Comparing Fractions with Like and Unlike Denominators: Concrete Level

PHASE 1: Initial Acquisition of Skill

Teach Skill with Authentic Context

Description: Two meaningful contexts are used to explicitly model these concepts/skills. For comparing fractions with like and unlike denominators (area model), the context is a pizza party. “Pizzas” and “pizza slices” made out of construction paper/tag board/cardboard are used. For comparing fractions with like and unlike denominators (sets model), the context is a visit to an amusement park where each student has a specified number of tickets to ride their favorite amusement rides.

Build Meaningful Student Connections

Purpose: to help students build meaningful connections between what they know about fractions/fractional parts and the process of comparing fractional parts.

*The following description is an example of how you might implement this instructional strategy for Learning Objective 1. A similar process can be used for the other learning objectives in this plan.

Learning Objective 1: Compare fractions with like denominators using concrete materials that represent the area model.

Materials:

Teacher -

• Concrete materials representing the area model (e.g. circle pieces.)
• A written statement reflecting the learning objective that is visible to all students.
• A candy bar and utensil for cutting it into two pieces.
• A platform for displaying concrete objects that allows all students to see.

Description:

1) L ink to students’ prior knowledge of/experiences with fractions and with previous area model concrete materials used to develop understanding of fractions.

For Example:

(Display circle pieces that represent a “whole, one-half, one-fourth, and one-eighth; or, display a “whole” circle, a complete set of “one-half” pieces, a complete set of “one-fourth” pieces, and a complete set of “one-eighth” pieces. * When displaying the sets of “one-halves,” “one-fourths,” and “one-eighths,” leave some space between the individual pieces) Who remembers what these materials represent? (Elicit appropriate responses.) Yes, these materials represent fractions/fractional parts. What does this one represent? (Point to the circle and elicit the response, “a whole.”) Yes, this circle represents a whole. What does this piece represent? (Point to the “one-half” piece and elicit the response, “one-half.”) Yes, this piece represents “one-half” because it is “one-half” of the circle. (Place the “one-half” piece on top of the circle to illustrate this point; or use complete sets of circle pieces and show that two “one-half” pieces make a whole circle.) *Continue this process for each circle piece or set of circle pieces.

2) I dentify the skill students will learn: Compare two fractions to determine which fraction is greater, which one is less, or if they are equal.

For Example:

(Display a written statement of the learning objective that is visible to all students. – e.g. “Compare two fractions to determine which fraction is greater, which one is less, or if they are equal.”) Today we are going to use materials like these (point to the displayed circle pieces) to compare two fractions to determine which fraction is greater, which fraction is less, or whether they are equal. (Point to the written learning objective as you say this.) What are we going to learn today? (Point to the written learning objective and elicit the response, “compare two fractions to determine which fraction is greater, which one is less, or if they are equal.”) Yes, we are going to learn how to compare two fractions to determine which fraction is greater, which one is less, or if they are equal.

To get an idea of what we will learn to do with more complex fractions, we can compare two of the fraction pieces we have displayed here. Let’s compare our “one-half” piece and our “one-fourth” piece. (Place the “one-half” piece and the “one-fourth” piece side-by-side.) Which fraction is greater than the other? (Elicit the response, “one-half.”) Yes, the “one-half” piece is greater than the “one-fourth” piece. We can show that this is so by placing the “one-fourth” piece on top of the “one-half” piece. (Place the “one-fourth” piece on top of the “one-half” piece.) We can see that the “one-fourth” piece is less than the “one-half” piece, because part of the “one-half” piece is still showing. (Point to the portion of the “one-half” piece that is exposed.) This is the kind of thing we will be doing today, except we’ll be learning how to compare many different fractions. We’ll learn how to compare fractions that are even more complex than these.

3) P rovide rationale/meaning for learning how to compare two fractions to determine which fraction is greater, which one is less, or if they are equal.

For Example:

Being able to compare fractions can be very helpful. Let me show you an example of why this is so. I have a candy bar here. (Show students a candy bar.) Now, say both Sharon and I put in the same amount of money to buy this candy bar. It would be important that we both got an equal part of the candy bar when we cut it into two parts to eat. (Call the student to the front of the room.) Now, say I cut the candy bar like this. (Cut the candy bar so that one piece is about one-fourth in size and the other piece is the remaining three-fourths of the candy bar. This piece (hold up the “one-fourth” piece) is about one-fourth of the whole candy bar. This piece (hold up the “three-fourth” piece next to the “one-fourth” piece) is about three-fourths of the whole candy bar. Now, would it be fair for me to give Sharon the “one-fourth” piece? (Hand the student the smaller piece of the candy bar, and elicit the response, “no.”) That’s right, it wouldn’t be fair. Why? (Elicit the response, “because she paid the same amount for the candy bar as you did so she should get an equal amount.”) Excellent thinking! In this case, it is important for both Sharon and I to know that the candy bar needs to be cut into two “one-half” pieces so they are equal in size. Since we both paid the same amount for the candy bar, both of us should get an equal fraction of the whole candy bar. If we didn’t know about fractions and how to compare them, we wouldn’t be able to share the candy bar fairly.

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Provide Explicit Teacher Modeling

Purpose: to provide students a clear teacher model of how to compare fractions with like and unlike denominators using concrete materials.

Learning Objective 1: Compare fractional parts with like denominators using concrete objects that represent the area model (e.g. circle pieces, fraction bars/strips, Cuisenaire rods).

Materials:

Teacher -

• Story problems/contexts that represent situations comparing fractional parts with like denominators (greater, less, equal). *Story problems should be color coded (e.g. number phrases that represents denominator is color-coded blue and number phrases that represent the numerators are color-coded red.)

For Example:

Ms. Gray and several of her students had a pizza party to celebrate the wonderful job they did learning math by ordering both a cheese pizza and a pepperoni pizza that were both the same size! After the party was over, some pizza was left over. Each whole pizza the group ordered had a total of four pieces. There was one slice of cheese pizza left and two slices of pepperoni pizza left. What fractional part of pizza left over is greater, cheese or pepperoni?

• Concrete materials that represent the area model including representations for the denominator as well as for the numerator –
• Pizzas and pizza slices made from cardboard. Plain cardboard circles separated into the appropriate fractional parts should be used to represent the denominator. The numerator is represented by equal size pizza slices cut out of tag board or construction paper - e.g. four equal pieces cut out for “fourths.” *Cheese slices can be colored “yellow” and pepperoni slices can have pepperoni slices drawn on them.)

• Circles/circle pieces - overhead pens or dry-erase markers can be used to draw lines that separate circles into appropriate fractional parts to represent the denominator

• Fraction bars/strips

• Cuisenaire rods
• Three “language cards:” one language card with “denominator” written; one language card with “numerator” written; and one language card with “like denominator” written. *Color code the language cards for “denominator” and “numerator” to match the color-coding used in the story problems.
  

Description:

A. Break down the skill of comparing fractional parts with like denominators using concrete objects that represent the area model.

1) Introduce a story problem/context.

2) Read the story problem aloud.

3) Read the story problem with your students.

4) Model finding what needs to be solved for in story problem.

5) Model finding the important information in the story problem

6) Represent the fractional parts with concrete objects.

7) Compare the fractional parts (greater, less, equal).

8) Solve the story problem.

Learning Objective 2: Compare fractional parts with unlike denominators using concrete materials that represent the area model.

Materials:

Teacher -

• Story problems/contexts that represent situations comparing fractional parts with unlike denominators (greater than, less than, equal to). *Story problems should be color coded (e.g. number phrase that represents denominator is color-coded blue and number phrases that represent the numerators are color-coded red.)

For Example:

Ms. Gray and several of her students had a pizza party to celebrate the wonderful job they did learning math by ordering a cheese pizza and a pepperoni pizza that were the same size! Although pizzas were the same size, the cheese pizza had a total of six slices and the pepperoni pizza had a total of eight slices. After the party was over, some pizza slices were left over. There were two slices of cheese pizza left and two slices of pepperoni pizza left. What fractional part of pizza left over is greater, cheese or pepperoni.

• Concrete materials that represent the area model including representations for the denominator as well as for the numerator –

• Pizzas and pizza slices made from cardboard. Plain cardboard circles separated into the appropriate fractional parts should be used to represent the denominator. The numerator is represented by equal size pizza slices cut out of tag board or construction paper - e.g. four equal pieces cut out for “fourths.” *Cheese slices can be colored “yellow” and pepperoni slices can have pepperoni slices drawn on them.)

• Circles/circle pieces - overhead pens or dry-erase markers can be used to draw lines that separate circles into appropriate fractional parts to represent the denominator
• Fraction bars/strips
• Cuisenaire rods
• Three “language cards:” one language card with “denominator” written; one language card with “numerator” written; and one language card with “unlike denominator” written. *Color code the language cards for “denominator” and “numerator” to match the color-coding used in the story problems.
  

Description:

A. Break down the skill comparing fractional parts with unlike denominators using concrete materials that represent the area model.

1) Introduce a story problem/context.

2) Read the story problem aloud.

3) Read the story problem with your students.

4) Model finding what needs to be solved for in story problem.

5) Model finding the important information in the story problem.

6) Represent the fractional parts with concrete objects.

7) Compare the fractional parts (greater, less, equal).

8) Solve the story problem.

Learning Objective 3: Compare fractional parts with like denominators using concrete materials that represent the sets model (e.g. sets of counting chips, beans, unifix cubes, tickets.).

Materials:

Teacher -

• Story problems/contexts that represent situations comparing fractional parts with like denominators represented by sets (greater, less, equal). Story problems should be color coded (e.g. number phrase that represents denominator is color-coded blue and number phrases that represent the numerators are color-coded red.)

For Example:

Jenny and Carlos went with their parents to Kings Dominion Amusement Park this past weekend. Both Jenny and Carlos had a total of eight tickets each to spend riding their favorite rides. Jenny used four of her tickets and Carlos used six of his tickets. Who used a greater fractional part of the total number of tickets given to them, Jenny or Carlos?

• Discrete concrete objects that can be grouped to sets model (e.g. “tickets” made from construction paper, counting chips, beans, unifix cubes, etc.)
• Paper plates to place each set of objects being compared.
• Pieces of string to encircle objects in sets that represent the numerator.
• Three “language cards:” one language card with “denominator” written; one language card with “numerator” written; and one language card with “like denominator” written. Color code the language cards for “denominator” and “numerator” to match the color-coding used in the story problems.
• Markers for writing and highlighting.

Description:

A. Break down the skill of comparing fractional parts with like denominators using concrete materials that represent the sets model.

1) Introduce a story problem/context.

2) Read the story problem aloud.

3) Read the story problem with your students.

4) Model finding what needs to be solved for in story problem.

5) Model finding the important information in the story problem.

6) Represent the fractional parts with concrete objects.

7) Compare the fractional parts (greater, less, equal).

8) Solve the story problem.

Learning Objective 4: Compare fractional parts with unlike denominators using concrete materials that represent sets model (e.g. sets of counting chips, beans, unifix cubes, tickets).

Materials:

Teacher -

• Story problems/contexts that represent situations comparing fractional parts represented by sets (greater, less, equal). *Story problems should be color coded (e.g. number phrase that represents denominator is color-coded blue and number phrases that represent the numerators are color-coded red.)

For Example:

Latrisa and David went with their parents to Kings Dominion Amusement Park this past weekend. Latrisa had a total of six tickets to spend riding her favorite rides. David had a total of eight tickets to spend riding his favorite ride. Latrisa used four of her tickets and David used four of his tickets. Who used a greater fractional part of the total number of tickets given to them, Latrisa or David?

• Discrete concrete objects that can be grouped to sets model (e.g. “tickets” made from construction paper, counting chips, beans, unifix cubes, etc.)
• Paper plates to place each set of objects being compared
• Pieces of string to encircle objects in sets that represent the numerator
• Circles made from cardboard/tag board that are separated into halves, fourths, sixths, eighths, etc. (*Use a marker to draw the fractional parts on each cardboard circle.)
• Three “language cards:” one language card with “denominator” written; one language card with “numerator” written; and one language card with “unlike denominator” written. *Color code the language cards for “denominator” and “numerator” to match the color-coding used in the story problems
• Markers for writing and highlighting.

Description:

A. Break down the skill of comparing fractional parts with unlike denominators using concrete materials that represent the sets model.

1) Introduce a story problem/context.

2) Read the story problem aloud.

3) Read the story problem with your students.

4) Model finding what needs to be solved for in story problem.

5) Model finding the important information in the story problem.

6) Represent the fractional parts with concrete objects.

7) Compare the fractional parts (greater, less, equal).

8) Solve the story problem.

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Scaffold Instruction

Purpose: to provide students the opportunity to build their initial understanding of how to compare fractions with like denominators using concrete materials, and to provide you the opportunity to evaluate your students’ level of understanding after your initial modeling of these skills.

*The steps for Scaffolding your instruction are the same for each concept you have explicitly modeled and with each Fraction Model you teach (Area and Sets). This teaching plan provides you a detailed example of scaffolding instruction for Learning Objective 1. A similar process can be used for the other learning objectives in this plan. You should Scaffold your instruction with each skill/concept you model.

Learning Objective 1: Compare fractions with like denominators using concrete materials that represent the area model.

Materials:

Teacher -

• · Visual display of story problem for comparing fractions with like denominators (area model); number phrases representing denominator and numerators are color-coded.
• Concrete materials suitable for representing the denominator and numerators described in the story problem. (e.g. circles cut our of construction paper separated into equal parts that represent the denominator; circle pieces that represent halves, thirds, fourths, sixths, eighths and that correspond in area to the circles.)
• Visual displays for the words “denominator,” “numerator,” and “like denominator. Color code “denominator” and “numerator” with the phrases in the story problem that represent these two concepts. Also have available non-color coded language cards when such color-coding is no longer needed during the scaffolding process.
• Markers/chalk for writing.

Students -

• Copy of the story problem for comparing fractions with like denominators (area model) - *For Scaffolding Using a Low Level of Teacher Support.
• Concrete materials suitable for representing the denominator and numerators described in the story problem. (e.g. circles cut our of construction paper separated into equal parts that represent the denominator; circle pieces that represent halves, thirds, fourths, sixths, eighths and that correspond in area to the circles.)
• Pencils/pens/markers for writing.

Description:

1) Scaffold Using a High Level of Teacher Direction/Support

a. Choose one or two places in the problem-solving sequence to invite student responses. Have these choices in mind before you begin scaffolding instruction.

Explicitly relate this action to the story context used during Explicit Teacher Modeling. Re-emphasize any new language introduced during Explicit Teacher Modeling (e.g. “like denominators”).

Introduce a story problem/context and read the story problem aloud with your students.

“Here’s another story problem that involves comparing fractional parts. I’d like for you to read the story problem with me.” (Read the story problem aloud with your students.)

Model how to identify what you are solving for (e.g. Look for question marks, circle the question mark and then underline the question.)

“What do I look for when I want to find what I am solving for? (Elicit the response, “the question/question mark.”) Yes, I look for the question. What do I do when I find the question? (Elicit the response, “underline the question and circle the question mark.) Yes, first I circle the question mark because it tells me that this statement is a question and then I underline the question.” (Circle the question mark and then underline the question.)

Model finding the important information in the story problem that will help you solve the story problem (e.g. Read each sentence and ask, “Is there a number phrase in this sentence?” – Circle the number phrases.).

Model finding the number phrases that represent the “denominator” and the “numerators.” - “After we’ve found what we are solving for, what important information do we look for next? (Elicit the response, “the number phrases.”) Yes. What strategy can I use to find the number phrases? (Elicit the response, “read each sentence and ask, “is there a number phrase in this sentence?”) Good. Let’s do that now.” (You and the students read through the story problem using this strategy to find the number phrases.)

Model deciding if all the important information is identified. - “ When we’ve finished finding the important information, what do we need to check for? (Elicit the response, “to be sure we’ve found all of the important information.”) That’s correct. What two important things do I look for? (Point to the phrases on the board and elicit the response, “question” and “number phrases.”) Good. First, I looked to find what I am solving for. How did I find what I was solving for? (Elicit the response, “you looked for the question, circled the question mark and underlined the sentence.”) How did I find the number phrases? (Elicit the response, “you read each sentence and asked, ‘is there a number phrase in this sentence?’”) Yes. How many number phrases did I find? (Elicit the response, “two.”) Great. I found two number phrases. (Point to the two number phrases that you circled and then write a check beside “number phrases” written on the board.) Have we found all of the important information?” (Point to the checked off phrases and elicit the response, “yes.”)

Represent the fractional parts with concrete objects.

Model how to represent the denominator. – “Now that I know what I am solving for and have circled the important information, then I can “act” out my story problem with these concrete objects. How can I represent my first number phrase? Well, it says…… I can represent it with…… What is the name for the total number of parts in a fraction problem. Oh, I remember. It’s called the denominator. (Display “denominator” language card.) So, there were a total of ____________ slices in both the cheese pizza and the pepperoni pizza. I can represent the total number of slices for each pizza with these __________________. See, each circle/pizza has a total of ____________ pieces/slices. These _______________ pieces/slices represent my denominator. (Point to “denominator” language card. Hmm, are the denominators for each pizza the same? Well, yes they are. Each whole pizza had ____________ total pieces. I know there is a name for denominators that are the same. They are called “like denominators.” (Display the “like denominators” language card.) What do we call denominators that are the same? ( Elicit the response, “like denominators.”) Yes, we call denominators that are the same, ‘like denominators.’”

Model how to represent the numerator. – “How can I represent my second number phrase? Well, it says…… I can represent it with…… What is the name for the number of parts of a whole in a fraction problem? Oh, I remember. It’s called the numerator. (Display “numerator” language card.) So, there were____________ slices of cheese pizza left over. (Count the number of pieces representing the cheese slices left.) What is the next number phrase? I can represent it with….. So, there were ____________ slices of pepperoni pizza left. . (Count the number of pieces representing the pepperoni slices left.) This number is also called a “numerator” since it represents parts or slices of a whole pizza. So, I have two numerators represented here. There are ____________ number of cheese slices left and _____________ number of pepperoni slices left.”

Model representing each “part” in fractional form.

“Now I need to represent the fractional parts the left over slices make. I know I can do this by placing the pieces left over on top of the whole circles that represent my denominator. (Model placing the pieces/slices that represent the fractional parts on top of the circles that represent the denominator.) Ok, I have _________ cheese slices/pieces out of a total of ________________ cheese slices/pieces. (Point to the pieces and then to each of the parts of the whole circle.) I also have _______________ pepperoni slices/pieces out of a total of _________________ pepperoni slices/pieces. Doing this will help me compare the fractional parts represented by the left over slices/pieces of cheese and pepperoni pizza.”

Model finding which fractional part is greater by comparing the concrete representations.

“How can I determine which fractional part is greater? Well, I can look at how much area/space the left over slices in the both my cheese and pepperoni pizzas slices take up. I know the slices that take up the most area/space will be the fractional part that is greatest. To help me, I can run my finger around the area taken up by my cheese slices. (Run your finger around the area covered by “cheese” slices.) Ok, I have a good idea of how much space the cheese pizza slices take up. I can do the same thing with my pepperoni slices. (Run your finger around the area covered by “pepperoni” slices.) Now I have a good idea of how much space the pepperoni pizza slices take up. Based on “seeing” and “feeling” the area taken up by the cheese and pepperoni slices, I can tell that the fractional part represented by the _____________ slices is greater. __________________ (name the fraction) is greater than _________________ (name the other fraction)

Model how to solve the story problem.

“Now that I know which fractional part is greatest, how can I solve the story problem? I need to find my question again. (Point to the question.) It says……. Well, I know the fractional part represented by my _____________ slices is greater than the fractional part represented by my _______________ slices. Therefore the __________________(name fraction) of ______________ pizza left is greater than the ___________________ (name fraction) of ___________________ left.”

b. Maintain a high level of teacher direction/support for another example if students demonstrate misunderstanding/non-understanding; move to a medium level of teacher direction/support if students respond appropriately to the selected questions/prompts.

2) Scaffold Using a Medium Level of Teacher Direction/Support

a. Choose several more places in the problem-solving sequence to invite student responses. Have these choices in mind before you begin scaffolding instruction.

Introduce a story problem/context and read the story problem aloud with your students.

“Here’s another story problem that involves comparing fractional parts. I’d like for you to read the story problem with me.” (Read the story problem aloud with your students.)

Model how to identify what you are solving for (e.g. Look for question marks, circle the question mark and then underline the question.)

“What do I look for when I want to find what I am solving for? (Elicit the response, “the question/question mark.”) Yes, I look for the question. What do I do when I find the question? (Elicit the response, “underline the question and circle the question mark.) Yes, first I circle the question mark, because it tells me that this statement is a question and then I underline the question.” (Circle the question mark and then underline the question.)

Model finding the important information in the story problem that will help you solve the story problem (e.g. Read each sentence and ask, “Is there a number phrase in this sentence?” – Circle the number phrases.).

Model finding the number phrases that represent the “denominator” and the “numerators.” – “After we’ve found what we are solving for, what important information do we look for next? (Elicit the response, “the number phrases.”) Yes. What two names do we use for the number phrases in a fraction story problem? (Elicit the response, “the “numerator” and “denominatorr.” *Point to the word cards if needed.) What strategy can I use to find the number phrases? (Elicit the response, “read each sentence and ask, “is there a number phrase in this sentence?”) Good. Let’s do that now.” (You and the students read through the story problem using this strategy to find the number phrases.)

Model deciding if all the important information is identified. – “When we’ve finished finding the important information, what do we need to check for? (Elicit the response, “to be sure we’ve found all of the important information.”) That’s correct. What two important things do I look for? (Point to the phrases on the board and elicit the response, “question” and “number phrases.”) Good. First, I looked to find what I am solving for. How did I find what I was solving for? (Elicit the response, “you looked for the question, circled the question mark and underlined the sentence.”) How did I find the number phrases? (Elicit the response, “you read each sentence and asked, ‘is there a number phrase in this sentence?’”) Yes. How many number phrases did I find? (Elicit the appropriate response) Great. Have we found all of the important information?” (Point to the checked off phrases and elicit the response, “yes.”)

Represent the fractional parts with concrete objects.

Model how to represent the denominator. – “Now that I know what I am solving for and have circled the important information, then I can “act” out my story problem with these concrete objects. What is the first number phrase? (Elicit the appropriate response.) How can I represent my first number phrase? (Elicit the response, “with a circle separated into ______ equal parts.) Excellent thinking! (Display the concrete materials that represent the denominator for each pizza/whole.) What is the name for the total number of parts in a fraction problem? (Display “denominator” language card and elicit the response, “denominator.”) How many total slices/pieces are there in each pizza? (Point to the circle and the regions that represent the individual slices/pieces and elicit the appropriate response.) Are the denominators for each pizza the same? (Elicit the response, “yes.”) What do we call denominators that are the same? (Display the “like denominators” language card and elicit the response, “like denominators.”) Yes, we call denominators that are the same, ‘like denominators.’”

Model how to represent the numerator. – “What is the second number phrase? (Elicit the appropriate response.) How can I represent my second number phrase? (Elicit the appropriate response.) Great thinking! (Display the concrete materials that represent the first numerator.) What is the name for the number of parts of a whole in a fraction problem (Display “numerator” language card and elicit the response, “numerator.”) (*Repeat this process for the third number phrase.) How many cheese slices were left? (Elicit the appropriate response.) How many pepperoni slices were left? (Elicit the appropriate response.) So what is the numerator for cheese slices? (Elicit the appropriate response.) What is the numerator for pepperoni slices?” (Elicit the appropriate response.)

Model representing each “part” in fractional form.

“Now I need to represent the fractional parts the left over slices make. I know I can do this by placing the pieces left over on top of the whole circles that represent my denominator. (Model placing the pieces/slices that represent the fractional parts on top of the circles that represent the denominator.) Ok, I have _________ cheese slices/pieces out of a total of ________________ cheese slices/pieces. (Point to the pieces and then to each of the parts of the whole circle.) I also have _______________ pepperoni slices/pieces out of a total of _________________ pepperoni slices/pieces. Doing this will help me compare the fractional parts represented by the left over slices/pieces of cheese and pepperoni pizza.”

Model finding which fractional part is greater by comparing the concrete representations.

“How can I determine which fractional part is greater? Well, I can look at how much area/space the left over slices in the both my cheese and pepperoni pizzas slices take up. I know the slices that take up the most area/space will be the fractional part that is greatest. To help me, I can run my finger around the area taken up by my cheese slices. (Run your finger around the area covered by “cheese” slices.) Ok, I have a good idea of how much space the cheese pizza slices take up. I can do the same thing with my pepperoni slices. (Run your finger around the area covered by “pepperoni” slices.) Now I have a good idea of how much space the pepperoni pizza slices take up. Based on “seeing” and “feeling” the area taken up by the cheese and pepperoni slices, I can tell that the fractional part represented by the _____________ slices is greater. __________________ (name the fraction) is greater than _________________ (name the other fraction).”

Model how to solve the story problem.

“Now that I know which fractional part is greatest, how can I solve the story problem? I need to find my question again. (Point to the question.) It says……. Well, I know the fractional part represented by my _____________ slices is greater than the fractional part represented by my _______________ slices. Therefore the __________________(name fraction) of ______________ pizza left is greater than the ___________________ (name fraction) of ___________________ left.”

b. Maintain a medium level of teacher direction/support for another example if students demonstrate misunderstanding/non-understanding; move to a low level of teacher direction/support if students respond appropriately to the selected questions/prompts.

3) Scaffold Using a Low Level of Teacher Direction/Support

a. When students demonstrate increased competence, do not model the process. Ask students questions and encourage them to provide all responses. Direct students to replicate the process with appropriate concrete materials at their desks as you work together.

Introduce a story problem/context and read the story problem aloud with your students.

“You have in front of you a copy of a story problem that involves comparing fractional parts. I’d like for you to read the story problem with me.” (Read the story problem aloud with your students.)

Model how to identify what you are solving for (e.g. Look for question marks, circle the question mark and then underline the question.)

"What do we look for when we want to find what we’re solving for? (Elicit the response, “the question/question mark.”) Yes, we look for the question. What do we do when we find the question? (Elicit the response, “underline the question and circle the question mark.) Yes, first we circle the question mark, because it tells me that this statement is a question and then we underline the question. Everyone circle the question mark and underline the question.”

Model finding the important information in the story problem that will help you solve the story problem (e.g. Read each sentence and ask, “Is there a number phrase in this sentence?” – Circle the number phrases.).

Model finding the number phrases that represent the “denominator” and the “numerators.” – “After we’ve found what we are solving for, what important information do we look for next? (Elicit the response, “the number phrases.”) Yes. What strategy can we use to find the number phrases? (Elicit the response, “read each sentence and ask, “is there a number phrase in this sentence?”) Good. Do that on your own now. (Monitor students as they do this.) What is the first number phrase?” (Elicit the appropriate response. *Continue this process for the remaining number phrases.)

Model deciding if all the important information is identified. – “When we’ve finished finding the important information, what do we need to check for? (Elicit the response, “to be sure we’ve found all of the important information.”) That’s correct. What two important things do I look for? (Elicit the response, “question” and “number phrases.”) Good. First, we looked to find what are solving for. How did we find what we are solving for? (Elicit the response, “we looked for the question, circled the question mark and underlined the sentence.”) How did we find the number phrases? (Elicit the response, “we read each sentence and asked, ‘is there a number phrase in this sentence?’”) Yes. How many number phrases did I find? (Elicit the appropriate response.) Great. we found ______ number phrases. Have we found all of the important information? (Elicit the response, “yes.”)”

Represent the fractional parts with concrete objects.

Model how to represent the denominator. – “Now that we know what we’re solving for and have circled the important information, then what do we do? (Elicit the response, “act” out our story problem with these concrete objects.) Yes. What is the first number phrase? (Elicit the appropriate response.) What is the name we use for total number of parts in a fraction problem? (Elicit the response, “denominator.”) How can we represent my first number phrase? (Elicit the response, “with a circle separated into ______ equal parts.) Excellent thinking! Place the appropriate circle in front of you. (Monitor students as they do this and check to see that all students use the appropriate concrete material.) How many total slices/pieces are there in each pizza? (Elicit the appropriate response.) Are the denominators for each pizza the same? (Elicit the response, “yes.”) What do we call denominators that are the same? (Elicit the response, “like denominators.”) Yes, we call denominators that are the same, ‘like denominators.’”

Model how to represent the numerator. – “What is the second number phrase? (Elicit the appropriate response.) What is the name for the number of parts of a whole in a fraction problem (Elicit the response, “numerator.”) How can we represent the numerator described by the second number phrase? (Elicit the appropriate response.) Great thinking! (Display the concrete materials that represent the first numerator. (Repeat this process for the third number phrase.) How many cheese slices were left? (Elicit the appropriate response.) How many pepperoni slices were left? (Elicit the appropriate response.) So what is the numerator for cheese slices? (Elicit the appropriate response.) What is the numerator for pepperoni slices? (Elicit the appropriate response.)

Model representing each “part” in fractional form.

“Now that we have our numerators identified and we know what our denominator is, what do we need to do in order to represent the fractional parts represented by the left over cheese and pepperoni slices? (Elicit the response, “place the slices/pieces that represent the left over cheese and pepperoni pizza on top of the circles that represent the whole pizzas.”) Excellent thinking! Everyone do that now. (Monitor students as they do this and check understanding.) How many cheese slices are there? (Elicit the appropriate response.) How many total cheese slices were there in the whole pizza? (Elicit the appropriate response.) How many pepperoni slices are there? (Elicit the appropriate response.) How many total pepperoni slices were there in the whole pepperoni pizza? (Elicit the appropriate response.) Why do we need to place our cheese and pepperoni slices on top of the whole pizzas? (Elicit the response, “to compare the fractional parts represented by the left over cheese slices and pepperoni slices.”) Wonderful thinking!”

· Model finding which fractional part is greater by comparing the concrete representations.

“How can we determine which fractional part is greater? (Elicit the response, “we can look at how much area/space the left over slices in the both our cheese and pepperoni pizzas slices take up) Yes. What else can you do? (Elicit the response, “run our finger around the area taken up by the cheese and pepperoni slices to feel the area.) Good thinking! Everyone do that now. (Monitor students and check for understanding.) Based on “seeing” and “feeling” the area taken up by your cheese and pepperoni slices, which fractional part is greatest, the fractional part represented by the cheese slices or the fractional part represented by the pepperoni slices? (Elicit the appropriate response.) What is the fraction name for the left over cheese slices? (Elicit the appropriate response.) What is the fraction name for the left over pepperoni slices? (Elicit the appropriate response.) Which fraction is greatest? (Elicit the appropriate response.)”

Model how to solve the story problem.

“Now that we know which fractional part is greatest, how can we solve the story problem? (Elicit the response, “answer the question we underlined.”) What is the question? (Elicit the appropriate response.) What is the answer? (Elicit the appropriate response.) Write the answer.” (Monitor students and check for understanding. *Write the answer on the chalkboard/dry-erase board after students have had the opportunity to respond.)

b. When you are confident students understand, ask individual students to direct the problem solving process or have the class direct you: Students ask questions and you and the students respond/perform the skill.

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Videos

Learning Objective 1: view  Clip 1, Clip 2, Clip 3

Compare fractional parts with like denominators using concrete objects that represent the area model (e.g. circle pieces, fraction bars/strips, Cuisenaire rods).

Learning Objective 2: view  Clip 1

Compare fractional parts with unlike denominators using concrete materials that represent the area model.

Learning Objective 3: view  Clip 1, Clip 2

Compare fractional parts with like denominators using concrete materials that represent the sets model (e.g. sets of counting chips, beans, unifix cubes, tickets.).

Learning Objective 4: view  Clip 1

Compare fractional parts with unlike denominators using concrete materials that represent sets model (e.g. sets of counting chips, beans, unifix cubes, tickets).

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