**Ratios** and Proportional Relationships Understand **ratio** concepts and use **ratio** reasoning to solve problems. (6.RP.A) 1.Understand **the** concept of a **ratio** and use **ratio** language to describe a **ratio** relationship between two quantities. For example, "**The** **ratio** of wings to beaks **in** **the** bird house at **the**/**the** distance between numbers **in** **the** complex plane as **the** modulus of **the** difference, and **the** midpoint of a segment as **the** average of **the** numbers at its endpoints. (N-CN.B.6) (DOK 1) Use complex numbers **in** **polynomial**/

**MATH**: how many ways can two letters be arranged from **the** four letters M, A, T, and H? 12 possible permutations: MA, AM, MT, TM, MH, HM, AT, TA, AH, HA, TH, HT CAT: How many permutations are there of **the** letters C A T ? 6 possible permutations: CAT, CTA, ATC, ACT, TAC, TCA **Polynomial** An expression that is **the**/ Slope A measure of **the** steepness of a line **The** **ratio** of **the** vertical change (rise) to **the** horizontal change (run) **The** change **in** y over **the** change **in** x (-2,3) rise = -2 run = 4 **The** symbol for slope is /

**in** Numerical Linear Algebra www.cs.berkeley.edu/~demmel Optimizing Krylov Subspace Methods Jim Demmel EECS & **Math**/, GMRES, Lanczos, Arnoldi, … Goal: minimize communication **in** Krylov Subspace Methods –Assume matrix “well-partitioned,” with modest surface-to-volume **ratio** –Parallel implementation Conventional: O(k log p) messages,/ from power method, precision lost! 32 33 How to make CA-GMRES Stable? Use a different **polynomial** basis for **the** Krylov subspace Not Monomial basis W = [v, Av, A 2 v, …], instead [v/

function is a function whose rule is a quotient of **polynomials** **in** which **the** denominator has a degree of at least 1. **In** other words, there must be a variable **in** **the** denominator. **The** inverse variations you studied **in** **the** previous lesson are a special type of rational function./ **The** sides of Fabio ’ s tree house are equal to **the** width of Calvino ’ s tree house. a. What is **the** **ratio** of **the** area of Calvino ’ s tree house to **the** area of Fabio ’ s tree house? b. Use this **ratio** to find **the** **ratio** of **the** areas if **the** /

? False Visual Representations What do we need to do to make this true? False or Flip **the** numbers Flip **the** inequality symbol Opportunities to Build Conceptual Understanding **in** MS Fractions, **ratios** and proportions Integer operations Simplifying expressions Solving equations and inequalities Linear functions Geometric concepts Multiplying **polynomials** (**Math** I) Factoring **polynomials** (**Math** I) Statistics and Probability Build procedural fluency from conceptual understanding. Mat h Tea chin g Pra/

is **the** golden **ratio**; Since τ+ 2≈ 3.6183 it was close to … 4. Royle (2001): there are planar graphs with **the** chromatic roots arbitrarily close to 4 from **the** left... **The** Four Color Problem has slipped away… again! 5/18/2015People + Ideas = History39 **The** roots/ thanks to Birkhoff Thanks to George David Birkhoff for his ingenious idea of **the** chromatic **polynomial** and everybody mentioned **in** this talk who explicitly and implicitly helped me **in** my **math** career If there was no Birkhoff with his paper 100 years ago, we/

include greatest common factor, difference of two squares, and trinomials Content Limits: All monomials **in** items will have, at most, two variables. Coefficients must be integers. **In** items requiring first factoring **the** GCF and then factoring **the** remaining **polynomials**, **the** remaining **polynomials** must have a maximum degree of two. **In** items that require simplifying algebraic **ratios**, **the** following factoring methods may be used: GCF, difference of two squares, and/or/

**polynomial** Higgs combination model Design goal: Separate building of Likelihood model as much as possible from statistical analysis using **the** Likelihood model More modular software design ‘Plug-and-play with statistical techniques Factorizes work **in**/ that together build a likelihood function Gauss(x,μ,σ) **Math** RooWorkspace (keeps all parts together) RooGaussian g RooFit/combined models **the** Higgs discovery plots were produced… LATLAS(μ,θ) = CMS Neyman construction with profile likelihood **ratio** test Wouter/

and application of infinite series, convergence, divergence, **ratio** and comparison tests power series, expansion of functions **in** series. Prerequisite: **Math**. 303 [Calculus 3] 503. Introduction to Modern Algebra. (3 hours) This is a study of **polynomials** and fundamental properties, linear dependents, matrices, invariants, bilinear forms, and selected topics from **the** theory of numbers and finite groups. Prerequisite: **Math**. 303. **The** 1952-53 East Tennessee State College Bulletin/

note- taking materials. Section 7.2 Operations on Rational Expressions Yesterday, we showed that we can simplify **ratios** of **polynomials** (rational expressions) **in** **the** same way that we simplify **ratios** of integers (fractions). **The** process **in** both cases involves completely factoring **the** numerator and denominator, and then canceling any factors that appear **in** both. We can also perform **the** operations of multiplication, division, addition and subtraction on rational expressions using/

**The** PowerPoint slides will be available on my school website: http://207.239.98.140/UpperSchool/**math**/stephensd/StephensFirstPa ge.htm, listed under “PCTM October 2005” http://207.239.98.140/UpperSchool/**math**/d n go as n gets large? Since this is **the** **ratio** of two sequences, each of which approaches infinity, explain /**the** ideas of sequences of series with Maclaurin and Taylor **polynomials**. a.Show simultaneously **the** graphs of y = sin x y = Put increasingly more terms of **the** series **in** **the** calculator to see how **the**/

continues, what will **the** enrollment be **in** **the** year 2016? c. What is **the** slope of **the** line you found **in** part (a)? d. Explain, using an English sentence, **the** meaning of **the** slope. e. If **the** trend continues, when will there be 3500 students? Responses **in** Traditional Class 1. **The** meaning of **the** slope is **the** amount that is gained **in** years and students **in** a given amount of time. 2. **The** **ratio** of students to/

27.4%26.1% Personal Finance (C) 29.6%32.8%34.4% **Math** For Everyday Life (W) 10%11.7%12.7% Consultation with Universities Initial /–Models of Sinusoidal Functions Tools for Operating and Communicating with Functions –**Polynomials**/Rational Expressions and Exponential Expressions –Inverses/Transformations/Function Notation –Mathematical Reasoning /**the** sine law and **the** cosine law (e.g., compare, using dynamic geometry software, **the** **ratios** of a/sin A, b/sin B and c/sin C **in** triangle ABC, while dragging one of **the**/

to **the** different topics of **the** ACT **Math** exam Understanding common problems for each topic Identifying strategies for **the** ACT **Math** section Pre-Algebra (23%) **The** following pre-algebra topics: Number Problems; **Ratios**; Proportions/**polynomials**, you need to know how **the** sign, power, and roots of a **polynomial** affect its graphical shape. For lines, **the** slope and y-intercept are **the** two most quantities things to know for a graph. For circles, **the** radius is included **in** **the** equation and will determine **the** size of **the**/

about Zernike **polynomials**? A little history about their development. Definitions and **math** - what are they? How do they make certain questions easy to answer? A couple of practical applications What will Zernikes do for me? Widely used **in** industry outside/) Z 9 8^(1/2) (p^3) * SIN (3A) FRINGE order, P-VStandard order, normalized Normalization coefficient is **the** **ratio** between P-V and normalized One unit of P-V coefficient will give an rms equal normalization factor Zernike coefficients Addition (subtraction) /

** Ex. 51-62** And finally **the** LCD with **polynomials** (factor first!) pg 400 Ex 5a)/**in** pond. **The** **ratio** 30/x is **the** **ratio** of tagged catfish to **the** total population. **The** **ratio** of 3/500 is **the** **ratio** of tagged catfish **in** **the** sample to **the** sample size. If catfish are really well mixed and **the** sample is random, **the** **ratios** should be equal. Ex 6 continued So there are 5000 catfish **in** **the** pond. ** Ex. 45-48** Example 7 page 429 now for a proportion **In** a conservative portfolio **the** **ratio** of **the** amount invested **in**/

(2005; 2006) Hong and Raudenbush used **the** rich covariates **in** **the** ECLS-K to estimate **the** effect of kindergarten retention on academic outcomes **in** **math** and reading Provided subset of data used **in** original analysis which included students who attended schools where at least some students were retained **in** kindergarten 10,726 students **in** 1,080 schools 144 pretreatment covariates Mean Unadjusted **Math** Scores for Retained and Promoted Students Purposes/

CAS Solution Lin McMullin **MATH** & SCIENCE TECHNOLOGY CONFERENCE January 18, 2008 Norman, Oklahoma Cubic Symmetry Show that any cubic **polynomial** has a point of rotational symmetry. Cubic Symmetry A Cubic’s Roots Show that **the** tangent line to a cubic at **the** point where x = **the** average of two of its roots, intersects **the** cubic at its third root. **Ratios**, We got **Ratios** Analytic Geometry Perpendicular bisector theorem/

**math** “rules” **The** Big Five NeSA practice test – take test identify concepts and vocabulary Vocabulary – videos, why, strategies, work session, S.O.M. Writing **in** **math** – strategies, websites, student work Lunch and Sharing – websites, writing # song, doubles song (Stephanie), card game for adding **polynomials** (Julia), Saxon **math**/and so on Chart and discuss **in** your groups: a. good things students are doing b. misunderstandings/mistakes c. how students are showing work Determine **the** **ratio** of “naked” problems to “/

continues, what will **the** enrollment be **in** **the** year 2016? c. What is **the** slope of **the** line you found **in** part (a)? d. Explain, using an English sentence, **the** meaning of **the** slope. e. If **the** trend continues, when will there be 3500 students? Responses **in** Traditional Class 1. **The** meaning of **the** slope is **the** amount that is gained **in** years and students **in** a given amount of time. 2. **The** **ratio** of students to/

**The** side and **the** diagonal of **the** square produced by **the** proof result from two iterations of **the** algorithm: d, s => s, d - s => 2s - d, d - s And **the** original **ratio** /**the** want." --Lincolns Autobiography of 1860 New **Math** against Euclid **In** 1959, at a conference on **the** teaching of mathematics **in** Réalmont, France, Jean Dieudonné rose and hurled **the**/t = m % n; m = n; n = t; } return m; Simon Stevin: **polynomial** gcd(**polynomial** m, **polynomial** n) { while (n != 0) { **polynomial** t = m % n; m = n; n = t; } return m; 3x2+2x-/

scatter science read || fpfit science read fpfit: fractional **polynomial** . scatter science write || fpfit science write . scatter science **math** || fpfit science **math** Notice that, **in** these examples, graphing yhat (i.e. y’s predicted /: for every unit increase **in** education (treated as a quantitative-interval variable), consumption increases by 16% on average, holding **the** other variables constant. . twoway qfitci ltotalconspc edhh, blc(maroon) Only quantitative, **ratio**-level variables with positive values/

) 1/2. Is this **the** right metric? **In** most applications **the** important factor is how well-behaved **the** triangulation is, not how short. **In** **polynomial** time one can find triangulations that (a)Maximize **the** minimum internal angle over all triangles (b)Minimize **the** maximum angle (c)Minimize **the** maximum aspect **ratio** (d)Minimize **the** maximum edge length (e)Minimize **the** maximum triangle height Closed “Open Problems” from **the** Columns “**The** NP-Completeness Column: An/

: **The** effect of **math** training on **math** scores was larger when participants could self-select into **math** training. **The** 4.19 point effect (out of 18 possible points) **in** **the** randomized experiment was overestimated by 19.6% (.82 points) **in** **the** nonrandomized/ X or transformations thereof, e.g. **polynomials**, log, interactions) Logistic model: Estimated PS logit: Estimated PS : PS (logit) is **the** predicted value from **the** logistic regression **The** distribution of **the** estimated PS-logit shows how similar/different /

balanced. Hence one loses less **in** **the** greedy correction phase. Provable approximation **ratio** above 0.7. Can a different rounding technique or analysis lead to much better bounds? Min bisection Min-cut solvable **in** **polynomial** time. Min-bisection: n/2/ Alon and Kahale [**Math**. Programming 98]. Vertex cover: Charikar [SODA 02]. Max 3SAT: Karloff and Zwick [FOCS 97]. Many more works on SDP. Hardness results: Hastad [JACM 01]. Summary SDP produces **the** best known approximation **ratios** for many maximization problems/

of Four: Graphical, Numerical, Algebraic and Verbal Representations Realistic Applications via **Math** Modeling Non-routine problems and assignments Algebra **in** Context – Not Just Drill Common Themes Families of Functions – Linear, Exponential, Power, Logarithmic, **Polynomial**, and Sinusoidal **The** significance of **the** parameters **in** **the** different families of functions Limitations of **the** models developed – **the** practical significance of **the** domain and range Common Themes Data Analysis Connections to Other Disciplines/

mathematical concepts. For instance, What is **the** significance of **the** base (growth or decay factor) **in** an exponential function? What is **the** meaning of **the** power **in** a power function? What do **the** parameters **in** a realistic sinusoidal model tell about **the** phenomenon being modeled? What is **the** significance of **the** factors of a **polynomial**? What is **the** significance of **the** derivative of a function? What is **the** significance of a definite integral? Further Implications/

**ratio** **in** which **the** denominator is 100. Proportions are used **in** every-day contexts, such as speed, recipe conversions, scale drawings, map reading, reducing and enlarging, comparison shopping, and monetary conversions. **The** HOW Classroom Instruction That Works Research-based Strategies Give One … Get One … On **the** Think Pad, write one way that being a **math** teacher **in**/70% correct) Given **the** equation for a conic section, describe its graph.50 Given **the** x-intercepts of a **polynomial** function, identify its factors/

Most people stay **in** **the** shallow end…we now need to venture to **the** unknown depths of knowledge. Develop connections from 6 th grade **math** through pre-calculus/ Problems involving Fractions, Decimals and Percents 5. **The** Percent Equation 6. Interest, Discounts, Sales Tax, etc 7. **Ratio** and Rates 8. Mean, Median, Mode, and/ Expressions 3.FCAT 4.Adding and Subtracting **Polynomials** 5.Multiplying Binomials 6.Special Products of **Polynomials** 7.Factoring **Polynomials** 8.Factoring Special Products What Materials will I/

Derivation – Numerical Integration – Root Finders – Minimization – Interpolation – Chebyshev **polynomials** (for function approximation) ROOT **Math** Libraries Wouter Verkerke, NIKHEF ROOT – Fitting & Minimization Fitting involves –Data /of MC templates, efficiencies from **ratios** (F. Filthaut) Wouter Verkerke, NIKHEF Examples of imported/interfaced external software **in** ROOT Fast Fourier Transforms – / Idea: represent **math** symbols as C++ objects –Result: 1 line of code per symbol **in** a function (**the** C++ constructor)/

**in** **the** 70’s) (as a teacher - I was a student here **in** **the** 70’s) Ten years as an “at home mom” Ten years as an “at home mom” **Math** tutoring **Math**/**Ratios**, Proportions, and Percents **Ratios**, Proportions, and Percents Geometry/Spatial Thinking Geometry/Spatial Thinking Area and Volume Area and Volume Right Triangle Applications **in** Algebra Right Triangle Applications **in** Algebra Topics Covered **in**/Operations with Exponents and **Polynomials** Operations with Exponents and **Polynomials** Quadratic Equations Quadratic Equations /

mathematical concepts. For instance, What is **the** significance of **the** base (growth or decay factor) **in** an exponential function? What is **the** meaning of **the** power **in** a power function? What do **the** parameters **in** a realistic sinusoidal model tell about **the** phenomenon being modeled? What is **the** significance of **the** factors of a **polynomial**? What is **the** significance of **the** derivative of a function? What is **the** significance of a definite integral? Further Implications/

**the** unit of time must match **the** unit of **the** interest rate. For example, if you measure time **in** months, you must divide an annual interest rate by 12 **in** order to calculate interest appropriately. **Ratios** and Proportions A **ratio** describes **the** relationship between two quantities. For example, **the** **ratio**/+ 2 + 6x 4 + 3x 2 + 5x – 2 = 9x 4 + 5x 2 – 5y 2 – 2x To multiply a **polynomial** by a number, multiply each term **in** **the** **polynomial** by **the** number: 5 x (x 2 - 4xy + 3y 2 –2) = 5x 2 – 20xy + 15y 2 -10 www.topmath.info//

**The** complex number system (performs arithmetic operations with complex numbers, represent complex numbers and their operations on **the** complex plane, use complex numbers and **polynomial** identities and equations Models for LRE for **math** specially designed instruction **in**/of multiplication and division to multiply and divide fractions) **Ratios** and proportional relationships (understand **ratio** concepts and use **ratio** reasoning to solve problems) **The** number system (apply and extend previous understanding of /

**MATH** 3 Warm-Up Using **the** data given: 23, 45, 23, 45, 67, 54, 34, 89, 56, 76, 12, 76 Give **the** Minimum, Lower Quartile, Median, Upper Quartile, Maximum, and standard deviation. (Use Calculator) Rational & Irrational Numbers Rational Numbers repeated **in** a pattern, terminate, or can be expressed as a **ratio**/? Long Division Synthetic Division Warm-Up Classify **the** following numbers as irrational or rational: √2 2.4 5.99999999999999……. Simplifying Rational Expressions Warm-Up Add **the** **polynomials**: (2x 2 + 5x – 2) /

numbers Fractions and Mixed Numbers Decimals Introduction to Algebra and **the** Real Number System **Ratio**, Proportion and Percent Linear Equations and Inequalities Introduction to Graphing Exponents and **Polynomials** Factoring **Polynomials** 6 Mat 007 Description (continued) **The** course is self paced, not lectured **The** course is pass/fail All work follows **the** students if they do not complete **the** course **in** 8 weeks 7 Mat 007 Procedure Students take a/

of Four: Graphical, Numerical, Algebraic and Verbal Representations Realistic Applications via **Math** Modeling Non-routine problems and assignments Algebra **in** Context – Not Just Drill Common Themes Families of Functions – Linear, Exponential, Power, Logarithmic, **Polynomial**, and Sinusoidal **The** significance of **the** parameters **in** **the** different families of functions Limitations of **the** models developed – **the** practical significance of **the** domain and range Common Themes Data Analysis Connections to Other Disciplines/

Marchetti, Shoreline CC Intro Statistics, **Math**& 146 Melonie Rasmussen & David Lippman, Pierce CC Precalculus 1 and 2, **Math**& 141/142 Dale Hoffman, Bellevue College Calculus 1, 2, and 3, **Math**& 151/152/153 Precalc 1 and/**Ratio** Test 10.8 Power Series 10.9 Representing Functions as Power Series 10.10 Taylor and Maclaurin Series 10.11 Approximation Using Taylor **Polynomials** 11.0 Introduction: Moving Beyond Two Dimensions 11.1 Vectors **in** **the** Plane 11.2 Rectangular Coordinates **in** Three Dimensions 11.3 Vectors **in**/

**Ratio** > 0.878 Integrality Gap = “algorithm achieves **the** gap’’ Inapproximability Is 0.878 **the** best possible approximation **ratio** for MaxCut? Satisfiable Unsatisfiable MaxCut value = K MaxCut value < K 10 15 3 7 1 1 1 1 1 3-SAT Instance **Polynomial**/of **the** proof!! Analogy to **Math** Proofs Could you check **the** proof of a theorem with any reasonable confidence by reading only 3 bits of **the** proof/S d |S| vertex set S A random neighbor of a random vertex **in** S is outside of S with probability expansion(S) Ф G = expansion(/

1.1, pp 409-412) Completely general spec always **the** “best fitting” model (lowest Deviance) Constant spec always **the** “worst fitting” model (highest Deviance) Use of ONE facilitates programming **Polynomial** specifications As **in** growth modeling, a systematic set of choices Choose centering constant/ Only n=132 (3.5%) took a **math** class for all of **the** 5 periods! RQs: When are students most at risk of dropping out of **math**? What’s **the** effect of gender? Does **the** gender differential vary over time? Data source: Suzanne/

**polynomial**-time constructible families of asymptotically good expander codes **in** which a constant fraction of error can be corrected **in** a circuit of size and depth. **The** action of this circuit can be simulated **in** linear time on a Pointer Machine or a RAM under **the**/**in** block length n of **the** code C(B,S). –# variable errors decreases by constant **ratio** each round –Only constraints containing these variables appear **in**/sub-codes C i, i 0,... 1 http://www.**math**.uiuc.edu/~jozef/math475/description.pdf Some questions to /

can apply normalization if we need to. We are only interested **in** finding **the** **ratio** F*F / A*A. We can always solve a system of N equations **in** N+1 unknowns for a **ratio** of any two of **the** coefficients. How would you solve this system for A/F? You/a **math**- ematician would say it is simple. Just a bunch of numbers, an exponential function, and **the** Hermite **Polynomials** H n. **Polynomials** are simple. H 0 (y) = 1, H 1 (y) = 2y, and other **polynomials** are given **in** Table 5.2 of Beiser. More important, we find that **the** wave/

value of **the** second component C ONCEPTUAL & R EFORM IDEAS **IN** M ATH 092 Distinguish between **Ratios** and Rates Average Rate of Change rate of change & slope slope slope & equations of lines Function basics lesson plan functions I.pdf functions II.pdf absolute value function.pdf Outside Class Project end of year project 092.pdf C ONCEPTUAL & R EFORM I DEAS **IN** **MATH** 101 course/

Simplify a n = a 1 * r n-1 Let’s try one Find **the** 10 th term **in** **the** sequence of 1, -6, 36, -216... a 10 = 1(-6) 9 = -10,077,696 Common **ratio** = -6 a 10 = 1 (-6) (10-1) Start with **the** sequence formula Find **the** common **ratio** between **the** values. Plug **in** known values Simplify a n = a 1 * r n-1 Exponential Growth & Decay/

Tested 4. Derive **the** formula for **the** sum of a finite geometric series (when **the** common **ratio** is not 1), and use **the** formula to solve problems. For example, calculate mortgage payments. ★ A-APR DOMAIN Arithmetic With **Polynomials** And Rational Expressions A-APR Cluster: Perform arithmetic operations on **polynomials**. New **in** Common CoreSame Common CoreOld Kansas Standard 1. Understand that **polynomials** form a system analogous to **the** integers, namely, they/

5 th order **polynomials** are rarely adopted, but help give you a sense of whether you should stick with **the** completely general specification. **Polynomial** Specifications: As **in** growth modeling, /1, pp 444-447) Estimated odds **ratios** for **the** 4 possible prototypical individuals **In** comparison to a White child who had not been abused, **the** odds of 1 st arrest are: /took a **math** class for all of **the** 5 periods! RQs: When are students most at risk of first dropping out of **math**? What’s **the** effect of gender? Does **the** gender /

**Polynomial** Applications GHSGT **Math** Review This diagram shows **the** dimensions of a cardboard box. Which expression represents **the** volume, **in** cubic feet, of **the** box? A) 3x 3 +2 B) 5x 3 +2 C) 3x 3 +6x 2 D) 5x 3 +6x 2 **Polynomial** Applications GHSGT **Math**/No Solution Other Equations GHSGT **Math** Review Sequences Arithmetic = adding or subtracting **the** same number each time Geometric = multiplying by a common **ratio** to get to **the** next term **in** **the** sequence GHSGT **Math** Review Find **the** 200 th number **in** **the** sequnce: 8, 10, 12/

Exploring relationship between fractions and division – Three dimensional Geometry – **The** coordinate grid Mathematical Topics – Basic Algebra and equations (including an/ Explored – Mastery of multiplication, division and fractions lead to study of **ratios**, proportional relationships, and unit rates. – Introduction of variables, algebraic expressions/learned **in** Pre-Algebra to **polynomials**. – Exploration of factoring **polynomials**. – Introduction of quadratic functions and equations. 5 th Grade *6 th Grade **Math** /

to study of **ratios**, proportional relationships,/**polynomials**. – Introduction of quadratic functions and equations. 8 Mathematical Practices – Make sense of problems and persevere **in** solving them. – Reason abstractly and quantitatively. – Construct viable arguments and critique **the** reasoning of others. – Model with mathematics. – Use appropriate tools strategically. – Attend to precision. – Look for and make use of structure. – Look for and express regularity **in** repeated reasoning. 5 th Grade *6 th Grade **Math**/

**THE** STUDENTS ARE TO BRING THEIR **MATH** TEXTBOOK TO CLASS EVERY DAY TO WORK ON ASSIGNMENTS. 7 TH GRADE **MATH** All operations of whole numbers, decimals, and fractions Number Theory facts Percentage and consumer **math** topics Algebraic expressions Equations and inequalities Order of operations Absolute value Computation with integers Squares and square roots **Ratio** and proportion Exponents, powers of ten, and scientific notation **Polynomials**/, State and Local government **in** **the** United States Types of governments/

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