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MATHEMATICS
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Curriculum Resource Guide Components
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At a Glance
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STANDARDS:Grades 7 & 8 Alignment
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Individual Behaviors/Classroom Practices
NYS Mathematics Indicators |
NYC Mathematics Indicators |
Standards 3: Mathematics |
NYC Performance Standards |
3.5 Measurement – Students use measurement in both metric and English measures to provide a major link between the abstractions of mathematics and the real world in order to describe and compare objects and data.
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• Estimate, make and use measurement in real-world situations.
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M6c The student estimates numerically and spatially.
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•Select appropriate standard and nonstandard measurement units and tools to measure to the desired degree of accuracy.
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M6d The student measures length, area, volume, weight, time and temperature accurately.
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• Develop measurement skills and informally derive and apply formulas in direct measurement activities.
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M2g The student measures angles, weights, capacities, time and temperature using appropriate units.
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| • Use statistical methods and measures of central tendency to display, describe and compare data. |
M4c The student analyzes appropriately central tendencies of data by considering mean and median.
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• Explore and reproduce graphic representations of data using calculators and computers.
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M6h The student uses recall, mental computations, pencil and paper, calculators, computers and advice from peers, as appropriate, to achieve solutions.
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• Develop critical judgment for the reasonableness of measurement.
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M5d The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining patterns, making conjectures and explaining why they seem true and by making sensible, justifiable statements.
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LESSON STRUCTURE
Essential Question:
An essential question is open-ended to engage students in focused and active inquiry/study. Students address the essential question within each lesson/session because the question is a focus that is woven throughout the Learning Experience.
Context:
The context identifies the title, the grade level and the subject of a Learning Experience. It also describes prerequisite knowledge (information that students must have in advance of the Learning Experience), which enables students to fully understand the Learning Experience.
Rationale:
The rationale describes the significance of this learning experience that includes an explanation of why a teacher selected this specific content and skills and its importance for students.
Standards:
The standards that are addressed are highlighted in each Learning Experience. At a glance, a teacher will know specifically what standards and skills have been addressed and embedded in the lessons/sessions.
Time:
The time segment includes both how many lessons/sessions and their length as well as the total number of lessons/sessions per week.
Instructional Resources:
All resources needed to replicate this Learning Experience are stated so that a teacher may replicate it. |
LEARNING EXPERIENCE
Learning Experience 9: Finding the Area of Rectilinear Figures and Complex Figures
Session: 3
Focus: To extend the use of area formulas for triangles and special quadrilaterals to rectilinear figures and complex figures.
Whole Group/Direct Teaching
Show all rectilinear and complex polygons on a blackline master via overhead projector. Have students brainstorm places that these figures might exist. Start the thinking by modeling real- world examples, such as:
Figure A - the front of a desk.
Figure B – the base of a chandelier.
Figure C – a dining room chair back.
Figure D – the side of a van.
Figure E – a piece of guardrail on a highway.
…Figure U – a picture frame.
Record all of the students’ responses, leaving them visible for later in the lesson.
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Reflection/ Thinking
Previously in this Learning Experience, the context for all of the problems was given. Here the students determine the context in the real world. These contexts will later support their composition of situational story problems.
Students will need a way to recall the brainstormed ideas.
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Small Group/ Independent
Distribute rectilinear and complex figures (blackline masters). Ask students to cut out all six.
Students explore ways to find the areas of each figure. Encourage students to fold or cut the figures and label all cut pieces with the letter of the original figure.
Distribute activity sheet with the same figures with dimensions now included. Have students find the area of 3 of the 6 figures. |
Reflection/ Thinking
This will help the teacher reduce preparation time. The pictures become manipulatives for the lesson. Having no measurements will encourage students to focus on strategy, not a numerical solution.
Labeling teaches students how to organize their work.
Possible strategies:
• Cut figures into quadrilaterals (rectangles, squares).
• Draw a perimeter around the entire figure so that all of the figure’s concave areas are included.
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Whole Group/Reflection/Share
Share strategies for finding area. Have students compare and contrast the strategies.
Use this time to question and nudge students’ thinking, being mindful to stretch ideas being shared.
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Reflection/ Thinking
Multiple solutions remind students that there is more than one way to solve a problem and using another person’s strategy sometimes increases understanding. Remember to share observations made during the small group session to extend the whole class sharing about strategies and ideas. |
Extensions
Have students go on a community walk and take pictures of rectilinear and complex figures in the real world. A throwaway camera could be used. Find the area of two or three of the figures by taking actual measurements or estimating the measures.
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Reflection/ Thinking
Students will be able to observe cultural and period architecture which is filled with geometric concepts studied in this Learning Experience. This extension could be done as a culminating project. |
UNDERSTANDING MATHEMATICS
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A manipulative is a concrete representation of a mathematical concept. Manipulatives are physical objects that students can move around, group, sort and use in a variety of ways to model mathematical concepts and develop deeper understandings.
Manipulatives help create an environment for discovery and provide an added degree of comfort for the visual and the kinesthetic learner. Whenever we are using manipulatives the overall objective is to help students build a concrete model of a situation, and in the process, reflect, analyze and test hypotheses. As a result, an activity that requires the use of manipulatives is not complete without discussion, the whole class sharing their findings, generalizing their conclusions, applying or connecting these to other situations or problems.
• Manipulatives are used in the math classroom:
• To provide for different learning styles and learning modalities.
• To help students make connections between the world they live in and the abstract representations they encounter in school.
• To validate students prior knowledge.
• To allow students to build basic concepts.
• To provide tools for problem solving.
• To provide a concrete approach for abstract concepts.
• To help students gain a feeling of confidence.
• To allow students to deal with more challenging types of problems.
• To encourage verbalization by providing a concrete representation.
• To check for understanding rather than memorization.
• To prepare students for higher levels of representation with numerals and symbols. |
GLOSSARY
Odd numbers. The odd numbers are : 1,3,5,7,9,… When an odd number is divided by 2, the remainder is 1. That is, an odd number can be named by the expression 2x+ 1, when x is a whole number.
Examples:
1 = 2x + 1, when x = 0.
3 = 2x + 1, when x = 1.
25 = 2x + 1, when x = 12.
One- to- one correspondence. Relationship between sets so that their members can be paired off, one from each set.
Open sentence. A number sentence which is unfinished; for example, 3 + 5 =
Operation. Addition, subtraction, multiplication and division.
Operational signs. +, -, x |
INTERNET SITE BIBLIOGRAPHY
http://unite.ukans.edu
A comprehensive collection of resources for K- 12 math and science.
http://www.mathgoodies.com
Math Goodies has lesson plans that use a problem-solving approach, homework help, message boards, puzzles and more.
http://math.rice.edu/~lanius/Lessons/index.html
Fun Math Lessons has a variety of active math lessons.
http://www.wnet.org/wnetschool/origlessons/indexr.html
Original lessons that were developed by a master teacher.
http://www.col-ed.org/cur/math.html
Math lesson plans from the Columbia Education Center.
http://www.aaamath.com/index.html
AAA Math has hundreds of lessons in all math areas, grades K- 8.
http://www.enc.org
Has math and science standards, TIMMS data, resources for teachers, including a set of internet sites. |
ASSESSMENT - RUBRICS
The use of a variety of assessment tools will give a complete picture of where students are in meeting the standards. In order to create a reflective instructional program, teachers should use a variety of ways to collect information. Teachers can use their observational skills, make use of checklists, keep anecdotal records, use portfolios, review entries in math journals and gather information through individual conferences.
How to Get an “A” On:
A Feast Fit for a Little More Than a Few!
An Exemplary Performance Includes:
• all parts of the project completed exceptionally well;
• an invitation that is creative and original and makes your guests want to come to your dinner;
• a mouth-watering description of the menu;
• an accurate increase in all the recipes;
• a timetable that is in a logical order and has most foods cooked in time to eat by 5 p.m.;
• the table is set in an organized fashion and done attractively;
• an essay that tells a story with descriptive details of what the group learned, (like in Amelia’s Diary!).
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A Good Performance
Includes:
• all parts of the project completed well;
• an invitation that is creative and original;
• a clear description of the menu;
• an accurate increase in all the recipes;
• a timetable that is in a logical order and has most foods cooked in time to eat by 5 p.m.;
• the table is set in an organized fashion and done attractively;
• an essay that tells a story with descriptive details of what the group learned.
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A Fair Performance
Includes:
• most parts of the project
completed;
• an invitation that is neat;
• a description of the menu;
• an accurate increase in all the recipes;
• a timetable that is not in a logical order and has some food not cooked in time to eat by 5 p.m.;
• the table is set in an organized fashion;
• an essay that tells a story with only a few details of what the group learned.
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A Novice Performance
Includes:
• few parts of the project completed;
• an invitation that is sloppy;
• a description of the menu that is not clear;
• many errors in the increasing of the recipes;
• a timetable that is not logical and has most food not cooked in time to eat by 5 p.m.;
• the table is setting is sloppy and not well organized;
• an essay that tells a story with very few details of what the group learned. |
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