Understanding by Design Lesson TemplatePreAssessment.docx & SummativeAssessment.docx

Grade Level: 3rd Grade

CCSS Domain: Number and Operations- Fractions

Desired ResultsCCSS Mathematics Standard(s):

3.NF.1

Understand a fraction 1/b as the quantity formed by 1

part when a whole is partitioned into b equal

parts; understand a fraction a/b as the

quantity formed by a parts of size 1/b.

Mathematical Practice(s):

MP.4 Model with mathematics.

MP.7 Look for and make use of structure.

MP.8 Look for and express regularity in repeated reasoning.

TransferStudents will be able to independently use their learning to…

T1: Recognize, frame and solve never-before-seen real life mathematical problems involving fractions.

MeaningUNDERSTANDINGSStudents will understand that…

U1: A fraction can be an equal part of a set or an equal part of a whole.

U2: Fractions represent the relationship between a part and the whole.

U3: In fractions the value of a digit depends on where it is placed in a fraction.

ESSENTIAL QUESTIONSStudents will keep considering...

E1: How do fractions apply to real life?

E2: Why do you need to know about fractions?

E3: Why do we need fractions?

Acquisition of Knowledge and SkillStudents will know…

K1: Key vocabulary relating to fractions such as: fractions, part, whole, composed, decomposed, numerator, denominator, equal parts.

K2: The numerator tells the number of equal parts being described relative to the whole.

K3: The denominator tells the total number of equal parts in the whole.

K4: A fraction can be decomposed of smaller fractions, and smaller fractions can be composed into a larger fraction.

Students will be skilled at…

S1: Representing common fractions such as 1/8, ¼, ½, 1/6, 1/3, and ¾ using words, numerals, and physical models.

S2: Identifying the numerator and denominator in a fraction.

S3: Explaining how a fraction 1/b a times can be composed into a/b. (i.e. 1/8 + 1/8 + 1/8 = 3/8) using words, equations, and models.

S4: Explaining how a fraction a/b can be decomposed into 1/b a times (i.e. 5/8 can be

decomposed into 5 pieces that are 1/8 in size) using words, equations, and models.

EvidenceEvaluative Criteria

Students will be able to partition objects from a whole.

Students will be able to justify why parts should be equal

Students will know how to compose/decompose fractions.

Students will correctly identify the numerator and denominator.

Student will see how fractions apply to real life.

PERFORMANCE TASK(S):Students will show that they really understand by evidence of…

Creating their own fractions based on specific shapes and partitions

Correctly justifying other student approaches.

Evaluating the use of various tools to model fraction relationships.

Creating a poster & Graffiti board with important fraction vocabulary and concepts.

Create unit fractions given specific shapes and partitions.

Creating their own fraction problems based on real-life examples.

Students will be able to partition objects from a whole.

Students will be able to justify why parts should be equal

Students will know how to compose/decompose fractions.

Students will correctly identify the numerator and denominator.

OTHER EVIDENCE:Students will show they have achieved goals by…

Scoring well on the summative assessment. Journal entries Class discussion

Learning Plan/ Summary of Key Learning Events and InstructionConcepts/Background Knowledge for teacher:

Prior to learning this concept the students will be exposed to standards:

2.G.2 - Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

2.G.3 - Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Students might have also had some exposure with standard 3.G.2 which states that students will know how to partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.

This means that the students will already know how to:

Define partition, equal area, part, whole and fraction as well as other fractional mathematics terminology such as halves, thirds, and fourths. Students will also have experience with portioning shapes into equal shares.

Vocabulary:

composed decomposed denominator equal parts fraction numerator one-eighth; 1/8 one-fourth; 1/4 one-half; 1/2 one-sixth; 1/6 one-third; 1/3 partitioned unit fraction whole

Literature: My Half Day by Fisher, Sneed, and Lee

Technology: Video: (Numerator and Divisor Song) http://www.youtube.com/watch?

v=j7WhRMvlQwo Virtual manipulative: (Fractions part of a whole)

http://nlvm.usu.edu/en/nav/frames_asid_104_g_2_t_1.html Virtual manipulative: (Fraction Bars)

http://nlvm.usu.edu/en/nav/frames_asid_203_g_2_t_1.html

Manipulatives: Shape cut-outs Fraction tiles/bars

Math centers based upon pre-assessment:

During first center day the students will be grouped together based on the results of their pre-assessments. Students will be grouped together based on their weaknesses on the pre-assessment. During the center day the students will meet with me to go over their areas of weakness.

On the second center day the students will be grouped based on their strengths. These students will receive an extension activity for one of the centers. These higher level activities are mentioned below.

Higher level activities:

Day One: Students will get additional shapes to partition into equal portions, including triangles. The students will have to also divide shapes multiple ways into equal partitions. So, a circle might have to be divided into ½, 1/3 and 1/7. Using manipulatives they will have to prove that these are equal.

Day Two: Students will use these additional shapes from the previous to represent different fractions that they are given. Using manipulatives they will have to prove that the portions are equal. Students can also chose to create their own fractions to represent and write this fraction in numeral form and also model it.

Day Three: Students can again use additional shapes and model how they built units from unit fractions. Students can also create their own equations for how they build units to form fractions.

Day Four: Students can create their own fraction examples that they have witnessed in the book or in real life.

Day Five: Students will have to prove how they know that there are multiple ways to view a fraction when no whole is specified.

Other activities (brainstorm other ideas):

Students can read the book Give me Half and find the fraction examples demonstrated in the book and model this. A link to this activity can be found at: http://mrsriccaskindergarten.blogspot.com/2012/11/yummy-fractions-freebie.html

Students can make a town based off of fractions. A link to this activity can be found at: http://www.ashleigh-educationjourney.com/search?updated-max=2011-06-28T07%3A32%3A00-07%3A00&max-results=7

Students can make a gumball machines and use fractions to represent the different color gumballs that they have in their jar. Here is a link to that activity:

http://2.bp.blogspot.com/-97OKYQS4kv0/T34SI6k2r1I/AAAAAAAADzQ/dmwg9rA1gt8/s1600/IMG_3263.jpg

Day One Lesson Plan (60 minutes)Materials:

Math Journals Paper and pencils for every student.

Video: http://www.youtube.com/watch?feature=player_embedded&v=pTbCbMOmVFc

ObjectivesO1: The student will understand that a fraction is part of a set or a part of a whole.

O2: The student will understand that a fraction is composed of equal sized parts.

Vocabulary equal part fraction partitioned whole

Formative Assessment

The students will write a journal entry in which they answer the following prompt?

“What have you learned about fractions?”

The students will fill out this entry in their mathematics journals that they have been keeping all semester.

ProceduresBefore Phase 1. TW play the following video which serves as an introduction to

fractions:http://www.youtube.com/watch?feature=player_embedded&v=pTbCbMOmVFcThe teacher will then say to students, “Well, as the video explains today we will be learning all about fractions!”

2. TW ask the students, “What is a fraction?” and wait for student responses. TW explain to students, “You seem to have some knowledge on what a fraction is and I got the impression that as a class definition we have come up with the idea that a fraction is a piece or part of something bigger. Although this is a good definition after today you guys will all have a much better understanding of what a fraction is”.

3. TW then ask students to name some examples of fractions that they know, “So now that we know that a fraction is a piece of something what are some fractions that you have heard of

before?” The teacher will then record some of these examples on the board for the class to look at.

4. TW then ask the students “When do we use fractions?” and record these examples on a list on the board. The idea should be established that we use fractions when we divide up food, such as dividing up cookies for everyone in the class or cutting up pizza so everyone can have a slice.

5. TW then say, “I have a funny story about fractions to tell you guys! To help you guys visualize this can everyone grab a piece of paper and a pen, I will make sure to let you know what you will be doing with these as we go on!

During Phase1. TW say, “Well, unfortunately my boyfriend Jacob does not have

the best understanding of fractions. Anyways, this weekend we decided to go to All American Pizza to get dinner this weekend. We decided to get a pizza, some bread sticks, and then we went over to Ridleys and picked up a chocolate cake. Why don’t you go ahead and draw these on your piece of paper.”

2. SW draw the shapes on their pieces of paper. TW then say, “Well after we got all of our food and got back to our house my boyfriend decided that we would each get half of the pizza since we were both equally hungry. Well, here is a picture that shows how my boyfriend divided up the pizza.” TW show the drawing to the students.

3. TW say (if the students don’t already call out), “Does this seem like it was very fair?” TW wait for student responses until the idea is established that it isn’t fair. TW probe the students by saying, “What do you mean it isn’t fair? He cut the pizza into two parts” TW wait for student responses until the idea is established that the two parts are not the same size and therefore it isn’t

fair”4. TW say, “So, you mean that ½ has to be two equal parts? How

should he have split the pizza then, please show me on the pizza that you drew”

5. SW draw on their paper pizza’s what splitting the pizza in half should look like.

6. Teacher will then say, “Well after we ate the pizza we decided to split up the breadsticks. There were four breadsticks in our order and my boyfriend was so nice and again decided to give me half of the breadsticks even though they are his favorite food. Here is how he split them up. Without saying anything out loud I want you guys to decide if this was fair and if it isn’t fair how you would split the breadsticks to make it fair, you can show me this on your paper with the breadsticks drawing. Here is how my boyfriend divided up the breadsticks.” TW show students the following picture:

After Phase 1. TW say, “Now for dessert our neighbor decided to come over to eat some of the cake with us. My boyfriend decided to split this cake up for us into three parts. So, that each of us got 1 of the 3 pieces” TW write this on the board 1/3.

2. TW then say, “Now help refresh what we have learned so far…Fractions have to have certain kind of parts. Can you help me? Fractions must have….” TW wait for student responses until the idea is reached that fractions must have EQUAL PARTS!

3. TW display the following picture and ask the students, “So why isn’t this 1/3?” TW wait for student’s responses until the idea is established that the parts aren’t equal. TW probe the students to

justify their responses by proving that the parts are not equal.

4. TW ask students to display how they decided to split up the cake into equal parts and justify how they know that they are equal.

5. TW then say, “Well, my boyfriend might still be skeptical. He thinks that half means just splitting something into two parts. So, one part is his and the other part is mine for two parts. How would you explain to him that this isn’t true?”

6. TW then direct the students to respond in their journal with what they have learned about fractions today.

Questions to Develop

Mathematical Thinking/Practic

es

How would you explain to him that this isn’t true?Does this seem like it is fair?

How would you split up the pizza/breadsticks/cake?Can you show me how these parts are equal?

Supportive and Advancing Questions

Fractions have to have what kind of parts? Can you help me?How come if you split something into two parts it isn’t always fair?

How should he have divided it so that it was fair?Higher level

questions/activities

Are there any other ways you could split up the cake?What if another person came over for desert, then how would you split

up the cake?How could you show my boyfriend that he didn’t split up the pizza

fairly?How would you explain this to him?

Anticipated Student

Approaches/Possible

Misconceptions

The biggest misconception that the students might have is the idea that just because an object is split into two pieces that makes it a

fraction. The students don’t realize that the fractions must be composed of equal portions. Luckily, this is the misconception that “my

boyfriend has with his pizza” so it is addressed in the lesson itself.

Day Two Lesson Plan (60 minutes)

Materials: Shape Cut Out’s for every student Markers for every student

ObjectivesO1: In fractions the value of a digit depends on where it is placed in a fraction.

O2: The student will understand that a fraction is composed of equal sized parts.

Vocabulary composed equal parts fraction numerator denominator partitioned unit fraction whole one-eighth; 1/8 one-fourth; 1/4 one-half; 1/2 one-sixth; 1/6 one-third; 1/3

Formative Assessment

The teacher will pass out papers with the word fraction written on the middle in big letters. The students will complete their own graffiti board for everything that they have learned about fractions so far.

ProceduresBefore Phase 1. As children are walking into class the following vide will be

playing called the numerator and denominator song: http://www.youtube.com/watch?v=j7WhRMvlQwo-

2. TW will then have the word fraction written in big letters on the board and encourage students to come up to the board and write key words that they hear during the song. This is the

graffiti board strategy.3. After listening to the song a couple of times TW say to the students, “We have learned some valuable words that relate to fractions today. Let’s put all these words into a poster that we

can display!”4. TW will work with students to create a poster that includes the

vocabulary that the students have written on the graffiti board. The poster should look something like this

During Phase 1. TW say, “Remember how yesterday we looked at how to divide

up the pizza, bread sticks, and cake so that my boyfriend and I both had equal shares. Well we are going to do something

similar today. Today we are going make fractions based off of different shapes in our centers. In the first center we will be

splitting circles into equal parts, in the third center we will be splitting squares into equal parts, in the third we will be splitting rectangles into equal parts, and finally in the fourth center you

will get to work one on one with me!”2. TW divide the students based on their strengths as

demonstrated in the pre-assessment. There will be about 4-6 students in each group. TW have the students go to the first

center that they will be working at. TW explain to the students, “The rules

3. The SW rotate between the four centers. They will remain in each center for about ten minutes depending on how efficiently

they are working. Here is how the centers work:

In center one the students will be given circle shapes. They must divide the circles into 2, 3, and 6 parts. The students will then have to color in what ½, ¼, and 1/6 looks like on these shapes.

In center two the students will be doing a similar activity and must divide squares into 2, 3, and 6 parts. The students will then have to color in what ½, ¼, and 1/6 looks like on these shapes.In center three the students again do a similar activity but this time they must divide rectangles 2, 3, and 6 parts. The students will then have to color in what ½, ¼, and 1/6 looks like on these shapes.

In center four the students get to work one on one with me. The activities that we work on will be based on the strengths that they demonstrated during the pre-assessment. Some students might move on to dividing up triangles into certain parts, or extra practice with splitting shapes into different parts such as 4, 7, or 9 pieces, other students might get extra support if they need additional help or have misconceptions that need to be addressed.

After Phase 1. SW gather in the carpet area for a discussion. During the discussion the TW start about by having students come and

display how they split up the various shapes. TW take multiple examples to see how students might have split the shapes in different ways. If the teacher notices any misconceptions she

will address these misconceptions during the discussion.2. TW then have the students show examples of how they

displayed the fractions on their papers. The TW collect multiple examples and place these on the board. The teacher will then ask the students if these all represent the same fraction even

though different portions are highlighted. This sets up the lesson for tomorrow when the teacher explains to students how

fractions can be composed of smaller fractions.Questions to

Develop Mathematical

Thinking/Practices

How would you show various fractions using the cut out shapes?How do you know that the shapes have been partition equally?

How could you prove that all of the parts are equal?

Supportive and Advancing Questions

What do you need to keep in mind when portioning a whole into fractions?

Higher level questions/activit

ies

How would you partition a shape for 7, 9, or 11 parts?How many different ways can you show the same fraction?

Anticipated Student

Approaches/Possible

Misconceptions

Students might not divide the shapes into equal portions, but teacher will have them reflect to the previous day’s lesson.

The students might get the numerator and denominator confused. So when asking to represent 1/3 for example they might be confused on

how to demonstrate this.

Day Three Lesson Plan (60 minutes)

ObjectivesO1: Students will know that a fraction 1/b a times can be composed into a/b. (i.e 1/8 + 1/8 + 1/8 = 3/8) using words, equations, and models.

Vocabulary composed decomposed denominator equal parts fraction numerator one-eighth; 1/8 one-fourth; 1/4 one-half; 1/2 one-sixth; 1/6 one-third; 1/3 partitioned unit fraction whole

Formative Assessment

Exit pass: The students will have to write “What was the most important thing you learned today?” on a slip of paper before leaving

the class.Procedures

Before Phase 1. As students enter the classroom they will play hop-scotch like the one in the following picture. They will have to throw a bean bag on one of the squares and tell the teacher what fraction the

bean bag landed on.

2. After the students have settled into their seats the lead teacher will state to the students that we will again be working in centers using the same shapes as yesterday. This time we are going to break apart these shapes. Does anyone what it is called when we break apart fractions into smaller fractions?” TW wait for responses until the idea is established that it is called decomposing. TW then say, “Today we are going to be using some manipulatives where we examine how the parts of a fraction relate to the whole including composing and decomposing fractions.”

3. TW say, “So, today we are going to be working in centers. In once center you will work with virtual fraction blocks, in another you will work with normal fraction blocks, in one you can use paper and pencil to represent fractions, and in another you can use an applet for guessing and checking fractions. For each of these centers I want you to keep a journal entry for what you learned specifically from each tool and how you would rate this tool and why.

During Phase 1. TW divide the groups into their centers. There should be about 4-6 students per group and they will spend about ten minutes in each center with 7 minutes devoted to exploration and 3 minutes devoted to reflection. The centers are as follows:

Center One: Students are given paper and pencil to demonstrate various fractions given such as ½, ¼, 1/8, ¾ etc.Center Two: In this center students are given fraction manipulatives to explore with such as fraction bars and fraction circles.

Center three: In this center the students are directed to use the following applet: http://nlvm.usu.edu/en/nav/frames_asid_103_g_2_t_1.html

Center four: In this center the students are directed to use the following tool: http://nlvm.usu.edu/en/nav/frames_asid_203_g_2_t_1.html

After Phase 1. We will have a class discussion where the students go over what they learned in each particular center. We will make a list on the

board for each center and what was learned.2. TW also probe students

Questions to Develop

Mathematical Thinking/Practic

es

What tool was the best for showing fractions?What tool was the best for showing the relationship between the part

and the whole?Which tool do you think was the most useful/ least useful

Supportive and Advancing Questions

What do you need to keep in mind when using this tool?What is the most confusing part about using this tool?

How can this tool be used to model fractions?How can this tool be used to show how fractions are composed and

decomposed?Higher level

questions/activities

How could this tool be used to demonstrate the importance of equal parts to my boyfriend?

How else could you use this tool to demonstrate fraction concepts?

Anticipated Student

Approaches/Possible

Misconceptions

Students might have a hard time figuring out the instructions on some of the websites, so I will have separate instructions posted at each of

the center hubs for students to look over.

Day Four Lesson Plan (60 minutes)Materials:

My Half Day

Objectives O1: Students will identify the numerator and denominator in a fraction.

O2: Students will know that a fraction a/b can be decomposed into 1/b a times (i.e. 5/8 can be decomposed into 5 pieces that are 1/8 in size) using words, equations, and models.

Vocabulary composed decomposed denominator equal parts fraction numerator one-eighth; 1/8 one-fourth; 1/4 one-half; 1/2 one-sixth; 1/6 one-third; 1/3 partitioned unit fraction whole

Formative Assessment

The students will create a journal entry in their math journals in which they answer the following prompt:

How are fractions used in real-life? Can you give an example?Procedures

Before Phase 1. TW say to the students, “Remember to day one of our fractions unit when we made a list of how we used fractions in real life?

Well let’s see if we can recall some of these instances”.2. SW respond with instances of how fractions are used in real-life

and the TW write this list on the board. TW will then probe students to think of other ways we see fractions on a daily basis. TW write these instances on the board so that we have a longer

list.3. TW then say, “Imagine that every part of your life was a

fraction…”During Phase 1. TW explain to students that “We will be reading a story of a

young boy that see’s everything as fractions. I want you to keep in mind all the instances that you see fractions while we read

along, feel free to jot these in your journals for future reference.2. TW read My Half Day to students.

3. As the teacher reads along TW point out fraction instances for the students. The TW write these on the board for the students

to look at.

4. After reading the book the class should have a list of fractions on the board. The TW have students point out the numerator and denominator of the fractions that we have written on the

board.After Phase 1. Students will create their own problems with fractions from

daily life and have a partner solve it.2. TW have some of these students come to the board to share

their problems with their classmates.

Questions to Develop

Mathematical Thinking/Practic

es

What are some examples of fractions in real life?What are some different ways to make the same fraction?

How can you show the same fraction in different ways?Explain how you can think of a fraction as being composed of smaller

fractions?Supportive and

Advancing Questions

What does the bottom number represent?What does the top number represent?

Which number is the numerator?Which number is the denominator?

Higher level questions/activit

ies

How can you think of different ways fractions are in our lives?Why is it important to know how fractions affect our daily lives?

Anticipated Student

Approaches/Possible

Misconceptions

Students might not think of how fractions can be broken down into smaller fractions, or how fractions can be found in everyday life.

Students might not also see the correlation between money and time with fractions.

Day Five Plan (60 minutes)

ObjectivesO1: Students will know the importance of specifying the whole.

Vocabulary composed decomposed denominator equal parts fraction numerator one-eighth; 1/8 one-fourth; 1/4 one-half; 1/2 one-sixth; 1/6 one-third; 1/3 partitioned unit fraction whole

Formative Assessment

Students will be asked to reflect in their journal on why it is crucial to specify the whole when looking at fractions.

ProceduresBefore Phase 1. TW draw three circles on the board and divide each of the circles

into thirds. TW then shade in the three different ways to represent 1/3 in the circles. (TW explain what they are doing as the process goes on) TW then say, “So what have we learned

about what I just did?” TW wait for student’s responses until the idea is established that there are different ways to represent the

same fraction”.2. TW then explain that today we are going to see how fractions

can be viewed in different ways & ask if any students might want to predict what this means.

During Phase 1. TW then put the following picture on the board:

2. TW then ask the students what this picture represents. TW record the common answers on the board.

3. TW then write the following three questions on the board:

Emily said that the picture represents 2/6 . Label the picture to show how Emily's answer can be correct.

Raj said that the picture represents 2/3 . Label the picture to show how Raj's answer can be correct.

Alejandra said that the picture represents 2 . Label the picture to show how Alejandra's answer can be correct

4. TW then direct the students to prove their responses by writing them in their journals.

After Phase 1. TW start by asking the students to share their responses with the class.

2. Teacher will then ask the students if they can think of any other fractions that people might see in the drawing.

3. TW then ask the students, “Who do you think is correct?” and wait for responses.

4. TW then explain to the students that all of the answers are correct because the whole was not specified. TW lecture the

students on the importance of specifying the whole.Questions to

Develop Mathematical

Thinking/Practices

How can you show me the students thinking?Can you show me how you labeled each of the students thinking?

Can you show me how you approached the problem?

Supportive and Advancing Questions

What student do you think is correct? Why?Why is it important to specify the whole?

How is (student’s) thinking correct/wrong?

Higher level questions/activit

ies

Are there any other fractions that you think people might see?

Anticipated Student

Approaches/Possible

Misconceptions

Students might not be able to justify the other student’s responses when they don’t think about what the whole might be representing.

Sources:1.)http://www.dpi.state.nc.us/docs/acre/standards/common-core-tools/

unpacking/math/3rd.pdf2.) Pinterest.com3.)http://luckeyfrogslilypad.blogspot.com/2012/04/introducing-fractions-with-

story.html4.)http://nlvm.usu.edu/en/nav/category_g_2_t_1.html 5.) http://s3.amazonaws.com/illustrativemathematics/illustration_pdfs/

000/000/833/original/illustrative_mathematics_833.pdf?1349968234