Multivariate poisson-lognormal model for modeling related factors in crash frequency by severity
Mehdi Tazhibi1, Iraj Kazemi2, Somaye Momenyan1, Hossein Haghshenas3
1 Department of Biostatistics and Epidemiology, School of Public Health, Isfahan University of Medical Sciences, Isfahan, Iran
2 Department of Statistics, University of Isfahan, Isfahan, Iran
3 Department of Transportation, School of Transportation, Isfahan University, Isfahan, Iran
|Date of Web Publication||31-Jan-2013|
Department of Statistics, University of Isfahan, Isfahan
Source of Support: None, Conflict of Interest: None
Aims: Traditionally, roadway safety analyses have used univariate distributions to model crash data for each level of severity separately. This paper uses the multivariate Poisson lognormal (MVPLN) models to estimate the expected crash frequency by two levels of severity and then compares those estimates with the univariate Poisson-lognormal (UVPLN) and the univariate Poisson (UVP) models. Materials and Methods: The parameters estimation is done by Bayesian method for crash data at two levels of severity at the intersection of Isfahan city for 6 months. Results: The results showed that there was over-dispersion issue in data. The UVP model is not able to overcome this problem while the MVPLN model can account for over-dispersion. Also, the estimates of the extra Poisson variation parameters in the MVPLN model were smaller than the UVPLN model that causes improvement in the precision of the MNPLN model. Hence, the MVPLN model is better fitted to the data set. Also, results showed effect of the total Average annual daily traffic (AADT) on the property damage only crash was significant in the all of models but effect of the total left turn AADT on the injuries and fatalities crash was significant just in the UVP model. Hence, holding all other factors fixed more property damage only crashes were expected on more the total AADT. For example, under MVPLN model an increase of 1000 vehicles in (average) the total AADT was predicted to result in 31% more property damage only crash. Conclusion: Hence, reduction of total AADT was predicted to be highly cost-effective, in terms of the crash cost reductions over the long run.
Keywords: Bayesian method estimation, crash severity, markov chain monte carlo simulation, multivariate lognormal distribution
|How to cite this article:|
Tazhibi M, Kazemi I, Momenyan S, Haghshenas H. Multivariate poisson-lognormal model for modeling related factors in crash frequency by severity. Int J Env Health Eng 2013;2:30
|How to cite this URL:|
Tazhibi M, Kazemi I, Momenyan S, Haghshenas H. Multivariate poisson-lognormal model for modeling related factors in crash frequency by severity. Int J Env Health Eng [serial online] 2013 [cited 2022 Jan 22];2:30. Available from: https://www.ijehe.org/text.asp?2013/2/1/30/114193
| Introduction|| |
Deaths and injuries related to the road crash are main worry of many governments. Globally, the number of people killed in the road traffic accidents annually is estimated almost 1.2 million, while the number of people injured could be as large as 50 million.  Due to these problems there has been extensive investigation on the prediction models and traffic crash analysis. ,,,,, The accident data are usually collected by severity (e.g., fatal, injury and property damage only).
Severity levels are correlated because the unobservable or omitted variables are expected to be common at all severity levels for a particular segment of roadway or for one intersection. Hence, crash data by severity are basically multivariate in nature. ,,,
Traditionally, analyses have used univariate distributions to model crash data at different levels of severity separately such as Poisson and negative binomial model. ,,,, The application of the univariate model ignores the correlation that exists across crash rates at different levels of severity. This issue leads to the inaccurate estimates for effects of factors.
Crash data usually have excess zeroes rather than those expected with a Poisson model. , In other words, if the corresponding variance is greater than its expectation in a Poisson model then the over-dispersion problem occurs. To takes into account this extra Poisson variation a random effect term with the gamma or the lognormal distribution enters in the model. Karlis, Ma and Tsionas used the multivariate Poisson (MVP) model to evaluate the effects of different covariates on the collision counts at different levels of severity. Karlis, Ma and Tsionas ,, However, the assumption of non negative covariance component and the over-dispersion problem still exist in the MVP model. The multivariate negative binomial (MVNB) model may solve the over-dispersion problem  for the non negative correlations. While the multivariate Poisson-lognormal (MVPLN) model accounts for the over-dispersion and also for several correlation structures. ,,,,
Then, MVPLN model is more ideal than the MVP and MVNB models, since (i) it accounts for over-dispersion (extra Poisson variation) and (ii) it takes into consideration general correlation structures.  The execution of MVPLN models is not straightforward. The parameter estimation is done within the Bayesian model, using a Markov chain Monte Carlo (MCMC) simulation method.  In this paper, we use the MVPLN model for the analysis of crash data by two levels of severity (Property damage only crash and Injuries plus fatalities crash). The parameters estimation is done in a Bayesian perspective by the Gibbs Sampler and the Metropolis-Hastings (M-H) algorithms that can be carried out using freely available Open BUGS software version 3.2.1. The data collected at the intersection in Isfahan for 6 months.
This paper is organized as follows. First, there is a statistical methodology is reviewed and then data set is described. The paper concludes with a results, discussion and recommendations for future research.
| Methodology|| |
Let yik (i = 1,...,n;k = 1,...,K) denotes the number of collision at the ith intersection and kth category. Assume that observations are uncorrelated across intersections but correlated over categories.
Let further Y be an n × K matrix and their elements Yik follow distributed as Poisson distribution with the mean rate parameter λik .
Assume that the Poisson rates are modeled by a function of covariates following a lognormal distribution, that is
where x′i = (xil,…,x ij)indicates the J-dimensional row vectors of covariate. Let we define X′ = (x l ,…,xn) the βk be the J-dimensional vectors β′k = (β lk,…,βjk) and B = (β l ,…,β K ) be the matrix of regression coefficients.
We model the correlation between yi1 ,…, yiK by incorporating the unobserved heterogeneity components εi 1 ,…,εiK in the model and assume that the vector εi is distributed as K-variate normal with mean zero and variance-covariance matrix ∑, so that ∑ is matrix with general structure to accommodate the correlation among the εik .
This model specification implies that the vector λi follows a K-variate lognormal distribution and the response vector Yi follows a K-variable Poisson-lognormal distribution.
The probability density function of Yi is given by
where ɸk is the K-variate normal distribution. The marginal distribution of the crash counts yi cannot be obtained by direct computation. Obtaining the marginal distribution requires the evaluation of a K-variate integral of the Poisson distribution with respect to the distribution of εi .
The expectation and the variance of the marginal joint distribution of yi and the covariance between the counts can be derived directly. Chib and Winkelmann  show that:
Hence this model allows for the over-dispersion because the variance yik excess the expectation as long as σkk > 0 for k = 1,…, K and the correlation structure is unrestricted because sign of σkj may be positive or negative. In the UVP model for testing over-dispersion we may use the Pearson's Chi-Square statistic divided by the degrees of freedom, i.e., . The value of χ2 greater than 1 indicates the over-dispersion problem.
The Markov Chain Monte Carlo (MCMC) method under the Bayesian framework can be used to compute the integrals and subsequently to utilize the parameter estimation.
At level one, we suppose that the parameters (B,∑) independently have the prior distributions:
where R is scale matrix and k usually assumed to be equal to the rank of matrix R.
In the second level according to Bayes' theorem (posterior ˜ prior × likelihood) the posterior distribution of parameters is obtained. Now the posterior distributions can be obtained using MCMC sampling.
The ∑−1 can be sampled using a Gibbs sampler. In contrast, the posterior distributions of the B and the εi are not in standard forms and thus the Metropolis-Hastings algorithm is used. The MCMC techniques are available in Open BUGS version 3.2.1. MCMC approach generates values of the parameters from approximate distributions and which are distributions that converge to the target posterior distributions. After converging, we can do those Bayesian inferences of the parameters based on the sample mean and sample standard deviation and also based on the sample 2.5 th percentile and the 97.5 th percentile
The interval between the 2.5 th percentile and the 97.5 th percentile are considered as the 95% credible intervals.
The Deviance Information Criteria (DIC) is a standard criterion in the Bayesian models comparison. As a goodness-of-fit measure, DIC is a Bayesian generalization of Akaike's Information Criteria (AIC) that penalizes complex models with larger parameter.
Here, we fit the MVPLN model to the crash data that are classified into K = 2 groups: The property damage only, k = 1, and the injuries plus fatalities, k = 2. This data set collected from 65 intersections in Isfahan city. Although, the original data contained the crash data from 280 intersections, we only used those where the traffic volume data were available. Thus, the sampling method is simple random sampling and size of our sample is 65 intersections. The traffic volume and crash data were provided for a 6-month period from calendar years 2011.
The term "crash", as defined by traffic police of Iran, refers to reportable on-street crash that includes at least one motor vehicle, and results in injury, at least $1,000 in property damage, or both. The data set does not include non-vehicular crash (e.g., cyclist hitting a pedestrian) and includes crash that is forwarded to the police service within a specified period. Total AADT is defined as entering traffic flow for total legs of intersection and total left turn AADT is defined as entering traffic flow for total curb left turn of intersection.
In all of models, crash counts at two levels of severity are considered as dependent variables and total AADT and total left turn AADT variables as independent variables.
There were 182 accidents of property damage only and 66 accidents of injuries plus fatalities at those 65 intersections for 6 month. [Table 1] contains summary statistics of variables of interest.
| Results|| |
We emphasize that while the MVPLN can handle more than two collision categories, here we use only two severity levels leading to a bivariate Poisson-lognormal.
Posterior distributions of parameters were estimated using the OpenBUGS software version 3.2.1. For the underlying models, 10,000 iterations were discarded as burn-in sample to eliminate the influence of starting values and then 100,000 iterations followed to obtain Bayes estimates (posterior means and standard deviations) of the regression coefficients β. Convergence was assessed by visual inspection of the Markov chain for all parameters. Trace plots of the model parameters and Mont Carlo errors was checked. As a rule of thumb, ratios of the Monte Carlo errors relative to the respective standard deviations of the estimates should be less than 0.05.
Results of [Table 2], [Table 3] and [Table 4] show that there are small differences between estimation of the parameters under three models: MVPLN, UVPLN and UNP. Also, they show that some of the parameter estimates are significant in the 95% credible interval. For example the effect of total AADT on the property damage only crash is significant in the all of models. Also, the effect of total left turn AADT on the injuries plus fatalities crash is significant just in the UNP model. Hence, holding all other factors fixed more property damage only crashes are expected on more total AADT. For example, under MVPLN model an increase of 1000 vehicles in (average) total AADT (rising from 4053 to 5053 vehicles) is predicted to result in 31% more property damage only crash, and under the UNP model an increase of 100 vehicles in (average) the total left turn AADT (rising from 1205 to 2205 vehicles) is predicted to increase injuries plus fatalities crash count by 10%.
Results of [Table 2] show that the value of Pearson's Chi-Square divided by degrees of freedom for two levels of severity is also greater than 1. The UNP model is not able to overcome the problem while the MVPLN model can take in to account it. [Table 4] shows that the estimates of the extra Poisson variation parameters in the MVPLN model are greater than zero and as Chib and Winkelmann (1995) show when those parameters are positive the variance will be greater than the expectation. Hence, the MVPLN model can capture over-dispersion.
It is known that the over-dispersion makes the standard errors to be underestimated. In which case standard errors are not accurate estimates of true uncertainties and the confidence interval estimates will not capture the true parameter values. We emphasize that for the unbiased estimates the small standard errors it is mean more precise estimates. This is particularly crucial in cases that involve the most severe crashes: Those that involve fatalities or major injuries. Because the final objective of any highway safety analysis is to reduce the frequency and severity of crashes, it follows that obtaining the most precise estimates of crash frequency at the highest severity levels is highly desirable. By comparing the standard deviation of parameters in [Table 2], [Table 3] and [Table 4], we can see that under UVP model the values of standard deviation of parameters are underestimated and that is why more variables got significant under the UVP rather than the MVPLN and the UVPLN model.
Also, the estimates of the extra Poisson variation parameters in the MVPLN model were smaller than the UVPLN model. As Basyouny and Sayed  showed:
Since the second term in the above equation dominates the first, we obtain
where are the estimates of the extra Poisson variation parameters under the MVPLN and the UVPLN, respectively. As precision is inversely proportional to the variance of expected collision frequency, it is estimated that the MVPLN model is more precise than the UVPLN model. This is because the MVPLN model takes into account the correlation among two levels of severity.
The correlation was estimated 0.712, which is significant. In real meaning, the higher property damage only crash rates are related to the higher injuries plus fatalities crash rates that even after controlling for the covariates, there is extra correlation between these two levels of severity that is not explained and is due to same deficiency in intersection design and/or other unobserved variables. In terms of goodness-of-fit, the Deviance Information Criteria (DIC) quantifies the relative goodness-of-fit of the models; therefore, it is useful for comparing models. As are presented in [Table 2], [Table 3] and [Table 4] the MVPLN model provides a better fit over the UVPLN and the UVP model as DIC for the MVPLN model is smaller than the sum of DICs for two levels of severity under the UVPLN and the UVP model.
| Discussion|| |
Traditionally, univariate models are used to analyze crash data for different levels of severity separately. This research used a MVPLN model to jointly analyze a sample of crash counts classified by two levels of severity at 65 intersections in Isfahan city. Furthermore, for comparison the UVPLN and the UVP models were fitted for each of severity levels. Results showed that the MVPLN model preferred rather than the UVPLN and the UVP model. Because univariate models neglect the correlation of crash counts at different levels of severity while the MVPLN model allows for a general correlation structure among different levels of severity, as well as handing over-dispersion. Parameters estimates of all models are achieved within the Bayesian framework that is implemented by using the OpenBUGS software.
Both, model parameter estimates and their variances were discussed. As expected, the results provide several recommendations for urban intersection safety treatments. For example, reduction of total AADT is predicted to be highly cost-effective, in terms of the crash cost reductions over the long run. Findings of this paper confirm with those already illustrated by other authors. For example in the study that is implemented in Washington State on rural two-lane Roadways by Ma et al., (2006) an increase of 1000 vehicles in average annual daily traffic level (rising from 3757 to 4757 vehicles) was predicted to increase total crash count by 16.4% and the disabling crash count by 40% at MVPLN model.  Park and Lord (2007) applied three models of the MVPLN, the UVP and the UVNB on the crash count data of five different severity levels collected from unsignalized intersections in California. They showed that effects of minor and major AADT of the intersection are significant at the five severity levels for three models. In their study the variables such that the Painted Left Turn and the Curb Med Left Turn were checked just two variables minor and major AADT were significant at two levels of severity. Thus, the reduction of traffic volume at the intersection is more considerable than other variables to reduce accident at two levels of severity. They also illustrated the over-dispersion problem in their data and thus fitting the UVP model leads to underestimation of standard deviations.  Also, Aguero-Valverde (2012) showed that effect of AADT on crash for rural two-lane segments that located in Central Pennsylvania is significant when fitting Poisson gamma, Poisson lognormal, and zero inflated random effects models.  In another study, Aguero-Valverde applied the MVPLN model for five levels of severity. He examined the ADT variable simultaneously with variables such that the Shoulder width, the Lane width and the Speed limit. Among all variables only ADT was significant at all levels of severity. El-Basyouny and Sayed (2009) used the MVPLN and the UNPLN for the purpose of developing collision prediction models relating the safety of urban 4-leg intersections to their traffic flows. They showed the effects of minor and major AADT are significant at two levels of severity (Property damage only and Injuries and fatalities crash) under two models MVPLN and UVPLN. they further showed the MVPLN model offers more precise estimates than the UVPLN model.
The correlation among two levels of severity was estimated 0.71, which this correlations may be caused by omitted variables (such as painted left turn, curb med left turn, Speed limit and surrounding land use), which can influence crash occurrence at all levels of severity. Essentially, higher crash rates of one type are associated with higher crash rates of other types. The MVPLN model accounts this correlation. That is the reason why it offers more precise estimates than the UVPLN model. results showed the estimates of the extra Poisson variation parameters in the MVPLN model were smaller than the UVPLN model. Since variance of expected collision frequency is function of the extra Poisson variation parameters and the precision is inversely proportional to the variance of expected collision frequency then precision of the MVPLN model is larger than the UVPLN model.
In terms of goodness-of-fit, the MVPLN model provides a better fit over the two other models as the DIC for the MVPLN model is less than the sum of DICs for the UVPLN and the UVP models. This paper is based on a small dataset and for a short period but despite the limitations of these data results of this paper conform to those in literatures. Nevertheless, there are challenges for further research, for example crash data usually have low sample mean values, small sample size, and statistical models become very unstable when they are estimated using this kind of data. , Hence, stability of models should be investigated along with the sensitivity analysis of models for different hyper-prior specifications. Furthermore, the changes in the signs of coefficients between different levels of severity need to be investigated. AADT models suffer from omitted variables bias but such models are well accepted in the traffic safety literature.
In this paper, two severity levels were investigated. However, different levels of severity could be included (e.g., fatality, severe injury, light injury, PDO, etc.), different crash types could be considered (e.g., angle, head-on, rear-end, sideswipe, etc.), or geometric design feature that are not available in our dataset could be incorporated in the model. By means of improvement in statistical tools, which allows for a more precise analysis and the appropriate dataset transportation engineers are able to better understand the relationship between crash counts and related factors in its occurrence.
| Acknowledgments|| |
The authors would like to thank the contribution of the traffic control and traffic police center of Isfahan, which provided the data for this analysis. (Project Number: 390592).
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[Table 1], [Table 2], [Table 3], [Table 4]


Gradient descent is an optimization algorithm that finds the local minimum of a function. The algorithm will take iterative proportional steps toward the negative gradient of the function at the current point. The algorithm usually starts with parameters (weights and bias) and improves them slowly as it tries to get a sense of the value of the cost function for weights that are similar to the current weights (by calculating the gradient). Then it moves in the direction which reduces the cost function by repeating this step thousands of times. The algorithm will continually minimize the cost function.
Next: Natural Language Processing


ERATOSTHENES of Cyrene – Greek: Ἐρατοσθένης, c. 276 BC– c. 195/194 BC is now known as a significant Greek mathematician. There are no known written works that exist today directly attributed to Eratosthenes. All that is known about Eratosthenes are antdotal references to him and his alleged theories.
ERATOSTHENES is now considered to have been a mathematician, a geographer, a poet, an astronomer, and also a music theorist. He is also credited with being the chief librarian at the Library of Alexandria.
In his day Eratosthenes may have been called Beta amongst his peers as recorded by the Byzantine encyclopedia of the ancient Mediterranean world known as the Souda or Suda  Entry ε 2898 of Suda. The reason was he was often second in scholarly competitions, and not first. Thus Beta was the second letter of the Greek alphabet and often used in Greek Math or Gematria as a number. Eratosthenes was also credited with the nickname of PentathlosChambers, James T. “Eratosthenes of Cyrene.” Dictionary Of World Biography: The Ancient World (January 1998): 1–3.
He is often credited as being the first person to calculate the circumference of the Earth, but his calculation was not modernly precise. However, today it is understood that much of the Greek so-called wisdom was drawn from older lost sources, such as Sumerian mathematics.
The Creator Formula work of Dr. Sol Adoni aka Sollog from 1995https://www.sollog.com/discoveries1/creator.shtml proved that the creation mythos in major religions hid a simple exact mathematical formula in their creation myths proving the EXACT CIRCUMFERENCE OF THE EARTH was known thousands of years before Eratosthenes lived and hidden inside the iconography of creation mythos, thus proving Eratosthenes was not the first to accurately measure the earth.
As the Chief Librarian of Alexandria, if he were even that, Eratosthenes would have had access to source material from Sumeria that is now lost today.
Eratosthenes is also credited with discovering the Sieve of Eratostheneshttps://paideias.com/sieve-eratosthenes/Introduction to Arithmetic by Nicomachushttps://www.archive.org/stream/nicomachigerasen00nicouoft#page/28/mode/1up. A prime number algorithm method to generate prime numbers that most likely was known well before Eratosthenes lived, since such advanced mathematical ideas are now known to have been understood by the Sumerians.
Recently the 8 Prime Spiralshttps://paideias.com/8-prime-spirals/ dethroned the Sieve of Eratostheneshttps://paideias.com/sieve-eratosthenes/ as the fastest simple prime number generator.
|↑1||Entry ε 2898 of Suda|
|↑2||Chambers, James T. “Eratosthenes of Cyrene.” Dictionary Of World Biography: The Ancient World (January 1998): 1–3.|
|↑5||Introduction to Arithmetic by Nicomachus|
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Taurasi’s stats helping teach math
Twitter user @Alexx_TheGreat posted early Wednesday morning about seeing WNBA star Diana Taurasi in her math book:
Diana Taurasi is in my math book :-) that makes me happy
@Alexx_TheGreatÂ posted on Twitter a photo of the math problem. It uses statistics from Taurasiâ€™s 2006 season:
Diana Taurasi, of the Phoenix Mercury, was the WNBA’s top scorer for the 2006 regular season, with a total of 860 points. The number of two-point field goals that Taurasi made was 65 less than double the number of three-point field goals she made. The number of free throws (each worth one point) she made was 34 less than the number of two-point field goals she made. Find how many free throws, two-point field goals, and three-point field goals Diana Taurasi made during the 2006 regular season.
From Intermediate Algebra: A Graphing Approach by K. Elayn Martin-Gay, Margaret Greene


Omar Abdul Halim, Master’s Student (started in Fall 2020; defended: February 2022 — UNBC).
Thesis: Dynamics of integrodifference and reaction-diffusion equations in heterogeneous media. (Omar’s Thesis is available at UNBC Library Reserves)
Thomas Peterson (NSERC-USRA, Summer 2022). Project: Nonlinear integral equations: deterministic and numerical analysis.
Katherine Saunderson (NSERC-USRA, Summer 2021). Project: Integrodifference equations and dynamics in patchy landscapes.


Experts in Statistics have advised more women to develop interest in Statistics and Data Science in order to harness their skills for better performance in their careers.
The call was made by various speakers at a South-East regional symposium to mark the 2020 International Year of Women in Statistics and Data Science at the Michael Okpara University of Agriculture, Umudike (MOUAU) in Abia.
It was organised by the Laboratory for Interdisciplinary Statistical Analysis (LISA) 2020 Network, MOUAU chapter, in collaboration with the International Statistical Institute (ISI).
Joy Nwabueze, a Professor of Statistics, said that it was important for more women to embrace statistics and data science.
She said that the more women there are in the area, the better for the realisation of the sustainable development goals.
Nwabueze, who is the first female Professor of Statistics in Nigeria, spoke on the topic: “Statistics and Good Governance: the role of Women in Statistics.”
She said that women in statistics and good governance would lead to greater economic stability and prevention of violence against women, amongst others.
She said that women in statistics are also good for development, adding that it would encourage the most use of the nation’s resources.
Nwabueze, who is the immediate past Vice-Chancellor (Administration) of MOAUA, said, “When women are into statistics, the family and society will be better.”
She said that good governance is participatory, consistent with rule of law, responsive, consensus-oriented, equitable and inclusive.
She further said that good governance is effective and efficient as well as accountable.
Also, Dr Hope Mbachu, a Statistics lecturer at the Imo State University (IMSU), encouraged more women to embrace statistics and data science, in order to be involved in social transformation.
According to her, social transformation is the main economic factor that moves countries forward.
Mbachu, who is the IMSU LISA Coordinator, spoke on “Strengthening Statistics and Data Science for Structural Development, the role of Female Statisticians.”
She said that the often neglected side of data science or statistics was the bane of using data erroneously.
Mbachu pointed out that where there was more gender equality, there would be more peace, among others.
“Women have made tremendous strides in Science, Technology, Engineering and Mathematics but there is still work to be done to achieve equality in what have been traditionally male-dominated disciplines,” she said.
The organiser and MOUAU LISA Coordinator, George Uchechukwu, said the programme was to commemorate the 200th anniversary of Florence Nightingale, who made landmark achievements in statistics and data science.
Uchechukwu, a lecturer in the Department of Statistics, MOAUA, advised female students to consider a career in statistics and data science, adding that opportunities abound in the field.
Earlier, the Vice-Chancellor of MOUAU, Prof. Francis Otunta, said the programme was important because there was a dearth of women in Mathematical Sciences.
Otunta, who was represented by Prof. Donatus Igbokwe, Dean, College of Physical and Applied Science, said the event would encourage more women to develop an interest in mathematical sciences, statistics and data science.
The News Agency of Nigeria (NAN) reports that the theme of the event is, “Connecting to build capacity to transform evidence into action and celebrate the International Year of Women in Statistics and Data Science.”


Dr Mahinthakumar is interested in large scale modeling of subsurface flow and transport, parallel and distributed computing, optimization and inverse problems, water distribution system analysis
Dr. Mahinthakumar's (referred to as Dr. Kumar by his students) long term goal is to develop efficient algorithms and tools to solve large scale civil and environmental engineering problems.
Dr. Kumar is currently focused on 1) real time optimization and inverse modeling for water distribution systems analysis, 2) large scale modeling and characterization of groundwater flow and transport systems, 3) parallel and distributed computing algorithms and tools for environmental applications.
In CCEE, Dr. Kumar collaborates with Dr. Arumugam, Dr. Brill, Dr. DeCarolis and Dr. Ranjithan.
At the graduate level, Dr. Kumar teaches Introduction to Numerical Methods for Civil Engineers (CE 536), High Performance Computing for Civil Engineers (CE 791A), and Inverse Modeling for Civil Engineers (CE 791B). In Numerical Methods, he teaches application of common numerical methods to civil engineering problem solving. The high performance computing course is a project based course focused on parallel and distributed computing algorithms for large scale civil engineering applications. The inverse modeling course focuses on heuristic and gradient based search techniques as well as Markov Chain Monte Carlo methods for the solution of civil engineering parameter estimation and system identification problems. The graduate students who work with Dr. Kumar enjoy computer programming and are interested in developing innovative methods and tools for large scale civil engineering problem solving.
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As I sat in the elementary art class, on assignment as a substitute at the school, I listened to the teacher’s lesson to the children on concentric circles. The exercise would begin with a dot being drawn. A small circle would be drawn around the dot, then a larger one around that, and then a larger one, and so on, and so on. Each circle shared the same center. In this particular exercise, several series of concentric circles would be drawn on a large sheet of paper, overlapping each other. The different color patterns made quite a cool effect.


CBSE Board exams are not easy to score as understanding of exam pattern is very important. To make it easy for our students, we have consolidated all previous year question papers and sample papers so that students can practice and get familiar with exam pattern.
On enrolling in this course, you will get access to 12-15 previous years papers and some CBSE sample papers for each subject. This is a good amount of material for practicing and checking your preparation for the exam.
Math Daily Practice for Class – I to VIII
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Workbook for Math is created based on Learner Solution approach of Learn, Practice and Test. Using high-tech teaching methods, giving options for personalized learning augmented with unlimited attempts the students will gain by using the app. Detailed analytics of students’ attempts and performance is provided to identify the weak and strong areas to help focus on the right topics.
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The Iowa Core curriculum provides a guide to delivering challenging and meaningful content to students that prepares them for success in life.
The Iowa Core curriculum identifies essential concepts and skills for kindergarten through 12th grade in literacy, mathematics, science, social studies, and 21st century skills. It also includes direction for teachers regarding effective instruction and assessment. It takes learning to a deeper level by moving students beyond superficial knowledge to deep conceptual and procedural knowledge. It also enhances student engagement by emphasizing interesting, robust, and relevant learning experiences.


For eligibility necessities for army reductions at the doctoral level, please evaluation the online benefits web page. The pupil will document a 2-minute video introducing himself/herself to the class and professor. This free course appears at Scotsman John Napier, finest known to for his treatise on Protestant religion.
- Section 2 critiques ideas of conditional probabilities and introduces Bayes’ theorem and its use in updating beliefs a couple of proposition, when data are observed, or info turns into obtainable.
- After 40 years of dabbling in maths, he revealed his desk of logarithms within the early seventeenth century.
- Credits measure the coed workload required for the profitable completion of a module or qualification.
- Section 2 reviews and supplies a extra formal method to a powerful technique of proof, mathematical induction.
- They’ll break down advanced topics and use partaking visual aids.
- A math schooling also can give you a private and skilled edge.
A math training also can give you a personal and skilled edge. Advanced mathematical skills can enable you to calculate your on-line business’s profit margins or compare the employment charges for graduates of different faculties. A stable understanding of math might help you derive distinctive insights and obtain your targets. Learn the fundamentals of algebra—focused on frequent mathematical relationships, such as linear relationships. Learn pre-algebra—all of the essential arithmetic and geometry skills wanted for algebra.
You really do have assets like someone who is going to highschool on campus. OpenLearn works with different organisations by providing free programs and resources that support our mission of opening up educational alternatives to more people in more places. This free course is concerned with a few of the statistical strategies used in epidemiology and extra extensively in medical statistics. Section 1 introduces cohort research by which individuals are categorized in accordance with their exposure and adopted ahead in time to gauge disease outcomes. This free course is anxious with modelling and estimation and looks in particular on the binomial distribution.
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Enroll at present and explore newbie to superior programs across calculus, statistics, algebra, geometry, sequences, exam prep, and extra. With the rise in on-line courses in India, learning has turn out to be extra customized and flexible. It has been proven to be a more competent means of learning and all it needs is a few primary hardware like a laptop computer, microphone, webcam and a high pace internet connection. Nowadays, enhancing a child’s mathematical capability requires nothing greater than a primary set-up which can oftentimes be cellular.
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Learn third grade math aligned to the Eureka Math/EngageNY curriculum—fractions, area, arithmetic, and a lot more. This Arithmetic course is a refresher of place value and operations for whole numbers, fractions, decimals, and integers. Some students and not using a sturdy math background expressed apprehension about pursuing quantitative studies at Stanford previous to the introduction of MATH 18, math major Emily Dickey ’23 stated. Quantitative studies are academic disciplines that use mathematical evaluation to grasp the world, together with social sciences, pure sciences and engineering.
Each module is damaged down into 10-minute lectures, though some modules can last up to 2 hours. Given that there isn’t any feedback from the teacher throughout this time, it can be hard to keep up and stay motivated. This course consists of four.5 hours of on-demand movies, then permits you to check your knowledge with 510+ practice questions. It’s great for many who want to master the basics of math to improve run 3 hooda math their employment opportunities, especially as you receive a certificates upon completion. Khan Academy is a incredible online platform that teaches math by way of a various vary of methods, including flashcards, quizzes, and video lectures. From start to finish, Preply permits you to rapidly develop your math expertise through a completely individualized course for a very cheap worth.
You also need to search out out if the class will require you to finish assignments and assessments in your personal time. In short, work out how much you can dedicate to learning math earlier than registering for a course so as to achieve whatever targets you’ve set for yourself. The finest digital math classes offer you a quantity of explanations of inauspicious ideas in varying formats. In addition, they break the material down into digestible modules and short classes to maintain you motivated. Presented by edX and UC San Diego, this course is designed for intermediate math college students who are excited about laptop science. It covers various theories, including information sorting and the application of dynamic programming.
For instance, on a day either before break or at the end of 1 / 4, our class was taking part in Scattergories. We got the letter « S » and had been informed to give a reason why someone can be late for college. I, in my naive thoughts, regardless of already being a high school junior, put « stoned, » considering somebody can be late if rocks were being thrown at them. That’s not what my classmates, including my brother and my trainer, thought. Particularly who thinks twins must be separated in school is also fascinating. A January 2014 research additionally published the Educational Policy journal surveyed principals, lecturers, mother and father, and twins about what they consider twins being separated in kindergarten.
Learn skills and instruments that support information science and reproducible analysis, to guarantee you can trust your individual analysis outcomes, reproduce them yourself, and communicate them to others. Learn to use R programming to use linear fashions to analyze data in life sciences. Across the globe, 617 million kids are missing fundamental math and reading skills. We’re a nonprofit delivering the education they want, and we need your help. Learn Precalculus aligned to the Eureka Math/EngageNY curriculum —complex numbers, vectors, matrices, and more.
With edX, you presumably can study at your personal pace in math courses at each degree, from high school pre-algebra to school algebra and beyond. Get a refresher on primary math, from subtraction to exponents, or discover intermediate ideas similar to polynomials and integrals. Alison provides over 40 free on-line math programs throughout a variety of various subjects and ability ranges. Whether you’re excited about strengthening fundamental core math abilities, or diving into superior calculus or statistics, we have the right course for you.


saxon math 1 curriculum
In Math 1, a written assessment is given every five lessons, an oral assessment every ten lessons.
Jan 19, 2018 · Saxon Math 1 covers everything that Saxon K does, so using both is redundant. Youll want to teach number familiarity before kindergarten, though. You can read all about how I do that using fun games in Homeschool Your Kindergartener (for free!) in Just 20 Minutes a Day. Saxon Math 1 offers plenty of practice. A Detailed Comparison of 15+ Upper-Level Math Curriculum Oct 11, 2017 · It helps the teacher teach the program and allows older students to work independently. Curriculum Saxon Math (Grades 5/4 Calculus) Teaching Method Spiral / Independent. Includes:Homeschool Kit includes Student Book, Solutions Manual, Tests & Answer Key. Extras Available:Teaching DVDs for upper levels.
Dec 03, 2014 · First Grade Saxon Math Curriculum Guide Sections and Lessons Key Standards Addressed in Section Approximate Dates P r 15 26, 2014 Section 1:Lessons 1-10 Making Sets of Tens & Ones with Concrete Objects, Numerals, Comparing Numbers, Using Graphs to Organize Data 1.NBT.1 Count to 50, starting at any number less than 50. In this range, read and Homeschool Planet Online Lesson Plans for Saxon MathThe curriculum for this Saxon Math 1 lesson plan can be found by using the following link~ Saxon Math 1, Home Study Kit By Saxon / Saxon Publishing. The Math One Home Study Kit teaches children to skip count by one's, two's, five's and tens, identify ordinal position to tenth, master all basic addition and subtraction facts, measure using inches, feet, and centimeters.
Aug 22, 2018 · Kindergarten Saxon Math Review. After consulting a few classical homeschooling references, I decided to give Saxon a try. They said that the kindergarten curriculum was more on the preschool level, but because I wanted to use it with both my three and five-year-old, I thought it Math curricula reviews for homeschoolingSaxon Math Placement Tests. Learning Things has free placement tests for Saxon Math for all levels. WatchKnowLearn Videos for Math. WatchKnowLearn catalogs hundreds of free educational videos "Wiki-style" in both broad and sub-categories so it's easy to search. There are sub-categories for topics for all levels of math.
Reviews of Saxon Math curriculum. Time:4 years. The books for 1-3 are very similar, but then 4th grade changes greatly. In 4th grade the lessons are very short and then there are about 4-6 problems pertaining to the lesson. After that there are 28-30 problems that review the entire year. Saxon Algebra 1 Homeschool Kits 9781600329715 Lamp Saxon Algebra 1 covers all the topics in a first-year algebra course including proofs, statistics, probability, and algebra-based real-world problems. Features of Saxon Algebra 1 Course. The 3rd Edition of Algebra 1 is the last edition of Saxon Algebra 1 to include Geometry. We think the Solution Manuals are a must for Algebra 1!
Saxon Algebra 1, 4th Edition covers all of the traditional first-year algebra topics while helping students build higher-order thinking skills, real-world application skills, reasoning, and an understanding of interconnecting math strands. Saxon Algebra 1 focuses on algebraic thinking through multiple representations, including verbal, numeric, symbolic, and graphical, while graphing calculator labs Saxon Math - West Brooke CurriculumSaxon takes an incremental (little by little) approach to math, introducing a new skill or principle each day, then reviewing these concepts and skills day after day for weeks. This approach helps build students' confidence in their ability to "do" math successfully. Students who have used this program receive consistently high scores on
Saxon Math K-3 Manipulative Kit. $120.50. These manipulatives are an essential element of the Saxon K-3 math program. Each teacher's manual contains a lesson-by-lesson list of all necessary materials. Includes handy plastic case. Saxon Math Order Form Robinson CurriculumWe recommend the use of Saxon math books 54, 65, 76, 87, Algebra 1/2, Algebra 1, Algebra 2, Advanced Math, and Calculus. Any student who completes this math course, following the recommendations in the Robinson Curriculum, can be expected to have a firm foundation in Math.
May 20, 2021 · The Saxon Math curriculum works on what is called the spiral method. That means in each Saxon math book, you will find concepts introduced, taught, and then revisited over time. Each day your child will have a new lesson, practice with that lesson, and then practice sets in which the new concepts are worked with alongside concepts that were introduced in previous lessons. Third Grade Saxon Math Curriculum Guide - SAUSDDec 03, 2014 · Third Grade Saxon Math Curriculum Guide Sections and Lessons Key Standards Addressed in Section Approximate Dates P r 15 26, 2014 Section 1:Lessons 1-10 2-Digit Numbers & Place Value, Elapsed Time, Data Collection & Display, Odd & Even Numbers 3.NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100.
Saxon Math. Intervention Description. 1. Saxon Math . is a curriculum for students in grades K12. The amount of new math content students receive each day is limited and stu-dents practice concepts every day. New concepts are developed, reviewed, and practiced cumulatively rather than in discrete chapters or units. This review focuses on studies of . Saxon MathsSaxon Math 1 Curriculum - ChristianbookThe Math One Home Study Kit teaches children to skip count by one's, two's, five's and tens, identify ordinal position to tenth, master all basic addition and subtraction facts, measure using inches, feet, and centimeters. Also teaches how to add quarters, draw polygons, and observe information from real graphs, pictographs, and bar graphs.


MathBoost! +Hands on Science (grades 3-5)
One of the secrets of math success -- and even math love -- is the ability to work with numbers quickly and easily. No one likes work that takes too long or takes too much work. So, asking a student to do a complex operation, like long division or a complicated equation, that requires many steps, feels impossible for students who are still using their fingers to add and subtract or consulting a chart for times tables. MathBoost uses fun ways -- games, races, contests -- to push kids to become fast and fluent in their grade level math facts (and above grade level math facts if we can get there). For many kids, learning that math can be pain-free is a revelation. Providing that revelation in the beginning of the school years can make a huge difference in how those kids look at math. So, we take mornings to try to grow our students' excitement for and competence in math.
Then we put the papers away and focus on hands-on science. For this, our last science camp of the summer, we'll pull the experiments and and demonstrations that were the biggest hits over our summer of biology, chemistry, and physics camps. Because this camp will be a "greatest hits" camp, we don't know yet which specific experiements we'll do, but we'll likely cover the science spectrum from life science (growing some plants or brine shrimp) to physics (electricity, magnetism, or gravity) and chemistry (What kinds of polymers can we create and why do they behave like that?). Students will likely create machines, circuits, structures, and slimes. Best of all, they will learn the science behind those creations and we hope to inspire them to do more science on their own, using what they learn in camp as a launching board!
Not only will this camp be all kinds of academic fun, but students should leave the camp with a great foundation in math facts and science, the perfect fun but academic launch into the new school year.


Are you curious about how many milliliters are in half a pint? Well, we’ve got the answer for you! In the US customary system of measurement, there are 236.588 milliliters in half a pint. That’s a precise conversion that can come in handy when working with recipes or understanding liquid volumes.
Converting Pints to Milliliters
Let’s dive into the world of conversions and explore how pints can be converted into milliliters. Understanding these conversions can come in handy when you need to switch between different measurement systems or work with recipes that use different units of volume. So, let’s get started!
- The Conversion Factor:
To convert pints to milliliters, we need to know the conversion factor. A pint is equal to 473.176 milliliters (ml). This means that for every pint, there are approximately 473 ml.
- Converting Pints to Milliliters:
Now that we have our conversion factor, let’s put it into action. To convert a given number of pints into milliliters, simply multiply the number of pints by 473.176.
- If you have 2 pints, multiply 2 by 473.176: 2 x 473.176 = 946.352 ml
- If you have half a pint, which is equivalent to 0.5 pints: 0.5 x 473.176 = 236.588 ml
- Practical Examples:
Understanding conversions becomes easier when we see how they apply in real-life scenarios.
a) Cooking: Imagine you’re following a recipe that calls for half a pint of milk but your measuring cup only has milliliter markings. By using our conversion factor, we know that half a pint is equal to approximately 236 ml of milk.
b) Bartending: Aspiring mixologists often encounter recipes in both pints and milliliters when crafting delicious cocktails at home or behind the bar counter. Knowing how to convert between these units allows them to accurately measure and create their signature drinks.
How Many ML in Half a Pint
When it comes to understanding measurements, it’s important to have a clear grasp on the conversion factors between different units. In this section, we’ll focus on clarifying how many milliliters (ml) are in half a pint. By breaking down the numbers and providing relevant examples, we aim to make this topic easier for you to comprehend.
To begin with, let’s establish the basic conversions. One pint is equal to 473.176 milliliters (ml), while half a pint is simply half of that amount, which gives us 236.588 ml. It’s worth noting that these values may vary slightly depending on rounding conventions or regional differences in measurement standards.
Understanding these conversions can be particularly useful when dealing with recipes or beverage servings. For instance, if a recipe calls for half a pint of water and you prefer measuring ingredients in milliliters, you’ll need approximately 236.588 ml.
So, next time you’re faced with a recipe that calls for half a pint and want to convert it into milliliters, remember this: there are 236.588 milliliters in half a pint according to the US customary system of measurement. Armed with this knowledge, you’ll be able to confidently tackle any culinary endeavor that comes your way!


The following are some Julia packages for scientific computing where I have been quite involved.
IntervalArithmetic.jl: Rigorous floating-point calculations using interval arithmetic in Julia. D.P. Sanders and L. Benet [doi]
IntervalRootFinding.jl: Find all roots of a function in a guaranteed way with Julia. D.P. Sanders and L. Benet
ReachabilityAnalysis.jl: Reachability Computations for Dynamical Systems in Julia. M Forets, C. Schilling, D. Freire Caporale, D.P. Sanders, L. Benet and S. Guadalupe
TaylorIntegration.jl: ODE integration using Taylor's method, and more, in Julia. J.A. Pérez-Hernández and L. Benet [doi]
TaylorModels.jl: Rigorous function approximation using Taylor models in Julia. L. Benet and D.P. Sanders [doi]
TaylorSeries.jl: A Julia package for Taylor polynomial expansions in one and several independent variables. L. Benet and D.P. Sanders [doi]
ValidatedNumerics.jl: Umbrela package for rigorous floating-point calculations with interval arithmetic in Julia. D.P. Sanders and L. Benet [doi]
See also JuliaIntervals and JuliaReach web pages.


Updated: Dec 11, 2020
The first six months are the most crucial period. Since it is quite different from traditional school math, the first six months will be spent on the basic rules and formulae of abacus. Students must practice for 15 minutes everyday as teachers requested. The parents can help by checking the answers for them. If students do not answer correctly, ask them to redo the questions again. If they are still having troubles, they can ask the teachers who they go to the class. Students will improve very quickly if they receive the help from parents and will enjoy going to class. Do not ask teachers to speed up the lessons. Not only will doing so give teachers pressure, it will also take away the opportunity for students to practice the materials, affecting later lessons.
If students complete the homework during this period, they will gradually have the image of abacus in their head and have the foundation for mental arithmetic. Then, the teachers will begin mental arithmetic with hearing then slowly proceed to mental with viewing. Therefore, parent’s support and encouragement will be the best facilitator in learning.


Get a jump start on back to school shopping and score this Texas Instruments 2-Line Scientific Calculator for only $8.88 (Reg. $17). If you have a high school student, you probably need this.
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Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change .
Linear Nonlinear Graph Answers. Displaying all worksheets related to - Linear Nonlinear Graph Answers. Worksheets are Comparing linear and nonlinear functions, Graphing linear, Notes linear nonlinear functions, Patterns and nonlinear functions, Linear equations work, Graphing linear equations work answer key, Graphing lines, Function table t1l1s1.
Worksheet 39.1 Practice Comparing Functions in Different Formats Comparing Functions in Different Formats – Independent Practice Worksheet Complete all the problems. 1) Two functions are represented in different ways. Function 1: The input-output table shows the x- and y- values of a quadratic function. x y 1 1 3 9 4 16 5 25 6 36 7 49 8 64 Function 2: The graph of a linear function is shown.
Comparing Functions Name: ANSWER KEY Date: _____Period: ____ Read each situation, then answer the questions by analyzing and comparing the different linear situations. 1.11..1. The Metropolis Zoo recently celebrated the birth of two new baby pandas! Mochi the panda cub has been measured and weighed each week since she was born. Weeks Weight 0 1 ©7 v240 Y1x3J PKzuZt daN YSVopf9txw Ia MrSes L5L zC M.C f WAnl 4l D Frli kgjh Jt Asi Hr1eZs5emr3v Eeed m.m l EMpavdOeb Sw vi wtch3 GI3nXf ZiBn3iqtMeT BC2a 1l ac CuSl0uxs 5. k Worksheet by Kuta Software LLC


Since the 1970s, String Theory has been fiddled with and tooled around as a means of unifying all fundamental forces and all forms of matter. There’s always been a problem bringing together the calculations of how our world operates on a large scale and of what gravity can affect, and the small quantum world of calculations where things begin to get weird and nearly unpredictable when dealing with small particles like atoms and electrons. The two world’s seem incompatible. The main problem has always been figuring out how gravity fits in to both worlds.
Well, in 1997, theoretical physicist Juan Maldacena proposed a model where the mathematical world of String Theory operated as a hologram and the larger world of the cosmos operated on more of a flatter realm where there is no gravity.


Dear parents/carers, This week is the Times Tables Rockstars England Rocks competition. It started on the 8th November 2021 and finishes on Thursday 11th November 2021 at 7:30pm. It would be great if you could support your children with this at home.
This is a fantastic opportunity for children to practise their times tables while competing against other schools and against other classes at Lake Farm Park Academy. There will be prizes too!
Please do support your children with this.
Mr. McElwee - Maths lead


Last week in our classroom, we began using the app Explain Everything. At first, I hesitated about using the app because it had so many functions and I was worried about Year 1 students becoming confused. I had to give myself a lot of time to adjust to it. I'm glad I stuck with it, because the children are showcasing amazing ideas, understanding and learning through this app!
Before I began focussing on the learning I wanted to integrate using this app, I allowed the students exploration time (with some goals). They needed to find out how to record, draw something, type something and insert something. I gave them 5 minutes to have a go and see what they could discover.
At the end of the exploration time, the students and I discussed what we had discovered and how to do things. I asked students to share through Airplay to the SmartBoard and they taught the class how they had discovered different functions. Not only, did the class learn new ways to use the app, I did too! It is always interesting to see what the children can uncover and how their minds work.
So what on earth do you use this app for? Well, it is so diverse and there are many potential avenues you could go down. I decided to do a bit of app-smashing here (that's where you use multiple apps to create a piece of work)! I used an app called Number Pieces, where students built a two digit number with MAB blocks (tens and ones). I got students to take a screenshot of their number and insert it into the app, Explain Everything. After this, students created a recording and highlight/wrote the number, as well as explained how many tens and ones were in their two digit number.
This app is so great, because through a worksheet you can never grasp the exact thoughts a child is having when they answer question, you can assume, but never quite know what the root of understanding is. This app allows us, as teachers, to dig deeper and gauge whether the child has a true understanding of the concept, it also allows us to see if they have language to support their understanding.
Some other ideas of learning areas and activities you could use Explain Everything for?...
When you have finished in the app, you can convert the file type and save it as a video or export it. Students can easily submit their work to their Google Drive or even a dropbox account if you have one established.
Old fashioned teaching and learning has a time and place still in the classroom. However, our students are growing up in a world immersed in technology. Not only do we need to provide them with skills to cope in this growing tech-filled environment, but we can also enhance their ability to learn content through creative, digital formats.


Ms. ZhenzhenDong ( 董珍珍 ): Reservoir engineer at Data Consulting Service (DCS) of Schlumberger. Dong has focused her research in areas involving unconventional gas reservoirs, reserve and resource assessment, well testing and performance analysis. Dong graduated from Texas A&M University with her PhD in Petroleum Engineering. Dong holds an M.S. degree in Petroleum Engineering from Research Institute of Petroleum Exploration and Development (RIPED), PetroChina, and B.S. degree in Mathematics from Northeast Petroleum University, China.


Math Pathway at Aki Kurose
In middle school, students start on a path that will lead them all the way to college math. The class sequence goes like this for students’ standards (or grade level) math class:
- 6th grade math
- 7th grade math
- 8th grade math
- Algebra I
- Algebra II
In order to be set up to take 4 years of high school level math, you need to stay at or above grade level in middle school math. Most 4-year colleges require that students took 4 years of high school math.
Most students at Aki Kurose take two math classes: a grade-level standards class, and an enrichment or empowerment class that acts to accelerate students’ skills at an individualized pace. This is to help all of our student mathematicians grow as much as possible in math during the year. Students are placed in math classes at Aki based on their performance in math the previous year, their SBA scores from multiple years, and teacher recommendation. Teachers are continually assessing student progress over the course of the year to determine their “just-right” math class. As students accelerate and get ahead in middle school math, they can take Algebra I and sometimes also Geometry while in middle school. Acceleration class options are as follows:
- 6th, 7th & 8th Grade Empowerment Tier 3 – bolstering of grade level skills in a small class setting
- 6th, 7th & 8th Grade Empowerment Tier 2 – small group work on grade level skills
- 6th, 7th & 8th Grade Empowerment for students with support services – MLL & SpEd
- Accelerated Math 6 & Math 7 (for 6th graders) – two years of math in one school year
- Math 8 for 7th graders
- Algebra for 8th graders
- Geometry for 8th graders
In some cases, students may take only one math classes and add in a second exploratory class. This typically happens when 6th graders are taking the Math 6/7 accelerated class, 7th graders take Math 8, or 8th graders take Algebra only.
College Entrance Requirements for Math – Considerations for High School Preparation
Two year colleges:
If you plan to attend community college, it’s a good idea to take Algebra, Geometry, Algebra II and Pre-calculus. Otherwise, you will need to spend money paying for classes that you could have taken in high school. Also, you won’t be able to get college credit for classes until you get to college-level math, which begins at pre-calculus.
Four year colleges:
Most four year colleges require at least 3 years of high school math and recommend that you take more. It is a good idea to take at 4 years of high school math if you plan to attend a four-year college or university.


Introduction to Geometric Proportionality
This App features a series of games about geometric proportionality. The figure below shows a circle with an arc and the same curves unwrapped into lines. The points are located proportionally along their respective parents (the full line segment and the circle). Points B and G (in green) are located proportionally at t, while points C and F are at -t.
The lengths of the line segment and arc connecting these points are displayed below; notice that they are identical. Drag the slider and press the "Animate" button to see what happens when the value of t changes.


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Geek out with BART and celebrate a special Pi Day this year
Have you ever thought about pi on BART? Not the lemon meringue or deep-dish apple kind -- the mathematical one. Turns out pi is used in lots of ways at BART.
Pi is a mathemathical constant, the ratio of the circumference of a circle to its diameter. It's typically written as 3.14159 and is celebrated with Pi Day on March 14.
This year is being called a "once in a lifetime" Pi Day. If you extend the digits out to the hour, minute and second, you get 3.14.15 (March 14, 2015) at 9:26:53 (just before 9:30 in the morning and evening). If you happen to be on a BART train at that time, give a cheer for pi! (You can download a PDF of all the pi examples, and some quizzes inside, to enlarge or print if you want to challenge the math lovers in your home).
FIVE WAYS BART USES PI


Seven Ways to Compute the Relative Value of a U.S. Dollar Amount - 1774 to Present
In 2011, the relative worth of $800.00 from 1861 is:
|$21,100.00||using the Consumer Price Index|
|$16,100.00||using the GDP deflator|
|$134,000.00||using the unskilled wage|
|$265,000.00||using the Production Worker Compensation|
|$276,000.00||using the nominal GDP per capita|
|$2,670,000.00||using the relative share of GDP|
Data for consumer bundle only starts in 1900.
If you need help/ determining which result is most appropriate for you, see Choosing the Best Indicator to Measure Relative Worth.
Samuel H. Williamson, "Seven Ways to Compute the Relative Value of a U.S. Dollar Amount, 1774 to present," MeasuringWorth, April .
Please let us know if and how this discussion has assisted you in using our calculators.


The Quantitative section of the GMAT strikes fear into the hearts of many test takers who haven’t taken a math class in years. If you’re one of those test takers, you shouldn’t worry! You can totally handle the math in this section.
While the Quantitative section is challenging, it doesn’t test especially advanced concepts. This guide will show you what math is on the GMAT, along with key tips on how to prepare.
First, let’s discuss a general overview of GMAT math.
GMAT Math Section: An Overview
The GMAT Quantitative section is your third section on the GMAT. You’ll take it after the 30-minute Analytical Writing Assessment, 30-minute Integrated Reasoning section, and an optional eight-minute break.
The Quantitative section is the first adaptive section on the GMAT, meaning that the questions change to match your ability level. You’ll start out with some medium level GMAT math questions, and the problems will get easier or harder depending on how you’re doing.
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The GMAT math section will continue to give you customized questions to get a more and more accurate measure of your skills. This adaptive format allows for comparable scores across different tests, plus it enhances security since no two tests look exactly the same.
You’ll get 31 GMAT math questions in 62 minutes, leaving you with approximately two minutes per question. Unlike the Integrated Reasoning section, you won’t have access to a calculator. You will get some note boards and markers to write out your work.
While you’ll need to know a variety of math concepts for this section, you’ll only see two question types. Read on to learn what they are.
GMAT Math Questions: 2 Types
There are just two types of Quantitative questions, and they’re interspersed throughout the section in random order. The two types are problem solving questions and data sufficiency questions.
Typically, about two-thirds of GMAT math questions will be problem solving and one-third will be data sufficiency. However, this proportion could change depending on how many experimental questions you get of each type.
Problem solving questions are just like any other typical question you’d get on a math test. They may ask you to solve an equation, figure out a word problem, or answer questions on a graph.
Each question has five answer choices, and there’s only one right answer for each. Where these questions do get tricky is when they require multiple steps or abstract thinking.
Here’s one example of a problem solving question on the Quantitative section of the GMAT.
The second type of Quantitative question, data sufficiency problems, are more unusual. These questions present you with a math problem followed by two pieces of information. You don’t have to solve the problem, but rather indicate whether one or both statements give you “sufficient data.”
Here’s an example of a GMAT math data sufficiency problem.
The answer choices for data sufficiency questions are always the same as the ones you see above. You don’t need to solve the initial problem. You just need to know if you could solve the problem based on the given information.
While there are only two question types in the Quantitative section, there are quite a few more concepts that show up. Below, you’ll find an overview of the tested GMAT math concepts, along with sample problems for each one.
What Math Is on the GMAT Quantitative Section?
As you read above, the Quantitative section doesn’t require you to be a mathematician. Your GMAT math review won’t involve advanced calculus or trigonometry. In fact, the GMAT math topics don’t get much more advanced than high school-level algebra.
The main challenge for a lot of people, then, is reviewing concepts that they studied in the past but haven’t used in a long time. The main skills you need to answer both question types involve arithmetic, algebra, geometry, and word problems.
Let’s take a closer look at the topics within each of these skill areas as outlined by the official GMAT Prep Software, starting with arithmetic.
GMAT Math Topics in Arithmetic
You’ll get a variety of problems that require skills in arithmetic. You shouldn’t have to do particularly advanced calculations, since you won’t have use of a calculator, but the questions may require some strategic problem solving and complex thinking.
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For arithmetic math in the Quantitative section, you should be familiar with
- Properties of integers
- Real numbers
- Ratio and proportion
- Powers and roots of numbers
- Descriptive statistics
- Counting methods
- Discrete probability
The problem below, for example, is a problem solving GMAT math question that involves fractions and percents.
This second sample problem is a data sufficiency question that calls on your arithmetic and logic skills.
GMAT Math Topics in Algebra
In addition to arithmetic, another major skill area is algebra. As part of your GMAT math review, you definitely want to get work with algebraic expressions and solve for variables. You should be comfortable with
- Variables and algebraic expressions
- Manipulating algebraic expressions
- Solving linear equations with one unknown
- Solving two linear equations with two unknowns
- Solving equations by factoring
- Solving quadratic equations
- Absolute value
The sample question below is a typical problem solving question that involves an algebraic expression with two variables and an exponent.
Data sufficiency questions may also call on your algebra skills, like in the practice question comparing two variables below.
GMAT Math Topics in Geometry
You won’t find quite as many questions on geometry as on arithmetic and algebra, but they definitely still come up. To prepare, you should study the following concepts:
- Lines and angles
- Polygons (convex)
- Rectangular solids and cylinders
- Coordinate geometry
This problem solving question, for instance, asks about properties of triangles.
This data sufficiency question requires you to have some understanding of angles and how they relate to one another.
GMAT Word Problems
Word problems may call on arithmetic, algebra, or geometry skills, plus they require careful reading to identify exactly what the problem is asking you to do.
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You might be asked to calculate simple interest or compound interest, calculate rate, or measure profits. You also may have to work with data from a table, line chart, bar graph, scatter plot, or pie graph. Finally, a question may ask you to analyze sets, as presented in Venn diagrams, or analyze probability.
These are some typical concepts you need to understand to solve GMAT word problems.
- Work (these questions typically show the rates at which a person and a machine work alone, and you’ll have to compute the rate at which they work together, or vice versa)
- Mixture (in these problems, substances with different characteristics are combined, and you’ll need to determine characteristics of the resulting mixture)
- Data interpretation
This problem solving practice question asks you to calculate how fast the Earth travels around the sun. You need to convert miles per second into miles per hour.
This data sufficiency question is a word problem that calls on your arithmetic and logic skills.
Knowing what math is on the GMAT is an important first step, but what else can you do to prepare for the Quantitative section? Read on for six key tips on getting ready for GMAT math.
How to Review Math for the GMAT: 6 Key Tips
How can you get ready for the problem solving and data sufficiency questions on the Quantitative section? What can you do to review concepts that you may not have seen since high school?
Read on for six tips to guide your GMAT math review.
#1: Review the Fundamentals
As you read above, GMAT math doesn’t involve particularly advanced concepts. You won’t need calculus, trigonometry, or any college-level mathematics.
Your first step in preparing, then, should be to review fundamental math concepts. You need to have a solid foundation of key concepts in arithmetic, algebra, and geometry, plus a firm grasp of logic and analysis that will help you solve word problems.
Make sure your study materials offer a thorough review of all tested concepts, and take time to work through the lessons.
#2: Drill With Realistic Practice Questions
As you review key concepts, you should reinforce your understanding with GMAT math practice problems. Even though the math in the Quantitative section isn’t particularly advanced, the questions are still challenging.
They often require complex thinking and multiple steps. Even if you know how to calculate the area of a right triangle or solve a quadratic equation, you need to make sure you can apply that knowledge to GMAT math questions.
By drilling with realistic practice questions, you’ll learn how to apply your math knowledge to the GMAT. The best practice questions come from the official test makers, as on the GMAT Prep Software. Third party test prep companies, like Kaplan and Manhattan Prep, also offer useful practice questions and tests.
As you answer the questions, take notes on anything that confuses or trips you up. Take time to read answer explanations and find the source of your confusion. By understanding your mistakes, you can take specific steps to fix them.
#3: Take Timed Practice Tests
After you’ve done initial concept review and tried out some practice problems, you should set aside time to take a timed GMAT practice test. Find a quiet place, take a practice test with a computer adaptive format, and give yourself 62 minutes to complete the section.
After you’ve finished the practice test, take time to review any questions that you got wrong or felt unsure about. Try to locate the source of your error.
Were you running out of time? Did you misunderstand what a question was asking you to do? Did you make an arithmetic mistake? Were you unfamiliar with a concept? Did you simply make a careless error?
By figuring out where you went wrong, you can take the right steps to fix it for next time. The practice test will not only help you figure out what concepts you need to review, but it will also give you feedback on your pacing. Doing well on the Quantitative section requires you to work quickly and efficiently, and taking practice tests will help you improve your test taking rhythm.
#4: Memorize Data Sufficiency Answer Choices
Data sufficiency questions are weird; there’s no doubt about it. You won’t really see questions like them anywhere but on the GMAT.
To minimize confusion, you should familiarize yourself with the answer choices before test day. You’ll always see the same five answer choices that ask whether statement 1, statement 2, both, or neither give you sufficient information to solve the problem.
Remember that you don’t need to solve these problems, but rather indicate whether you have enough information to solve them. Memorize the answer choices before test day so you know exactly what you’re looking for in each data sufficiency problem and don’t have to waste time reading through or trying to differentiate between the answer choices.
#5: Write Out Your Work
As goes the mantra of high school math teachers everywhere, make sure to show your work. You can’t use a calculator in the Quantitative section, and mental math will only get you so far.
You’ll get note boards and markers in the testing center to write out your work. You should definitely use them to work through the challenging, often multi-step problems.
Try to be neat and organized so you don’t run out of room or make a careless error. If your answer doesn’t match any of the answer choices, then you may need to comb through your work to find the calculation mistake. Try to keep everything legible to make things easier on yourself.
These note boards won’t be graded and are entirely for your own use. You can use them on any of the GMAT sections, but they’re especially helpful when solving math problems.
#6: Practice, Practice, and Practice Some More
As a skill like any other, GMAT math demands a lot of dedicated practice. Whether or not you consider yourself a “math person,” you should set aside time to review concepts, answer sample questions, and take practice tests.
Not only will you sharpen your math skills, but you’ll also become a more efficient test-taker who can handle answering 31 questions in 62 minutes. Don’t get discouraged, but know that you can improve with effort, time, and a customized study plan.
GMAT Math Questions: Final Points
To prepare for the Quantitative section of the GMAT, you should focus on fundamental concepts of arithmetic, algebra, and geometry, plus you need to prepare for word problems. You’ll answer two types of questions, the straightforward problem solving questions and the more unusual data sufficiency questions.
After months of GMAT math review, you can go into the test familiar with these question types. You can hone your time management skills by taking timed practice tests. You’ll also find that the math section gets more and more manageable as you answer lots of GMAT practice questions.
You don’t need to be a math person to do well on the GMAT Quantitative section, but you do need to spend time preparing. With enough effort, you can sharpen skills that you first developed in high school and get yourself ready to conquer GMAT math.
Now you’re an expert in GMAT math, but do you know what the rest of the exam looks like? Check out our guide to the full GMAT structure and format, along with examples of each question type across the test.
The GMAT is a daunting test, but how hard is it, really? This article tackles that complicated question to show you just how challenging you can expect the GMAT to be.


Fuzzy soft sets: a model for making choices in an intertemporal framework
Fuzzy soft set
Fecha de publicación
Alcantud, J.C.R., Muñoz Torrecillas, M.J. (2018). Fuzzy soft sets: a model for making choices in an intertemporal framework. XXth Edition of the Mathematical Modelling Conference Series at the Institute for Multidisciplinary Mathematics, Valencia, Spain, 16-18 July 2018
[EN]This paper introduces a model where the options are characterized by one fuzzy soft set in each of an indefinite number of periods. This model extends the standard case of fuzzy soft sets. We explain how to associate a characteristic fuzzy soft set with each model. Finally, a decision making procedure for the selection of alternatives is proposed. The target applications include portfolio selection in finance, environmental issues, et cetera.
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Chesterfield Learning Center-Reading tutoring, math tutoring, reading comprehension, SAT Prep, ACT Prep, homework help
At its most basic, calculus is the study of change, and gives us a way to model problems and predict solutions in a variety of settings. This is an exciting time in history to be studying calculus (although some students may disagree). Computers have greatly increased the potential for mathematical modeling, and students can easily see applications of calculus in action. From the space shuttle to the NASDAQ, calculus shapes the modern world.
For some students, calculus is a gateway to higher math and science. For others, this is the most advanced and difficult math course they will ever take, and represents the end of their formal math education. For either kind of student, a calculus tutor can make a world of difference in their ability to understand and master the crucial principles of calculus.
Students who challenge themselves with this advanced course will be learning skills and information discovered by the greatest minds in history. But with derivatives, functions, interval notation, and more, it’s understandable that many students feel overwhelmed by their calculus assignments. The daily homework can feel overwhelming, particularly for high school upperclassmen with a heavy course load of AP or other advanced subjects. Parents, who perhaps never took the course or have not used their calculus knowledge in years, often feel unprepared to help.
That’s why we have experienced calculus tutors who know how to build confidence in their students and keep them motivated. Our calculus tutors know how to teach this complicated material in clear and understandable ways, so students can feel enthusiasm rather than confusion and excitement rather than fear. As experienced educators, our calculus tutors are the kind of instructors who will not do a student’s work for them. Rather, with patience and clarity, they teach concepts and study habits that empower students to be effective, independent learners.
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All items in Mathematics
Software to assist with expressing and solving mathematical equations
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Interactive tool for Windows and Macintosh that lets you create mathematical notation for word processing, web pages, desktop publishing, presentations, and for TeX, LaTeX, and MathML documents
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