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21.1: I1.01: Overview - Mathematics


Topic I—Linear and Quadratic Models

Objectives:

  1. Recognize when a dataset shows a relationship between the variables that is approximately linear.
  2. Use a spreadsheet to adjust the intercept and slope parameters of a linear formula so that the graph of corresponding points on the resulting line are close to the points graphed from a data set.
  3. Use linear formula that best fits the data as a model for the data, predicting the output y value for any specified input x value.
  4. Recognize when a dataset shows a relationship between the variables that is approximately quadratic.
  5. Use a spreadsheet to adjust the location and scale parameters of a quadratic formula so that the graph of corresponding points on the resulting parabola are close to the points graphed from a data set.
  6. Use the quadratic formula that best fits the data as a model for the data, predicting the output y value for any specified input x value.
  7. Distinguish between appropriate and inappropriate extrapolation of a model.

Overview

In previous topics we have dealt with numbers produced from formulas, and separately with datasets showing the relationships between two variables. Such formulas are models of the measurement data, and their graph will pass close to the data points.

The model formula is used to predict output values. In this topic we will examine models that are linear (that is, their graphs are straight lines), as well as one kind of non-linear model.

The models will not match the data exactly. There will always be some noise due to unavoidable random errors in the data-measurement process. Also, sometimes the actual pattern underlying the data will not match the model’s formula (e.g., if the data has a curved graph and the model is a straight line). In that case even the best linear model will have to go above the data in some areas and below it in others.

Just as we computed deviations from the average when we analyzed the noise in repeated measurements, we will compute deviations from the model when we are trying to decide how well a particular model fits a dataset. A standard deviation based on these deviation values will be a numerical measure of how good the model is. We can also at the deviations to see if the model is too simple, since in an over-simple model most adjacent deviation values will have the same sign, positive or negative (in the correct model, the data will be randomly above or below the model values).

The data variable you want your model to predict should be used for the output y values in the dataset. Thus the other variable should be used for the input x values. Occasionally it is reasonable to also make use of the inverse model, where the role of the data variables is reversed and the second variable is used to predict the first one. If a model is linear, the inverse model for that data is also linear.

Note that which data variable is modeled as output can be different for people with different goals. One person might want to use temperature measurements to predict how long a metal bar will be, while someone else might to use the measured length of the bar to estimate what the temperature is. Both people could use the same set of calibration data, but would assign different x and y roles to the data variables when they make their predictive models.

In this topic we will focus on two simple models (linear and quadratic formulas), but the techniques shown will work in almost exactly the same way for fitting any kind of mathematical model to data. Some other useful models will be discussed in later topics.

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  • Mathematics for Modeling. Authored by: Mary Parker and Hunter Ellinger. License: CC BY: Attribution

21.1: I1.01: Overview - Mathematics

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Methods for allergen analysis in food: a review

Food allergies represent an important health problem in industrialized countries. Undeclared allergens as contaminants in food products pose a major risk for sensitized persons. A proposal to amend the European Food Labelling Directive requires that all ingredients intentionally added to food products will have to be included on the label. Reliable detection and quantification methods for food allergens are necessary to ensure compliance with food labelling and to improve consumer protection. Methods available so far are based on protein or DNA detection. This review presents an up-to-date picture of the characteristics of the major food allergens and collects published methods for the determination of food allergens or the presence of potentially allergenic constituents in food products. A summary of the current availability of commercial allergen detection kits is given. One part of the paper describes various methods that have been generally employed in the detection of allergens in food their advantages and drawbacks are discussed in brief. The main part of this review, however, focuses on specific food allergens and appropriate methods for their detection in food products. Special emphasis is given to allergenic foods explicitly mentioned in the Amendment to the European Food Labelling Directive that pose a potential risk for allergic individuals, namely celery, cereals containing gluten (including wheat, rye and barley) crustaceans, eggs, fish, peanuts, soybeans, milk and dairy products, mustard, tree-nuts, sesame seeds, and sulphite at concentrations of at least 10 mg kg −1 . Sulphites, however, are not discussed.


Just as we can perform operations such as addition - on two numbers to obtain a new number, set theory operations are used to form a set from two other sets. There are a number of operations, but nearly all are composed from the following three operations:

    – A union signifies a bringing together. The union of the sets A and B consists of the elements that are in either A or B. - An intersection is where two things meet. The intersection of the sets A and B consists of the elements that in both A and B. - The complement of the set A consists of all of the elements in the universal set that are not elements of A.

Using the series expansion we have:

If $x$ is positive it is immediately obvious that there can be no equality.

If $x<0$ then the RHS is greater than 1 and $e^<1$.

This is not strictly an "algebraic" solution, but with the term in $e^x$ we do not expect anything purely algebraic.

"Lambert W" is a hint for "algebraic solution".
The solution for $mathrm^x + x = 1$ is $1-mathrm W(mathrm)$,
to find ALL complex solutions, use all branches of the Lambert W .

$ egin &dots 1 - mathrm_<-4>(mathrm) &= 3.159947300 + 23.47017395 i 1 - mathrm_<-3>(mathrm) &= 2.849014724 + 17.17149358 i 1 - mathrm_<-2>(mathrm) &= 2.393982241 + 10.86800606 i 1 - mathrm_<-1>(mathrm) &= 1.532092122 + 4.597158013 i 1 - mathrm_<0>(mathrm) &= 0.000000000 1 - mathrm_<1>(mathrm) &= 1.532092122 - 4.597158013 i 1 - mathrm_<2>(mathrm) &= 2.393982241 - 10.86800606 i 1 - mathrm_<3>(mathrm) &= 2.849014724 - 17.17149358 i 1 - mathrm_<4>(mathrm) &= 3.159947300 - 23.47017395 i 1 - mathrm_<5>(mathrm) &= 3.396557044 - 29.76478701 i &dots end $

explanation

You can see this very easily graphically. The equation is $e^x=1-x$ and the two sides of the equation are plotted here (from Wolfram Alpha):

The intuition for a formal proof also follows directly from the picture (the functions are both monotonic but in opposite directions), if that's your aim.

Let $f(x) = e^x + x - 1$. Then, for any given $x$, $f(x) = 0$ if and only if $e^x + x = 1$.

You have already noticed that $f(0) = 1 + 0 - 1 = 0$, so it is a solution.

Now, we turn to calculus, not algebra. We have $f'(x) = e^x + 1$. Since $e^x > 0$ for all $x$, we know that $e^x + 1 > 0$ as well. In other words, $f'(x)$ is positive for all $x$ which tells us that $f(x)$ is an increasing function on the entire real line. Therefore, it could only possibly be 0 at one point, and you already found that point.

Now, if you haven't had calculus, you could still get the same basic idea. For example, you know $y = x$ is increasing. That is something you should know. Perhaps you have learned that $y = e^x$ is always increasing as well, because even in an algebra class, they would probably give you a bunch of properties of $y = e^x$ when they introduce it. Add these two functions together, and it's still increasing. Subtract 1, and the function is simply translated downward 1 unit, so it's still increasing everywhere. Again, the conclusion is the same.


Domain connection shows as &ldquounauthenticated&rdquo

I have seen various different questions for this problem floating around but either the circumstances arent the same or the solution doesnt work so thought i would post it to see if anybody has any suggestions.

Various domain PCs and laptops appear to randomly give the connection name of "lewis.local 2(Unauthenticated)" - lewis.local being our domain - and provides an exclamation mark where the network type logo is normally shown.

This also appears to happen every time connecting via vpn.

  • 2 servers both running windows server 2003 R2 (x32)
  • main server has AD, DNS and DHCP installed
  • IPv4 on approx 30 client machines (some wired, some wireless)

If anybody has any thoughts on solutions i would appreciate it. I have tried removing all but AD server roles, resetting all of the systems and nothing.

It doesnt prevent anything from working just like a domain connection most of the time however it is getting fustrating!

Also dont know if it could have anything to do with it but the DHCP server seems to have quite a long lead time on issuing the IP address to the client.


Sources and Further Reading

Regular readers of this website would know that I believe you should always cite quality academic articles when writing about topics at university.

Below are some useful sources that you can cite if you are writing an essay on this topic.

My Recommendations for Further Reading

I strongly recommend the Kentli and Alsubaie articles which are the easiest to understand introductions to the topic that I have come across. Both are freely available online if you click the following links:

And here are all the articles I recommend you cite in your essay:

  • Alsubaie, M. A. (2015). HC as One of Current Issue of Curriculum. Journal of Education and Practice, 6(33): 125 – 128. Retrieved from: https://files.eric.ed.gov/fulltext/EJ1083566.pdf
  • Apple, M. W. (2004). Ideology and Curriculum. London: Routledge & Kegan Paul.
  • Boostrom, R. (2010). HC. In: Kridel, C. (Ed.) Encyclopaedia of Curriculum Studies. (pp. 440 – 441). Los Angeles: SAGE.
  • Cubukcu, Z. (2012). The Effect of HC on Character Education Process of Primary School Students. Educational Sciences: Theory & Practice, 12(2): 1526-1534. Retrieved from: http://files.eric.ed.gov/fulltext/EJ987859.pdf
  • Durkheim, E. (1961). Moral Education. New York: Free Press.
  • Giroux, H. A. (2001). Theory and Resistance in Education. London: Bergin & Garvey.
  • Jackson, P. (1968). Life in Classrooms. New York: Holt Rinehart and Winston Publishers.
  • Kentli, F. (2009). Comparison of HC Theories. European Journal of Educational Studies, 1(2): 83 – 88.
  • Margolis, E. (2001). The HC in Higher Education. New York: Routledge.
  • Morris, E. (2005). “Tuck in that shirt!” Race, class, gender, and discipline in an urban school. Sociological Perspectives, 48(1): 25-48. Doi: https://www.jstor.org/stable/10.1525/sop.2005.48.1.25
  • Thornberg, R. (2009). The moral construction of the good pupil embedded in school rules. Education, Citizenship and Social Justice,4(3): 245-261. Doi: https://doi.org/10.1177%2F1746197909340874
  • Walton, G. (2005). The hidden curriculum in schools: Implications for lesbian, gay, bisexual, transgender, and queer youth. Alternate Routes: A Journal of Critical Social Research, 21(1): 18-39. Retrieved from: http://www.alternateroutes.ca/index.php/ar/article/view/20362

The above citations are in APA format. If you need some guidance on converting the citations to another format, read my advice here.

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Mathematical Content

The abundant use of digital technology does not only raise the need for skills that complement what computers (can) do, but it also influences what mathematics is or becomes relevant in our society. A clear example is that of statistics. A large part of the information in our society is statistical information, and we may point to “big data” as a growing field, as a result of the unprecedented access to data and computer power. People therefore need some basic understanding of statistical processing and analyses, which has been called statistical literacy (Gal, 2002). Another topic that gains new significance is space-geometry, as a consequence of the growing role of 3D imaging and 3D printing. Among the most relevant topics, Hoyles et al. (2010) mention measurement, data collection, variables and co-variation, reading and interpreting data, graphs, and charts. Brady et al. (2015) also point to the importance of co-variation and functions, which they see as building blocks for modeling aspects of systems.

In a more general sense, it may be argued that the software embedded in computerized apparatus will have the character of mathematical models of reality, which will consist of systems of interconnected mathematical relations. Thus, individuals will need an understanding of variables, co-variation, and functions. In a similar manner, we may argue that working with computers requires that phenomena from reality are translated into numerical quantities. This points to the need for a deep understanding of the process of quantifying reality, including the awareness that quantifying reality goes with a reduction of information, and that quantification might even result in meaningless numbers in some cases. Understanding the process of quantifying reality requires a broad understanding of measuring and measures comprising notions of uncertainty and repeated measurement, mean, and measurement error. Similarly, concepts such as data creation and sampling come to the fore.

A specific point concerns the ability to work with computer tools. Next to the computerization of all sorts of apparatus, the twenty-first century has also brought a variety of computer tools which are available in the form of handheld calculators, spreadsheets, computer algebra systems, graphing tools, and so forth. This means that students will have to learn to work with these types of computer tools. Often, this will not only concern technical instructions—as in the case of spreadsheets—but may also involve complex processes. As an example of such a process, we may refer to what researchers of “computer algebra systems” (CAS) call “instrumentation” (Drijvers & van Herwaarden, 2000). The user of a CAS has to develop an instrumentation scheme, which not only consists of a series of actions but also involves mathematical objects and strategies. With other computer tools, a similar interconnectedness with learning mathematics may be expected.

Mathematics in Everyday Life

When considering the goals of mathematics for the future, we also have to think of the use of mathematics in everyday life. Here too, the demands in terms of “the knowledge and skills required to effectively manage and respond to the mathematical demands of diverse situations” (Gal, Groenestijn, M van, Manly, Schmitt & Tout, 2003, p. 4) grow as a consequence of the increasing digitalization of our society. In relation to this, the terms numeracy and quantitative literacy are used, both with varying and overlapping meanings. Typically, those concepts do not discriminate between work and everyday life. Still, the main orientation appears to be towards everyday life and citizenship. In relation to the latter, Steen (2001) points to the importance of quantitative literacy for democratic discourse and civic decision-making, which may involve quantitative information, such as interest-rate cuts by the Federal Reserve, changes in gasoline prices, trends in student test scores, election results, and risks of dying from colon cancer.

Studies in numeracy, as well as mathematical literacy, are of significance for our investigation, as they offer reference points for what all adults need to thrive in the twenty-first century society. With this in mind, we will briefly describe the topics that are mentioned in the ALL Numeracy Framework (Gal et al., 2003). Dealing with quantity and number requires not only common measures, such as length area, weight, time, money, etc., but also measures such as humidity, air pressure, population growth rates, and profits of companies. In the context of dimension and shape, one has to grasp the dimensions of real objects and abstract things and visualizations thereof (maps, projections, etc.). Dealing with the world mathematically further asks for recognizing, interpreting, and creating patterns, functions, and relationships, while using tools such as tables, graphs, symbols, and words, functions, and relationships between variables are considered essential for understanding (basic) economic, political, and social analyses. The pervasive role of statistics in the digital society asks for the ability to deal with data and chance, encompassing big ideas such as variability, sampling, error, prediction, and the distinction between signal and noise. Related aspects are data collection and data displays (graphs, frequency tables, and pie charts). The ALL framework entails change as a separate category, which includes how organisms grow, populations vary, prices fluctuate, and traveling speed may vary. Also rates of change may need attention, as in the context of compound interest, for instance.

Not surprisingly, there is a large overlap with what is required in the workplace, although there seems to be an even stronger priority on mathematical contexts used in the ALL framework. Contextual problems are often the core of adult education, while the participants are supported in preserving the connection with the problem context and the solution in their reasoning. As another difference, we may point out that the overriding concern in the literature on adult life skills concerns self-reliance and self-confidence.

Considerations

Before concluding this stocktaking, we want to reiterate that the objective of this paper is not to produce an exhaustive inventory of goals and societal demands. The aim of this reconnaissance is to create a basis for discussion, a discussion which, in our view, is long overdue. We will therefore briefly address some topics that have been underexposed in our exploration. A self-evident issue here is the role of more formal mathematics. Our focus has mainly been on the practical value of mathematics in the world outside school. The goal of mathematics education, however, is also to prepare students for further education to which we may add the importance of understanding and appreciating mathematics as a goal in and of itself. Thus, a proper balance will have to be sought.

This does not necessarily entail an opposition formal abstract mathematics may also support practical applications. We may illustrate this with number theory. Number theory concerns number systems, properties and relations of numbers, special numbers (triangular numbers, square numbers, perfect numbers, and prime numbers), divisibility, etc. These are issues that are relevant in the digital society in connection to modern phenomena like coding, hacking, etc. Further, it can be taught on a very elementary level number theory is very suitable to let children explore, experiment, and ask themselves questions. This topic is also easy to link to history of mathematics and to our human heritage, which in itself can be seen as a goal of mathematics education. Finally, it may be noted that prime numbers offer unsolved problems in mathematics and are thus related to advanced mathematics.

The need to prepare students both for further education and for life means that a balance has to be found with the goal of preparing students for life outside school. This encompasses issues such as canonical versus noncanonical forms of mathematics, understanding mathematics on a generic level and coming to grips to a rigorous underpinning, and the tension between grasping mathematics in contexts or on a formal level.

Topics that Lose Relevance

While discussing topics that need to be incorporated in the curriculum, we have circumvented the tricky issue of what topics should receive less attention. One of the mathematics educators who put this topic on the agenda is Zalman Usiskin. He makes the following arguments. We continue to see what he calls “phony traditional word problems” in textbooks and instruction, even though realistic settings and modeling are important for future mathematics learning (Usiskin, 1980, 2007). Traditional word problems must be replaced by problems situated in realistic settings with real goals and implications. Further, he goes on to say, since computer algebra systems on handheld calculators can conduct trinomial factoring efficiently and most quadratics observed in realistic settings are solved by the quadratic formula, he advocates that manual factoring be deleted from the curriculum (Usiskin, 2004). Finally, in his view, the role of geometric proof should be in question as well (Usiskin, 2007). He argues that proofs are fundamental to understanding how all mathematics works, not just in geometry, so we should leave some proving in the curriculum. However, he notes that there are multiple dynamic geometry tools that allow a different type of mathematical exploration and proof that should be integrated into instruction. Mathematics educators and policymakers should consider equally what mathematics calculations are important for the curriculum and what abstract mathematics is essential.


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A deeper look at digital resources and functionality of the platform between student and teacher
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July 303:00 PM - 4:30 PM PST CA Inspire Science K–5 – Program Overview and Intro to Digital
Overview of instructional pathway, program components and digital resources
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August
August 53:00 PM - 4:30 PM PST CA Inspire Science K–8 - Program Overview and Intro to Digital
Overview of instructional pathway, program components and digital resources
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August 63:00 PM - 4:30 PM PST CA Inspire Science High School – Program Overview and Intro to Digital
Overview of instructional pathway, program components and digital resources
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August 63:00 PM - 4:30 PM PST CA Inspire Science K–5 - In-Depth Digital and Q/A
Overview of instructional pathway, program components and digital resources
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August 93:00 PM - 4:30 PM PST CA Inspire Science 6–12 - In-Depth Digital and Q/A
A deeper look at digital resources and functionality of the platform between student and teacher
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August 173:00 PM - 4:30 PM PST CA Inspire Science K–8 - Program Overview and Intro to Digital
Overview of instructional pathway, program components and digital resources
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August 203:00 PM - 4:30 PM PST CA Inspire Science Transitional Kindergarten – Program Overview and Digital
Overview of instructional pathway, program components and digital resources
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August 203:00 PM - 4:30 PM PST CA Inspire Science K–5 - In-Depth Digital and Q/A
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August 233:00 PM - 4:30 PM PST CA Inspire Science High School – Program Overview and Intro to Digital
Overview of instructional pathway, program components and digital resources
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August 253:00 PM - 4:30 PM PST CA Inspire Science K–8 - Program Overview and Intro to Digital
Overview of instructional pathway, program components and digital resources
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August 263:00 PM - 4:30 PM PST CA Inspire Science 6–12 - In-Depth Digital and Q/A
A deeper look at digital resources and functionality of the platform between student and teacher
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August 263:00 PM - 4:30 PM PST CA Inspire Science TK – Program Overview and Digital
Overview of instructional pathway, program components and digital resources
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August 273:00 PM - 4:30 PM PST CA Inspire Science High School – Program Overview and Intro to Digital
Overview of instructional pathway, program components and digital resources
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August 303:00 PM - 4:30 PM PST CA Inspire Science 6–12 - In-Depth Digital and Q/A
A deeper look at digital resources and functionality of the platform between student and teacher
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August 313:00 PM - 4:30 PM PST CA Inspire Science K–5 - In-Depth Digital and Q/A
A deeper look at digital resources and functionality of the platform between student and teacher
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September
September 23:00 PM - 4:30 PM PST CA Inspire High School – Program Overview and Intro to Digital
Overview of instructional pathway, program components and digital resources
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September 73:00 PM - 4:30 PM PST CA Inspire Transitional Kindergarten – Program Overview and Digital
Overview of instructional pathway, program components and digital resources
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September 83:00 PM - 4:30 PM PST CA Inspire 6–8 – Program Overview and Intro to Digital
Overview of instructional pathway, program components and digital resources
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September 83:00 PM - 4:30 PM PST CA Inspire K–5 – Program Overview and Intro to Digital
Overview of instructional pathway, program components and digital resources
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September 133:00 PM - 4:30 PM PST CA Inspire K–5 - In-Depth Digital and Q/A
A deeper look at digital resources and functionality of the platform between student and teacher
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September 133:00 PM - 4:30 PM PST CA Inspire 6–12 - In-Depth Digital and Q/A
A deeper look at digital resources and functionality of the platform between student and teacher
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September 223:00 PM - 4:30 PM PST CA Inspire K–5 - In-Depth Digital and Q/A
A deeper look at digital resources and functionality of the platform between student and teacher
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September 233:00 PM - 4:30 PM PST CA Inspire 6–12 - In-Depth Digital and Q/A
A deeper look at digital resources and functionality of the platform between student and teacher
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September 293:00 PM - 4:30 PM PST CA Inspire 6–12 - In-Depth Digital and Q/A
A deeper look at digital resources and functionality of the platform between student and teacher
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September 303:00 PM - 4:30 PM PST CA Inspire K–5 - In-Depth Digital and Q/A
A deeper look at digital resources and functionality of the platform between student and teacher
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June
June 101:30 PM - 3:30 PM PST CA My Math K–5 – Program Overview and Digital
Overview of instructional pathway, program components and digital resources
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June 171:30 PM - 3:30 PM PST CA Math 6–8 – Program Overview and Digital
Overview of instructional pathway, program components and digital resources
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July
July 143:00 PM - 4:30 PM PST CA My Math K–5 – Program Overview and Digital
Overview of instructional pathway, program components and digital resources
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July 153:00 PM - 4:30 PM PST CA Math 6–8 – Program Overview and Digital
Overview of instructional pathway, program components and digital resources
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July 233:00 PM - 4:30 PM PST CA Math 6-8 – Program Overview and Digital
Overview of instructional pathway, program components and digital resources
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August
August 163:00 PM - 4:30 PM PST CA My Math K-5 – Program Overview and Digital
Overview of instructional pathway, program components and digital resources
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August 243:00 PM - 4:30 PM PST CA Math 6–8 – Program Overview and Digital
Overview of instructional pathway, program components and digital resources
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August 253:00 PM - 4:30 PM PST CA Math 6–8 – Program Overview and Digital
Overview of instructional pathway, program components and digital resources
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August 263:00 PM - 4:30 PM PST CA My Math K–5 – Program Overview and Digital
Overview of instructional pathway, program components and digital resources
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September
September 83:00 PM - 4:30 PM PST Glencoe High School Mathematics 2014
Overview of instructional pathway, program components and digital resources
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September 143:00 PM - 4:30 PM PST CA Math 6–8 – Program Overview and Digital
Overview of instructional pathway, program components and digital resources
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September 153:00 PM - 4:30 PM PST CA My Math K–5 – Program Overview and Digital
Overview of instructional pathway, program components and digital resources
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September 203:00 PM - 4:30 PM PST CA My Math K–5 – Program Overview and Digital
Overview of instructional pathway, program components and digital resources
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September 223:00 PM - 4:30 PM PST CA Math 6–8 – Program Overview and Digital
Overview of instructional pathway, program components and digital resources
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Watch the video: Topic Part 1 (October 2021).