Concept attainmentA Math example: First the teacher chooses a concept to developed. (i.e. Math facts that equal 10) Begin by making list of both positive "yes" and negative no" examples: The examples are put onto sheets of paper or flash cards. Positive Examples: (Positive examples contain attributes of the concept to be taught) i.e. 5+5, 11-1, 10X1, 3+4+4, 12-2, 15-5, (4X2)+2, 9+1 Negative Examples: (for examples choose facts that do not have 10 as the answer) i.e. 6+6, 3+3, 12-4, 3X3, 4X4, 16-5, 6X2, 3+4+6, 2+(2X3), 16-10 Designate one area of the chalkboard for the positive examples and one area for negative examples. A chart could be set up at the front of the room with two columns - one marked YES and the other marked NO. Present the first card by saying, "This is a YES." Place it under the appropriate column. i.e. 5+5 is a YES Present the next card and say, "This is a NO." Place it under the NO column. i.e. 6+6 is a NO Repeat this process until there are three examples under each column. Ask the class to look at the three examples under the YES column and discuss how they are alike. (i.e. 5+5, 11-1, 2X5) Ask "What do they have in common?" For the next tree examples under each column, ask the students to decide if the examples go under YES or NO. At this point, there are 6 examples under each column. Several students will have identified the concept but it is important that they not tell it out loud to the class. They can however showthat they have caught on by giving an example of their own for each column. At this point, the examples are student-generated. Ask the class if anyone else has the concept in mind. Students who have not yet defined the concept are still busy trying to see the similarities of the YES examples. Place at least three more examples under each column that are student-generated. Discuss the process with the class. Once most students have caught on, they can define the concept. Once they have pointed out that everything under the YES column has an answer of 10, then print a new heading at the top of the column (10 Facts). The print a new heading for the NO column (Not 10 Facts).Concept Attainment Lesson on Fractions Learner Objective: Students will demonstrate their understanding of a fraction by defining and classifying examples and non-examples with 100% accuracy. Learner Background: Students have prior knowledge of whole numbers. Modifications: Teaching Model/Strategy: The teaching model for this lesson is concept attainment. This strategy was chosen for this lesson because the students are being introduced to fractions for a first time. Using a concept attainment model for this lesson enables students to form a clear definition of fractions by using examples and non-examples.

Subject Area: MathematicsSpecific Content: Prime and compositenumbersGrade Level:5thLength of Lesson: 35-40 minutesProceduresList each procedure according to stages of Concept Attainment1.Mental math warm-up:6 x 4Divided by 2Times 3Plus 4Times Times 8Divided by 4Minus 4Minus 24Divided by 42.Working with critical attributes of each set toady YES and NO examples3.Scratch paper provided4.First set of YES/NO examples several offered5.Speak with a partner6.First hypothesis comments?7.Additional YES/NO examples8.Additional comments on first hypothesis9.Additional YES/NO examples10.New hypotheses11.Comments12.Additional YES/NO examples13.Elicit thinking, comments on responses14.Clarify vocabulary as needed based on student responses15.Review concepts of prime and composite numbers directlyInclude each question you are planning to ask students in the appropriate place in your lesson plan 1.What is your hypothesis? (This is repeated throughout the lesson.)2.Why do you think this is so? (This is repeated throughout the lesson.)3.What do you think about what ______ just said? (This repeated throughout the lesson.)4.Do you see anything new in these examples?Can you draw your path to help explain your thinking?Did any other tables?7.How can you explain your thinking?8.Can you provide some new examples of YES/NO numbers?

Relationship to Standards: Content Standard 1: Number Sense; Develop a sense of magnitude of numbers by ordering and comparing whole numbers, commonly used fractions, decimals and money amounts. Methods of Assessment: Teacher observation, and whole group discussion. Materials and Instructional Arrangements: The Hershey's Milk Chocolate Bar Fractions Book Chart Paper Markers Bag of Examples/Non-Examples of fractions Students will be seated at their designated table while I give examples and non-examples of fractions. Their attention will be directed to the chart paper hanging in front of the room. When it is time to read The Hersheys Milk Chocolate Bar Fractions Book, students will be called to sit on the carpet in their special spots. Lesson Set-up: To get the childrens attentions I will tell them that I am looking for star listeners. They will be told that they need to keep their eyes on me; otherwise they will miss the mystery that they as detectives have to figure out. The children will turn and talk to their neighbor about what they think the magic word is. Initiation: Explain to students that you would like them to be "detectives". They must examine the "clues" to figure out what math word I am thinking about. I will give examples of a fraction by pulling out Leggo pieces that can be split into many pieces, and other toys that cannot be split. I will explain before I pull out the toys that some are examples and some are not, that define a fraction. We will list them on the chart paper in front of the room. Sequence of the learning activities:1.) I will explain that the students are detectives and they need to figure out the mystery math word.2.) I will tell them that in my secret bag I have examples and non-examples of the math word I am thinking about.3.) I will write on the board examples and non-examples, and as I pull out the first toy have the students turn and talk to a neighbor and describe the toy.4.) Then I will call on students to tell me how they described the toy. Once we agree on the description I will do the first example and place it under the correct heading, either example/non-example.5.) As we continue with the lesson I will call on students to tell me whether they think its an example/non-example.6.) The teacher will write down the characteristics of the examples and non-examples on the chart paper.7.) Leave all toys in plain sight for all to see.8.) Students will eventually notice that all the examples can be broken into small pieces, ex) the Leggo toys as compared to a teddy bear.9.) Guide students through process; pull out more examples than non-examples.10.) After the secret bag is empty steer childrens thought process to writing a definition of the word.11.) The definition should be something in this mannerism, The math word means something that was once a whole can be broken down into parts.12.) Once the students have come up with a definition tell them the secret word, by introducing The Hershey's Milk Chocolate Bar Fractions Book.13.) Students will then recognize the strong relationship between the examples and the book. Closure: I will ask the students to tell me what they learned. Together we will review the definition of a fraction. I will tell them that todays lesson will help them out as they learn a lot more about fractions over the next few weeks. Explain that they will learn how to write, identify, and name fractions. Congratulate for a job well done! Reflection on Practice: Student Achievement: A majority of the students in the class understood that the examples can be broken down into pieces. Once they got that idea, they predicted that a fraction is a part of a whole, just like the leggo toys. The students that understood the concepts helped their friends seated at their tables. I had to keep reinforcing the examples and non-examples for the students who did not fully understand the concept. Teacher Efficacy: The initiation worked well for this lesson. I was enthusiastic about teaching the concept to the children. The kids noticed that I was excited and that too became excited about investigating the examples and non-examples to write a definition of the new math word. Some of the students did not catch on to the concept attainment model; this might be because my cooperating does not use this model often. My directions may have been confusing to some students because instead of telling them facts and definitions, they had to form them on their own. I will be sure to reinforce the definition of a fraction to the children immediately. I will explain that this lesson was a fun way to learn about fractions. In the long range I will be sure to carefully pick out examples and non-examples that are clear to all of my students. I thought toys would be a good resource because all children are familiar with them. One reasonable alternative approach would be to teach fractions as a direct instruction lesson. I would give the students the definition, and then we would explore the world of fractions. This would be a clear approach to teaching this lesson.UNIT 1Concept Attainment: Triangular NumbersSubject Area:MathematicsSpecific Content:Number TheoryGrade Level:4thTiming:45-55 minutesInstructional Objective:1.To introduce triangular numbers.2.Learners list the first 5 triangular numbers and their pictorial representations.Long-Term Objective:1. Learners to derive the formula for the nth triangular number.2.Learner to see geometric ideas used to discover properties of triangular numbers.Prerequisite Knowledge and Skills:1. Experience with Concept Attainment2. Basic arithmetic3. Elementary geometric factsWhy is the Content of This Lesson Relevant?This content uses geometric and counting ideas to derive the formula for the nth triangular number. Thereby illustrating the potential of using geometric ideas to discover elementary properties of triangular numbers.Materials:1.Graph paper, pencil, eraser2.Paper to record which the first 5 triangular number3.Table of example and non-examples ( See pictures 2,3,4 and 5 at the beginning of the article.)Slide 2 is the top of the the table of examplesand Slide 3 is the remainder of the table of examplesModel ofTeaching:Concept AttainmentProceduresThe procedures according to the phases of Concept Attainment1. State the objectives of the lesson. "I'm thinking of numbers that have a certaingeometric property. You are to discover the numbers from the examples representing them."2. The examples are presented in a two-column table. The first column is labeled the YES. It contains examples of the concept. The second column is labeled NO. The examples in the NO list do not have the properties of the concept.3. Teacher: After the first line of the table, present only one line at a time.4. Direct learners to read the first YES and NO examples.5. Instruct learners to formulate a hypothesis of the concept using the examples in the YES list.Say: "The examples in the YES list are a representations of numbers I am thinking about. Do you see a common property they have? What name would you use to classify this number?"Also state: Use the examples in the NO list to help you to fine tune you thoughts of the concept?6. Direct learners to count the objects that make up the example. And ask, "Can you make the NO examples look like the Yes examples?"7. Introduce another YES/No example from the table and repeat 4, 5, 6.8. Continue to exhaust the table. Add more if needed. When the students get the concept, direct them to add five additional YES examples to the table.9. Ask: Do you see a ways to express the YES examples as numbers? If so, what numbers represents the first 7 examples in the YES list?10.State: It is the convention to make the first number of YES list to be one.11. How would you explain your arrival at the name and property of the numbers in the YES column?Counseling Aids:Use these comments to redirect struggling students. Remind learners that the geometric examples in the YES list represent a number.To ensure learners do not think the table contains all triangular numbers, ask, "Is there a largest YES number?"Assessment Criteria:What evidence will demonstrate your students'learningtoday?Learners who give the correct geometric pattern for a given triangular number will have learned the concept.Identify cultural concerns and describe how you will address them.To ensure learners understand what a triangle is: Review triangles and rectangles before introducing this unit.Technology - What technology might enhance this lesson?Small white boards can be helpful.UNIT 2The Formula for the NthTriangular NumbersSubject Area:MathematicsSpecific Content:Number TheoryGrade Level:5thTiming:45-55 minutesInstructional Objective:1.Derive the formula for the nth triangular number.2.Learners compute five triangular numbers using the formula.Long-Term Objective:Learners discover properties of triangular numbers using geometric and algebraic properties.Prerequisite Knowledge and Skills:1.Basic algebraic properties and2.Elementary geometric factsWhy is the Content of This Lesson Relevant?It demonstrates a geometric method to compute the nth triangular number. Further, it presents a useful approach to explore other properties of triangular numbers.We will use Tnto represent the nthtriangular number.The Approach1. Discuss whether the properties of the image representing a triangular number are affected if its representation is rotated?"2. Direct learners to draw the third triangular number.3. Ask them to make a copy of the geometric representation ofT3on the same page butrotated through 180 degrees.Learners should have the image of Slide 4.4. Direct learners to merge the original representation ofT3and the rotated representation of T3intoa rectangle.Learners should have the image of Slide 5.So 2 timesT3is the count in the rectangle, which is 3 times 4 =12.SoT3= 3(3+1)/2Ask: Do you think this formula will hold for the 4th triangular number? That is, if the "3" is replaced with a "4" in the formula, will the formula give a correct answer?Direct them to make the substitution and verify.Ask them to replace the "3" in the formula with a "5". Does the formula give the correct count forT5?Ask: "Do you think that replacing "3" with "15", will gives the correct count for the 15thtriangular numbers?"Ask for an explanation of their belief.Discuss whether replacing the "3" with an "n" limits in the derivation of the formula limits the conclusion.UNIT 3SomeApplications1. If three hundred people gather and each person shakes the hand of every other person in the group exactly once, the total number of handshakes will be T299.Teacher: To help learners understand this fact, lead them through this example. It clearly reveals the secrets of the solution.The Example: Consider a group of four people. Call them Able, Baker, Charlie and Dan. A visual representation of the group is four dots labeled A, B, C and D.Represent a handshake between two people in the group by a line segment joining the points that represent the two people. Thus, a line segment connecting A and B means Able and Baker shook hands. We will count the number of handshakes when all group members shake hands only once.Direct students to connect A to B, A to C and finally A to D. Ask the question: What do these connections represent? These connections account for three of the total number of handshakes when these group members shake hands once.Have students to connect B to C and B to D. Ask for the meaning of these connections. Then ask them, "Why don't we connect B to A?" These connections account for 2 more connections within the total handshakes.Finally have students to connect C to D. This connection accounts for the final handshake in the group.The total number of connections among the four points is 3+2+1 =T(4-1)= T3.So the total number of handshakes that takes place in the group when each person shakes the hand of every other person exactly once is T3.Let's go a little deeper.If a group of "n" objects have a single connection among themselves, then the number of connections is Tn-1. This is just a restatement of the handshake problem.The important concepts in the handshake problem are:1. The interpretation of the word " handshake",2. The number of objects in the group, "n" and3. Each pair has only one connection.To complete the generalization of the situation, have students to complete the following table.Then instruct them to add their contribution of "objects in a group " and "an action to represent a handshake" to the table. See the first four table entries for ideas.Remind them that the action can only take place once between the objects.Direct them to state the meaning of these substitutions.Remind them that under all circumstances, the total number of actions taking place in the group is Tn-l.Objectsin a groupAction among oncePeople------------------------------------touch each other oncePeople------------------------------------telephone each other onceTrees------------------------------------- in a grove grow to each otherCancer cells in an organ-------------stick to another in one placeTowns-------------------------------------------------?Airports-----------------------------------------------?2. Read Ezekiel 1: 4 - 8. How many connections are there between these four creatures? To see the scripture:Go tohttp://www.biblegateway.comand in the space for scripture lookup, supply Ezekiel 1:4-83. Certain birds tend to migrate in the geometric pattern used to represent triangular numbers.[1] John Sweller and Graham Cooper, The Use of Worked Examples as a Substitute for Problem Solving in Learning Algebra,Cognition and Instruction 1985.[2] Romelia Morales,Valerie Shute and James Pellegrino,Developmental Differences in Understanding and Solving Simple Mathematics Word Problems,Cognition and Instruction 1985.Concept Formation Model Math LessonGrade/Level3

Students23 students, 13 girls, 10 boys8 students are Advanced, 11 students are Proficient, and 4 students are Basic in Math.4 ELs, one being at risk3 Attention Deficit (not officially identified)

Subject Area(s)Mathematics

Concept(s)Students will observe, descibe and classify geometric shapes using their own criteria, and label the groups based on the concept formation model.

StandardsCalifornia Academic Content StandardsMathematics, Grade 3Measurement and Geometry | 2.0 Students describe and compare the attributes of plane and solid geometric figures and use their understanding to show relationships and solve problems:- Standard 2.1: Identify, describe, and classify polygons (including pentagons, hexagons, and octagons).

Objective(s)Using solid geometric figures, students will observe the shapes, make a list of the different attributes they observe, and group similar objects together, and then label the groups. Students will be graded based on a simple rubric.Students will listen to directions, speak with classmates in a group to discuss attributes of the shapes, and write a list of attributes and then write groups with labels.To conclude the lesson, students will discuss as a class what groups they formed and how they labeled them, and watch a short video from Brainpop.com about geometric shapes.

Prerequisite Background Skills/Knowledge

Vocabulary/Language Skillspolygon - many sidesattributes or characteristics - distinguishing features, what you can observe with your eyes that make that shape different

Classroom ManagementI will begin the lesson with the students seated at their desks. I will explain what we are going to do and review vocabulary. I will then explain that we are going to work in groups and I will explain what each person's job in that group will be.I have identified 3 different jobs for the groups, so I am making 8 groups of 3 students (1 group will have 2 students since there are 23 students). I will choose groups by name sticks, althoughI will try to spread the boys out since there are more girls than boys. When I call the groups up by name,I will hand them each a "job" card with the description of their responsibilities printed on it. I will tell the group which area of the room I want them to sit on the floor and work. The 3 jobs are as follows:Materials - You are responsible for getting the materials for your groups and returning the materials at the end of the lesson.Making a List - You are responsible for making a list of all the characteristics your group observes about the shapes.Groups - You are responsible for placing the objects in groups and labeling the groups.When group time is over, students will return materials to the front of the room and return to their desks. We will discuss what groups they formed and how they labeled them, and then watch a short video about geometric shapes.

Models of InstructionConcept Formation and direct instruction

MATERIALS AND TECHNOLOGY

Materials8 bags of 10 geometric shapes of assorted shapes, colors, and sizes, 16 pieces of lined paper, 8 pencils and clipboards (one for each group) , job description cards, tickets.

Technology ResourcesComputer with internet, an account with Brainpop.com, projector, LCD screen, printer

Technology AppliedI will use the computer to type out and print "job" cards for the students in their groups.I will use the computer to access a short video about polygons on brainpop.com and project it onto the LCD screen.

PROCEDURE

OpenToday we are going to explore different shapes called polygons. Let's break down the word and see if we can figure out what it means. Have you hear the word hexagon or pentagon? What do you think the word segment "gon" means? (I will call on someone.) Right! It means sides. Does any know what poly means? Right! It means many. So, today we are going to work in groups to observe different attributes of polygons. Does anyone know what the word attributes means? (I will write it on the board and call on someone). Attributes means distinguishing features, or what you can observe with your eyes that make that shape different from another shape, or unique.

InputI will talk to them about the different attributes I can observe about the students in the room. Some people have dark hair, brown eyes, white tennis shoes. I will also talk about how I could group them together. All the students with brown eyes, all the students with white shoes, and we will discuss how students can fit into more than one group.I will explain to them that they are going to make a list of all the different things they can observe about the shapes and write a list. I will give examples and begin a short list on the board as an example:4 sides, red, 6 facesThen I will explain the them how I want them to find shapes that have similar attributes and put them in a group and give the group a name or a label. "What goes together?How can we label this group?" When I feel that the students understand the process of List, Group, Label, I will break them into groups and hand out materials so they can begin their observations.

Guided PracticeWhen the students are seated on the floor with their materials, I will tell them to dump the shapes onto the floor and begin to make a list of all the different attributes they can observe.I will walk around the room and make sure the students understand and are making a list of all the different attributes they can observe about the shapes.When the groups have a good sized list of attributes, I will direct them to look at the list and see if you can find some similar things. What goes together? How can we put the shapes in groups? I will walk around and encourage the students and make sure they are understanding and working on grouping the shapes and labeling the groups.When the students seem to be done with this activity, I will ring a bell and have them return the materials to the front of the room and return to their desks.

Independent Practice

CloseI will ask the students to share how they groups the objects and what labels they gave the groups.We will then watch a short video clip from brainpop.com introducing polygons.

ASSESSMENT/REFLECTION

AssessmentDuring the lesson I will be walking around to the groups to make sure they are understanding the lesson and Iwill give guidance and feedback as needed.I will collect the papers at the end of the lesson and use group assessment. To assess student achievement, I have created a 4 point rubric:4: Made an extensive list of attributes that the shapes have. Has 3 or more groups. Has groups labeled.3: Has a short list of attributes. Has 2 groups. Has groups labels.2: Has a short list of attributes. Has only 1 group labeled.1: Has a short list at attributes. Has 1 or no groups and groups are not labeled.