walker “Ah-ha” Moment: • Students are excited by the cups and their precariousness—the potential to fall.

• A student notes that the cups look like stairs.

• Students shout out the pattern as they realize how many cups to add each time. • Students realize, as Ms. Walker holds the stick, that the angle is steeper when 2

cups are added at a time.

Discussion

Mathematical content knowledge will vary, but may include the following:

• Count, read, and write whole numbers. • Compare sets of objects (i.e., greater than, less than, equal to).

• Use objects to model addition and subtraction problems.

• Recognize when an answer is reasonable. • Sort objects by one attribute.

Motivational activities will vary, but may include the following:

• Use games to review prerequisite skills. • Prepare warm-up activities to activate prior knowledge.

• Have students work on initial assignments in pairs to take advantage of peer

teaching. • Set up learning centers to provide high-success activities using previous-years’ skills.

Documents

California Professional Development for McGraw-Hill ...€¦ · Classroom Management and Differentiated Instruction Whole-Class Differentiation Workshop Activity 12 Whole-Class Differentiation