Description: A problem context of ordering and sharing pizza is used.
Purpose: to help students make meaningful connections between what they have experienced with ordering and sharing pizza and identifying and making equivalent fractions.
*The following is a description of how you 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: Identify and represent equivalent fractions using concrete objects with an area model.
Materials:
Teacher –
Cardboard pizzas representing a whole, halves, fourths and eighths
Whiteboard or other visual display or word cards with half, fourth, eighth
Description:
1.) L ink to students’ prior knowledge of identifying and representing fractions.
For Example:
How many of you like to eat pizza? What is your favorite kind of pizza? When we share a pizza with others, we are each getting a part of the whole pizza. We can divide the pizza into halves (point to word half on board) or fourths (point to word) or even eighths (point to word). We know that a half, a fourth and an eighth are called fractions. They mean that we have taken a whole, like this whole pizza and divided it equally into parts (show with cardboard pizza slices).
2.) I dentify the skill students will learn: Identifying and representing equivalent fractions using concrete objects.
For Example:
Today we are going to learn how to tell if each person is getting the same amount of pizza by looking at the equal parts of pizzas. We are going to look at the different parts and see which ones show the same amount. We are going to study how to identify and make Equivalent (point to word on board) parts. Equivalent parts look different but represent the same amount of a whole.
3.) P rovide rationale/meaning for identifying and representing equivalent fractions.
For Example:
Knowing how to do problems like this will help us when we work with fractions. It will help you when you shop to decide how to share things. When you have to share and equally divide things like pizzas, or hershey bars, you can make sure that everyone gets the same amount.
Purpose: to provide students with a clear teacher model of how to identify and make equivalent fractions using concrete objects.
Learning Objective 1: Identify and represent equivalent fractions using concrete objects with an area model.
* This skill should first be taught using an area model, then a measurement model, and then a sets model. After completing all phases of the instructional plan with an area model, and measuring student mastery, the concept should then be taught using a measurement model. After completing all phases of the instructional plan with a measurement model, and measuring student mastery, the concept would then be taught using a sets model.
Materials:
Teacher –
Pizza boxes from different types of pizzas
Cardboard pizzas that are divided into fourths, eighths, halves, etc.
Visual display area
Description:
A. Break down the skill of identifying and representing equivalent fractions using concrete objects with an area model.
1.) Identify the fractional part shown by each object.
2.) Compare fractional parts.
Learning Objective 2: Identify and represent equivalent fractions using concrete objects with a measurement model.
* Identifying and representing equivalent fractions should first be taught using an area model, then a measurement model, and then a sets model. After completing all phases of the instructional plan with a measurement model, and measuring student mastery, the concept should then be taught using a sets model.
Materials:
Teacher –
Fraction strips or bars, or Cuisenaire rods
Visual display area
Description:
A. Break down the skill of identifying and representing equivalent fractional parts using concrete objects with a measurement model.
1.) Identify the fractional part shown by each object.
2.) Compare fractional parts.
Learning Objective 3: Identify and represent equivalent fractions using concrete objects with a sets model.
* This skill should first be taught after using an area model, and then a measurement model. After completing all phases of the instructional plan with an area and then a measurement model, and measuring student mastery, the concept should then be taught using a sets model.
Materials:
Teacher –
Set of mini pizzas (real or construction paper) placed in two horizontal rows of four.
Four pepperoni slices (real or construction paper)
String
Visual display area
Description:
A. Break down the skill of identifying and representing equivalent fractional parts using concrete objects with a sets model.
1.) Identify the fractional part represented by each part of the set.
2.) Compare fractional parts.
Purpose: to provide students an opportunity to build their initial understanding of identifying and representing equivalent fractions using concrete objects and to evaluate your students’ levels of understanding after you have initially modeled the skill.
* The following description is for Learning Objective 1: Identify and represent equivalent fractions using concrete objects with an area model. A similar process could be used for the other learning objectives in this plan.
Materials:
Teacher –
Cardboard pizzas that are divided into fourths, eighths, etc.
Word cards with ‘Equivalent’ written on it
Bags of fraction pieces – one bag has halves, another fourths, etc.
Visual display area
Students -
Bags of fraction pieces – one bag has halves, another fourths, etc. Each set of pieces is colored a different color
Index cards with equivalent written on them
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. (Examples of choices are shown in red.)
Identify the fractional part shown by each object.
Identify the whole(s).
We have looking at finding equivalent fractional parts. Now, I am going to give you another problem and have you help me with it. I have two pizzas here. I have cut up one pizza into six pieces and one pizza into eight pieces. First I need to decide if these two pizzas are the same size. How do you think we could do that? Right, _____ please come up and show us whether these two pizzas are the same size. Boys and girls, _____ put all the pieces for the pizza I cut up into sixths and laid them on top of the pizza that I cut into eighths. Are they the same size? Yes, they are.
Identify the number of parts that are in each fractional part
I said I cut this pizza into eighth equal pieces, and I cut this pizza into six equal pieces. I am going to give four of the eight pieces of this pizza to Jason. If I give four of the eight pieces to Jason, I wonder what fractional part of the pizza he will have? Well, four of eight pieces will be four eighths. And I am going to give three of the six pieces to Marisa. If I give three of the six pieces to Marisa. What fractional part of the pizza does she have? Right, three sixths. So, what fractional part of this pizza does Jason have? Right, four eighths. And what fractional part of this pizza does Marisa have? Correct again, three sixths. Well, they have different fractional parts, but I wonder if they have the same amount of pizza?
Compare fractional parts.
Manipulate the pieces to see if they take up the same amount of space.
Each person has a different number of pieces. Jason, let’s put your pieces down first. 1,2,3,4 of eight pieces. Jason has four eighths of the pizza. How can we see if Marisa has the same amount? Right, we can put her pieces on top of Jason’s. If they cover Jason’s and take up the same amount of space, then we know that they have the same amount of pizza. Well, let’s try it.
Label fractional parts as equivalent/nonequivalent
Look, Marisa’s pieces cover all of Jason’s pieces. They take up the same amount of space. That tells us that Jason and Marisa have the same amount of pizza. They have equivalent fractional parts (put word card next to display of pizza pieces). How did we know that Jason’s four eighths were equivalent to Marisa’s three sixths? Right, we put them on top of each other to see if they took up the same area, the same amount of space.
b. Maintain a high level of teacher direction/support for another example if students demonstrate mis../../../understanding/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.
Identify the fractional part shown by each object.
Identify the whole(s).
For the next problem, I want you to help me even more. This time I am going to use fraction circles. Look at these bags. In this bag I have twelve pieces that make a circle. In this bag I have six pieces that make a circle. One of these circles is divided into twelve pieces and one of them is divided into six pieces. What’s the first thing I need to do? Right, first I need to decide if these two circles are the same size. How do you think we could do that? Right, _____ please come up and show us how to do that. Are they the same size? Yes, they are.
Identify the number of parts that are in each fractional part
Let’s see if four pieces from this circle (point to circle divided into twelfths) is the same as two pieces from this circle (point to circle divided into sixths). What fractional part is shown by these four pieces? Well, the circle is divided into twelve equal parts, and I have four of them, so that is four twelfths. What fractional part is shown by these two pieces (point to sixths). Right, two sixths, because the circle is divided into six pieces and I have two of them.
Compare fractional parts.
Manipulate the pieces to see if they take up the same amount of space.
Now what do I need to do? Right, I need to put them on top of each other. Do they take up the same amount of space? Yes, they do.
Label fractional parts as equivalent/nonequivalent.
These fractional parts take up the same amount of space, so we can say that they are equivalent (label with word card). What is the fraction here? Right, four twelfths is equivalent to___ two sixths.
b. Maintain a medium level of teacher direction/support for another example if students demonstrate mis../../../understanding/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 at their desks as you work together.
Identify the fractional part shown by each object.
Identify the whole(s).
For the next problem, I am going to have you do it at your desks. Each of you has two bags that have fraction circle pieces in them. One of the bags has red fraction pieces and one of the bags has blue fraction pieces. What’s the first thing you need to do? Right, you need to decide if the two circles are the same size. Show me how you are going to do that? Right, you put all the red pieces down to make a circle and then put all the blue pieces down to make another circle on top of them. Each person’s circles are the same size. How many red fraction pieces are there ? Right, four. How many blue fraction pieces? Right, eight.
Identify the number of parts that are in each fractional part
Now I want you to pick 3 of the red fraction pieces and 6 of the blue pieces. What fractional part is shown by these three red pieces? Right, three fourths. What fractional part is shown by the six blue pieces? Right, six eighths.
Compare fractional parts.
Manipulate the pieces to see if they take up the same amount of space.
Now what do you need to do? Right, you need to put them on top of each other. Do they take up the same amount of space? Yes, they do.
Label fractional parts as equivalent/nonequivalent
If fractional parts take up the same amount of space, we say that they are ______? Right, equivalent. Label your equivalent fraction pieces with an index card that says equivalent. Tell me what are the two equivalent fractions you have shown? Right, three fourths is equivalent to six eighths.
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.
Purpose: to provide students multiple practice opportunities to identify equivalent fractions with concrete objects.
Learning Objective 1: Identify and represent equivalent fractions using concrete objects with an area model
Instructional Game/Cooperative Learning
Materials:
Teacher –
Bell or timer
Envelope with fraction pieces and answer sheet to model how to do activity
Sample cue sheet and response sheet to use when introducing and modeling the activity
Set of fraction pieces on overhead or other visual display to use when providing whole class review
Students -
A folder with 10 numbered envelopes with 2 sets of fraction pieces and an answer sheet.
Cue sheets with questions: “Did you place the parts together?”; “ Do the parts take up the same amount of space?”
Response sheets with two columns numbered 1-10, labeled equivalent/not equivalent.
Description:
Activity:
Students will work in pairs. Each student will have at turn being a coach and being a player. The coach will have a set of 10 envelopes with fraction pieces. The player will choose a number 1-10, take the corresponding envelope, and lay the fraction pieces from the envelope out on the table. For example, envelope # 1 might have the fraction pieces one half and two fourths. The player may manipulate the pieces if he/she needs to and then will put a check in the appropriate column on the response sheet. The coach then checks the answer sheet. If the player has checked the correct column, the coach will put a 2 next to the answer. If the player does not have the correct answer, the coach will use the cue sheet to remind the player to put the fraction pieces on top of each other and make sure that the fraction pieces take up the same amount of space. The player then can try the problem again. If he/she checks the correct column this time, the coach puts a 1 next to the answer. After the player has completed all ten problems, the coach will total the player’s score and record it at the top of the page. After all pairs have completed 10 problems, the teacher will review each problem with the whole class, asking selected players to come forward and show the rest of the class the answer to a specific problem. The coaches and players then switch roles using a different set of envelopes.
Structured Peer Tutoring Steps:
1.) Select pair groups and assign each pair a place to practice (try to match students of varying achievement levels if possible).
2.) Review directions for completing peer tutoring activity and relevant classroom rules. Practice specific peer tutoring procedures as needed (see step #4).
3.) Model how to perform the skill(s) within the context of the activity before students begin the activity.
4.) Divide the practice period into two equal segments of time. One student in each pair will be the “player” and will respond to the prompts. The other “player” will be the “coach” and will give the “player” the prompts and evaluate the player’s response. The coach will also provide positive reinforcement, corrective feedback, and assign points based on the player’s responses (e.g. two points for correct response the first time, one point for the correct response the second time.).
5.) Provide time for student questions.
6.) Signal students to begin.
7.) Signal students when it is time to check problems with the whole class and then switch roles.
8.) Monitor students as they work in pairs. Provide positive reinforcement for both “trying hard,” responding appropriately, and for students using appropriate tutoring behaviors. Also, provide corrective feedback and modeling as needed.
Purpose: to provide students multiple practice opportunities to identify and represent equivalent fractions with concrete objects.
Learning Objective 1: Identify and represent equivalent fractions using concrete objects with an area model
Instructional Game/Cooperative Learning
Materials:
Teacher –
Bell or timer
Sets of fraction pieces that can be visually displayed (overhead, flannel board, magnetic strips, etc.)
Sheet/chart to record team scores
Students -
Envelopes with fraction pieces
Description:
Activity:
Students will work in groups of 4 or 5 students. Each student will have envelopes with one or more set of fraction pieces (e.g. student A has an envelope with fourths, student B has thirds, student C has sixths, and student D has eighths). The teacher will display a fractional part (e.g. one half, three fourths, etc) and each team will need to show as many equivalent fractions as they can make in the time allotted. After the teacher rings the bell, one member of each team will share one of their solutions. Teams can earn points for each correct decision.
Cooperative Learning Groups Steps:
1.) Provide explicit directions for the cooperative group activity including what you will do, what students will do, and reinforce any behavioral expectations for the game.
2.) Arrange students in cooperative groups. Groups should include students of varying skill levels.
3.) Assign roles to individual group members and explain them:
a. Materials manager (gets the envelopes for the group.)
b. Time Keeper (makes sure that students are on task and complete each problem in time allotted.)
c. Turn taker (makes sure that each member of the group gets a chance to contribute to solving problem)
d. Encourager(s) (encourages each person)
4.) Distribute materials.
5.) Model one example of skill(s).
a. Listen to problem.
b. Show solution(s)
c. Make sure that the team agrees with the decision before time is called.
6.) Review/model appropriate cooperative group behaviors and expectations.
a. Agree or disagree with a teammate’s decision.
b. Listen while teams are sharing responses.
7.) Provide opportunity for students to ask questions.
8.) Teacher monitors and provides specific corrective feedback & positive reinforcement.
a. Circulate around the tables and check on children’s responses.
b. Make sure that each child receives feedback on his/her decision.
c. Ask each child in the class to share his/her decisions at least once either with the entire class or individually with the teacher.
Purpose: to provide you with continuous data for evaluating student learning and whether your instruction is effective. It also provides students a way to visualize their learning/progress.
Materials:
Teacher –
Appropriate prompts if they will be oral prompts
Appropriate visual cues when prompting orally
Student –
Appropriate response sheet/curriculum slice/probe
Graph/chart
Description:
Steps for Conducting Continuous Monitoring and Charting of Student Performance:
1.) Choose whether students should be evaluated at the receptive/recognition level or the expressive level.
2.) Choose an appropriate criteria to indicate mastery.
3.) Provide appropriate number of prompts in an appropriate format (receptive/recognition or expressive) so students can respond.
* Based on the skill, your students’ learning characteristics, and your preference, the curriculum slice or probe could be written in nature (e.g. a sheet with appropriate prompts; index cards with appropriate prompts), or oral in nature with visual cues (ask students to tell you which of several visually displayed solutions is the correct solution for a problem), or a combination of written curriculum slices/probes and oral prompts with visual cues (e.g. ask students to demonstrate solution to given oral problem).
4.) Distribute to students the curriculum slice/probe/response sheet/concrete materials.
5.) Give directions.
6.) Conduct evaluation.
7.) Count corrects and incorrects/mistakes (you and/or students can do this depending on the type of curriculum slice/probe used – see step #3).
8.) You and/or students plot their scores on a suitable graph/chart. A goal line that represents the proficiency (for concrete level skills, this should be 100% – 5 out of 5 corrects) should be visible on each students’ graph/chart).
9.) Discuss with children their progress as it relates to the goal line and their previous performance. Prompt them to self-evaluate.
10.) Evaluate whether student(s) is ready to move to the next level of understanding or has mastered the skill at using the following guide:
Concrete Level: demonstrates 100% accuracy (given 3 to 5 response tasks) over three consecutive days.
11.) Determine whether you need to alter or modify your instruction based on student performance.
Flexible Math Interview
Purpose: to provide you with additional diagnostic information in order to check student understanding and plan and/or modify instruction accordingly.
Materials:
Sets of fraction pieces envelopes
Description:
With individual students or in small groups, the teacher will have the student to solve an equivalence problem. The teacher will show the student(s) two sets of fraction pieces (e.g. one half and three sixths or one half and one third, etc.) .The teacher will ask the student(s) to identify the given fractions as equivalent or not equivalent and explain the decision. The teacher should note errors and/or misconceptions while the student is teaching, but the teacher should not stop the student for correction purposes. By having the student complete the entire explanation, the teacher will gain a better understanding of the student’s thinking. The teacher should confer with the student regarding specific errors or misconceptions after the activity is over.
Purpose: to provide periodic student practice activities and teacher directed review of this skill after students have mastered it.
1. Problem of the Day
Materials:
Concrete objects that depict fractional parts displayed near an area that has two columns with words equivalent, not equivalent
Description:
The teacher will present a problem of the day verbally and by displaying the items in a designated area. Students will decide if the objects show equivalent or non-equivalent fractions and will record their responses by putting their initials under the appropriate column. Teacher will review the problem after all students have responded. This should initially be done each day, then 2 times/week, weekly, bi weekly, and then intermittently.
2. Classroom Routines
Materials:
Class roster
Description:
Students will be told to line up for lunch and show equivalent fractions (e.g. one half and three sixths).
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