Order of Operations and Algebraic Thinking

Transcription

1 Ohio s New Learning Standards 5 th Grade Gifted Math Vocabulary: Order of Operations and Algebraic Thinking ü Brackets-mathematic symbols that show what operation goes first in the order of operations ( ) or [ ] ü Braces-symbols used to show that the numbers written between them are part of a set { } ü Evaluate Expressions-to replace the variable in an expression with a number and then solve it. ü Expression-a variable or combination of variables, numbers, and symbols that represents a mathematical relationship d + 8 ü Interpret Expression-to state what an expression means without solving the expression: 4 x (2 + 3) = two plus three times four ü Numerical Expression-a mathematical statement including numbers and operations 7 x 8 ü Numerical Pattern-an ordered set of numbers, the order helps you predict what will come next 2,4,6,8,10 ü Operations-a mathematical calculation, the four basic operations are addition, subtraction, multiplication, and division ü Order of Operations-an agreed upon order for performing operations to simplify expressions: Parentheses Exponents Multiplication/Division Addition/Subtraction ü Parenthesis-the symbol ( ) in an expression that shows which operation goes first Write and Interpret Numerical Expressions 5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols What are the grouping symbols in an expression and what do they mean? What is the order of operations for evaluation expressions? 1

2 5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, add 8 and 7, then multiply by 2 as 2 x (8+7).Recognize that 3 x ( ) is three times as large as , without having to calculate the indicated sum or product. What is an expression? What is the order of operations for evaluating expressions? Analyze Patterns and Relationships 5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. From ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule Add 3 and the staring number 0, and the given rule, Add 6 and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. How can I show the relationship of patterns in numbers? 2

3 Numbers and Operations in Base-Ten Vocabulary: ü Area Model- a model of multiplication that shows each place value product or a model of division that helps keep track of how much of the dividend is left to divide ü Array-an arrangement of objects in rows and columns ü Algorithm-a process or set of rules to be followed in calculations ü Base-Ten Numerals-a system of numbers that is based in units of 10. This is the system that is commonly used. ü Compare-to describe whether numbers are less than, greater than, or equal to one another ü Concrete Models-an object that represents a number or math concept that is used to help understand the concept ü Decimal-a number with one or more numbers to the right of the decimal point ü Divisor-the number that another number is divided by ü Dividend-the number that is divided by another number ü Equation-a statement that two mathematical expressions are equal ü Expression- a variable or combination of variables, numbers, and symbols that represents a mathematical relationship d + 8 ü Expanded Form-a way to write numbers that shows the place value of each digit = x (1/10) + 9 x (1/100) + 8 x (1/1000) ü Exponent-the number that tells how many times the base is used as a factor ü Hundredth- one part in a hundred = 1/100 or 0.01 ü Multiple- the product of a number and any whole number ex: multiples of 6: 6,12,18,24 ü Multiply-a process to find the total number of items in equal sized groups ü Place Value-the value of the place of a digit in a number ü Power of Ten-uses a base number of 10 with an exponent 10 = 100; 10 = The number of zeros corresponds with the exponent. ü Product-the answer in a multiplication problem ü Quotient-the answer in a division problem ü Round-to find the nearest value of a number based on a given place value ü Tenth-one part in a ten 1/10 ü Thousandth-one part in a thousand 1/100 3

4 Understand the Place Value System 5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. How does a digit s position affect its value? 5.NBT.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. How do you estimate and find products by using multiples of ten? 5.NBT.3 Read, write, and compare decimals to thousandths How do I read, write, and compare decimals to the thousandths place? 5.NBT.3(a) Read and write decimals to thousandths using base-ten numerals, number names, and expanded form. For example, = 3 x x x 1+ 3 x (1/10) + 9 x (1/100) + 2 x (1/1000) How do I read and write decimals using base ten numerals, number names, and expanded form? 5.NBT.3(b) Compare two decimals to thousandths based on meaning of the digits in each place using >,=,< symbols to record the results of the comparisons. How do you compare decimals to thousandths using >,<, and = signs? 5.NBT.4 Use place value understanding to round decimals to any place. How do I round decimals? 4

5 Perform operations with multi-digit whole numbers and with decimals to hundredths. 5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm. How do I multiply large numbers and explain the process? 5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividend and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. How do I divide multi-digit whole numbers and explain the process? How do I illustrate and explain the process of dividing multi-digit whole numbers? 5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. How do I add and subtract decimals? How do I multiply decimals? How do I divide decimals? How do I explain the computation using models or drawings? Activities/Projects: Differentiating Instruction With Menus Ø Place Value List p.39 Ø Decimals List p.54 Ø Whole Numbers Baseball Menu p Ø Money Tic-Tac-Toe p. 50 Ø Coordinate Graphs p.48 Math Art Ø Copycat Coordinates 5

6 Numbers and Operations Fractions Vocabulary: ü Benchmark Fraction-fractions that are used to help understand the relative size of other fractions. The common core benchmark fractions are: 0. 1/4, ½, ¾, and 1 ü Denominator-the number below the fraction bar-it tells how many equal parts in all ü Equivalent Fractions-fractions with different names, but equal values ü Factors-a number that is multiplied by another number to find a product. The factors of 18 are 1,2,3,6,9,18 ü Fraction-a number that names part of a whole or part of a group ü Improper Fraction-a fraction in which the numerator is larger than the denominator ü Mixed Number-a number represented by a whole number and a fraction ü Numerator-the number above the fraction bar that tells how many parts of a group ü Scaling-resizing of a fraction through multiplication ü Unit Fraction-a fraction that has a numerator of 1 Use equivalent fractions as a strategy to add and subtract fractions. 5.NF.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4= 8/ /12 = 23/12 How do I add and subtract fractions with unlike denominators? How do I add and subtract mixed numbers with unlike denominators? 5.NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. How do I solve word problems with fractions? How can I estimate using benchmark fractions to see if my answer makes sense? 6

7 Apply and extend previous understanding of multiplication and division to multiply and divide fractions. 5.NF.3 Interpret a fraction as division of the numerator by the denominator (a/b=a b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. How do fractions represent division? Why is it helpful to think of a fraction as division? 5.NF.4 Subtract multiples of 10 in the range from multiples of 10 in the range of (positive or zero differences), using concrete models of drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction: relate the strategy to a written method and explain the reasoning used. 5.NF.4(a) Interpret the product (a/b) x q as a parts of a partition of q into equal b parts; equivalently, as the result of a sequence of operations a x q b. How do I multiply a whole number by a fraction? How do I multiply a fraction by a fraction? 5.NF.4(b) Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. How do I multiply fractional side lengths to find the areas of rectangles? How do I use unit squares to find the area? 5.NF.5 Interpret multiplication as scaling (resizing) How does the size of the factor impact the size of the product? 7

8 5.NF.5(a) Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. How does the size of the factor impact the size of the product? 5.NF.5(b) Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b=(n x a)/(n x b)to the effect of multiplying a/b by 1. What happens to the product when you multiply a given number by a fraction less than 1? What happens to the product when you multiply a given number by a fraction greater than 1? 5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers e.g., by using visual fraction models or equations to represent the problem. How can I use visual fraction models or equations to represent real word problems involving multiplication of fractions and mixed numbers? How do I solve, explain, and illustrate problems that multiply fractions and mixed numbers? 5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. How do I divide by unit fractions by whole numbers? How do I divide whole numbers by unit fractions? 5.NF.7(a) Interpret division of a unit fraction by a non-zero whole number, and compute such quotients? How do I divide unit fractions by whole numbers? What does the quotient mean? 5.NF.7(b) Interpret division of a whole number by a unit fraction and compute such quotients 8

9 How do I divide a whole number by a unit fraction? What does the quotient mean? 5.NF.7(c) Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions e.g., by using visual fraction models and equations to represent the problem. How do I solve real world problems involving the division of fractions? How do I explain the solution to the problems? Activities/Projects: Differentiating Instruction with Menus Ø Adding and Subtracting Fractions Tic-Tac-Toe p. 67 Ø Multiplying and Dividing Fractions List Menu p.69 9

10 Measurement and Data. Vocabulary: ü Area- the number of square units needed to cover a flat surface ü Convert-to change from one form or system to another ü Cubic Unitsü Data Set-a group of collected information ü Formula-a set of symbols that express a mathematic rule ü Line Plot-a graph that uses marks (often Xs) above a number line to show data ü Rectangular Prism-a solid three-dimensional figure in which all 6 faces are rectangles ü Solid Figure-a three-dimensional figure ü Unit Cube-one unit that has a length, width, and height of 1 unit ü Volume-the amount of space a solid figure takes up. Volume = length x width x height Convert like measurements unit within a given measurement system 5.MD.1-Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5cm to 0.05 m) and use these conversions in solving multi-step, real world problems. How do I convert between different standard measurement units? How do I use these conversions in solving multi-step, real world problems? 5.MD.2-Make a line plot to display a data set of measurements in fractions of a unit (1/2, ¼, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. How do I create a line plot with fraction measurements? How do I solve problems using the data from a line plot? Geometric Measurement: understand concepts of volume and relate volume to multiplication and addition 5.MD.3-Recognize volume as an attribute of solid figures and understand concepts of volume measurement What is volume? 5.MD.3(a)- a cube with side length 1 unit, called a unit cube, is said to have one cubic unit of volume and can be used to measure volume 10

11 How do I find the volume of a solid figure using a unit cube? 5.MD.3(b)-a solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n unit cubes. How do I find the volume of a solid figure by using unit cubes? 5.MD.4-measure volumes by counting unit cubes, using cubic cm, cubic in, cubic feet, and improvised units? How do I use unit cubes to measure volume in cubic cm, cubic in, and cubic feet? 5.MD.5-relate volume to the operations of multiplication and addition and solve real world mathematical problems involving volume. How is volume related to multiplication and addition? 5.MD.5(a)-find the volume of a right rectangular prism with whole number side lengths by packing it with unit cubes, and show the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height and area of the base. How can you show that the volume is the same by using unit cubes and multiplying edge lengths? 5.MD.5(b)-apply the formulas V = l x w x h and V = b x h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems. How do I use a formula to find the volume of right rectangular prisms? 5.MD.5(c)- recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. How do I find the volume of solid figures made up of non-overlapping parts? 11

12 Projects/Activities: Differentiating Instruction With Menus Ø Measuring Capacity Tic-Tac-Toe p.92 Ø Measuring Weight Tic-Tac-Toe p.104 Ø Measurement Game Show Menu p

13 Geometry Vocabulary: ü Polygon ü Regular Polygon ü Triangle ü Hexagon ü Octagon ü Equilateral Triangle ü Isosceles Triangle ü Scalene Triangle ü Right Triangle ü Acute Triangle ü Obtuse Triangle ü Parallelogram ü Trapezoid ü Square ü Rhombus ü Rectangle ü Generalization ü Coordinate Grid ü X-axis ü Y-axis ü Origin ü Ordered Pair ü X-coordinate ü Y-Coordinate Graph points on the coordinate plane to solve real-world and mathematical problems 5.G.1-Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point the plane located by using an ordered pair of numbers, called its coordinates. Essential questions: How do I construct a coordinate system and plot points on it? How do I name the location of points on a coordinate grid? 5.G.1(a)- Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate). 13

14 How do the names of the two axes and the coordinates correspond? 5.G.2- Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. How do I plot points on a coordinate grid? What is the mathematical relationship between coordinates? Classify two-dimensional figures in categories based on their properties. 5.G.3-Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angels and squares are rectangles, so all squares have four right angles. What attributes do I consider when classifying a two-dimensional figure? 5.G.4-Classify two-dimensional figures in a hierarchy based on properties. Considering a two-dimensional figure and all its attributes, what geometric families does it belong to? Projects/Activities: Differentiating Instruction with Menus Ø Lines and Congruency 2,5,8 p.81 Ø Game Show Menu p.88 Ø Shapes Tic-Tac-Toe p.83 Resources: offers a variety of math games that align with standards Envision Math Triumph online Khan Academy Brain Pop Study Island 14

15 Simple Solutions 5th Grade Common Core Math Math Core 2k: Problem of the day, mental math strategies, and model lessons and student activities that are common core aligned 15

16 Fourth Grade Suggested Pacing Plan: This is a grade accelerated suggested pacing plan for the fourth grade gifted students. Due to pretesting and compacting, the teacher may need to adjust this schedule to suit the needs of the students. The teacher may also need to supplement with other resources. The Simple Solutions Common Core 5th Grade Math, MathCore2k, a variety of teaching and learning strategies, projects, math journals, essential questions, and formative assessments will also be implemented daily into this curriculum. Quarter 1: Envision Math Topic 1: Place Value Ø 5.NBT.1 Ø 5.NBT.3 Ø 5.NBT.3a Ø 5.NBT.3b Envision Math Topic 2: Adding and Subtracting Decimals Ø 5.NBT.4 Ø 5.NBT.7 Envision Math Topic 3: Multiplying Whole Numbers Ø 5.NBT.2 Ø 5.NBT.5 Ø 5.NBT.6 Ø 5.OA.1 Ø 5.OA.2 Envision Math Topic 4: Dividing by 1-Digit Divisors Ø 5.NBT.6 Ø 5.OA.2 16

17 Envision Math Topic 5: Dividing by 2-Digit Divisors Ø 5.NBT.6 17

18 Quarter 2 Envision Math Topic 6: Multiplying by Decimals Ø 5.NBT.1 Ø 5.NBT.2 Ø 5.NBT.7 Envision Math Topic 7: Dividing Decimals Ø 5.NBT.1 Ø 5.NBT.2 Ø 5.NBT.7 Envision Math Topic 8: Numerical Expressions: Patterns and Relationships Ø 5.OA.1 Ø 5.OA.2 Ø 5.OA.3 Envision Math Topic 9:Adding and Subtracting Fractions Ø 5.NF.1 Ø 5.NF.2 Envision Math Topic 10: Adding and Subtracting Mixed Numbers Ø 5.NF.1 Ø 5.NF.2 18

19 Quarter 3 Envision Math Topic 11:Multiplying and Dividing Fractions and Mixed Numbers Ø 5.NF.3 Ø 5.NF.4 Ø 5.NF.4a Ø 5.NF.4b Ø 5.NF.5 Ø 5.NF.5a Ø 5.NF.5b Ø 5.NF.6 Ø 5.NF.7 Ø 5.NF.7a Ø 5.NF.7b Ø 5.NF.7c Envision Math Topic 12:Volume of Solids Ø 5.MD.3 Ø 5.MD.3a Ø 5.MD.3b Ø 5.MD.4 Ø 5.MD.5 Ø 5.MD.5a Ø 5.MD.5b Ø 5.MD.5c Envision Math Topic 13: Units of Measure Ø 5.MD.1 Envision Math Topic 14: Data Ø 5.MD.2 Ø 5.G.2 19

20 Envision Math Topic 15: Classifying Plane Figures Ø 5.G.3 Ø 5.G.4 Envision Math Topic 16: Coordinate Geometry Ø 5.G.1 Ø 5.G.2 Ø 5.OA.3 Quarter 4: Test prep, review and on to 6 th grade material Possible math project to accompany our study of the Underground Railroad Resources to use within Envision: v Diagnostic Tests v Topic Tests v Performance Assessments v Quick Checks v Common Core Questions v Tiered Worksheets 20