Helping Your Children Learn in the Middle School Years MATH


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1 Helping Your Children Learn in the Middle School Years MATH Grade 7 A GUIDE TO THE MATH COMMON CORE STATE STANDARDS FOR PARENTS AND STUDENTS
2 This brochure is a product of the Tennessee State Personnel Development Grant (SPDG). It was researched and compiled by Dr. Reggie Curran, University of Tennessee, Knoville, and reviewed by Ryan Mathis, Tennessee Department of Education, through a partnership between SPDG and the University of Tennessee s Center for Literacy, Education and Employment (CLEE). Nathan Travis, Grant Director Tennessee Department of Education 710 James Robertson Parkway Nashville, TN Donna Parker, Project Manager Ryan Mathis, Content Reviewer Margy Ragsdale, Copyeditor THREADS OF INNOVATION: PROFESSIONAL DEVELOPMENT COLLABORATIONS FAMILY INVOLVEMENT HIGHER EDUCATION INVOLVEMENT USE OF TECHNOLOGY INITIATIVE RTI Literacy INITIATIVE Teacher Equity INITIATIVE School and Instructional Climate INITIATIVE Early Childhood INITIATIVE RTI Math For more information about the Common Core State Standards and Child Development, check out this website: Tennessee Common Core at This project is supported by the U.S. Department of Education, Office of Special Education Programs (OSEP). Opinions epressed herein are those of the authors and do not necessarily represent the position of the U.S. Department of Education. UT PUBLICATION NO.: R
3 Purpose of this Booklet CONTENTS What are the Common Core State Standards? 4 Overview of Goals for Standards for Mathematical Practice 5 What are the Tennessee Focus Clusters? Math Standards by Grade Helping Your Children Learn in the Middle School Years 6 Seventh Grade Math Standards Descriptions Student Checklist Grade 7 9 Helping Students Succeed in Math Tennessee RTI Model RTI DecisionMaking Process 6 Additional RTI Resources for Parents 7 Strategies for Families to Help Struggling Learners 40 Citations and Acknowledgements A GUIDE TO THE MATH COMMON CORE STATE STANDARDS FOR PARENTS AND STUDENTS 1
4 Purpose of This Booklet This booklet has two goals: n to help parents understand more about what their children are learning in school, and n to help students know if they have mastered the skills their teachers epect them to know in each grade Teachers work from a set of standards that tell them what to teach. Each state has created its own standards, and those standards have not been the same across our country. However, most states have recently agreed to use the same set of standards the Common Core State Standards. More information is included about the Common Core State Standards in the following pages. This booklet will eplain what the Common Core State Standards are, and about the skills on which Tennessee teachers will focus math instruction while transitioning to the Common Core State Standards. You will find general information that will give you an overview of what the standards are and why states are using them. At the end of the list of standards and eplanations, you will find a bo with an I know this! checklist. These are short statements about the skills your children will be epected to have mastered by the end of the year. Ask your children to look at them to see if they feel they have mastered those skills, or if they need some etra help in specific areas. We hope you will find this booklet helpful in your effort to be a partner in your child s education and development. If you come across a math term and don t remember what it is or what it means, check out the Math is Fun dictionary at
5 What are the Common Core State Standards? Academic standards are statements that describe the goals of schooling what children should know or be able to do at the end of the school year. For eample, the second grade math standards state that by the end of the school year, a second grader should be able to count to 10 and understand what each digit in a threedigit number represents. However, standards have not been the same across the United States. In the past, states have had their own sets of standards. This means that children in one state may be learning different things at different times (and in different years) than children in another state. Many states have recently agreed to use a common set of standards for learning that takes place in their classrooms; these are the Common Core State Standards (CCSS). One major benefit of having common standards across states is that children are being required to learn the same information in the same years in each of those states, so that a child moving from one state to another will not be behind the children in the new location. A common set of standards ensures that all students, no matter where they live, are focused on graduating from high school prepared for postsecondary education and careers. The Common Core State Standards for Math have two components: Standards for Mathematical Practice and Standards for Mathematical Content. The Practice Standards describe the kind of math teaching and learning that will produce the most successful learning and that will help students dig deeper and better understand math. The Content Standards outline the concepts and skills to be learned in each grade; teachers will balance procedural skills with understanding by finding ways to engage students in good practices that will help them understand the math content as they grow in math maturity and epertise throughout the elementary, middle, and high school years. The Common Core State Standards will provide students, teachers, and parents with a shared understanding of what students are learning. With students, parents, and teachers all on the same page and working together for shared goals, we can increase the likelihood that students will make progress each year and will graduate from school prepared to succeed and to build a strong future for themselves and the country. Parents: In this booklet, you will find an overview of the standards for the seventh grade, showing you what your children should be able to do by the end of the school year. At the end of the section, you will find a bo with this I can do it! symbol. Discuss these items with your child to see if he/she is able to complete these tasks. Students: Find the I know this! bo at the end of each section and check yourself to see if you can do all those things. I know this!
6 Overview of Goals for Standards for Mathematical Practice The Standards for Mathematical Practice describe skills and behaviors that all students should be developing in their particular grades. These practices include important processes (ways of doing things) and proficiencies (how well we do things), including problem solving, reasoning and proof, communication, representation, and making connections. These practices will allow students to understand and use math with confidence. Following is what children will be working to be able to do with increasing ease: Make sense of problems and persevere in solving them Find the meaning in problems Analyze, predict, and plan the path to solve a problem Verify answers Ask themselves the question: Does this make sense? Reason abstractly and quantitatively Be able to translate the meaning of each math term in any equation Interpret results in the contet (setting) of the situation Construct arguments and evaluate the reasoning of others Understand and use information to build arguments Make and eplore the truth of estimates and guesses Justify conclusions and respond to arguments of others Model with mathematics Apply math to problems in everyday life Identify quantities (amounts, numbers) in a practical situation Present, show, or eplain the problem and solution in an understandable way Use appropriate tools strategically Consider the available tools when solving problems, and know which tool is most appropriate in the situation Be familiar with tools appropriate for their grade level or course (pencil and paper, concrete models, ruler, protractor, calculator, spreadsheet, computer programs, digital content on a website, and other technological tools) Be precise Be able to communicate accurately with others Use clear definitions, state the meaning of symbols, and be careful when specifying units of measure and labeling aes (the and y lines that cross at right angles to make a graph) in math figures Calculate accurately and efficiently Look for and make use of structure Recognize patterns and structures Step back to find the big picture and be able to shift perspective See complicated things as single objects, or as being made up of several objects Look for and identify ways to create shortcuts when doing problems When calculations are repeated, look for general methods, patterns, and shortcuts Be able to evaluate whether an answer makes sense The major domains included in the math standards for grades 68 are listed below. In each grade, students build on what they learned previously to form a progression of increasing knowledge, skill, or sophistication. MAJOR DOMAINS FOR GRADE MATH STANDARDS Ratios and Proportional Relationships The Number System Epressions and Equations Functions Geometry Statistics and Probability 4
7 Tennessee Focus Clusters The Common Core State Standards in Mathematics present an opportunity to engage Tennessee students in deeper problem solving and critical thinking that will build the math and reading skills students will need for success. The new core standards will allow teachers to provide focus, coherence, and rigor (difficulty and thoroughness). Students will think more deeply and know more than how to just get the answer or read the words on the page they will understand! Teachers will link major topics within grades math includes reading and reading includes math (and other subjects as well). Finally, teachers will provide more challenge to students so they will understand how to apply what they are learning to the real world. While teachers will teach all of the standards, they will focus instruction on specific areas that will build stronger understanding. To help teachers ease into the move from the Tennessee State Standards to the Common Core State Standards, educators in the state of Tennessee have created a list of clusters (TNCore Focus Clusters) on which teachers will focus instruction in the net two years. Clusters are groups of standards that connect needed concepts and skills. The table below shows the focus areas for each grade for school years and In addition, teachers will still be teaching some of the information outlined in the Tennessee State Standards. Eventually, Tennessee teachers will be moving fully to the Common Core State Standards. TNCore FOCUS CLUSTERS FOR MATH and th Grade 7th Grade 8th Grade Understand ratio concepts and use ratio reasoning to solve problems. Apply and etend previous understandings of arithmetic to algebraic epressions. Apply and etend previous understandings of numbers to the system of rational numbers. Reason about and solve onevariable equations and inequalities. Analyze proportional relationships and use them to solve realworld and mathematical problems. Solve reallife and mathematical problems using numerical and algebraic epressions and equations. Apply and etend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. Use properties of operations to generate equivalent epressions. Understand the connections between proportional relationships, lines, and linear equations. Define, evaluate, and compare functions. Analyze and solve linear equations and pairs of simultaneous linear equations. Use functions to model relationships between quantities. + Mathematics 5
8 Seventh Grade Math Focus Standards for Seventh Grade Teachers will focus on specific skills in each grade. For seventh grade, the instructional focus will center on these skills: Analyze proportional relationships and use them to solve realworld and mathematical problems. Solve reallife and mathematical problems using numerical and algebraic epressions and equations. Apply and etend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. Use properties of operations to generate equivalent epressions. Skills that focus on these areas appear in the shaded bo below. While these skills are priority areas, students will be learning all of the skills listed in the following sections. Ratios and Proportional Relationships For seventh graders, the math standards outline the skills that should be developing, so that a student can say, I can (insert math goal). For eample, a student might say, I can add, subtract, multiply, and divide rational numbers. Your child will be working on the following skills this year. Ratios and Proportional Relationships Analyze proportional relationships and use them to solve realworld and mathematical problems. 1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For eample: If 1/ gallon of paint covers 1/6 of a wall, then how much paint is needed to cover the entire wall? The amount of paint needed for the entire wall can be computed by 1/ gallon divided by 1/6 wall. This calculation gives gallons. 6
9 . Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. For eample: The table below gives the price for different numbers of books. Do the numbers in the table represent a proportional relationship? If the amounts from the table below are graphed (number of books, cost), the pairs (1, ), (, 6), and (, 9) will form a straight line through the origin (0 books cost 0 dollars), indicating that these pairs are in a proportional relationship. The ordered pair (4, 1) means that 4 books cost $1. However, the ordered pair (5, 1) would not be on the line, indicating that it is not proportional to the other pairs. Number Cost COST NUMBER OF BOOKS blue cups total cups Equivalent Ratios Versus Equivalent Fractions Equivalent Ratios 4 6 more parts, same size parts more total cups, more blue cups 6 9 Equivalent Fractions _ = 4_ = 6 more parts, smaller parts 6_ 9 same whole amount, same portion b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Cost of Bananas This graph represents the price of the bananas at one store. What is the constant of proportionality? From the graph, it can be determined that 4 pounds of bananas is $1.00; therefore, 1 pound of bananas is $0.5, which is the constant of proportionality for the graph. The cost of bananas at a store can be determined by the equation: P = $0.5n, where P is the price and n is the number of pounds. PRICE (cents) BANANAS POUNDS c. Represent proportional relationships by equations. For eample: If total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can 7
10 be epressed as t = pn. A student is making trail mi. Create a graph to determine if the quantities of nuts and fruit are proportional for each serving size listed in the table. If the quantities are proportional, what is the constant of proportionality or unit rate that defines the relationship? Eplain how you determined the constant of proportionality and how it relates to both the table and graph. y Serving Size Cups of Nuts () Cups of Fruits (y) The relationship is proportional. For each of the other serving sizes there are cups of fruit for every 1 cup of nuts (:1). The constant of proportionality is shown in the first column of the table and by the slope of the line on the graph. FRUIT (cups) NUTS (cups) d. Eplain what a point (, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. y Number of Packs of Gum (g) Cost in Dollars (d) The graph to the right represents the cost of gum packs as a unit rate of $ dollars for every pack of gum. The unit rate is represented as $/pack. Represent the relationship using a table and an equation. COST (dollars) PACKS OF GUM Equation: d = g, d is the cost in dollars and g is the number of packs of gum.. Use proportional relationships to solve multistep ratio and percent problems. Eamples are simple interest, ta, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. For eample: After eating at a restaurant, your bill before ta is $5.50. The sales ta rate is 8%. You decide to leave a 0% tip for the waiter 8
11 based on the preta amount. How much is the tip you leave for the waiter? How much will the total bill be, including ta and tip? Total bill will be food + ta + tip, or $ (.0 $5.50) + (.08 $5.50). The amount paid = Tip = (0.0 $5.50) = $10.50 Ta = (0.08 $5.50) = $4.0 Total Bill = $ $ $4.0 = $67.0 The Number System Apply and etend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 1. Represent addition and subtraction on a horizontal or vertical number line diagram. + = a. Describe situations in which opposite quantities combine to make 0. On this number line, the numbers a and b are the same distance from 0. What is the sum of a + b? b 0 a b. Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contets. For eample: John earns $ for raking the leaves, but he owes his brother $. How much money will John have after he pays his brother? ( + ) = 0.  and are shown to be opposites on the number line because they are equal distance from zero and therefore  0 have the same absolute value. The sum of the number () and its opposite () is zero. c. Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contets. For eample: If one of the integers is negative, 9
12 subtract the absolute value of it from the other number. Eample: 14 + (6) = 14 6 = 8 If both of the integers are negative, add their absolute values and prefi the number with a negative sign. Eample: (14) + (6) = = ( 0) d. Apply properties of operations as strategies to add and subtract rational numbers. (See Properties Charts in the following section.). Apply and etend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a. Understand that multiplication is etended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations (see charts below), particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contets. b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with a nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing real world contets. Integer a number with no fractional parts Rules for Multiplying and Dividing Signed Numbers MULTIPLYING The product of two numbers with the same signs is positive. The product of two numbers with different signs is negative. DIVIDING The quotient of two numbers with the same signs is positive. The quotient of two numbers with different signs is negative Integers include the counting numbers (1,, ), zero (0), and the negative of the counting numbers (1, ,  ). An rational number is any number that can be made by dividing one integer by another. The term comes from the word ratio. EXAMPLES: 1/ is a rational number (1 divided by, or the ratio of 1 to ) is a rational number. 1 is a rational number (1/1). is a rational number (/1)..1 is a rational number (1/100) is a rational number (66/10). Rational Number a b c. Apply properties of operations as strategies to multiply and divide rational numbers. (See Properties Charts in the following section.) 10
13 d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number eventually terminates in zero or repeats. Terminating decimal numbers can easily be written in decimal form. For eample: 0.67 is 67/100 For eample:.4098 = 4098/ Nonterminating decimal numbers can also be rational numbers. For eample, 1/9 converted into a decimal is (doesn t end), but since it can be written as a fraction (1/9), it is a rational number. In fact, every nonterminating decimal number that REPEATS a certain pattern of digits is a rational number. An irrational number is a real number that cannot be written as a simple fraction. Irrational means not rational. For eample, pi (π) is a famous irrational number: (and more...). You cannot write down a simple fraction that equals pi. The popular approimation of /7 = is close but not accurate.. Solve realworld and mathematical problems involving the four operations with rational numbers. For eample: Your cell phone bill is automatically deducting $ from your bank account every month. How much will the deductions total for the year? = 1 () = $84 Properties of Addition and Subtraction Commutative Property. The commutative property says that the positions of the numbers in a mathematical equation do not affect the ultimate solution. Five plus three is the same as three plus five. This applies to addition, regardless of how many numbers you add together. The commutative property allows you to add a large group of numbers together in any order. The commutative property does not apply to subtraction. Five minus three is not the same as three minus five. Associative Property. The associative property applies to more complicated equations that use parentheses or brackets to separate groups of numbers. The associative property says that numbers you are adding together can be grouped in any order. When you are adding numbers together, you can move the parentheses around. For eample, ( + 4) + = + (4 + ). The associative property also does not apply to subtraction since (  4)  does not equal  (4  ). This means that if you are working on a subtraction equation, you cannot move the brackets around. Identity Property. The identity property says that any number plus zero equals itself. For eample, + 0 =. The identity property also applies to subtraction since  0 =. Zero is known as the identity number because in addition and subtraction it does not affect other numbers. When a child is adding or subtracting large groups of numbers, remind her that the number zero does not affect other numbers in the equation. Inverse Operations. In addition to the properties that affect addition and subtraction separately, addition and subtraction also relate to each other. They are inverse operations, which is similar to saying that addition and subtraction are opposites. For eample, five plus three minus three equals five because adding and then subtracting the threes cancels both of them out. Encourage your child to look for numbers that cancel each other out when he is adding and subtracting. 11
14 Properties of Multiplication Commutative Property. The order of the numbers doesn t change the result (answer to the problem). p q = q p Associative Property. The grouping of the factors doesn t change the answer. (p q) r = p (q r) Distributive Property. Multiplying the sum (or difference) by a number is the same as multiplying each number in the sum (or difference) by the number and adding (or subtracting) the product. 9 (0  ) = (9 0) (9 ) 8 (40 + 5) = (8 40) + (8 5) Zero Property. When any number is multiplied by zero, the answer is zero. 98,756,4 0 = 0 Epressions and Equations Use properties of operations to generate equivalent epressions. 1. Apply properties of operations as strategies to add, subtract, factor, and epand linear epressions with rational (whole numbers that may or may not be epressed as fractions, for eample, 4 or 4/1) coefficients. For eample: We can use the commutative and associative properties to add linear epressions with rational coefficients (e.g., 4 + ( + ) = 4 + ( + ) = (4 + ) + =  + ). We can use the distributive property to add and/or subtract linear epressions with rational coefficients (e.g., 1/5 + /5 = (1/5 + /5) = /5). We can use the distributive property to factor a linear epression with rational coefficients (e.g., = ( + ). We can use the distributive property to epand a linear epression with rational coefficients (e.g., /(9 + 6) = (/ 9) + (/ 6) = 6 + 4).. Understand that rewriting an epression in different forms in a problem contet can shed light on the problem and how the quantities in it are related. For eample: If you know that your rent is $500 now and is going up 5% net year, how much will the new rent payment be? 5% is the same as.05, so a a = 1.05a. Increase by 5% is the same as multiply by 1.05, so $500 multiplied by 1.05 equals $55.00 your new rent. 1
15 Solve reallife and mathematical problems using numerical and algebraic epressions and equations. 1. Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For eample: If a woman making $5 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $.50, for a new salary of $7.50. If you want to place a towel bar 9 /4 inches long in the center of a door that is 7 1/ inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the eact computation. 7 1_ inches 7 1_ 9 _ = 17 _ 17 _ = 8 7_ _ inches 8 9 _ inches 8 7_ inches 4 8 estimate of 9 inches correct. Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. For eample: The youth group is going on a trip to the state fair. The trip costs $5. Included in the price is $11 for a concert ticket and the cost of passes, one for the game booths and one for the rides. Each of the passes cost the same price. Write an equation representing the cost of the trip and determine the price of each of the passes = cost of one pass + 11 = 5 = 41 = $0.50 1
16 a. Solve word problems leading to equations of the form p + pq = r and p( + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For eample: The perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? p = l + w 54 = ( 6) + ( w) Width = 1 centimeters 54 = 1 + w 54 1 = w 4 = w 1 = w 54 = ( 6) + ( 1), also 54 = (6 +1) b. Solve word problems leading to inequalities of the form p + q r or p + q r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the contet of the problem. For eample: As a salesperson, you are paid $50 per week plus $ per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions. p + q r = 16.66, so if you want to make at least $100, you must make at least 17 sales ( 17) = = 101 Salary Salary for Week
17 Geometry Draw, construct, and describe geometrical figures and describe the relationships between them. 1. Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Computations with rational numbers etend the rules for manipulating fractions to comple fractions. 7 in. Perimeter is (7 ) + ( ) = 18 Area is (7 ) = 14 in in. For eample: if the rectangle above is enlarged using a scale factor of 1.5, what will be the perimeter and area of the new rectangle? The perimeter is linear or onedimensional. Multiply the perimeter of the given rectangle (18 in.) by the scale factor (1.5) to get an answer of 7 in. We could also increase the length and width by the scale factor of 1.5 to get 10.5 inches for the length (7 1.5) and in. for the width ( 1.5). The perimeter could be found by adding to get 7 in. The area is twodimensional so the scale factor must be squared. The area of the old rectangle is 14 but the scaled up rectangle would be , or 14.5, which is Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. Eample 1: Which of these are quadrilaterals with at least one set of parallel sides and no right angles? has parallel sides and right angles has parallel sides and no right angles has parallel sides and no right angles has parallel sides and right angles has no parallel sides and no right angles Eample : Will three sides of any length create a triangle? Possibilities to eamine are: Answer: A will not work; B and C will a. 1 cm, 5 cm, and 6 cm work. Students should recognize that the sum b. cm, cm, and cm of the two smaller sides must be larger than c. cm, 7 cm, 6 cm the third side. 15
18 . Describe the twodimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Cuts made parallel will take the shape of the base; cuts made perpendicular will take the shape of the lateral (side) face. a. If a pyramid is cut with a plane (green) parallel to the base, the intersection of the pyramid and the plane is a cross section (red). b. If the pyramid is cut with a plane (green) passing through the top verte and perpendicular to the base, the intersection of the pyramid and the plane is a triangular cross section. c. If the pyramid is cut with a plane (green) perpendicular to the base but not through the top verte, the intersection of the pyramid and the plane is a trapezoidal cross section (red). Solve reallife and mathematical problems involving angle measure, area, surface area, and volume. 1. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. Students understand the relationship between radius and diameter. Students also understand the ratio of circumference to diameter can be epressed as pi (π). Building on these understandings, students generate the formulas for circumference and area. The circumference of a circle (the distance around the circle) is times the radius (the distance from the center to the edge of a circle) times π (.14). ( c = πr). Radius Diameter Circumference The ratio of a circle s circumference (distance around the circle) to the diameter is π, or (the digits go on forever without repeating, so we use.14 as an approimate number in calculations). The area of a circle is circle is π times the radius squared. r CIRCLE Area = π r Circumference = π r r = radius 16
19 The illustration shows the relationship between the circumference and area. If a circle is cut into wedges and laid out as shown, a parallelogram results. Half of an end wedge can be moved to the other end and a rectangle results. The height of the rectangle is the same as the radius of the circle. The base length is ½ of the circumference, which is ( π r). The area of the rectangle (and therefore the circle) is found by the following calculations: Area = Base Height Area = ½ ( π r) r Area = π r r Area = π r r πr The base of the rectangle is made of half the outer wedges of the circle. The other half forms the top edge of the rectangle. Real life eample: If a circle is cut from a square piece of plywood, how much plywood would be left over? The area of the square is 8 8 or 784 in. The diameter of the circle is equal to the length of the side of the square, or 8'', so the radius would be 14''. The area of the circle would be approimately in. The difference in the amounts (plywood left over) would be ( ) or in. 8''. Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure. Complementary angles: Two angles are complementary if they add up to 90 degrees (right angle). 40º 50º These two angles (40º and 50º) are complementary angles because they add up to 90º. Notice that together they make a right angle. 7º 6º However, the angles don t have to be together. These two are complementary because 7º and 6º = 90º. 17
20 Supplementary angles: Two angles are supplementary if they add up to 180 degrees. 40º 140º These two angles (40º and 140º are supplementary angles because they add up to 180º. Notice that together they make a straight angle. 60º 10º However, the angles don t have to be together. These two angles are supplementary because 60º + 10º = 180º. Vertical angles: Angles that are opposite each other when two lines cross. In this eample, aº and bº are vertical angles. aº bº The interesting thing here is that vertical angles are equal: aº = bº Adjacent angles: Two angles are adjacent if they have a common side and a common verte (corner point) and don t overlap. What is and isn t an adjacent angle? aº aº bº b º aº bº These are adjacent angles. They share a verte and a side. These are not adjacent angles. They only share a verte, not a side. These are not adjacent angles. They only share a side, not a verte. 18
21 A 6º C º Angle ABC is adjacent to angle CBD because they have a common side (line CB) and they have a common verte (point B). B D Eample: Write and solve an equation to find the measure of angle. Solution: The right angle at the bottom 40º is 90, and the top angle is 40. Since º the angles of a triangle add up to 180, the equation would be ( ), or 50. The missing angle is 50. The measure of angle is supplementary to 50, so subtract 50 from 180 to get a measure of 10 for.. Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Students will know some formulas for finding area, volume, and surface areas, but knowing the formula does not mean memorization of the formula; it means to have an understanding of why the formula works and how the formula relates to the measure (area and volume). For eample: Students can build on their work with nets in the 6th grade, recognizing that finding the area of each face of a threedimensional figure and adding the areas will give the surface area. In addition, students understanding of volume can be supported by focusing on the area of base times the height to calculate volume, and understanding of surface area can be supported by focusing on the sum of the area of the faces. For eample: The surface area of a cube is 96 in. What is the volume of the cube? Solution: The area of each face of the cube is equal. Dividing 96 by 6 gives an area of 16 in for each face. Because each face is a square, the length of the edge would be 4 in. The volume could then be found by multiplying or 64 in. 19
22 Statistics and Probability Use random sampling to draw inferences about a population. 1. Understand that statistics can be used to gain information about a population by eamining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. Random sample: A sample in which each individual or object in the population has an equal chance of being selected. For eample: The school food service wants to increase the number of students who eat hot lunch in the cafeteria. The student council was asked to conduct a survey of the student body to determine the students preferences for hot lunch. They have determined three ways to do the survey. The three methods are listed below. Determine if each survey option would produce a random sample. Which survey option should the student council use and why? 1. Write all of the students names on cards and pull them out in a draw to determine who will complete the survey.. Survey the first 0 students that enter the lunchroom.. Survey every rd student who gets off a bus. The student council should use solution #1; it would be a true random sample because every student has the same chance to have his/her name pulled to complete the survey. In #, only those students who get to lunch early have a chance, and in #, only those students who ride the bus have a chance.. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. Eamples: Estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. For eample: Below is the data collected from two random samples of 100 students regarding students school lunch preference. Make at least two inferences based on the results. Student Sample Hamburgers Tacos Pizza TOTAL # # Solution: Most students prefer pizza. More people prefer pizza than hamburgers and tacos combined. 0
23 Draw informal comparative inferences about two populations. 1. Informally assess the degree of visual overlap of two numerical data distributions with similar variability, measuring the difference between the centers by epressing it as a multiple of a measure of variability. For eample: Jason wanted to compare the mean height of the players on his favorite basketball and soccer teams. He thinks the mean height of the players on the basketball team will be greater but doesn t know how much greater. He also wonders if the variability of heights of the athletes is related to the sport they play. He thinks that there will be a greater variability in the heights of soccer players as compared to basketball players. He used the rosters and player statistics from the team websites to generate the following lists. Basketball Team Heights of Players in inches for Season 75, 7, 76, 78, 79, 78, 79, 81, 80, 8, 81, 84, 8, 84, 80, 84 Soccer Team Heights of Players in inches for 010 Season 7, 7, 7, 7, 69, 76, 7, 7, 74, 70, 65, 71, 74, 76, 70, 7, 71, 74, 71, 74, 7, 67, 70, 7, 69, 78, 7, 76, 69 To compare the data sets, Jason creates two dot plots on the same scale. The shortest player is 65 inches and the tallest players are 84 inches. In looking at the distribution of the data, Jason observes that there is some overlap between the two data sets. Both teams have some players between 7 and 78 inches tall. Jason decides to use the mean and mean absolute deviation to compare the data sets. Jason sets up a table for each data set to help him with the calculations. The mean (the number found by the dividing the sum of a set of numbers by the number of addends, sometimes called the average) height of the basketball players is inches as compared to the mean height of the soccer players at 7.07 inches, a difference of 7.68 inches. The mean absolute deviation (MAD) is calculated by taking the mean of the absolute deviations (Deviation just means how far from the normal, and absolute deviation is a positive number) for each data point. The difference between each data point and the mean is recorded in the second column of the table. Jason used rounded values (80 inches for the mean height of basketball players and 7 inches for the mean height of soccer players) to find the differences. The absolute deviation, the absolute value (positive number) of the deviation, is recorded in the third Height of Soccer Players (in.) Height of Basketball Players (in.) 1
24 Heights of Players on Soccer and Basketball Teams Soccer Players (n = 9) Basketball Players (n = 16) Height (in.) = 090 Deviation from Mean (in.) Absolute Deviation (in.) = 6 Height (in.) = 176 Deviation from Mean (in.) Absolute Deviation (in.) = 40 Mean = = 7 in. MAD = 6 9 =.1 in. Mean = = 80 in. MAD = =.5 in. column. The absolute deviations are summed and divided by the number of data points in the set. The mean absolute deviation is.14 inches for the basketball players and.5 for the soccer players. These values indicate moderate variation in both data sets. There is slightly more variability in the height of the soccer players. The difference between the heights of the teams is approimately times the variability of the data sets ( =.04).
25 . Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For eample: The seventh grade s essays were scored on specific traits. The scores for ideas Scores were,, 6, 5,, 5,, 4, 6, 5, 5, 6, 4,, 4,, 4, 5, for Ideas 4, 6, 5, and the scores for organization were, 6,, 5, 4, 0,, 1,, 4,, 6, 5,, 5, 4, 4,. Which trait got higher scores? Scores for Organization Showing the two graphs side by side helps students make comparisons. For eample: Students would be able to see from the display of the two graphs that the ideas scores are generally higher than the organization scores. One observation students might make is that the scores for organization are clustered around a score of whereas the scores for ideas are clustered around a score of 5. Investigate chance processes and develop, use, and evaluate probability models. 1. Understand that the probability of a chance event is a number between 0 and 1 that epresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/ indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. 0 1_ 1 The closer the fraction is to 1, the greater the probability the event will occur. Larger numbers indicate greater likelihood. Students also recognize that the sum of all possible outcomes is 1. Eample 1: There are three choices of jellybeans grape, cherry and orange. If the probability of getting grape is /10 and the probability of getting cherry is 1/5, what is the probability of getting orange? /10 + 1/5 + = 1 /10 + /10 + = 1 5/10 + = 1 = 1 5/10 = 5/10, or 1/ There is a 1/ chance you will get orange.
26 Eample : The container at the right contains gray, 1 white, and 4 black marbles. Without looking, if Eric chooses a marble from the container, will the probability be closer to 0 or to 1 that Eric will select a white marble? A gray marble? A black marble? Solution: You have a better chance at choosing a black marble Gray has /7 chance which is closer to 0 White has a 1/7 chance, which is closer to 0 Black has a 4/7 chance, which is closer to 1. Approimate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approimate relative frequency given the probability. For eample: A baseball player comes up to bat. He has hit the ball 118 times in his last 4 times at bat. What is the probability that he will hit the ball this time? One way to interpret what this probability number means is what people call long run relative frequency. The phrase long run is an important part of this interpretation. Let s take the eample of the baseball player. We re partway into the season and he has batted 4 times. He s gotten a hit 118 times. Another way of saying this is that the frequency of at bats is 4 and the frequency of successful at bats, that is the frequency of hits, is 118. Relative frequency is defined by taking the total hits and dividing them by the total at bats 118/4 (which doesn t reduce evenly below 59/171, but is a little more than 1/. The relative frequency is.45 minus the proportion of total at bats that resulted in a hit. People who are involved with playing baseball tend to multiply that relative frequency or proportion by a thousand and say the batting average is 45. EXAMPLE: A baseball player gets 118 hits in 4 times at bats. The frequency of opportunities to get a hit (at bats) is 4. The frequency of successes (hits) is 118. EXAMPLE: A baseball player gets 118 hits in 4 times at bats. The frequency of (at bats) is 4. The frequency of (hits) is 118. The RELATIVE FREQUENCY of hits per at bats is. RELATIVE FREQUENCY: 118 =.45 4 A common and useful interpretation of probability is long run relative frequency. The baseball player s relative frequency of hits is.45. We could think of the player s probability of getting a hit as the player s long run relative frequency. 4
27 . Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, eplain possible sources of the discrepancy. Probability is such a natural part of your life that you rarely think about it. However, every time you use a word like might, may, undoubtedly, without fail, or maybe, you are voicing a probability that an event will occur. Scientists and mathematicians like to epress probability more accurately. a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For eample: If a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. Jane has a 1/0 chance of being selected. A female has an 8/0 chance of being selected. Students in class 0 Male Students 1 b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For eample: Find the approimate probability that a spinning penny will land heads up or that a tossed paper cup will land openend down. Do the outcomes for the spinning penny appear to be equally likely, based on the observed frequencies? If you toss a penny in the air, the probability (P) that it will land heads up can be epressed: P = # of times it lands heads/total number of coin tosses. The number will be 5/10, which reduces to ½. This means that you have a 1 out of chance that the penny will land heads. Female Students 8 The graph (at right) shows the results of an eperiment in which a coin was tossed thirty times. Find the eperimental probability of tossing tails for this eperiment. P (tails) = number of times tails occurs number of possible outcomes 18 = 0 = 5 The eperimental probability of tossing tails is. 5 Compare the eperimental probability you found in eample to its theoretical probability. 1 The theoretical probability of tossing tails on a coin is, so the eperimental probability is close to the theoretical probability. Number of Tosses You call it! Heads Tails 5
28 4. Find probabilities of compound events (an event made of two or more simple events) using organized lists, tables, tree diagrams, and simulation. a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. b. Represent for compound events using methods such as organized lists, tables and tree diagrams. Eample 1: How many ways could the three students, Amy, Brenda and Carla, come in 1st, nd and rd place? Solution: Making an organized list will identify that there are si ways for the students to win a race: A, B, C A, C, B B, C, A B, A, C C, A, B C, B, A Eample : Students conduct a bag pull eperiment. A bag contains five marbles. There is one red marble, two blue marbles and two purple marbles. Students will draw one marble without replacement and then draw another one. What is the sample space for this situation? Eplain how the sample space was determined and how it is used to find the probability of drawing one blue marble followed by another blue marble. Eample : A fair coin will be tossed three times. What is the probability that two heads and one tail in any order will result? (Adapted from SREB publication Getting Ready for Algebra I: What Middle Grades Students Need to Know and Be Able to Do) Solution: HHT, HTH, and THH, so the probability would be /8. Eample 4: Show all possible arrangements of the letters in the word FRED using a tree diagram. If each of the letters is on a tile and drawn at random, what is the probability of drawing the letters FRED in that order? What is the probability that a word will have an F as the first letter? Solution: There are 4 possible arrangements (4 choices choices choices 1 choice). The probability of drawing FRED in that order is 1/4. The probability that a word will have an F as the first letter is 6/4 or 1/4. Start F R E D R E D F E D F R D F R E E D R D R E E D F D F E R D F D R F R E F E F R D E D R E R D R D F E F D R D F F R E R E F R F 6
29 c. Design and use a simulation (an eperiment that models a reallife situation) to generate frequencies for compound events. Simulations require the use of random numbers. Random numbers have no pattern; they cannot be predicted in any way. Knowing a random number in no way allows you to predict the net random number. An eample of a random number generator is a standard number cube or dice, which will randomly generate a number from 1 to 6 when you roll it, or a scientific calculator that may or may not generate a larger range of numbers. COMPOUND EVENTS The combination of two or more events EXAMPLES Roll two dice Deal two cards When the mathematics becomes too complicated to figure out the theoretical probability of certain events, statisticians often use computer simulations instead. Comple simulations like modeling the weather, traffic patterns, cancer radiation therapy, or stock market swings require tens of billions of random numbers. For those simulations a large computer running for hours or even days is needed. 7
30 Seventh Grade Student SelfCheck List I know this! Students: You have been working on learning these things this year. Are you able to do these things? Check the bo net to the skill if you can do it. Seventh grade students deepen their understanding of proportional relationships to solve complicated problems. They etend their understanding of rational numbers to include computation (add, subtract, multiply, and divide). Irrational numbers are introduced in seventh grade. Algebraic foundations are practiced and etended. Students continue to etend their understanding of probability and statistics by describing populations based on sampling, and investigate chance to develop, use, and evaluate probability models. Use proportional relationships to solve multistep operation and percent problems. If a person walks ½ mile in each ¼ hour, what is her speed per hour? Compute unit rates. Add, subtract, multiply, and divide rational numbers. Know irrational numbers (numbers that are not rational) and approimate them with rational numbers. [Writing (an irrational number) as a decimal is Understand that is between 1 and, then between 1.4 and 1.5. we can approimate it in a fraction, but it will not be eact.] Use properties of operations to solve algebraic equations. Use square root and cube root symbols to represent solutions to equations. Evaluate square root and cube roots (of small perfect square roots and cube roots). Know that is irrational. Use numbers multiplied by a power of ten to estimate very large or very small quantities (the population of the United States is 10 8 ). Add, subtract, factor, and epand linear epressions. Construct simple equations and inequalities to solve problems. Draw, construct, and describe geometrical figures and describe the relationships between them. Solve problems involving angle measure, area, surface area, and volume (cylinders, cones, and spheres). Know formulas for volumes of cones, cylinders, and spheres. Know the formulas for area and circumference of a circle. Use random sampling to describe and compare populations. Find, calculate, and eplain the probability of a chance event. For eample, if a student is selected from a class, find the probability that Jane will be selected and the probability that a girl will be selected. Or if 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? 8
31 Helping Students Succeed in Math Response to Intervention: A New Way of Teaching that Helps All Students Learn, and a New Way of Identifying Possible Learning Disabilities Response to Intervention (RTI) RTI is a tiered process of instruction that allows schools to identify struggling students early and provide appropriate instructional interventions. Early intervention means more chances for success and less need for special education services. RTI would also address the needs of children who previously did not qualify for special education. Tiers in RTI RTI is a delivered to students in tiers or levels. There is much discussion about how many tiers should be in RTI models. The threetiered model is the most common. This means there are different levels of intervention, based on the needs of the student. The level of intervention increases in intensity if a child does not respond to instruction. Tier 1: This is the core curriculum and instruction that all students receive. A key question to ask is Does the core curriculum meet the needs of most (8090%) students? Whole Group/Core Instruction is differentiated (effective teaching that involves providing students with different ways to acquire content, often in the same classroom) to meet ALL students needs and is implemented by the general education teacher. Tier : This tier involves small group interventions for some students (515%) needing MORE instruction or supplementary interventions. For eample, intervention time may be 0 minutes per day in addition to the Core (Tier 1) and can be implemented by the general education teacher or other trained and qualified staff (math specialist teacher, related services staff, paraprofessional). These interventions are usually provided to a small group of students with similar skill needs. Tier : At this tier, intensive interventions are provided for a few students (15%) needing the MOST help. For eample, intervention time may be 60 minutes per day in addition to the Core (Tier 1). These interventions are typically implemented by other trained and qualified staff that could include the special education teacher. For some students, this could also be a replacement core curriculum. Some students may need a replacement core when they are significantly discrepant (more than two years behind) from typical peers and the general core is not eplicit enough to meet their intense skill needs. Key Terms in RTI Universal Screening is a step taken by school personnel early in the school year to determine which students are at risk for not meeting grade level standards. Universal screening can be accomplished by reviewing recent results of state tests, or by administering an academic screening test to all children in a given grade level. Those students whose test scores fall below a certain cutoff are identified as needing more specialized academic interventions. Student Progress Monitoring is a scientifically based practice that is used to frequently assess students academic performance and evaluate the effectiveness of instruction. Progress monitoring procedures can be used with individual students or an entire class. Scientific, ResearchBased Instruction refers to specific curriculum and educational interventions 9
32 that have been proven to be effective; that is, the research has been reported in scientific, peerreviewed journals. RTI in Tennessee Schools pdf Introduction The role of the public education system is to prepare ALL students for success after high school. The Tennessee Department of Education (TDOE) believes that the framework surrounding positive outcomes for ALL students in Tennessee is the Response to Instruction and Intervention (RTI²) model. This framework integrates Common Core State Standards, assessment, early intervention, and accountability for at risk students in the belief that ALL students can learn. What is RTI²? The RTI² framework is aligned with the department s beliefs and allows for an integrated, seamless problemsolving model that addresses individual student needs. This framework relies on the premise of highquality instruction and interventions tailored to student need where core instructional and intervention decisions are guided by student outcome data. In Tennessee, the education system will be built around a tiered intervention model that spans from general education to special education. Tiered interventions in the areas of reading, math, and/ or writing occur in general education depending on the needs of the student. If a student fails to respond to intensive interventions and is suspected of having a Specific Learning Disability, then the student may require special education interventions (i.e. the most intensive interventions and services). As always, parents reserve the right to request an evaluation at any time. Historically, the primary option available to students who were not successful in the general education classroom was a placement in special education. Often, these students did not demonstrate significant discrepancies between their achievement and intellectual ability until the third grade; therefore, leaving use of the discrepancy model coined the wait to fail model. In 004, the Individuals with Disabilities Education Act (IDEA) was reauthorized to reflect an important change in the way schools meet individual student need(s). An emphasis was placed on early intervention services for children who are at risk for academic or behavioral problems. Schools can no longer wait for students to fail before providing intervention. Instead, they should employ a problemsolving model to identify and remediate areas of academic concern. It is important to the Tennessee Department of Education that the RTI² framework represents a continuum of intervention services in which general education and special populations work collaboratively to meet the needs of all students. This includes shared knowledge and commitment to the RTI² framework, its function as a process of improving educational outcomes for ALL students, and its importance to the department to meet requirements related to the Individuals with Disabilities Education Act (IDEA) and the Elementary and Secondary Education Act (ESEA). Individuals with Disabilities Education Act (IDEA), as reauthorized in 004, states that a process that determines whether the child responds to scientific, researchbased interventions may be used to determine if a child has a specific learning disability. IDEA also requires that an evaluation include a variety of assessment tools and strategies and cannot rely on any single procedure as the sole criterion for determining eligibility. A Response to Instruction and Intervention (RTI²) method will now be used (effective July 014) to determine whether a child has a specific learning disability (SLD) in basic reading skills, reading comprehension, reading fluency, mathematics calculation, mathematics problem solving, 0
33 or written epression for students in grades K1. Other areas of SLD including listening comprehension and oral language, in addition to behavioral concerns, may be added in the future. The RTI² Framework is a model that promotes recommended practices for an integrated system connecting general and special education by the use of highquality, scientifically researchbased instruction and intervention. The RTI² framework is a Tier model that provides an ongoing process of instruction and interventions that allow students to make progress at all levels, particularly those students who are struggling or advancing. The Tennessee RTI² Model (on the following page) is a picture of a wellrun RTI² system. It represents the goal of what an RTI model will look like. When Tier I instruction is functioning well, it should meet the needs of 8085% of the student population. Only 1015% of the student population should need Tier II interventions and only 5% should need Tier III interventions. Parent contact is an essential component of RTI² and reinforces the culture of collaboration. A variety of means to reach parents may be used, including: automated phone systems, electronic mail, US Mail, and studentdelivered communications. Local Education Agencies (districts and schools) must designate a person to coordinate and/or make contact with parents at the school level. This person must contact parents for each of the following reasons: before initiating or discontinuing tiered interventions, to communicate progress monitoring data in writing every 4.5 weeks for students receiving tiered interventions, in the event there is a referral to special education, and regarding the dates and duration of universal screenings. 1
34 Tennessee RTI Model Guiding Principles Leadership Culture of Colloboration Prevention and Early Intervention TIER I ALL 8085% All students receive researchedbased, highquality, general education instruction using Common Core Standands in a positive behavior environment that incorporates ongoing universal screening and ongoing assessment to inform instruction. TIER II SOME 1015% In ADDITION to Tier I, interventions are provided to students that fall below the 5% percentile on universal screening and are struggling academically and/or behaviorally. Researchbased interventions will be provided to students within their specific area(s) of deficit. These students are progress monitored using a tool that is sensitive to change in the area of deficit and that provides a Rate of Improvement (ROI) specific to the individualdeficit. EVALUATE DEFINE DataBased Decision Making IMPLEMENT ANALYZE INCREASING SUPPORT FOR STUDENTS TIER III FEW 5% In ADDITION to Tier I, interventions are provided to students who have not made significant progress in Tier II, are grade levels behind or are below the 10th percentile. Tier III interventions are more eplicit and more intensive than Tier II interventions. Researchedbased interventions will be provided to students within their specific area(s) of deficit. These students, who are struggling academically and/or behaviorally, are monitored for progress using a tool that is sensitive to change in the area of deficit and that provides a Rate of Improvement (ROI) specific to the individual deficit.
35 RTI DecisionMaking Process Does not meet gradellevel epectations (below 5th percentile) UNIVERSAL SCREENING Ready for gradelevel instruction Eceeds advanced epectations TIER I Core Instruction 8085% High quality instruction aligned to Common Core Standards Instructional decisions driven by ongoing formative assessment High quality professional development and support If student is more than 1.5 years behind, may need Tier III intervention Does not meet gradelevel epectations Ongoing Assessment Meets gradelevel epectations Eceeds gradelevel epectations TIER II Targeted Intervention 1015% Addresses the needs of struggling and advanced students Additional time beyond time allotted for the core instruction HIgh quality intervention matched to studenttargeted area of need Provided by highly trained personnel Progress monitoring required for databased decision making Does not meet gradelevel epectations Meets gradelevel epectations Provide enrichment TIER II Targeted Intervention 5% Addresses small percentage of struggling students More eplicit and more intensive intervention targeting specific area of need Intervention provided by highly trained professionals Progress monitoring required for databased decision making Does not make significant progress Makes significant progress Consider possible need for Special Education referral after Tier II and Tier III interventions and fails to make adequate progress based on gap analysis.
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