Emily&Herbst& 1&

*&I&think&it&important&to&note&that&my&task&was&to&teach&students&how&to&identify&

mistakes&within&problems.&&These&students&are&very&behind&in&their&mathematics&

skills,&and&as&a&result,&I&believe&the&lessons&are&below&what&is&considered&high&school&

standards.&&The&closest&standard&I&could&find&that&was&related&to&this&material&is&&

CCSS.MATH.CONTENT.6.NSA.1-Apply-and-extend-previous-understandings-of-

multiplication-and-division-to-divide-fractions-by-fractions.-

Interpret&and&compute&quotients&of&fractions,&and&solve&word&problems&involving&

division&of&fractions&by&fractions.&

&

Part&1:&Planning&for&Instruction&and&Assessment&

& This&lesson&was&given&to&an&English&Language&Learners&class&at&Waterloo&

West&containing&ninth,&tenth,&and&eleventh&graders.&&The&class&contains&ten&students&

total.&&A&class&of&this&size&is&not&typical&at&Waterloo&West&but&is&appropriate&to&give&

these&students&more&oneGonGone&attention.&&&

The&lesson&involves&students&looking&at&problems&and&identifying&mistakes.&&

Up&until&this&point,&the&students&seem&to&have&had&a&difficult&time&grasping&

mathematical&content.&&The&students&have&been&learning&at&an&extremely&slow&rate&

such&that&it&is&typical&to&take&a&lesson&that&would&be&covered&in&one&day&and&

extending&it&so&it&covers&two&to&three&days.&&&

In&the&classes&leading&up&to&this&lesson,&students&have&been&learning&about&

fractions.&&More&specifically,&the&students&have&been&learning&how&to&multiply&

fractions,&how&to&divide&fractions,&and&the&relationships&between&whole&numbers&

and&fractions.&&The&goals&of&this&lesson&include:&students&will&understand&how&to&set&

Emily&Herbst& 2&

up&equations&for&word&problems&and&students&will&be&able&to&identify&mistakes&in&the&

work&of&others,&correct&it,&and&explain&why&the&equation&was&wrong&to&begin&with.&

To&launch&the&lesson,&students&will&be&given&the&opportunity&to&solve&two&

problems&that&are&similar&to&problems&they&have&seen&in&previous&lessons.&&These&

problems&act&as&a&preview&to&the&activity&for&the&day.&&Below&are&the&two&examples&of&

launch&problems&used&along&with&questions&that&might&be&asked:&

Example&1:&There%are%12%students%in%my%class,%and%my%teacher%bought%3%pizzas%(24%

slices%total).%%How%many%slices%does%each%person%get?%

Questions&for&example&1:&What%do%we%want%to%know?%%Why?%%How%do%we%want%to%set%up%

the%problem%to%assure%that%we%are%solving%for%this?%

Example&2:&If%I%want%to%make%3%pies,%and%I%need%2.5%apples%per%pie,%how%many%apples%do%

I%need%total?%

Questions&for&example&2:&What%do%we%want%to%know?%%Why?%%How%do%we%want%to%set%up%

the%problem%to%assure%that%we%are%solving%for%this?%%What%would%our%equation%be%for%

two%pies?%%What%would%our%equation%be%for%one%pie?%%What%would%our%equation%be%for%

half%a%pie?%

These&two&examples&are&of&relevance&to&students&because&these&problems&or&similar&

problems&can&arise&in&the&future.&&Again,&these&examples&are&also&reflective&of&what&

they&have&recently&learned.&&My&questions&encourage&students&to&deeply&think&about&

how&to&set&up&equations&and&understand&what&each&term&in&the&equation&means.&&

These&are&focal&points&that&will&be&revisited&throughout&the&lesson.&

To&help&students&achieve&the&goals&previously&stated,&students&will&be&given&a&

worksheet&(see&Appendix&A)&that&contains&six&different&word&problems&with&

Emily&Herbst& 3&

solutions.&&Three&of&the&solutions&are&correct,&and&three&of&the&solutions&are&wrong.&&

Students&will&be&given&class&time&to&identify&the&three&solutions&that&have&errors&and&

fix&them.&&To&help&students&understand&what&was&expected&of&them,&we&will&walk&

through&the&first&two&problems&together.&

The&first&problem:&

a. A&car&travels&120&miles&in&3.6&hours.&

What&is&its&average&speed&in&miles&per&hours?&

120÷ 3.6&

To&help&students,&I&would&ask&them&to&just&focus&on&the&question.&&Questions&I&might&

ask&include:&What%are%we%supposed%to%find?%%How%do%we%know%this?%%Does%anyone%have%

any%ideas%as%to%how%to%set%up%this%problem?&&Students&will&then&work&with&students&at&

their&table&to&create&an&equation&that&matches&what&is&asked.&&Students&should&be&

able&to&identify&that&we&are&creating&an&equation&that&reflects&miles&per&hour.&&From&

here,&we&will&use&information&in&the&example&to&create&an&equation&that&pairs&the&

correct&number&with&the&correct&term.&&Students&will&then&look&at&the&solution&that&

was&paired&with&the&problem&(120÷ 3.6)&and&identify&if&the&equation&they&created&

matches&this&equation.&&For&this&problem,&the&equations&will&be&equal.&&&

& The&second&problem:&

a. A&snail&travels&0.8&miles&in&40&hours.&

What&is&its&average&speed&in&miles&per&hour?&

40÷ 0.8&

To&help&students,&I&might&ask&questions&such&as:&What%does%per%mean?%So%if%we%have%

miles%per%hour,%how%would%we%set%up%the%problem?%How%does%Angela%have%the%problem%

set%up?%What%is%Angela%really%solving%for?%%So%what%is%a%similar%problem%that%we%can%

come%up%with%that%uses%these%numbers?%%Students&will&work&with&each&other&to&

Emily&Herbst& 4&

identify&the&equation&that&matches&the&question&being&asked.&&Students&will&then&

compare&this&equation&with&the&equation&Angela&had.&&Since&the&equations&do&not&

match,&students&will&identify&the&wrong&calculation,&correct&the&calculation,&write&the&

correct&answer&with&units,&create&a&new&problem&that&uses&these&numbers,&and&solve&

the&new&problem.&

& Students&will&then&be&given&the&rest&of&the&class&period&to&analyze&the&four&

other&problems&on&the&worksheet.&&I&will&be&floating&around&the&room,&observing&

student&progress,&and&aiding&students&as&needed.&&Students&will&then&go&over&the&rest&

of&the&problems&the&following&day&in&class.&

&

Part&2:&Instructing&and&Engaging&Students&

& Overall,&I&think&the&delivering&of&my&lesson&went&over&very&well.&&Students&we&

participating,&we&were&able&to&make&the&problems&fun,&and&students&had&the&

opportunity&to&work&with&a&partner&of&their&choice.&

& My&introduction&involved&students&solving&two&questions&that&required&the&

use&of&information&that&they&had&learned&throughout&the&week.&&I&provided&students&

with&a&sufficient&amount&of&time&to&think&about&and&solve&each&problem.&&For&these&

problems,&students&needed&to&provide&the&correct&answer&as&well&as&explain&to&the&

other&classmates&why&the&answer&was&right.&&For&the&first&example,&multiple&students&

participated&and&offered&up&different&ideas&as&to&how&to&solve&the&problem.&&For&the&

second&problem,&students&were&struggling.&&Although&the&students&had&seen&similar&

problems&throughout&the&week,&the&students&were&struggling&to&identify&the&amount&

of&apples&per&pie&as&well&as&how&to&put&this&information&into&an&equation.&&To&help&

Emily&Herbst& 5&

students&identify&the&relationship&between&the&numbers,&I&asked&them&to&solve&for&

2.5&pies,&1&pie,&3&pies,&2&pies,&and&.5&of&a&pie.&&&

Although&I&believe&that&walked&through&these&different&examples&helped,&I&

wish&I&had&drawn&visual&images&on&the&board.&&So&for&the&pizza&example&I&could&draw&

the&pizza&slices&and&the&people,&and&for&the&apple&pie&example&I&could&draw&the&

apples&and&piecrust.&&I&think&it&would&have&been&especially&helpful&to&draw&the&

images&in&this&class&since&all&of&the&students&are&English&Language&Learners.&&They&

are&still&struggling&with&English,&so&the&pictures&could&act&as&a&bridge&that&provides&

them&the&opportunity&to&understand&and&succeed.&

After&going&through&the&introductory&examples,&I&asked&the&question,&“Are&

there&any&questions?”&&This&question&is&often&seen&as&a&filler&and&not&so&encouraging&

of&students&to&ask&questions.&&In&the&future,&I&hope&to&replace&this&will&“What&

questions&do&you&have?”&&This&helps&encourage&students&to&generate&a&question&

about&the&material&as&well&as&elicits&student&thinking.&&In&different&situations,&I&could&

also&ask&another&question&that&would&allow&me&to&check&student&understanding.&&For&

example,&I&could&as&“How&about&if&we&have&5&pies.&&Then&how&many&apples&do&we&

need?”&&This&provides&me&with&the&opportunity&to&see&that&students&understand&the&

content&without&asking&“Are&there&any&questions?”&

For&the&activity,&I&read&aloud&the&directions&myself.&&This&allowed&me&the&

opportunity&to&emphasis&key&points&that&were&considered&important&within&the&

assignment.&&Because&I&was&doing&so&much&talking,&I&feel&that&I&started&to&lose&student&

attention.&&In&the&future,&it&might&be&better&that&I&ask&students&to&read&the&directions&

aloud.&&This&participation&can&keep&students&focused&and&on&task.&

Emily&Herbst& 6&

I&think&going&through&two&of&the&problems&was&good.&&It&provided&students&

with&the&chance&to&see&what&was&expected&of&them&for&the&rest&of&the&assignment.&&

For&the&first&problem,&I&asked&students&a&series&of&questions&and&recorded&their&

results&on&the&board.&&During&the&second&problem&that&we&went&through&as&a&class,&I&

gave&students&time&to&think&about&possible&solutions&and&if&Angela’s&answer&was&

right&or&not.&&The&class&agreed&that&Angela&had&the&wrong&answer,&so&I&asked&a&

student&to&come&put&their&answer&on&the&board.&&I&asked&students&what&they&thought&

of&the&new&answer,&and&there&was&some&disagreement.&&The&answer&that&was&put&on&

the&board&was&wrong.&&As&a&class&we&discussed&what&this&answer&was&actually&saying,&

and&another&student&went&to&the&board&to&provide&the&correct&answer.&&&

In&my&own&classroom,&I&want&students&to&understand&that&having&mistakes&is&

ok.&&What&is&not&ok&though&is&not&trying.&&It&was&clear&that&the&student&that&presented&

the&wrong&answer&clearly&thought&about&the&problem.&&Although&this&answer&was&

wrong,&it&provided&me&with&an&opportunity&to&address&this&solution&strategy&to&the&

class.&&I&was&able&to&dig&deeper&into&the&material&to&help&students&truly&understand&

why&the&wrong&answer&was&wrong&and&why&the&right&answer&was&right.&

& Now&that&we&had&identified&that&Angela’s&solution&was&wrong&and&identified&

the&correct&equation,&it&was&time&to&come&up&with&a&new&problem&and&answer.&&

During&this&portion&of&the&lesson,&I&asked&students&to&help&create&different&ideas&for&

what&the&problem&could&be&about.&&Students&decided&that&the&problem&should&deal&

with&students&looking&at&sale&signs&in&the&mall.&&Although&I&was&thrilled&that&I&was&

gaining&student&attention&by&asking&for&their&continued&participation,&I&feel&that&I&

could&have&conducted&this&part&a&bit&better.&&I&seemed&to&stumble&through&this&part&in&

Emily&Herbst& 7&

which&I&was&unsure&of&how&to&set&up&this&problem.&&My&mentor&teacher&came&to&the&

rescue&and&we&team&taught&this&portion.&&&

& Through&the&rest&of&the&class&period,&students&were&given&time&to&work&on&the&

rest&of&the&worksheet.&&I&was&a&floater&around&the&room,&so&I&was&given&the&chance&to&

see&each&students’&progress.&&I&aided&students&when&they&were&stuck&without&leading&

them&to&the&answer.&&Students&then&had&the&opportunity&to&go&through&the&

assignment&with&my&mentor&teacher&the&following&day&of&class.&

&

Part&3:&Assessment&of&Student&Learning&

& Students&were&still&working&on&the&assignment&at&the&end&of&class,&so&it&then&

became&homework.&&Although&I&did&not&have&any&physical&copies&of&student&work&

that&allowed&me&to&see&student&learning,&I&was&able&to&informally&assess&student&

learning&throughout&the&class.&

& Through&the&introductory&examples&and&the&assignment,&I&was&able&to&walk&

around&the&classroom&and&monitor&student&learning.&&By&walking&around&the&room&I&

could&look&at&student&work,&ask&students&to&clarify&their&markings,&and&encourage&

students&to&explain&to&me&their&answers.&

& During&class,&I&was&also&able&to&identify&understanding&through&student&

participation&and&nonGverbal&feedback.&&This&helped&direct&my&teaching&so&I&would&

either&stay&on&a&topic&in&which&students&needed&help&or&move&on&to&other&material.&

PBIT Lesson Planning Guidelines with Instructional Formative Assessment

Title: Looking at Problems

Grade: 9th, 10th, 11th

Iowa Core Characteristics of Effective Instruction

Iowa Core Standards for Mathematical Practices

Iowa Core Standards for Mathematical Content: Domain/Cluster/Standards

Prerequisite Knowledge Understand what it means to divide Understand what it means to multiply

Learning Goals

Understand: 1. how to set up equations to word

problems 2. how to solve word problems

Success Criteria I can: 1. find errors in other people’s

work. 2. set up equations to word

problems

Evidence of Mathematical Practices linked to Success Criteria

(e.g. Problem Solving – Success Criteria 1)

Focus Question !

POSSIBLE MISCONCEPTIONS, ERRORS, OR POTENTIAL TROUBLE SPOTS

Possible Student Misconceptions, Errors, or

Potential Trouble Spots

Teacher Questions and Actions to Resolve Misconceptions, Errors, or

Trouble Spots:

1.

SUCCESS CRITERIA: INSTRUCTIONAL FORMATIVE ASSESSMENT STRATEGIES

Success Criteria Questions and/or

Strategies

Possible student responses or actions

Follow-up Questions/ Actions

· · 1.

· 1.

· 1.

· 1.

· 1.

LESSON SEQUENCE (Include plans for adjustments to accommodate all learners)

Launch

Activities Students will solve two problems that will act as a review of what they should know and be a preview to the activity for the day.

Notes There are 12 students in my class, and my teacher bought 3 pizzas (24 slices total). How many slices does each person get? What do we want to know? Wanted to know slices per person.

24 ÷ 12 If I want to make 3 apple pies, and I need 2.5 apples per pie, how many apples do I need total? What do we want to know? Wanted to know apples per pie. Here is our equation for three pies:

2.5!×!3 What would our equation be for

Materials Promethean Board

two pies? 2.5!×!2!

What would our equation be for one pie?

2.5!×!1 What would our equation be for ½ a pie?

2.5!× 12 =2.52 =

54 = 1.25

Explore

Activities Students will have the class time to identify the three problems with errors and fix them. As a class we will do a and b together.

Notes Talking points for:

a. “Right now just focus on the question. What are we supposed to find? Does anyone have any ideas as to how to set up this problem?”

b. “What does per mean? So if we have miles per gallon, how would we set up the problem? How does Angela have the problem set up? Now that we have fixed the problem, create a new problem to give Angela more practice working with similar numbers in a similar situation. So what is a similar problem that we can come up with that uses these numbers?”

Materials Promethean Board

Calculators

Summarize

Activities Students will finish the activity as homework if they do not complete

Notes

Materials Promethean Board

it in class.

Looking'at'Problems! ! ! ! ! Name:____________________!!

1. Angela!has!written!calculations!to!show!how!she!would!solve!each!of!the!six!problems!below.!!Your!task!is!to!check!her!work!because!she!has!made!some!errors!in!three!of!the!problems.!!For!each!of!the!problems!below:!

• Identify!the!problem!and!wrong!calculation.!• Think!about!why!Angela!might!have!made!the!mistake.!• Provide!the!correct!calculation!and!correct!answer!with!units.!• Create!a!new!problem!to!give!Angela!more!practice!working!with!

similar!numbers!in!a!similar!situation.!• Provide!a!correct!answer!to!the!problem!you!created.!

!a. A!car!travels!120!miles!in!3.6!hours.!

What!is!its!average!speed!in!miles!per!hours?!120÷ 3.6!

b. A!snail!travels!0.8!miles!in!40!hours.!What!is!its!average!speed!in!miles!per!hour?!

40÷ 0.8!

c. Mrs.!Abir!buys!some!apples!at!$1.50!per!pound.!!!She!spends!$3.50.!!How!many!pounds!does!she!buy?!

3.50÷ 1.50!

d. Mario!buys!some!tomatoes!at!$0.90!per!pound.!!He!Spends!30!cents.!!How!many!pounds!does!he!buy?!

90!÷ 30!

e. Lena’s!motorcycle!gets!62.5!miles!per!gallon.!!!How!far!can!she!go!on!3.4!gallons?!

62.5÷ 3.4!

f. Mr.!Okawa’s!car!gets!20!miles!per!gallon.!He!only!has!0.4!gallons!left!in!the!tank.!How!far!will!he!travel!before!he!runs!out!of!gas?!

20÷ 0.4!

!Angela’s!wrong!calculation:!!!!

Correct!calculation!with!units:!

Correct!answer!with!units:!

New!problem!for!Angela:!!!!!

Correct!calculation!with!units:!

Correct!answer!with!units:!

Angela’s!wrong!calculation:!!!!

Correct!calculation!with!units:!

Correct!answer!with!units:!

New!problem!for!Angela:!!!!

Correct!calculation!with!units:!

Correct!answer!with!units:!

!!Angela’s!wrong!calculation:!!!!

Correct!calculation!with!units:!

Correct!answer!with!units:!

New!problem!for!Angela:!!!!!

Correct!calculation!with!units:!

Correct!answer!with!units:!

!

Looking'at'Problems! ! ! ! ! Name:____________________!!

1. Angela!has!written!calculations!to!show!how!she!would!solve!each!of!the!six!problems!below.!!Your!task!is!to!check!her!work!because!she!has!made!some!errors!in!three!of!the!problems.!!For!each!of!the!problems!below:!

• Identify!the!problem!and!wrong!calculation.!• Think!about!why!Angela!might!have!made!the!mistake.!• Provide!the!correct!calculation!and!correct!answer!with!units.!• Create!a!new!problem!to!give!Angela!more!practice!working!with!

similar!numbers!in!a!similar!situation.!• Provide!a!correct!answer!to!the!problem!you!created.!

!a. A!car!travels!120!miles!in!3.6!hours.!

What!is!its!average!speed!in!miles!per!hours?!!Miles'per'hour'

120÷ 3.6! Correct!

b. A!snail!travels!0.8!miles!in!40!hours.!What!is!its!average!speed!in!miles!per!hour?!!What'does'per'mean?'So'if'we'have'miles'per'gallon,'how'would'we'set'up'the'problem?'How'does'Angela'have'the'problem'set'up?'(Hours'per'mile)'

40÷ 0.8! Wrong!

c. Mrs.!Abir!buys!some!apples!at!$1.50!per!pound.!!!She!spends!$3.50.!!How!many!pounds!does!she!buy?!

3.50÷ 1.50! Correct!

d. Mario!buys!some!tomatoes!at!$0.90!per!pound.!!He!Spends!30!cents.!!How!many!pounds!does!he!buy?!!What'does'$0.90'represent?'(Price'per'pound)'What'does'30'cents'represent?'So'if'$0.90'is'for'one'pound'and'we'buy'30'cents'worth,'how'would'we'set'up'our'problem?'

90!÷ 30! Wrong!

e. Lena’s!motorcycle!gets!62.5!miles!per!gallon.!!!How!far!can!she!go!on!3.4!gallons?!!What'does'62.5'represent?'(Miles'per'gallon)'So'if'62.5'miles'is'for'one'gallon,'how'would'we'find'out'how'far'Lena'can'go'on'3.4'gallons?'

62.5÷ 3.4! Wrong!

f. Mr.!Okawa’s!car!gets!20!miles!per!gallon.!He!only!has!0.4!gallons!left!in!the!tank.!How!far!will!he!travel!before!he!runs!out!of!gas?!

20÷ 0.4! Correct!

!!

Angela’s!wrong!calculation:!b.!!40÷ 0.8!!!

Correct!calculation!with!units:!Miles!per!hour!0.8!÷ 40!

Correct!answer!with!units:!!.02!

Angela’s!wrong!calculation:!d.!!90!÷ 30!!!

Correct!calculation!with!units:!. 30!÷ ! .90!

Correct!answer!with!units:!!.3333!

Angela’s!wrong!calculation:!e.!!62.5÷ 3.4!!!

Correct!calculation!with!units:!62.5!×!3.4!

Correct!answer!with!units:!!212.5!

!!