texas professional development for mcgraw-hill mathematics ...€¦ · classroom management and...

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Texas 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 ....................................................................20

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

• equal sets

Discussion

Possible answers may include the following: • Students have difficulty 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

Possible answers may include the following: • 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 or seeing its use in journal writing are

some informal ways 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

Possible answers may include the following:

• 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.

• Strategies to help second language learners develop fluency with academic language include:

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

• compare

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. (If the numerator is 1,

then, the greater the denominator, the smaller the fraction piece.)

Discussion

Possible answers may include the following: • 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 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 pre-written 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 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 die 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 2 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: • identifying assignments on a board using pictures or words

• 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

• pre-making 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 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 him or her 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 completed 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 can not 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 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

Possible answers may include the following: • Organized space minimizes disruptions during transitions.

• The least amount of 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 the students need to use them will minimize noise and

disruptions.

• Students 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 include 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 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 the following:

• standardized tests • prior-years’ grades and 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 the following:

• 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.

Possible goals 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, operation, and quantitative reasoning Strength • TEKS K.1C The student is expected to use numbers to describe how many objects are

in a set (through 20) using verbal and symbolic descriptions.

Grade 1 Number, operation, and quantitative reasoning Challenge

• TEKS 1.1A The student is expected to compare and order whole numbers up to 99

(less than, greater than, or equal to) using sets of concrete objects and pictorial

models.

Grade 2 Number, operation, and quantitative reasoning Critical Need

• TEKS 2.3A The student is expected to recall and apply basic addition and subtraction facts (to 18).

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 of the following reasons:

• 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 of the following reasons:

• 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. • It provides a good indicator of where students have improved and where there still

exists a critical need because it is used in multiple years.


Next Steps Workshop Activity 19

Next Steps Workshop Activity

[Sample data; participant patterns and standards will vary with data.]

Identify any pattern or trend of weakness in your student data: Students tend to have trouble with mental math and memorized facts.

Choose a representative area of weakness and write one associated standard: • Area of Weakness:

memorizing addition facts to 18

memorizing subtraction facts

• Associated Standard: TEKS 2.3A The student is expected to recall and apply basic addition and

subtraction facts (to 18).

Make a detailed list of the concepts students must understand the skills they must

demonstrate to be proficient in this standard:

• Represent and name whole numbers. • Understand the meaning of addition.

• Understand the meaning of subtraction.

• Name fact families and describe their relationships.

• Memorize one-digit addition facts to 18. • Memorize related subtraction facts.

Discussion

Possible answers for remediation may include the following:

• Create learning centers to provide additional practice activities. • Use small groups and learning pairs to allow for peer-teaching opportunities.

• Have small groups work on reteaching exercises while other students complete

projects or center work.

• Use teacher’s aides or parent helpers to work with individual students.

Possible answers for using patterns and trends of data to balance teaching time may

include the following: • Identify where students have had difficulty in the past and focus more teaching time on

those standards.

• Note prerequisite skills that need additional instruction before teaching new skills.

• Recognize strands that show multiple critical needs and spend extra teaching time on standards within those strands.


Continuing the Cycle Workshop Activity 20

Continuing the Cycle Workshop Activity


Standard Addressed: TEKS 2.3A The student is expected to recall and apply basic addition and subtraction

facts (to 18).

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 solve a related problem, group ones to make tens, or solve 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.


Continuing the Cycle Workshop Activity 21

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.

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Mathematical Reasoning




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

Mathematics Topic or Strand: Number, operation, and quantitative reasoning TEKS 2.2A The student is expected to

use concrete models to represent and name fractional parts of a whole object (with

denominators of 12 or less).

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


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 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 using 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.

• 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: • 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.

• 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—Questions 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: Number, operation, and quantitative reasoning TEKS 2.2A The student is expected to

use concrete models to represent and name fractional parts of a whole object (with

denominators of 12 or less).

Prerequisite Skills:

• Count, read, and write whole numbers.

• Compare and order whole numbers. • Compare and order areas by size.

• Separate areas into equal parts.

• Separate the same size area into differently sized parts (e.g., a square into 2 equal parts or 4 equal parts).

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

• Slice a pizza or pie.

• Identify money examples, such as a quarter or half-dollar.

• Tell time examples, 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. Possible answers may include the following:

• 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


• 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


• 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 language 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 (e.g., using money, such as half dollars) or quarters • playing video games (e.g., 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 only 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 should 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.

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