Understanding By Design
Unit Design
for
Operations & Rational Numbers
Developed by
Carrie E. Johnson
Pontiac Academy for Excellence
[pic]
Understanding by Design
Unit Design Worksheet
|Unit Title: Developing an Understanding of Operations & Rational Numbers |Subject/Course: Mathematics |
|Topic: Operations and Rational Numbers |Grade: 6th |Staff Name: Carrie Johnson |
|Stage 1 - Desired Results |
|Established Goals: |
|6. NS. 2 Fluently divide multi-digit numbers using the standard algorithm. |
|6. NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation |
|6. NS.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. |
|Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of a number is the number |
|itself, e.g., -(-3) = 3, and that 0 is its own opposite. |
|Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by |
|signs, the locations of the points are related by reflections across one or both axes. |
|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. |
|6. NS.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. |
|For example, express 36 + 8 as 4 (9 + 2). |
|Understandings: |Essential Questions: |
|Students will understand | |
|how to use a standard algorithm in order to divide multi-digit numbers |What is the standard algorithm for dividing multi-digit numbers? |
|how to add, subtract, multiply, and divide multi-digit decimals. |1. When will I use division in the real world? |
|how to extend a number lines and coordinates axes to represent points on a number |Why do we need to know how to add, subtract, multiply, and divide decimals? |
|line and a coordinate plane using negative number coordinates. |How do I graph positive and negative numbers on a number line and coordinate |
|how to recognize and indicate numbers and its opposite located on opposite sides |grid? |
|of zero on a number line. |What does a negative number represent, and what is the opposite of the negative |
|how to indicate ordered pairs in their correct quadrants on a coordinate plan. |number? |
|Students will also understand the reflections of points across one or both axes |How do I find an ordered pair on a grid and its opposite/reflected pair? |
|are different only by their signs. |How will positioning numbers on a horizontal and vertical number lines and |
|how to position integers and rational numbers on a horizontal or vertical number |coordinate grids help me in the real world? |
|line. Students will also be able to plot pair of integers and rational numbers on |How can I express a sum of two numbers by using common factors as a multiple of |
|a coordinate plane. |a sum of two whole numbers? |
|how to find the greatest common factor of two whole numbers and find the least | |
|common multiple of two whole numbers. Students will also understand how to use the| |
|distributive property to express a sum of two whole numbers. | |
|Students will know |Students will be able to |
|how to use a standard algorithm in order to divide multi-digit numbers |explain how to divide multi-digit numbers |
|how to add, subtract, multiply, and divide multi-digit decimals. |explain how to add, subtract, multiply, and divide multi-digit decimals |
|how to extend a number lines and coordinates axes to represent points on a number |demonstrate how to extend a number line and coordinate axes to represent points |
|line and a coordinate plane using negative number coordinates. |on a number line/coordinate plane using negative numbers. |
|how to recognize and indicate numbers and its opposite located on opposite sides |identify numbers and the opposite on both sides of zero on a number line |
|of zero on a number line. |identify ordered pairs ordered pairs and their opposites in the correct |
|how to indicate ordered pairs in their correct quadrants on a coordinate plane. |quadrants on a coordinate plane. |
|Students will also understand the reflections of points across one or both axes |describe how to plot pairs of integers and rational numbers on a coordinate |
|are different only by their signs. |plane. |
|how to position integers and rational numbers on a horizontal or vertical number |explain how to find the greatest common factor and least common multiple of two |
|line. Students will also be able to plot pairs of integers and rational numbers on|whole numbers. |
|a coordinate plane. |describe how to use the distributive property to express a sum of two whole |
|how to find the greatest common factor of two whole numbers and find the least |numbers. |
|common multiple of two whole numbers. Students will also understand how to use the| |
|distributive property to express a sum of two whole numbers. | |
|Unit Enduring Understanding: |Unit Question: |
|Students will understand that there are rules to mathematical procedures and how |How do you use mathematical operations by using decimals and negative numbers |
|to apply them. |and plotting points on a coordinate grid? |
|Stage 2 - Assessment Evidence |
|Performance Tasks: |
|Goal: The student’s task is to solve mathematical operations involving rational numbers and plotting the numbers on a coordinate grid (students will be given two |
|operations to solve for each point). As they plot each plot they connect it to the previous point. When they are finished plotting the points it will make a |
|picture. Students will present their picture to the teacher and students. |
|Role: Investigator |
|Audience: Teacher and Students |
|Situation: Students want to discover the hidden picture. In order to discover the picture the students have to solve mathematical operations involving rational |
|numbers and plotting points on coordinate grids. Students have to connect the points as they go. When they have completed, they will have discovered a picture. |
|They will color their finished product. |
|Product: A discovered picture they found by solving mathematical operations and plotting the points on a coordinate grid. |
|Standards and Criteria for Success: Students will be graded on their ability to solve mathematical operations involving rational numbers. Students will also be |
|graded on their ability to plot the points on a coordinate grid and connecting them point by point. Finally, students will be graded on their ability to discover a|
|picture from the points they plotted and neatly coloring it. |
|Key Criteria: |
|CATEGORY |
|4 |
|3 |
|2 |
|1 |
| |
|Operations |
|90% - 100% of the mathematics operations are correct. |
|70% - 89% of the mathematics operations are correct. |
|50% - 69% of the mathematics operations are correct |
|Operations are not correct. |
| |
|Coordinate Grid |
|90% - 100% of the points plotted are correct. |
|70% - 89% of points plotted are correct. |
|50% - 69% of points plotted are correct |
|Points plotted are not correct. |
| |
|Illustrations |
|The plotted points make a detailed picture and it is colored. |
|Most of the plotted points make a picture and is partially colored. |
|Some of the plotted points make a picture and is partially colored. |
|The points plotted do not illustrate a picture and is not colored. |
| |
| |
| |
|Other Evidence: |
|BEFORE |DURING |AFTER |
|Pretest: teacher created |Think/Pair/Share: To see where students are |Entrance/Exit Slips: to see what students have |
|KWL |struggling |retained from previous day or what they learned after |
|Bellwork Questions |Observation |a lesson |
|Questioning: To get a quick grasp of where students |Daily Assignments: teacher created and book items |Observation |
|are at. |MBWA (Manage By Walking Around) |Quiz: teacher created |
| |Entrance/Exit Slips: to see what students have |Test: teacher created |
| |retained from previous day or what they learned after| |
| |a lesson | |
| |Drawings: Vertical and Horizontal Number Lines, | |
| |Coordinate Grids | |
|Describe the assessment/s and state the prompt if applicable. ( F ( S | | |
| | | |
|What type of scoring tools will be used for evaluation? | | |
|( Analytic rubric (Checklist | | |
|( Holistic rubric (Answer Key | | |
|(Criterion rubric ( Other | | |
| | | |
|Student Self-Assessment and Reflection: |
| |
|Each student will write about what they knew at the beginning of the unit and what they know now. The students will reflect on the real-world application and tell|
|how it helped them to understand operations and rational numbers. |
|Stage 3 - Learning Plan |
|Differentiated Instruction: |
|LAYER C: (50 points) |
|Vocabulary Journal of key terms from the Operations and Rational Numbers Unit * (15 points) |
|CHOICE 1: Write definitions of key terms in Vocabulary Journal |
|CHOICE 2: Draw a picture to represent key terms in Vocabulary Journal |
|CHOICE 3: Type definitions of key terms on a mini laptop (Electronic Vocabulary Journal) |
|Notebook of daily assignments (Checklist will be used to grade notebooks)* (20 points) |
|Bellwork Log from “Before Activities” * (15 points) |
| |
|LAYER B: (50 Points) |
|(Students need to pick one activity from each of the following groups: 1, 2, or 3; 4, 5, or 6; 7 or 8; 9, 10, or 11; 12 or 13) |
|Complete a 10 problem worksheet on dividing multi-digit numbers (10 points) |
|On a poster write the algorithm and show an example of how to divide multi-digit numbers (10 points) |
|Play “Dividing Multi-Digit Numbers” (10 points) |
|Complete a 10 problem worksheet on adding, subtracting, multiply, and dividing multi-digit decimals (10 points) |
|On a poster write the algorithm and show an example of how to add, subtract, multiply, and divide multi-digit numbers (10 points) |
|Play “Adding and Subtracting Multi-Digit Decimals” & “Multiplying and Dividing Multi-Digit Numbers” (10 points) |
|Complete a 10 problem worksheet on plotting points on a 4-quadrant plane. (10 points) |
|Play “Coordinate Graphs as Maps” (10 points) |
|Complete a 10 problem worksheet on finding the Greatest Common Factor and Least Common Multiple of two whole numbers. (10 points) |
|On a poster write about what the Greatest Common Factor and Least Common Multiple is, and show at least 3 examples of each showing how you found what the Greatest |
|Common Factor and Least Common Multiple is of two whole numbers. (10 points) |
|Play “Greatest Common Factor” and “Least Common Multiple” and |
| (10 points) |
|Complete a 10 problem worksheet on using the distributive property to express two whole numbers. (10 points) |
|Play “Distributive Property” (10 points) |
|LAYER A: (70 Points) |
|(Students must do 1 & 2 and have a choice between 3 and 4) |
|Hidden Picture (GRASP)* – Finding the hidden picture solving mathematical operations and graphing points on a coordinate plane (40 points) |
|Unit Test* (20 points) |
|Reflect Daily in Math Notebook (10 points) |
|Complete KWL Chart on Operations and Rational Numbers (10 points) |
| |
|*denotes Mandatory Activities |
|Learning Activities: |
| |
|W: Where are we going? To a deeper understanding of operations and rational numbers |
|Why? Because it allows students to apply their knowledge of operations and rational numbers to real world problems. |
|What is expected? Go over rubric and course overview. |
|H: I will hook student interest by becoming a “Discoverer” by solving simple mathematical operations to decode a phrase. I will tell the class, “Throughout |
|this unit you will be taking on the role of the ‘discoverer,’ except from this point forward your tasks will become more challenging. It will be your job at the |
|end of the unit to ‘Discover the Hidden Picture.’” |
|E: Students will be equipped to do well on this unit through journaling, a variety of lessons and assignments (written, artistic, & web-based), class |
|discussions, and the “Hidden Picture” activity. |
|R: Students will be asked to rethink and revise their work through their K-W-L charts at different times during the unit to rethink the use and understanding |
|of operations and rational numbers. Students will also have an opportunity to share their thinking during class discussions. |
|E: Students will self-evaluate through daily entrance and exit slips and through the, “End of the Unit Reflection.” |
|T: Learning will be tailored by using the differentiated instruction above. |
|O: The unit will be organized so that key concepts build upon one another. See calendar below for day-by-day schedule. |
|Essential Vocabulary |
|Add: to find the sum of two or more numbers |
|Algorithm: Steps to solving a problem |
|Axes: a central line that bisects a two-dimensional body or figure. |
|Coordinate Axis: one of the fixed reference lines of a coordinate system. |
|Coordinate Plane: a plane containing two perpendicular axes (x and y) intersecting at a point called the origin (0,0). |
|Coordinates: Shown as a pair of letters or numbers. Example (2,4) |
|Diagram: a graph, map, or plane |
|Distributive Property: a product can be written as, the sum of, or difference between, two products. |
|Divide: sharing or grouping a number into equal parts. |
|Factor: a whole number that divides exactly into another number or a whole number that multiplies with another number to make a third number. |
|Fluent: spoken or written fluently |
|Greatest Common Factor: The largest/biggest number that will divide two or more other numbers exactly |
|Horizontal Number Line: a number line that is parallel to the horizon. (Left to Right) |
|Integers: a positive number, a negative number or zero. NOT: a fraction or a decimal |
|Least Common Multiple: the smallest number that is the multiple of two or more other numbers. |
|Multi-Digit: a number that has two or more numbers. |
|Multiply: a mathematical operation where a number is added to itself a number of times. |
|Negative Number: a number less than zero. |
|Number Line: line marked with numbers, used to show operations. |
|Operation: there are four basic operations in arithmetic used to solve problems. They are Addition, Subtraction, Multiplication, and Division. |
|Opposite Numbers: Two numbers that have the same numeral but have opposite signs. |
|Ordered Pairs: a pair of numbers where order is important. It is often used to indicate a point on a coordinate plane, graph, or map. Example: (4,6) is different |
|to (6,4) |
|Plane: A flat surface |
|Quadrants: any quarter of a plane divided by an x and y axis. |
|Rational Numbers: A real number that can be written as: a ratio of two integers (fraction), excluding zero as a denominator, a repeating or terminating decimal, or|
|an integer. |
|Reflection: a mirror view |
|Subtract: to take one quantity away from another |
|Sum: the total or whole amount. It is the result of adding |
|Vertical Number Line: a number line that goes from top to bottom. It makes a right angle with the horizon. |
|Whole Numbers: a counting number from zero to infinity |
|Sequencing the Learning |
|Monday |Tuesday |Wednesday |Thursday |Friday |
|Pretest |Hook: “Discoverer” |Dividing Multi-Digit Numbers |Dividing Multi-Digit Numbers |Adding and Subtracting Decimals |
| | | | | |
| |Begin K-W-L |LEVEL C Activities |Entrance Slip |K-W-L |
| | |LEVEL B Activities | | |
| |Class Discussion | |LEVEL C Activities |LEVEL C Activities |
| | |Exit Slip |LEVEL B Activities | |
|Monday |Tuesday |Wednesday |Thursday |Friday |
|Adding and Subtracting Decimals |Multiplying and Dividing |Multiplying and Dividing |QUIZ: |Plotting Points on a 4-Quadrant |
| |Decimals |Decimals |Math Check-Up |Grid |
|Think/Pair/Share | | | | |
| |K-W-L |Entrance Slip | |K-W-L |
|LEVEL C Activities | | | | |
|LEVEL B Activities |LEVEL C Activities |LEVEL C Activities | |LEVEL C Activities |
| | |LEVEL B Activities | | |
|Exit Slip | | | | |
|Monday |Tuesday |Wednesday |Thursday |Friday |
|Plotting Points on a 4-Quadrant |Plotting Points on a 4-Quadrant|Greatest Common Factor & Least|Greatest Common Factor & Least|Distributive Property |
|Grid |Grid |Common Multiple |Common Multiple | |
| | | | |K-W-L |
|Think/Pair/Share |Entrance Slip |K-W-L |Entrance Slip | |
| | | | |LEVEL C Activities |
|LEVEL C Activities |LEVEL C Activities |LEVEL C Activities |LEVEL C Activities |LEVEL B Activities |
|LEVEL B Activities |LEVEL B Activities | |LEVEL B Activities | |
| | | | |Exit Slip |
|Exit Slip | | | | |
|Monday |Tuesday |Wednesday |Thursday |Friday |
|Distributive Property |“Hidden Picture” Level A |“Hidden Picture” Level A |Unit Test Review |Unit Test |
| |GRASP Activity |GRASP Activity | | |
|Entrance Slip | | |Finish K-W-L |Post-Unit Reflection |
| | | | | |
|Think/Pair/Share | | | | |
| | | | | |
|LEVEL C Activities | | | | |
|LEVEL B Activities | | | | |
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