Unit Name:



Unit Name: Number Theory and Fractions Grade Level: 6

Subject/Topic Area(s): Mathematics – Numbers to Algebra Time Frame: 4 weeks

Class Description/Developmental Needs of Students:

The class has 16 girls and 14 boys. There is one IEP student with language and speech needs who needs preferential seating and shortened assignments. He sits in the front row in front of the teacher desk. He is also given fewer problems on the homework assignments. He also receives a little extra one on one attention during teacher circulation of class. He is also offered the option to come in for extra help at lunch, but he has yet to do that. There are four EL students ranging in levels from Beginner to Early Advanced. Assignments are shortened for them and the CELDT 1 student sits next to the CELDT 3 student in case translations are required. There is one 504 student for a speech impediment that rarely presents in class. There is also one at-risk student that has difficulty focusing and has a history of not turning in assignments. He is a 7th grader in a 6th grade class. He is given extra attention during teacher circulation to show him support and encouragement and he is seated with other students who are responsible and would never give him a hard time for being in 7th grade.

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|STAGE ONE: DESIRED RESULTS |

|ESTABLISHED GOALS (State and/or National Content Standard (s)): |

|CA- California Common Core State Standards (2012) |

|Subject: Mathematics |

|Grade: Grade 6 |

|Domain: The Number System 6.NS |

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|Area: Compute fluently with multi-digit numbers and find common factors and multiples. |

|Standard: |

|2. Fluently divide multi-digit numbers using the standard algorithm. |

|Standard: |

|3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. |

|Standard: |

|4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use |

|the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. |

|example, express 36 + 8 as 4 (9 + 2). |

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|Area: Apply and extend previous understandings of numbers to the system of rational numbers. |

|Standard: |

|6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on|

|the line and in the plane with negative number coordinates. |

|c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational |

|numbers on a coordinate plane. |

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|UNDERSTANDINGS: |ESSENTIAL QUESTIONS: |

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|Different equivalent values can be written in a variety of ways |How can equivalency be determined between numbers represented in various ways, |

|Different representations of numbers can be converted into each other |i.e. fractions, decimals, mixed number, etc. |

|Varying values can be compared and ordered |How can numbers be represented in different ways? |

|Greatest Common Factors are the basis for fractional simplification and |How can we compare numbers of varying value? |

|Distributive Property |How do you find and use in real life the greatest common factor of two whole |

|Prime factors are building blocks to various numerical representations |numbers? |

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|KNOWLEDGE: |SKILLS: |

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|Fundamental Theorem of Arithmetic as it can be represented by prime factorization|Finding the prime factorization, greatest common divisor, and least common |

|Correlation between prime factorization and greatest common factor and fraction |multiple |

|simplification |Writing equivalent fractions, mixed numbers, and decimals |

|Know Equivalency, Mixed Numbers, Fractions, Decimals, Place Value, Terminating |Comparing and ordering rational numbers |

|Decimals, Repeating Decimals, Improper Fractions |Conversion between fractions, decimals, improper fractions and mixed numbers |

|Placement of non-negative numbers on the Number Line | |

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|STAGE TWO: ASSESSMENT EVIDENCE |

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|PERFORMANCE TASKS (STUDENTS WILL DEMONSTRATE STANDARD BY): |OTHER EVIDENCE (FORMATIVE): |

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|A Party With Palm Trees |Homework: 9 assignments from text and workbooks |

|Group activity to plan a party with 16 girls and 12 boys. Need to determine | |

|number of tables, seating arrangement, best recipes, and party favors according |Quizzes: Covers first 2 sections of text, then last 3 sections of text |

|to parameters that apply concepts from all sections of unit | |

| |Unit Test: Covers entire chapter of text, 6 sections |

|Fraction/Decimal Face-off | |

|Students work in pairs, one person is A, one is B. They race each other in A & B |Explore Greatest Common Factor |

|problems provided by teacher designed to show that sometimes fractions are more |Students work individually on 4 page worksheet on Greatest Common Factor. |

|convenient and sometimes decimals are. Students reflect and discuss as a class |Essential question posted on worksheet is “How do you find and use the greatest |

|after the Face-off when each is better or if they have a personal preference. |common factor of two whole numbers?” Worksheet is scaffolded with partially |

| |filled-in tables, answer prompts and graphics. Teacher models some problems, |

|Create Prime Factors Up to 100 Chart |especially word problems and adds illustrations. Class discussions are begun at |

|Everyone given 1-100 grid. As a class discuss Rules of Divisibility and go |different stages of worksheet. Last “Practice” page is assigned for homework. |

|through numbers up to 100 to determine all primes and all composites. Highlight | |

|primes and students will keep chart in their binders for use throughout unit. |Halloween Graph |

| |Students provided with graph paper and one of three separate sheets of over 100 |

| |coordinate points, i.e. (12,34). During class time students graph the points and |

| |connect the dots to reveal Halloween images. If not completed in class, students |

| |can finish it for homework. Students are to color images so they can be hung up |

| |on the wall. |

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| |Modified Tic-Tac-Toe |

| |The board has a row of nine squares numbered 1-9. Players take turns selecting |

| |squares. The goal of the game is for a player to select squares such that any |

| |three of the player’s squares add up to 15. |

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| |Assorted: In-class informal discussions, participation, student demonstrations, |

| |and assessments |

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|STAGE THREE: LEARNING PLAN |

|(UNIT SEQUENCE) |

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|LEARNING ACTIVTIES (WHERETO): (45 minute periods) |

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|Lesson 3.1 – Prime Factorization: Fundamental Thm of Arithmetic (Discuss how Prime Factorization saved the world – deciphering the Nazi codes in WWII), Primes, |

|Composites, Step Diagram Factoring, Discussion/Group Work (aids in reflections and differentiation with peer tutoring) on definition of Prime Factorization, |

|Homework – W, H, E, R, E2, T, O |

|Create Prime Factors Up to 100 Chart: Everyone given 1-100 grid (graphic organizer and modeling that aids in differentiation). As a class discuss Rules of |

|Divisibility and go through numbers up to 100 to determine all primes and all composites. Highlight primes and students will keep chart in their binders for use |

|throughout unit. E, R, E2, T, O |

|Explore Greatest Common Factor - Students work individually on 4 page worksheet on Greatest Common Factor. Essential question posted on worksheet is “How do you |

|find and use the greatest common factor of two whole numbers?” Worksheet is scaffolded with partially filled-in tables, answer prompts and graphics. Teacher models|

|some problems, especially word problems and adds illustrations. Class discussions are begun at different stages of worksheet. Last “Practice” page is assigned for |

|homework. - W, H, E, R, E2, T, O |

|Halloween Graph – What will the dots make? Students provided with graph paper and one of three separate sheets of over 100 coordinate points, i.e. (12,34). During |

|class time students graph the points and connect the dots to reveal Halloween images. If not completed in class, students can finish it for homework. Students are |

|to color images so they can be hung up on the wall. - H, E, T |

|Lesson 3.2 – Greatest Common Divisor/Greatest Common Factor: SWBAT find the greatest common divisor of two or more whole numbers either by using a list or prime |

|factorization; Modeling and graphic organizer. Homework - E, R, T, O |

|Quiz Review Handout – work in pairs and discuss answers with entire class - E, R, E2, T |

|Modified Tic-Tac-Toe: The board has a row of nine squares numbered 1-9. Players take turns selecting squares. The goal of the game is for a player to select |

|squares such that any three of the player’s squares add up to 15. Pairs are strategically grouped to assist ELs and IEP students - H, E, T |

|Quiz 4A, Homework - R |

|Lesson 3.5 – Equivalent Fractions and Decimals: SWBAT write fractions as decimals, and vice versa, and determine whether decimal is terminating or repeating; |

|Vocabulary Hunt: equivalent, terminating decimal, repeating decimal, words written on the board at beginning of class; Spiral Review: place values and long |

|division (graphic organizers and visual aids for differentiation); Fraction vs. Decimal Face-off: Students work in pairs, one person is A, one is B. They race each|

|other in A & B problems provided by teacher designed to show that sometimes fractions are more convenient and sometimes decimals are. Students reflect and discuss |

|as a class after the Face-off when each is better or if they have a personal preference; Homework - W, H, E, R, E2, T, O |

|Lesson 3.4 – Equivalent Fractions and Mixed Numbers: SWBAT identify, write, and convert between equivalent fractions and mixed numbers; Finding Equivalent |

|Fractions, Writing Fractions in Simplest Form, Determining Whether Fractions are Equivalent, Homework – Section 3.4 was taught out of sequence from the text and |

|after Section 3.5. The rationale for the switch was to focus completely on fractional amounts first, specifically between 0 and 1, and how they can be represented |

|as fractions or decimals. Mixed numbers and improper fractions include whole number amounts in addition to the fractional amounts, so it seemed more natural this |

|should follow a lesson on just fractional amounts with no whole numbers involved. - E, R, T |

|Lesson 3.6 – Equivalent Fractions and Ordering Fractions and Decimals on Number Line: Students will create number lines and locate fractions on the number line to |

|conceptualize fractional and decimal amounts and their relation to other rational numbers; Homework - W, H, E, R, E2, T, O |

|A Party with Palm Trees: Group activity to plan a party with 16 girls and 12 boys. Need to determine number of tables, seating arrangement, best recipes, and party|

|favors according to parameters that apply concepts from all sections of unit (groups designed to facilitate differentiation) - W, H, E, R, E2, T, O |

|Quiz Review from Text – work in pairs and discuss answers with entire class - E, R, E2 |

|Quiz 4B, Unit Test Study Guide Handout for Homework - W, E, R, T, O |

|Unit Test Review: Go over problems from Study Guide. Students use individual whiteboards to reveal answers as a group so teacher can assess who and how many got |

|answers correct. Determine how much time should be spent on reviewing problem based on student responses. Use varying instructional strategies depending on |

|responses to address any learning gaps, such as, model problem, call on students to explain, encourage class discussions, etc. Provide a lunch time study session |

|for students that want extra help. - W, E, R, E2, T, O |

|Unit Test- Tests are graded subjectively to account for ELs and IEPs - R, T |

|MONDAY |TUESDAY |WEDNESDAY |THURSDAY |FRIDAY |

|Lesson 3.1 |Prime Factorization Chart |Explore Greatest Common Factor |Halloween Graph |No School |

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|Homework: p. 130 #21-36, 41-48, | | | | |

|62, 63 | | | | |

|BASIC: 21-35 (odd) | | | | |

|MONDAY |TUESDAY |WEDNESDAY |THURSDAY |FRIDAY |

|Lesson 3.2 |Quiz Review Handout |Modified Tic-Tac-Toe (p. 156 |QUIZ 4A |Lesson 3.5 |

| | |print out directions from | | |

|Homework: p. 134 | |my.) |Homework: Workbook p. 20 #1-33 |Homework: p. 148 |

|#1,2,7-15,20-24,38,39 | | |(odd) and p. 21 #1-7,22,23 |#1-9,12,13,14-34 (even), OMIT 28|

|BASIC: #1,2,7-15 | | |BASIC: p.20 #1-15 (odd), p. 21 |BASIC: #1-9,12 |

| | | |#1-7 (odd) | |

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|MONDAY |TUESDAY |WEDNESDAY |THURSDAY |FRIDAY |

|No School |Lesson 3.4 (Examples 1,2,3) |Lesson 3.4 (Example 4) |Lesson 3.6 (Example 1 & 3) |A Party with Palm Trees |

| | | |Positive numbers only, emphasis on the | |

| | | |number line | |

| |Homework: p. 144 #1-13,21-36 |Homework: p. 144 | | |

| |BASIC: #1-12 |#37-44,46-59,60-66 (even) | |Homework: p. 154 |

| | |BASIC: #37-44,46-50 |Homework: p. 152 |#1-10,13-16,25,27,28 OR |

| | | |#1,4,8,11,14,15,25,26,30-33,39-46 |p. 155 as a group activity |

| | | |BASIC: #1,4,8,11,14,15,25 | |

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|MONDAY |TUESDAY |WEDNESDAY |THURSDAY |FRIDAY |

|Quiz Review |QUIZ 4B |Study for Unit Test |Unit Test | |

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|Homework: p. 159 #34-74 (even), |Homework: Unit Test Study Guide | | | |

|82 OMIT #62, 68 |Handout | | | |

REFLECTIONS

Describe the effectiveness of this lesson in helping students meet the learning goals.

This unit breaks down the desired learning objectives into small, manageable chunks that are organized to build each lesson upon the next, so a review of previous material is naturally incorporated into the teachings. There is also a range of activities and instructional strategies throughout designed to best relate the content, keep kids engaged and also to reach as many differentiated learning styles and needs as possible. Nearly everyday there is a mix of student work groups, whether individual, partner or small group. Discussions are utilized often and many informal assessments are implemented to gauge understanding. Concepts covered are often related to real world scenarios or to concepts covered previously, like number line and place value. Common Core requirements are addressed with the real world applications and explored deeper understandings, especially in worksheets and group work, but also with special attention paid to mathematical vocabulary and literacy in word problems and answers.

Aside from the learning objectives being broken into smaller units, there are also 3 formal testing opportunities, 2 quizzes and 1 unit test. The quizzes assess the students’ understanding of the first and second halves of the unit and are given less weight in grading than the final unit test. This allows the teacher and students to see if there are any areas or skills that need further reinforcement prior to the unit test while still allowing students to improve their grades if they don’t score well on the quizzes. For each quiz and test, students are given study guides as homework with problems representative of the actual tests. In that way, reviewing the student work on the study guides in class as groups, in discussion or through teacher modeling also gives another opportunity to catch any gaps in learning the students may have for the material. Students have an additional opportunity to review with a lunch study session the day of the test. And, as mentioned in the unit plan, the grading is slightly subjective according to differentiated student needs as well as taking into consideration the affective learning needs of all students.

How will you apply what you have learned in future instruction?

I used the text and my mentor teacher’s input as a guide and I was actually surprised, at first, how small the components of each lesson were. I was also surprised by what material was new or unfamiliar to 6th graders. It’s one thing to read the standards of what is expected by each grade level, but it’s another to see explicitly which concepts textbooks and teachers impart. I see I still have to acclimate to this age group as far as knowing what they already do and should understand and what will need to be more comprehensively taught.

I do feel that there was a good deal of accommodation made for differentiating to the needs of IEPs and ELs, but I see that this will be a difficult area for me as a teacher to ever feel like I’m doing enough. Especially with students like the CELDT Level 1 student, who is eager to work hard. Even when I was a tutor, if my students tried hard, I would work with them extra time for free and let them email me whenever they needed additional help. I admit, my first inclination is to find a way to work more with these students one-on-one, but that is likely unrealistic. I do think it will take time and even further research on my part to see what other avenues I can take to help meet the needs of my students most effectively and efficiently. Along the same lines, I don’t think I was as prepared as I should have been in this unit to meet the needs of higher achieving students. Part of this is getting to know this grade level group as I stated before, but I also saw it play out when I taught one of the unit lessons to the other 6th grade class. That class has some very high achievers and there were at least 5 kids that were done with all of my activities early. I was able to scramble something together for them, but I would like to better my designs to address the entire range of learners I may face in a classroom.

I think I would like to make quiz and test review the day before exams a little more procedural so students know what to expect and can focus solely on the review. I would like to continue to use individual white boards for student responses, but be sure that all students raise their boards for the answers to the same problems at the same time. I would like to make note (perhaps on a spare seating chart) of which students were missing which concepts, that way I could try to have a review day where the class is set up into stations. There would be one station for each of the major concepts covered. I could assign students to a particular station or allow them to choose whichever one they needed more help with and then the kids could rotate throughout the period. At each station there would be additional problems for the kids to work on and perhaps a peer tutor that excels in the area. And finally, I would also aim to have more regular lunch study groups.

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