Maths counter: Counters – Maths Materials

Counting pills | NZ Maths

Purpose

This problem solving activity has an algebra focus.

Achievement Objectives

NA4-9: Use graphs, tables, and rules to describe linear relationships found in number and spatial patterns.

AO elaboration and other teaching resources

Student Activity

Pharmacists sometimes use a triangular tray to quickly count pills.

You could make a tray like this out of cardboard that can be used to count up to 15 table tennis balls.

Imagine if the tray was larger and the balls were very small.

The 5th number on the tray is 15. The 6th number is 21. The 7th is 28.

What is the 99th number?

Specific Learning Outcomes

• Express rules in words.

• See more than one rule for a given pattern.

• Devise and use problem solving strategies to explore situations mathematically.

Description of Mathematics

This problem is about patterns, how to continue them, and how to find the general term of a pattern.  Finding the next term by looking at the recurrence rule, using a table, and incorporating number properties are all valuable skills that can be used in many situations. To successfully solve this problem, your students should have some prior experience with creating tables, noticing patterns in tables, and describing rules for given patterns in their own words. Note that students are not expected to use symbols when describing these rules in this problem.

This problem is one in a series of Algebra problems: Triangular Numbers, Level 3; Square and Triangular Numbers, Level 5; and Triangular Number Links, Level 6. The problems explore triangular numbers and lead to an algebraic formula for the nth triangular number.

See also Building Patterns Constantly and Building Patterns Incrementally, Algebra, Level 3.

Required Resource Materials

• Copymaster of the problem (Māori)
• Table tennis balls or counters
• Copymaster of the problem (English)

Activity

The Problem

Pharmacists sometimes use a triangular tray to quickly count pills.
You could make a tray like this out of cardboard that can be used to count up to 15 table tennis balls.

Imagine if the tray was larger and the balls were very small. The 5th number on the tray is 15. The 6th number is 21. The 7th is 28.

What is the 99th number?

Teaching Sequence

1. Introduce the problem to the class. Brainstorm ideas for approaching the problem and keeping track of what has been done.
2. Encourage them either to use a table and to see how the entries in the table are related, or to use a geometrical approach. (See Solution.)
3. As the students work on the problem in pairs you might ask the following questions to extend their thinking:
   What strategies might help you to find the answer?
   How can you use your knowledge about numbers here?
   Can you see any patterns that might help?
   Can you put those patterns into words?
   There are further scaffolding questions in the solution.
4. Share the students’ answers and the provided solutions. Ask them to explain their reasoning. Encourage students to think about both possible approaches (shown below).
   Which approach can you understand best?
5. Ask students to write up a solution.
6. Pose the extension problem as appropriate. Perhaps have them begin with a smaller number like 66. Encourage them to use the word rule for triangular numbers that they have found.
7. Discuss this with the whole class.

Extension

How many pills will there be along the side of the triangular stack if there are 6216 pills altogether?

Solution

This problem has been solved by a range of simpler methods at earlier levels (see Triangular Numbers, Level 3). 

Challenge the students to complete the table (you could leave the third column blank to increase the difficulty). What do they have to multiply by each time in order to go from the term to the triangular number? (The entries in the third column should be, in order, 1, 1.5, 2, 2.5, 3, 3.5, 4.)

What is the pattern for the numbers in the third column? How do they relate to the other two?

How can you now get the third triangular number? The 10th? Can you give a rule for the triangular numbers for any given term? (Take the term and multiply by half of the next term. )

Help them to see that the number in the third column is half of the next term (4 is half of 8). So the 7th triangular number is 7 x (8/2) = 28. If this is applied to the 99^th row of the table we will have to multiply by a half of 100. This leads to a solution of 99 x 50 = 4950.

To see this from a geometrical point of view, we can put two lots of the 99th triangular numbers together. Colour one red and the other blue (see below). How big are the sides of the rectangle that is produced? (The rectangle is 99 by 100.) So 2 x (99th triangular number) = 99 x 100. This means that

the 99th triangular number = (99 x 100)/2 = 4950.

Students should be encouraged to express in words the relationships involved. For example, to say,

“a triangular number is the term number x (the next number term)/2”; or

“a triangular number is (the term number x (the term number plus 1))/2”.

Can you see that both of these numbers are the same?

Can you write the expressions using algebra (e.g. using symbols in place of some numbers and the equals sign)? (This is not expected for Level 4 students but some of them might be able to understand it. You will need to use your judgement here.)

Extension to the Solution

There are 6216 pills. This is a triangular number and the term of that number is the length of the side of the pills. So

6216 = (the term number x (the term number plus 1))/2.

This means that:

12432 = the term number x the term number plus 1.

Can you find two factors of 12432 that differ by 1 and that multiply together to give 12432? By experimenting or using square roots you should be able to find that 111 x 112 = 12432. So 6216 is the 111th triangular number.

Attachments

Add to plan

Level Four

Adding a counter—Portal for ArcGIS | Documentation for ArcGIS Enterprise

The GeoPlanner dashboard counters show the current value of a key performance indicator (KPI). Measuring the value of KPI shows progress towards a goal or phenomenon. KPIs are dimensions that include attributes that are used in a script, constants, and simple mathematical operators. GeoPlanner dashboards can include multiple counters. The counters show KPI data that corresponds to the currently active scenario.

You can add one or more counters to the GeoPlanner Dashboard, each of which will show the numeric value of the KPI for your plan. The value shown in each counter will be calculated from the formula that is used to determine the current value of the performance measurement setpoint. These equations for calculating plan dimensions support a combination of individual object attribute values ​​from the Active Scenario, constants, and simple mathematical operators.

The steps below describe how to create a new counter.

1. On the map toolbar, click the Dashboard button.
2. In the upper left corner of the information panel, click the Global Settings icon and select Add Performance Indicator from the drop-down menu.
3. In the Performance Indicator dialog box, enter a title in the Title field. This text will be shown in the dropdown menu of the dashboard widget. This should be a short description that easily indicates what performance metric this counter will show.
4. Enter a performance indicator range in the Range fields. You must enter the upper and lower limits and, optionally, you can enter 2 additional averages. The range on the meter will be displayed as a color scale, by default from red to green. When you press the H-B switch, the color scale of the counter will be switched to the reverse.
5. Click the Scenario Layer drop-down menu and select the scenario layer from which the metrics will be calculated.
6. Optionally, click the Filter drop-down menu and select a subset of feature types to include in the equation. All types are used by default.
7. Click Equation Builder to display the corresponding dialog box. Equations can include fields (in brackets) from the script layer, constants, units of length or area, and simple mathematical operators. The equations are applied for each object. See Creating Counter Equations and Examples for more information.
8. Optionally, click the Sum drop-down menu and select Sum or Average. This will determine whether the values ​​will be summed or averaged for all features used in the meter. This KPI shows the total number of acres in the active scenario.
9. Click Update. A new counter has been created and added to the dashboard.

Meter Equation Builder

Creating equations to calculate quantitative performance indicators for plans requires at least one attribute or constant (for example, the number of acres of the selected land use type). It can also be calculated based on several attributes and coefficients (for example, potential tax revenue, which takes into account the percentage of each type of land use, the area-to-storey ratio, and the tax rates for each type). An example syntax is shown at the bottom of the Equation text box. The equations are applied for each object. The results are summarized or averaged across all objects to show a single KPI value in the counter. The following rules apply to equations:

Empty values ​​

When calculating, if a zero value is found in the equation, then the entire equation will be excluded from processing. For example, you create a meter that uses the equation ((([squarefeet]*[LotCoverage])*[stories])*[residentialfactor])/1000 to calculate available building space. There are 10 script objects. If 5 of these objects have zeros in any of the attribute values ​​(LotCoverage, stories or residentialfactor), then each of these 5 objects will be excluded from the calculation for the counter. For best results, recalculate any available zero values ​​to 0 (or whatever value equals zero in this example) and always provide a default value for attributes in the GeoPlanner template.

Examples

The examples below use the attributes found in the GeoPlanner template for land use planning. Each example shows a counter and its properties. These are just tutorial examples.

The following counters estimate the number of residential units in the plan by multiplying the living meters per acre attributes by the number of acres for each unit: [DWELLING UNIT PER ACRE] * [ACRES].[DWELLING UNIT PER ACRE] is an attribute alias.

The following meter estimates the daily water consumption per household by multiplying the residential meters per act for each facility by the number of acres for each facility, and then multiplying the result by the daily consumption attribute values: ([DWELLING_UNIT] * [ACRES])*[WATER_CONSUMPTION]. Field names are used instead of aliases. The target value is set to 50,000, and the value range is configured so that low values ​​are preferred over high values. The following figure shows how you can use the Equation Builder to build an equation to evaluate this KPI.

The counter below shows the average speed limit for all roads in the plan, based on the average value of the speed limit attribute: [SPEED_LIMIT]. Equation operation «Medium» selected and only 2 range values ​​(low and high).

Updating or deleting the counter

You can update the counter by clicking on the gear icon in the upper right corner of the counter. This will open the Performance Meter dialog box. It allows you to configure all the parameters of the indicator.

You can delete the counter by clicking on the trash can icon in the upper right corner of the counter. This will remove the counter from the project. But the counter will not be removed from the shared dashboard.

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Mathematical Center in Akademgorodok

World-class scientific centers according to the priorities of scientific and technological development are created within the framework of the national project «Science» on the basis of scientific organizations, universities or their associations in the form of a consortium. In 2019Three world-class genomic research centers and four international mathematical centers were created as part of the first phase. One of the latter will be discussed in our material.

World-class Mathematical Center «Mathematical Center in Akademgorodok» was created on the basis of Novosibirsk State University and the Institute of Mathematics. S.L. Sobolev of the Siberian Branch of the Russian Academy of Sciences practically.

The NCMU in Siberia sets an ambitious goal — the integration of Russian mathematical research into the international scientific agenda, increasing the attractiveness of work in the Siberian macroregion for Russian and foreign leading scientists and young promising researchers.

A fundamentally new system for integrating education, research and high-tech business in mathematics and related fields will be implemented within the center.

Science

Despite the fact that the Mathematical Center in Akademgorodok was established only in the middle of last year, the NCMU team actively conducts both research work and organizes conferences, schools and workshops to exchange experience and knowledge with Russian and foreign colleagues. More than 20 events are planned for 2020.

The Center implements three research programs, within which work is carried out on 11 projects: «Solving mathematical problems that arise in natural science», «Data processing, machine learning and cryptography» and «Efficient algorithms and theoretical issues of computational complexity».

For example, within the framework of the Inverse Problems of Natural Science project, refined scenarios for the development of COVID-19 in Moscow were obtained, taking into account the self-isolation index and the percentage of testing of detected cases, obtained by solving the parameter refinement problem for the SEIR-HCD mathematical model, consisting of 7 non-linear ordinary differential equations. For small data statistics (until May 1, 2020) and for larger statistics (until May 21, 2020), a peak of detected cases in Moscow is predicted with an error of two days and fewer detected cases by 174 people from the real number.

The most important scientific result of 2019, as the director of the Mathematical Center at Akademgorodok Evgeny Vdovin notes, is to obtain a polynomial algorithm for checking the isomorphism problem of the group G, which is a noticeable result in the study of one of the seven millennium problems about the equality of complexity classes P and NP (or the enumeration problem): whose answer can be quickly (from an algorithmic point of view) checked, has a fast solution. All modern public-key cryptographic systems are based on the assumption that the brute-force problem has a negative solution. One of the most recent topical problems, the complexity status of which is still not known, and in the progress on the answer about the complexity status of which a significant answer has recently been received, is the problem of graph isomorphism. Laszlo Babai and his students proved that the problem of graph isomorphism and the problem of isomorphism of permutation groups are fundamentally reducible to each other, i.e. have one difficulty status. The result obtained at the Mathematical Center in Akademgordk shows that for a very wide class of finite groups (including, in particular, all nilpotent groups), the isomorphism problem is solved quickly.”

The leadership pays special attention to solving applied problems using machine learning methods and, in particular, deep learning. So, in early July, the Scientific Council of the Faculty of Mechanics and Mathematics of NSU (MMF NSU) decided to create a «Laboratory of Applied Digital Technologies» at the center.

The Mathematical Center in Akademgorodok will also serve as a «one stop shop» for communication with external stakeholders on the organization of research in the field of mathematics and its applications by scientific organizations of Akademgorodok. A cooperation agreement has already been signed with Huawei and three institutes (the Sobolev Institute of Mathematics, the Institute of Computational Mathematics and Mathematical Geophysics, the A.P. Ershov Institute of Informatics Systems). Currently, work is underway to create a consortium strategy that will position these institutes and NSU as a single whole with the Mathematical Center in Akademgorodok as a single platform for launching research and educational projects.

Education

Education is another major area of ​​activity of the Mathematical Center. In 2019, a number of pilot projects were launched as part of the development of undergraduate studies. A new program «Research Group of Fundamental Mathematics» has been opened at MMF NSU. The goal is to involve students in scientific research in the field of geometry, topology, number theory and other fundamental areas. What is especially valuable, the guys from the very first years will work on research projects under the guidance of leading Russian mathematicians and study courses that combine fundamental foundations and modern results in their respective fields.

The second project — «Engineering School» — consists of two profiles: «Artificial Intelligence» and «Applied Engineering». The main objective of the program is to train engineers of a new type. Students are waiting for mandatory modules on the management of scientific projects and start-up projects in the field of high technologies, culminating in the participation of students in specialized competitions and business accelerators of the Academpark. The training is based on modern engineering problems and work on real projects.

The third project, the launch of which is scheduled for September 1, 2020, is a master’s program in bioinformatics, which is being developed jointly with the Kurchatov Genomic Center. Together with the master’s program, it is also planned to create a joint laboratory that will support the master’s program and train master’s students on real tasks in genome research. Thus, it is planned to take a leading position in genomic research, which is a strategic priority for the development of science in the world.

As part of the Center’s activities in 2019, a significant transformation of the classical master’s programs in the areas of «Mathematics» and «Mathematics and Computer Science» was carried out. The English-language master’s program «Artificial Intelligence and Big Data Analytics» is also being continuously improved. Also, an English-language bachelor’s program is being prepared for launch in 2021 at the Faculty of Mechanics and Mathematics of NSU.

At the Mathematical Center in Akademgorodok, attention is also paid to additional education programs for both schoolchildren and experienced specialists. For example, by order of Vimpelcom PJSC, two programs «Introduction to Project-Based Machine Learning» and «Project-Based Machine Learning.