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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