Planning for Differentiation and Best Strategies
Karla Shotts
OTL565-Cultural and Linguistic Diversity
Colorado State University- Global Campus
Tony Pellegrini
April 8, 2015
Your Name:
Karla Shotts
CSU-Global Course:
OTL565
Subject / Course:
Algebra
Topic:
Pythagorean Theorem
Lesson Title:
Analyzing the Wheel of Theodorus
Level:
Grade 8
Lesson Duration:
:66
Common Core or State Standard(s): Colorado Department of Education
Standards:
8.NS.A.2
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
8.G.A.5
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
8.G.B.6
Explain a proof of the Pythagorean Theorem and its converse.
8.G.B.7
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.B.8
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Description of Lesson as currently taught:
Do Now: Read Investigation 4 (pg 46) Answer: What could be some real world applications of the Wheel of Theodorus? (Review notes from yesterday)
Activity: Pick one REAL WOLRD problem. Complete.
Problem 4.1 (pg 47) We are exploring some interesting applications of the Pythagorean Theorem.
1) Copy the Wheel of Theodorus onto paper and label each hypotenuse with its length. Use the symbol for square root (√) to express lengths that are not whole numbers.
2) Use a ruler to measure each hypotenuse on the wheel.
3) For each hypotenuse that is not a whole number, give two consecutive whole numbers the length is between. IE: √2 is between 1 and 2.
4) Use a calculator to estimate the value of each length and compare the results to the approximations you found in #2
Exit ticket: Write out the 3 vocabulary terms and descriptions at the bottom of page 48 (Rational numbers, irrational numbers, and real numbers). In which category is √2?
Differentiated Instructional Approaches and Pre-Assessments
Climate
To establish a climate that fosters differentiation, we must first acknowledge that a classroom consists of multiple intelligences and learning styles. It starts with taking a close look at my philosophy of classroom management and maintaining a positive classroom climate. In addition, I should foster a climate that encourages team building, where students can take intellectual risks. Students are allowed to work in groups, or pairs. Even in Math, we debate current topics with the understanding that every student is allowed to express themselves with no judgment. Classroom expectations and behaviors are set from the very beginning. Each require practice and need to be taught and re-taught consistently during the year. Students that feel they are heard and respected tend to accomplish more in the classroom.
It is important to teach students that every student must make continuous progress no matter what their level of knowledge and skills. The environment should be conducive for students to become lifelong learners. In “Teacher’s Survival Guide”, written by Dr. Julia Roberts and Dr. Tracy Inman, the authors claim the most basic reason for differentiation, “The most basic reason to differentiate is that children differ. Because children are different in their readiness to learn specific content and skills, it is necessary to respond accordingly. Children of the same age who are in the same grade have a range of reading abilities, varied interests and experiences with the content being studied, and different levels of skills for thinking critically and creatively as well as in communicating via writing and speaking. Children seldom come to any class ready for learning at the identical rate and at the same level of complexity.”
The authors also say it is a matter of fairness, “it is unfair to have some children struggling with assignments that are too difficult, just as it is unfair to have some children waiting for something new to learn. Fairness means matching learning experiences to needs.”
Knowing the Learner
Instructional Support:
Differentiation:
Math Fellow takes 2 students to another room to work on same lesson plan that includes:
- one-to-one support
- sentence stems
- peer support
- leveled questions
- Differentiated concepts
SPED support (additional teacher) takes 1 student/day to work on same lesson plan that includes:
- One-to-One instruction
- Probing students during instruction and assessments.
- Assessing Students Knowledge based on daily lesson.
- Co-Teaching when appropriated based on providing students with content information.
Accommodations and Modifications:
Special Education and General Education teachers will work collaboratively to provide accommodations and modifications based on students' potential need during the delivery and assessment of the content material.
Student 1 – Ability level text, Check for understanding, Flexible setting/group, Extended time, Graphic organizer/learning tools, Flexible scheduling/breaks, Graphic organizer/learning tools, Preferential seating, Modify complexity of assignment, Use of calculator, and Use of manipulatives.
• Accommodations and modifications: Special Education and General Education teachers will work collaboratively to provide accommodations and modifications based on students’ potential need during the delivery of the content material.
Student 2- Check for understanding, Extended time, Graphic organizer/learning tools, Modify complexity of assignment, Preferential seating, Study guide, Use of calculator, Use of manipulatives, and Flexible scheduling/breaks.
Student 3- Extended time, Flexible scheduling/breaks, Flexible setting/group, Graphic organizer/learning tools, Oral testing, Read aloud/sign for directions, Read aloud/sign test items, and Use of calculator.
Pre-Assessment
At the start of each year, we are given IEP’s (Individualized Education Plan) and 504’s (for students with A-D-D and other learning barriers) This lets us know which students have familial, education, medication or language challenges. We use this information all year in planning a course for students.
In Algebra, we use many pieces of information to pre-assess students. Middle school students take the NWEA Measures of Academic Progress (MAPS) test three times a year, in the fall, winter and spring to get a snapshot of the students’ math skills. We can take a look at the score at each phase breaking it into small pieces. For this lesson plan, we look at testing results for square roots and other properties needed for Pythagorean Theorem. It sorts out those with successes and struggles and gives me the information to know if everyone is on the same page.
In addition, each day, the “Do Now” includes problems from the previous day. It lets me know if the student understands the topic and if we can move on. Depending on the topic, we also use KWL charts-K-What do the students know already? W-What do the students need and want to know? And L-what did the student learn?
While researching this topic, I also found some out-of-the-box pre-assessment techniques, such as boxing, graffiti wall, yes/no cards, and turn and talk. They can be found at this link!
Learning Target (Objectives, Student Set Goals, and/or Essential Questions):
Objectives:
In this lesson, you will learn how to do the following:
•
Examine different formal proofs of the Pythagorean theorem
•
Examine some applications of the Pythagorean theorem, such as finding missing lengths
•
Learn how to derive and use the distance formula
Essential Questions:
How can you prove the Pythagorean Theorem?
• What does the converse of the Pythagorean Theorem prove?
• What is the difference between a leg and the hypotenuse of a right triangle?
• How do you find the length of the hypotenuse given the lengths of both legs of a right triangle?
Learning Task (Remember to consider relevance and career/workforce readiness skills around what is being taught):
Number of Days: 4
Learning Task
In 8th grade, this unit is a critical area. Students take prior knowledge of vertical and horizontal distances on a coordinate plane and join it with constructing triangle from three measures of angles or sides. This module will further their knowledge explaining a proof of the Pythagorean Theorem and its converse. They will take a2 +b2 =c2 and determine missing side lengths in a right triangle in real-world and mathematical problems (with two and three dimensions).
Students also use the Pythagorean Theorem to find the distance between two points on the coordinate plane and area within a triangle.
Real Life Applications: Retrieved from Bright Hub Education:
1) Road Trip: Let’s say two friends are meeting at a playground. Mary is already at the park but her friend Bob needs to get there taking the shortest path possible. Bob has two way he can go - he can follow the roads getting to the park - first heading south 3 miles, then heading west four miles. The total distance covered following the roads will be 7 miles. The other way he can get there is by cutting through some open fields and walk directly to the park. If we apply Pythagoras's theorem to calculate the distance you will get:
(3)2 + (4)2 =
9 + 16 = C2
√25 = C
5 Miles. = C
Walking through the field will be 2 miles shorter than walking along the roads. .
2) Painting on a Wall: Painters use ladders to paint on high buildings and often use the help of Pythagoras' theorem to complete their work. The painter needs to determine how tall a ladder needs to be in order to safely place the base away from the wall so it won't tip over. In this case the ladder itself will be the hypotenuse. Take for example a painter who has to paint a wall which is about 3 m high. The painter has to put the base of the ladder 2 m away from the wall to ensure it won't tip. What will be the length of the ladder required by the painter to complete his work? You can calculate it using Pythagoras' theorem:
(5)2 + (2)2 =
25 + 4 = C2
√100 = C
5.3 m. = C
Thus, the painter will need a ladder about 5 meters high.
3) Buying a Suitcase: Mr. Harry wants to purchase a suitcase. The shopkeeper tells Mr. Harry that he has a 30 inch of suitcase available at present and the height of the suitcase is 18 inches. Calculate the actual length of the suitcase for Mr. Harry using Pythagoras' theorem. It is calculated this way:
(18)2 + (b)2 = (30)2
324 + b2 = 900
B2 = 900 – 324
b= √576
= 24 inches
4) What Size TV Should You Buy? Mr. James saw an advertisement of a T.V.in the newspaper where it is mentioned that the T.V. is 16 inches high and 14 inches wide. Calculate the diagonal length of its screen for Mr. James. By using Pythagoras' theorem it can be calculated as:
(16)2 + (14)2 =
256 + 196 = C2
√452 = C
21 inches approx. = C
5) Finding the Right Sized Computer: Mary wants to get a computer monitor for her desk which can hold a 22 inch monitor. She has found a monitor 16 inches wide and 10 inches high. Will the computer fit into Mary’s cabin? Use Pythagoras' theorem to find out:
(16)2 + (10)2 =
256 + 100 = C2
√356 = C
18 inches approx. = C
Student Differences
· Readiness: MAPS test and pre-assessment (questions, problems, samples, etc) will give indication of student readiness for base information needed to move to Pythagorean Theorem. Students will set goals that will focus their attention on instructional improvement and measure of academic progress.
· Once we have information on where students begin, we can vary the pace, the level of complexity, degree of independence as well as the amount of structure and support needed by each student.
· A group called “Teaching As Leadership” has outlined a guide for determining readiness that includes cognitive, developmental, diagnostic, tracking and other information.
· Goals are specific (but not to specific), challenging but reachable and interventions impact directly on the experience of the learner.
Varying Tasks
· Content: Students who need support with squares and square roots will first do IXL.com math support problems.
· All students have a chance to pick one REAL WORLD problem. They range from easy (#1) to difficult (#5). Students of varying abilities get support from 3 teachers in the classroom.
· Once problems are answered, students are split into groups for Think-Pair-Share exercise. Students get to compare answers and give explanations.
Feedback Strategy (Frequent checks for understanding):
1) Have students teach part of the lesson to small groups of their peers
2) Turn and Talk: students turn to partner and explain answers to a math problem
3) Add opportunity for students to have 2-3 problems to choose from to increase relevance and student buy-in. (real world application problems)
4) Application-learn by doing. Are students finding a deeper learning experience through experimentation?
Summative Assessment (Collect student data):
5 minutes: Student will have a Peer-to-peer review
Take notes while checking partners work
Looks for patterns of wrong answers
Start class conversations based on targeting those patterns
5 minutes: Homework
Review problem areas
Develop homework center targeted on specific skills identified in assessments
Add Re-teaching tools for students who need continued support on topic
SOURCES:
Roberts, J., Inman, T. (2013). Teacher’s Survival Guide: Differentiating Instruction in the Elementary Classroom. Prufrock Press. Retrieved from http://www.prufrock.com/Assets/ClientPages/pdfs/TSG_Diff_Elem_Sample.pdf
Principal Kendrick webpage. (2007). Pre-assessment tools. Retrieved from https://kendrik2.wordpress.com/2007/09/27/pre-assessment-strategies/
Bright Hub Education. (2015). Pythagoras’s theorem Used in Real Life Experiences. Retrieved from http://www.brighthubeducation.com/homework-math-help/36639-applications-of-pythagoras-theorem-in-real-life/
Teaching As Leadership. (2014). Differentiate your plans to fit your students. Retrieved from http://teachingasleadership.org/sites/default/files/How_To/PP/P-4/P4_Readiness.pdf