Investigative Skills Toolkit (Numeric) Student Task Sheet TI-Nspire Numeric Version

Size: px
Start display at page:

Download "Investigative Skills Toolkit (Numeric) Student Task Sheet TI-Nspire Numeric Version"

Transcription

1 Introduction Investigative Skills Toolkit (Numeric) Student Task Sheet TI-Nspire Numeric Version This activity will develop your ability to use the TI-Nspire Numeric Handheld to investigate and generalise results from numerical patterns. The TI-Nspire and Mathematical skills that will be developed are: defining functions using sigma notation generating sequences using statistical regression models to reveal algebraic generalisations. Mathematical Aim As a context for developing the above 4 skills, you will use them to find the algebraic formula for the sum: n You will then find the algebraic formula for the sum: n 2 and finally find the formula for: n 3 Note: There are many ways of going about deriving the formula for the sequence sums that are listed above. Some of these ways are quicker than the approach you will follow here. However, an important purpose of this task is to have you experience how a wide variety of technological skills can be brought together to work in powerful ways. Such techniques are able to be used even when the formulae being sought are not as well known as the above three sequence sums. Structure of Tasks Tasks 1, 2, 3 and 4 introduce each of the 4 individual skills, listed above. Task 5 merges these skills together in order to acheive the Mathematical Aim given above.

2 Task 1 Defining Functions A function has three things: a name, an input and an output. Create a New Document and insert a Calculator Page. Type in g(x):=x+2 and press. as shown on the right. Notice the colon in front of the equals sign. You can quickly obtain := by pressing / then t. This line reads as g of x is defined as equal to x plus two Now type g, and you will see that it is already displayed in bold. This means that the Nspire system knows that something named as g already exists in the current problem. Investigative Skills Toolkit (Numeric) Now continue to type in g(4) and press. You should be given the answer of 6. The function g took the input of 4, and after adding 2 to it, presented the output of 6. Another way of accessing the function of g is to press the h key. Do that now. Nice that it includes the brackets for you, isn t it? Now type g(-7) and before you press, see if you can correctly predict what it will display. You will see that the function g takes whatever it is given as an input, and generates an output that is simply the input plus two. Add a new calculator page by pressing / then ~ then Add Calculator Define a new function called h(x,y):=x+y 2, as shown on the right. This function has two inputs, and one output. Use the h key to type h(2,3) and before you press, see if you can correctly predict what it will display. Repeat for h(5,-4). Watch carefully what happens when you type in h(g,g) and be sure that you know why it gives the error message that it does when you press. Ask your teacher if you are unsure.

3 Add a new calculator page by pressing / then ~ then Add Calculator Define a new function called discrim, as shown below discrim(tom,dick,harry):=dick 2-4 tom harry This function has three inputs, and one output, and the variable names are no longer single letters. Notice the multiplication dots between 4 and tom and harry. Use the h key to type discrim(1,2,3) and before you press, see if you can correctly predict what it will display. Repeat for discrim(5,-2,7).

4 Task 2a Sigma Notation Investigative Skills Toolkit (Numeric) Σ is the Greek capital letter called sigma. A lowercase letter sigma looks like this: σ σ is used in Statistics to represent a measure of how spread out a data set is. Σ is used to mean the adding up of several terms. It requires three inputs : a lower bound, an upper bound and an algebraic expression. Here are two examples, which you shall shortly check on the Nspire. Don t type anything in yet. Example (i). 5 r 1 ( r ), which reads as the sum from r equals 1 to 5 of r This is a short-hand way of writing , which is equal to 15. Example (ii). 9 c 6 ( c 2 ), which reads as the sum from c equals 6 to 9 of c squared This is a short-hand way of writing , which is equal to 230. In each example expression, the letter under the sigma sign increases by one each time from the lower bound until it reaches the upper bound. For each value it takes on this journey, the algebraic expression is evaluated. You will now check the two results for examples (i) and (ii): Insert a new Problem by pressing: ~... Insert... Problem... Add Calculator. To obtain the sigma notation template, press t and select the sigma notation icon, as shown on the right. You will then have an empty template as shown, and the cursor is flashing in the region where you need to type in the variable will appear in your expression. Set about entering in the letters and numbers as shown in example (i), above, and then press. Repeat for example (ii). continued.../

5 You should now have a screen display like this: Investigative Skills Toolkit (Numeric) Now edit one of these expressions to generate the sigma notation to find the following sum: You should obtain the answer of 139. Can you now predict the values of each of the following summations, before you press? So, to finish, sigma notation can be used very effectively to add up lots of terms in a sequence for which you already know an algebraic expression for each term. You can consider the sigma notation to be a function with 3 inputs: the lower bound, the upper bound and a formula. It generates a single output of a value, or an expression, for the sum of all the terms it represents.

6 Task 2b (Extension) - Sigma Notation Equations You can use sigma notation with other Nspire functions... Consider the problem of determining the value of n such that n = 153. You need to solve for n, which may at first glance seem only to be able to be done by a method of trial and improvement. As can be seen on the right, the nsolve(...) command can be used. Find this by pressing: b... Algebra... Numerical Solve Consider another problem of determining the value of p such that 5 p +6 p +7 p +8 p = Before you look at the screenshot solution on the right, can you set up an nsolve(...) statement to obtain the value of p? Consider a final problem of finding out the sum of which 40 odd numbers equals a total of Before you look at the screenshot solution on the right, can you set up an nsolve(...) statement to obtain the answer to this puzzle? Hint: the 40 th odd number after the start is 39 odd numbers after the first odd number!

7 Task 3 Generating Sequences Investigative Skills Toolkit (Numeric) The sequence command can generate a sequence of terms, based upon a formula, a start value and end value. The default setting is that it increases its variable by one each time. You can hopefully appreciate that it has very similar syntax to the sigma notation. However, the sigma notation sums all terms in a sequence without letting you see them. The sequence command will show you all the terms, but not sum them. Insert a new Problem by pressing: ~... Insert... Problem... Add Calculator. The first example (left) generates the sequence of squared numbers from 1 2 to 4 2. The second example generates the sequence of cubed numbers from 4 3 to 6 3. Notice that the output of this command is not a single value, but a list of values, as denoted by the curly brackets, {...}. The syntax of the command is: seq( formula, variable in formula, start value, end value ) Type in the following commands, and see if you can predict what they will display, before you press. seq(4-k,k,2,7) seq(5a+3,a,19,25) seq(3 a,a,2,6) Now try to type the sequence commands that will display each of the following outputs: 1. the first five odd numbers 2. the sequence {75, 69, 63, 57, 51} 3. the sequence {4 5, 5 6, 6 7, 7 8 }... which will be displayed as {1024,15625,279936, }

8 Task 4 Using Statistical Regression Models When you obtain a sequence of numbers and you are not sure what formula generated them, it is often helpful to plot them on a graph. Shown below are typical graphs of the most commonly occuring types of formulae formula with n 2 in it formula with n 3 in it formula with n 4 in it As you can see, there are only subtle differences between them, so a bit of intelligent decision making is required when trying to apply these formulae. For a worked example, you shall create a sequence using a formula that you already know, plot it, and then use Statistical Regression to provide you back with the formula you used. This will act as a self-check process, in readiness for when you are using the technique to find an unknown formula. Insert a new Problem by pressing: ~... Insert... Problem... Add Lists & Spreadsheet. Then type into the very top left cell on the screen, above the greyed-out row the variable name n.value and press. Then move to the very top of column B, and type in term.value and press. Notice that you can use the decimal point as a valid character in the names of our variables, and it helps to make them more readable. You shall now generate the first 10 terms in the sequence whose n th term is given by 4n 2-3n As ever, there are many ways to do this... First, you shall generate the n.values from 1 to 10. You can use the sequence command for this: seq(n,n,1,10) To enter this command, move the cursor to the grey row, second from the top, in column A, and press =. The screen will then look like that shown on the right: Type in seq(n,n,1,10) and press. You should then see column A fill up with the numbers 1 to 10.

9 You shall repeat this process for column B, only this time the sequence command will be: seq(4n 2-3n,n,1,10) When you have done this, your screen should look like this: Now insert a new page with Data & Statistics, and click on where it says Click to add variable at the foot of the screen. Select n.value Move to the left of the screen, where it (used to) say Click to add variable and click again. Select term.value You should then have a screen looking like the one shown on the right. You already know that the formula for this sequence is a quadratic expression, but you would be forgiven for thinking it might be cubic or quartic if you didn t already know. Look back and compare this graph with the 3 screenshots at the very start of Task 1d. You will now instruct the TI-Nspire to fit the best formula it can to these plotted points. Press b... Analyze... Regression... Show Quadratic The regression formula will then be displayed as shown on the left. You were expecting 4n 2-3n and it has displayed something very similar to this, only with different variable names and surplus characters. This is to be expected when using Regression techniques. Move the cursor over the displayed equation and press / then b then select Remove Regression If you didn t know it was a Quadratic formula that you were after, you could have done: b... Analyze... Regression... Show Cubic or even b... Analyze... Regression... Show Quartic Do each of these now.

10 You will notice that when selecting Show Quartic, the displayed regression equation seems overly complex. The full equation looks like this (split up so that you can see each term more clearly): y= E 13 x 4 or, in a shortened form, E 12 x x x -4. E 11 y= 1.464E 13 x E 12 x x x -4. E 11 You can see something like y=4 x 2-3 x lurking in the middle of this expression, albeit in a slight disguise due to rounding errors from the regression process! And you have surplus terms which are very small indeed. For example, look at the coefficient of the x 4 term. It is 1.464E 13 This is the TI-Nspire s way of displaying the number Which is the number You can therefore conclude that the x 4 term is not there as its coefficient is such a small number. Similarly for the coefficient of x 3 which is And for the constant on the end, which is This leaves you with y= x x You can take this to mean y=4 x 2-3 x after adjusting for rounding errors. It is very important that you understand the above logical process, else you may make errors when interpreting results from Statistical Regression calculations. You can, of course, have the TI-Nspire help you with this interpretation process! You can instruct the regression formula to be displayed with slightly reduced accuracy, thereby revealing the most important terms. To do this, insert a new Calculator Page. Press h and select stat.results Note: all the other variables that start with stat. were automatically generated by the Statistical Regression process that you commanded it to perform.

11 Then press: b... Number... Approximate Fraction Investigative Skills Toolkit (Numeric) You will have a command line that reads as: stat.results approxfraction(5.e 14) Don t press yet! Edit the last characters to change it from 14 to 10: stat.results approxfraction(5.e 10) This instructs it to start rounding at about the 10 th decimal place, rather than the 14 th decimal place. The full answer that you will obtain is shown above, but you will have to scroll up and down to read it on the handheld. This display lists the values of each of the coefficients of a, b, c, d and e in the regression equation of y= a x 4 +b x 3 +c x 2 +d x +e The values of a, b and e are fractions with such large denominators, that they can be ignored, as their contribution will be so small. This leaves the value of c = 4 and d = -3 And so, you can now see more clearly the desired formula of y=4 x 2-3 x Statistical Regression Conclusion Statistical Regression is very powerful, but it requires careful interpretation to draw the correct conclusions. It is also worth remembering that its output is not proof that the formula works. It is merely the best approximation given the data that was processed and the type of regression equation that you were trying to fit to that data.

12 Task 4b - Statistical Regression Practice The previous example was based around the formula for the n th term to be 4n 2-3n. You will now change the setup so that the TI-Nspire system tries to detect the formula 5n 3 -n/7 Return to the Lists & Spreadsheet page and edit the seq(4n 2-3n,n,1,10) command in column B to read seq(5n 3 -n/7,n,1,10) All the other dependant pages will automatically update, and the new regression equation will be displayed on the graph. On the Calculator page you should re-execute the command line: stat.results approxfraction(5.e 10) If this does not obviously reveal the expression 5x 3 -x/7, then you will have to adjust the tolerance of the approxfraction command from 5.E 10 to something like 5.E 8 or 5.E 6 This adjustment is because of the fraction 1/7 that is in the original formula. This is further evidence that you must remain flexible when using Statistical Regression, as the complexity of the formula that you are seeking often affects the accuracy of the calculations. Task 4c - Find the Formula Tasks 4a and 4b were built around already knowing the generating formula. This is not really realistic, as why would you try to find out something you already knew?!? Consider the following sequence: 2, 14, 50, 130, 280 Your task will be to find a formula for the n th term. To do this, return to the Lists & Spreadsheets page and do the following: 1. move to the grey formula row of column B and press. to delete the sequence statement. 2. with the selected cell still in the grey formula row of Column B, press / then b then select Clear Data You should then have a screen similar to that shown on the right. Now, type in the 5 sequence values into column B, to give a screen similar to that shown on the right: You may have noticed that column A still has a sequence running from 1 to 10, but you do not have 10 numbers in column B. You will need to edit the formula in column A to generate only the numbers from 1 to 5. Do that now.

13 Move back to the Data & Statistics graph to verify the plausibility of fitting a quartic, cubic or quadratic equation to the data. You can choose to fit whichever type of equation you think best suits. Remember that if you opt for, say, a quartic, and the data is only quadratic, then you should expect the coefficients of the x 4 and x 3 terms to be neglible. However if you opt for a quadratic and the data is quartic, then the regression line will not fit the data well and the coefficients in the equation will most likely be ugly Once you have completed this process, and chosen the required amount of accuracy for the approxfraction command, compare your result to the screen on the right. The display still looks unappealing with all the fractional coefficients, but it s really just saying: which can be tidied up by manual techniques as follows You should now check that this formula does indeed generate the first 5 terms that you were given - why not use the factorised form that is given above, and work through the evaluation process on paper, without using the TI-Nspire to help you! If you encounter difficulty, have your teacher show you how to process the numerical calculation with the most efficient ease. To practice the above process, edit the data on your TI-Nspire to help you identify the formulae for the n th term for each of the following two sequences: a) 0, 4, 22, 70, 170 b) 0, 3, 11, 26, 50

14 Task 5a Combining Formulae + Sigma Notation + Sequences + Statistical Regression Insert a new Problem by pressing: ~... Insert... Problem... Add Calculator. Enter both of the statements as shown on the right, which define a variable called power and a function called total. As the variable power equals 1, if you were to enter in total(5), it would calculate the sum Try it now. You will be able to change the value of power to other numbers later on, and therefore all the calculations that you will design in the next few steps will be automatically updated. Now define a variable called terms:=6 (see right) Entering the command total(terms) ought to calculate the sum Try it now. You are now going to use all the techniques learnt so far to find the formula for n 1. Insert a new Lists & Spreadsheet page. 2. Type into the very top left cell on the screen, above the greyed-out row the variable name n.value and press. 3. Move to the very top of column B, and type in term.value and press. 4. Move the cursor to the formula row in column A, and press =. 5. Type in seq(r,r,1,terms) and press. 6. Move the cursor to the formula row in column B, and press =. 7. Type in seq(total(r),r,1,terms) and press. You should have a screen that now looks like the one given below: The purpose of using the variable terms, makes sure that both lists have the number number of terms in them. You should now: 1. insert a Data & Statistics page 2. plot the data 3. fit an appropriate regression line 4. insert a Calculator page 5. if required, use the approxfraction command to extract the desired formula from the regression equation (stat.regeqn)

15 Compare your results to the screenshot on the right. Investigative Skills Toolkit (Numeric) Using the quartic regression approach, you can disregard the values for a, b and e as they are so small. This leaves you with: Task 5b Sum of the first n Square Numbers As declared at the outset of this whole activity, you will now attempt to find the formula for n 2 This requires very little extra work, as you have already constructed a fully dynamic system of pages in the current problem. 1. Return to the calculator page where you defined power to equal Now redefine power:=2 3. This will automatically update the sequences and data plots. 4. You may have to try a different type of regression function (quartic/cubic/quadratic), depending upon what you previously used. 5. Use the approxfraction command and some manual factorising to obtain the nicest form for the formula. 6. Check your factorised formula by evaluating it for the first 5 terms, or so. Task 5c Sum of the first n Cube Numbers Finally, you will find the formula for n 3 1. Repeat the above 6 step process, redefining power:=3 Task 5d (Extension) - Sum of the first n Quartic Numbers It is possible to attempt to find the formula for n 4 However, you need to try it yourself to discover what happens. Can you find a way of using Statistical Regression to obtain a working formula? Can you identify the feature of this last challenge that causes the whole statistical regression process on the TI-Nspire to falter?

Standardized Tests: Best Practices for the TI-Nspire CX

Standardized Tests: Best Practices for the TI-Nspire CX The role of TI technology in the classroom is intended to enhance student learning and deepen understanding. However, efficient and effective use of graphing calculator technology on high stakes tests

More information

(Refer Slide Time 04:53)

(Refer Slide Time 04:53) Programming and Data Structure Dr.P.P.Chakraborty Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Lecture 26 Algorithm Design -1 Having made a preliminary study

More information

Number of Regions An Introduction to the TI-Nspire CAS Student Worksheet Created by Melissa Sutherland, State University of New York at Geneseo

Number of Regions An Introduction to the TI-Nspire CAS Student Worksheet Created by Melissa Sutherland, State University of New York at Geneseo 1 Name Number of Regions An Introduction to the TI-Nspire CAS Student Worksheet Created by Melissa Sutherland, State University of New York at Geneseo Below you will find copies of the notes provided on

More information

Maximizing the Area of a Garden

Maximizing the Area of a Garden Math Objectives Students will determine the relationship between the width and length of a garden with a rectangular shape and a fixed amount of fencing. The garden is attached to a barn, and exactly three

More information

Using Excel This is only a brief overview that highlights some of the useful points in a spreadsheet program.

Using Excel This is only a brief overview that highlights some of the useful points in a spreadsheet program. Using Excel 2007 This is only a brief overview that highlights some of the useful points in a spreadsheet program. 1. Input of data - Generally you should attempt to put the independent variable on the

More information

Computational Mathematics/Information Technology. Worksheet 2 Iteration and Excel

Computational Mathematics/Information Technology. Worksheet 2 Iteration and Excel Computational Mathematics/Information Technology Worksheet 2 Iteration and Excel This sheet uses Excel and the method of iteration to solve the problem f(x) = 0. It introduces user functions and self referencing

More information

Note that ALL of these points are Intercepts(along an axis), something you should see often in later work.

Note that ALL of these points are Intercepts(along an axis), something you should see often in later work. SECTION 1.1: Plotting Coordinate Points on the X-Y Graph This should be a review subject, as it was covered in the prerequisite coursework. But as a reminder, and for practice, plot each of the following

More information

Activity Overview A basic introduction to the many features of the calculator function of the TI-Nspire

Activity Overview A basic introduction to the many features of the calculator function of the TI-Nspire TI-Nspire Activity: An Introduction to the TI-Nspire Calculator Function By: Leigh T Baker Activity Overview A basic introduction to the many features of the calculator function of the TI-Nspire Concepts

More information

Macmillan TI-Nspire supplement

Macmillan TI-Nspire supplement February 2011 Contents Changes to the hardware the touchpad... 1 Changes to the software OS version 2... 2 The Home screen... 2 Page layout... 4 Calculator page menu changes... 5 Systems of equations...

More information

Mathematics Background

Mathematics Background Finding Area and Distance Students work in this Unit develops a fundamentally important relationship connecting geometry and algebra: the Pythagorean Theorem. The presentation of ideas in the Unit reflects

More information

Ingredients of Change: Nonlinear Models

Ingredients of Change: Nonlinear Models Chapter 2 Ingredients of Change: Nonlinear Models 2.1 Exponential Functions and Models As we begin to consider functions that are not linear, it is very important that you be able to draw scatter plots,

More information

What s Half of a Half of a Half?

What s Half of a Half of a Half? Overview Activity ID: 8606 Math Concepts Materials Students will use a physical model to determine what happens fractions TI-0XS when they repeatedly halve a piece of paper, and then they decimals MultiView

More information

DOWNLOAD PDF MICROSOFT EXCEL ALL FORMULAS LIST WITH EXAMPLES

DOWNLOAD PDF MICROSOFT EXCEL ALL FORMULAS LIST WITH EXAMPLES Chapter 1 : Examples of commonly used formulas - Office Support A collection of useful Excel formulas for sums and counts, dates and times, text manipularion, conditional formatting, percentages, Excel

More information

Implicit Function Explorations

Implicit Function Explorations Activities with Implicit Functions and Implicit Differentiation on the TI-89/Voyage 00 Dennis Pence Western Michigan University Kalamazoo, Michigan USA Abstract: Unfortunately the topic of implicit differentiation

More information

Algebra 2 Common Core Summer Skills Packet

Algebra 2 Common Core Summer Skills Packet Algebra 2 Common Core Summer Skills Packet Our Purpose: Completion of this packet over the summer before beginning Algebra 2 will be of great value to helping students successfully meet the academic challenges

More information

Some (semi-)advanced tips for LibreOffice

Some (semi-)advanced tips for LibreOffice Some (semi-)advanced tips for LibreOffice by Andy Pepperdine Introduction We cover several tips on special things in Writer and Calc and anything else that turns up. Although I use LibreOffice, these should

More information

First Steps. with. Technology for Learning Maths & Science. Seven introductory tutorials for teachers

First Steps. with. Technology for Learning Maths & Science. Seven introductory tutorials for teachers First Steps with Technology for Learning Maths & Science Seven introductory tutorials for teachers About this document This document includes seven introductory tutorials to take new users through many

More information

PR3 & PR4 CBR Activities Using EasyData for CBL/CBR Apps

PR3 & PR4 CBR Activities Using EasyData for CBL/CBR Apps Summer 2006 I2T2 Process Page 23. PR3 & PR4 CBR Activities Using EasyData for CBL/CBR Apps The TI Exploration Series for CBR or CBL/CBR books, are all written for the old CBL/CBR Application. Now we can

More information

The first thing we ll need is some numbers. I m going to use the set of times and drug concentration levels in a patient s bloodstream given below.

The first thing we ll need is some numbers. I m going to use the set of times and drug concentration levels in a patient s bloodstream given below. Graphing in Excel featuring Excel 2007 1 A spreadsheet can be a powerful tool for analyzing and graphing data, but it works completely differently from the graphing calculator that you re used to. If you

More information

Binary, Hexadecimal and Octal number system

Binary, Hexadecimal and Octal number system Binary, Hexadecimal and Octal number system Binary, hexadecimal, and octal refer to different number systems. The one that we typically use is called decimal. These number systems refer to the number of

More information

1

1 Zeros&asymptotes Example 1 In an early version of this activity I began with a sequence of simple examples (parabolas and cubics) working gradually up to the main idea. But now I think the best strategy

More information

Excel Primer CH141 Fall, 2017

Excel Primer CH141 Fall, 2017 Excel Primer CH141 Fall, 2017 To Start Excel : Click on the Excel icon found in the lower menu dock. Once Excel Workbook Gallery opens double click on Excel Workbook. A blank workbook page should appear

More information

EXCEL PRACTICE 5: SIMPLE FORMULAS

EXCEL PRACTICE 5: SIMPLE FORMULAS EXCEL PRACTICE 5: SIMPLE FORMULAS SKILLS REVIEWED: Simple formulas Printing with and without formulas Footers Widening a column Putting labels and data in Bold. PART 1 - DIRECTIONS 1. Open a new spreadsheet

More information

Introducing the new Casio fx-9860g AU

Introducing the new Casio fx-9860g AU Introducing the new Casio fx-9860g AU by Peter Karamoshos and Alastair Lupton. Introduction For students, teachers and schools who currently use the CFX-9850 GB PLUS, these new machines are very exciting.

More information

Algebra 2 Semester 1 (#2221)

Algebra 2 Semester 1 (#2221) Instructional Materials for WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the 2016-2017 Course Guides for the following course: Algebra 2 Semester

More information

L E A R N I N G O B JE C T I V E S

L E A R N I N G O B JE C T I V E S 2.2 Measures of Central Location L E A R N I N G O B JE C T I V E S 1. To learn the concept of the center of a data set. 2. To learn the meaning of each of three measures of the center of a data set the

More information

Introduction to Sequence

Introduction to Sequence for Version 3 Basic Edition Lesson 18 Introduction to Sequence Welcome What do you think of when you hear the word sequence? Do you think of a set of numbers listed in a special order? If we can find a

More information

Inequalities. Kim Mullins Grade level: 5-8 Subject: mathematics Time required: 90 minutes

Inequalities. Kim Mullins Grade level: 5-8 Subject: mathematics Time required: 90 minutes Kim Mullins minutes Activity overview This activity is designed to introduce students to the concept of inequalities. Students will discover how to graph inequalities on a number line. Concepts Inequalities

More information

Learning Objectives: Topic Karnaugh Maps. At the end of this topic you will be able to;

Learning Objectives: Topic Karnaugh Maps. At the end of this topic you will be able to; Topic.2.3 Karnaugh Maps Learning Objectives: t the end of this topic you will be able to; Draw a Karnaugh map for a logic system with up to four inputs and use it to minimise the number of gates required;

More information

Linear and Quadratic Least Squares

Linear and Quadratic Least Squares Linear and Quadratic Least Squares Prepared by Stephanie Quintal, graduate student Dept. of Mathematical Sciences, UMass Lowell in collaboration with Marvin Stick Dept. of Mathematical Sciences, UMass

More information

Teacher Activity: page 1/9 Mathematical Expressions in Microsoft Word

Teacher Activity: page 1/9 Mathematical Expressions in Microsoft Word Teacher Activity: page 1/9 Mathematical Expressions in Microsoft Word These instructions assume that you are familiar with using MS Word for ordinary word processing *. If you are not comfortable entering

More information

Teacher Notes. Introduction. Resources. The first lesson. Diameters of Polygons

Teacher Notes. Introduction. Resources. The first lesson. Diameters of Polygons Introduction Teacher Notes It s not only circles that have diameters: the OED defines a diameter as a straight line passing from side to side through the centre of a body or figure, especially a circle

More information

QUICK EXCEL TUTORIAL. The Very Basics

QUICK EXCEL TUTORIAL. The Very Basics QUICK EXCEL TUTORIAL The Very Basics You Are Here. Titles & Column Headers Merging Cells Text Alignment When we work on spread sheets we often need to have a title and/or header clearly visible. Merge

More information

Why Use Graphs? Test Grade. Time Sleeping (Hrs) Time Sleeping (Hrs) Test Grade

Why Use Graphs? Test Grade. Time Sleeping (Hrs) Time Sleeping (Hrs) Test Grade Analyzing Graphs Why Use Graphs? It has once been said that a picture is worth a thousand words. This is very true in science. In science we deal with numbers, some times a great many numbers. These numbers,

More information

Learning Packet THIS BOX FOR INSTRUCTOR GRADING USE ONLY. Mini-Lesson is complete and information presented is as found on media links (0 5 pts)

Learning Packet THIS BOX FOR INSTRUCTOR GRADING USE ONLY. Mini-Lesson is complete and information presented is as found on media links (0 5 pts) Learning Packet Student Name Due Date Class Time/Day Submission Date THIS BOX FOR INSTRUCTOR GRADING USE ONLY Mini-Lesson is complete and information presented is as found on media links (0 5 pts) Comments:

More information

STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I. 4 th Nine Weeks,

STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I. 4 th Nine Weeks, STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I 4 th Nine Weeks, 2016-2017 1 OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for

More information

Exemplar for Internal Achievement Standard. Mathematics and Statistics Level 1

Exemplar for Internal Achievement Standard. Mathematics and Statistics Level 1 Exemplar for Internal Achievement Standard Mathematics and Statistics Level 1 This exemplar supports assessment against: Achievement Standard (2.2) Apply graphical methods in solving problems An annotated

More information

Year 8 Mathematics Curriculum Map

Year 8 Mathematics Curriculum Map Year 8 Mathematics Curriculum Map Topic Algebra 1 & 2 Number 1 Title (Levels of Exercise) Objectives Sequences *To generate sequences using term-to-term and position-to-term rule. (5-6) Quadratic Sequences

More information

MAT 003 Brian Killough s Instructor Notes Saint Leo University

MAT 003 Brian Killough s Instructor Notes Saint Leo University MAT 003 Brian Killough s Instructor Notes Saint Leo University Success in online courses requires self-motivation and discipline. It is anticipated that students will read the textbook and complete sample

More information

U4L4B Box Problem - TI Nspire CAS Teacher Notes

U4L4B Box Problem - TI Nspire CAS Teacher Notes U4L4B Box Problem - TI Nspire CAS Teacher Notes You are provided with a sheet of metal that measures 80 cm by 60 cm. If you cut congruent squares from each corner, you are left with a rectangle in the

More information

Module Four: Formulas and Functions

Module Four: Formulas and Functions Page 4.1 Module Four: Formulas and Functions Welcome to the fourth lesson in the PRC s Excel Spreadsheets Course 1. This lesson concentrates on adding formulas and functions to spreadsheet to increase

More information

Detailed instructions for video analysis using Logger Pro.

Detailed instructions for video analysis using Logger Pro. Detailed instructions for video analysis using Logger Pro. 1. Begin by locating or creating a video of a projectile (or any moving object). Save it to your computer. Most video file types are accepted,

More information

Civil Engineering Computation

Civil Engineering Computation Civil Engineering Computation First Steps in VBA Homework Evaluation 2 1 Homework Evaluation 3 Based on this rubric, you may resubmit Homework 1 and Homework 2 (along with today s homework) by next Monday

More information

Excel Basic: Create Formulas

Excel Basic: Create Formulas Better Technology, Onsite and Personal Connecting NIOGA s Communities www.btopexpress.org www.nioga.org [Type Excel Basic: Create Formulas Overview: Let Excel do your math for you! After an introduction

More information

number Understand the equivalence between recurring decimals and fractions

number Understand the equivalence between recurring decimals and fractions number Understand the equivalence between recurring decimals and fractions Using and Applying Algebra Calculating Shape, Space and Measure Handling Data Use fractions or percentages to solve problems involving

More information

NUMBER 1 ALGEBRA 1 AUTUMN TERM YEAR 7

NUMBER 1 ALGEBRA 1 AUTUMN TERM YEAR 7 NUMBER 1 Know what even numbers, odd numbers, factors, multiples, primes, squares and square roots are and how to find them. Find the Highest Common Factor by listing factors and/or using Venn diagrams.

More information

Spreadsheet View and Basic Statistics Concepts

Spreadsheet View and Basic Statistics Concepts Spreadsheet View and Basic Statistics Concepts GeoGebra 3.2 Workshop Handout 9 Judith and Markus Hohenwarter www.geogebra.org Table of Contents 1. Introduction to GeoGebra s Spreadsheet View 2 2. Record

More information

The Real Number System

The Real Number System The Real Number System Pi is probably one of the most famous numbers in all of history. As a decimal, it goes on and on forever without repeating. Mathematicians have already calculated trillions of the

More information

Excel Basics: Working with Spreadsheets

Excel Basics: Working with Spreadsheets Excel Basics: Working with Spreadsheets E 890 / 1 Unravel the Mysteries of Cells, Rows, Ranges, Formulas and More Spreadsheets are all about numbers: they help us keep track of figures and make calculations.

More information

Excel Basics Fall 2016

Excel Basics Fall 2016 If you have never worked with Excel, it can be a little confusing at first. When you open Excel, you are faced with various toolbars and menus and a big, empty grid. So what do you do with it? The great

More information

Intro To Excel Spreadsheet for use in Introductory Sciences

Intro To Excel Spreadsheet for use in Introductory Sciences INTRO TO EXCEL SPREADSHEET (World Population) Objectives: Become familiar with the Excel spreadsheet environment. (Parts 1-5) Learn to create and save a worksheet. (Part 1) Perform simple calculations,

More information

Here is the data collected.

Here is the data collected. Introduction to Scientific Analysis of Data Using Spreadsheets. Computer spreadsheets are very powerful tools that are widely used in Business, Science, and Engineering to perform calculations and record,

More information

The O key will power the unit on. To turn the unit off, press the yellow L key, then O key.

The O key will power the unit on. To turn the unit off, press the yellow L key, then O key. fx-9860gii Quick Reference Guide Selecting the RUN Q icon will allow you to perform general computations and arithmetic. The function keys allow you to access the tab (soft key) menus that will come up

More information

1.4. Skills You Need: Working With Radicals. Investigate

1.4. Skills You Need: Working With Radicals. Investigate 1.4 1 Skills You Need: Working With Radicals 1 2 2 5 The followers of the Greek mathematician Pythagoras discovered values that did not correspond to any of the rational numbers. As a result, a new type

More information

The Mailbox MATH NSPIRED. Math Objectives. About the Lesson. TI-Nspire Navigator System. Activity Materials TEACHER NOTES

The Mailbox MATH NSPIRED. Math Objectives. About the Lesson. TI-Nspire Navigator System. Activity Materials TEACHER NOTES Math Objectives Students will use volume formulas to solve problems. Students will recognize that the area of a composite shape is the sum of the areas of its smaller parts. Students will determine the

More information

Quadratic Equations Group Acitivity 3 Business Project Week #5

Quadratic Equations Group Acitivity 3 Business Project Week #5 MLC at Boise State 013 Quadratic Equations Group Acitivity 3 Business Project Week #5 In this activity we are going to further explore quadratic equations. We are going to analyze different parts of the

More information

Notebook Assignments

Notebook Assignments Notebook Assignments These six assignments are a notebook using techniques from class in the single concrete context of graph theory. This is supplemental to your usual assignments, and is designed for

More information

Year 8 Key Performance Indicators Maths (Number)

Year 8 Key Performance Indicators Maths (Number) Key Performance Indicators Maths (Number) M8.1 N1: I can solve problems by adding, subtracting, multiplying and dividing decimals. Use correct notation for recurring decimals, know the denominators of

More information

SCHEME OF WORK YR 8 THETA 2 UNIT / LESSON

SCHEME OF WORK YR 8 THETA 2 UNIT / LESSON SCHEME OF WORK YR 8 THETA 2 UNIT / LESSON STEPS STEPS FROM TO 1 Number 4th 6th OBJECTIVES 1.1 Calculations 5th 5th Use written methods to add and subtract with decimals. Calculate mentally. Calculate with

More information

SLStats.notebook. January 12, Statistics:

SLStats.notebook. January 12, Statistics: Statistics: 1 2 3 Ways to display data: 4 generic arithmetic mean sample 14A: Opener, #3,4 (Vocabulary, histograms, frequency tables, stem and leaf) 14B.1: #3,5,8,9,11,12,14,15,16 (Mean, median, mode,

More information

FSA Algebra 1 EOC Practice Test Guide

FSA Algebra 1 EOC Practice Test Guide FSA Algebra 1 EOC Practice Test Guide This guide serves as a walkthrough of the Algebra 1 EOC practice test. By reviewing the steps listed below, you will have a better understanding of the test functionalities,

More information

Level 4 means that I can

Level 4 means that I can Level 4 means that I can Describe number patterns Find multiples Find factors Work out the square numbers Use word formulae Use co-ordinates in the first quadrant Multiply and divide whole numbers by 10

More information

I can solve simultaneous equations algebraically, where one is quadratic and one is linear.

I can solve simultaneous equations algebraically, where one is quadratic and one is linear. A* I can manipulate algebraic fractions. I can use the equation of a circle. simultaneous equations algebraically, where one is quadratic and one is linear. I can transform graphs, including trig graphs.

More information

Characteristics of Exponential Functions

Characteristics of Exponential Functions Math Objectives Students will identify the characteristics of exponential functions of the form f(x) = b x, where b > 1. Students will identify the characteristics of exponential functions of the form

More information

9. MATHEMATICIANS ARE FOND OF COLLECTIONS

9. MATHEMATICIANS ARE FOND OF COLLECTIONS get the complete book: http://wwwonemathematicalcatorg/getfulltextfullbookhtm 9 MATHEMATICIANS ARE FOND OF COLLECTIONS collections Collections are extremely important in life: when we group together objects

More information

The method of rationalizing

The method of rationalizing Roberto s Notes on Differential Calculus Chapter : Resolving indeterminate forms Section The method of rationalizing What you need to know already: The concept of it and the factor-and-cancel method of

More information

Getting Started with the TI- Nspire CX

Getting Started with the TI- Nspire CX Getting Started with the TI- Nspire CX Mr. Dana F. Morse dmorse@ti.com New York State Educational Technology Consultant Texas Instruments TI- Nspire CX for Beginners 1 Undo Like a Computer Touchpad Swipe

More information

Lesson 76. Linear Regression, Scatterplots. Review: Shormann Algebra 2, Lessons 12, 24; Shormann Algebra 1, Lesson 94

Lesson 76. Linear Regression, Scatterplots. Review: Shormann Algebra 2, Lessons 12, 24; Shormann Algebra 1, Lesson 94 Lesson 76 Linear Regression, Scatterplots Review: Shormann Algebra 2, Lessons 12, 24; Shormann Algebra 1, Lesson 94 Tools required: A graphing calculator or some sort of spreadsheet program, like Excel

More information

Lab1: Use of Word and Excel

Lab1: Use of Word and Excel Dr. Fritz Wilhelm; physics 230 Lab1: Use of Word and Excel Page 1 of 9 Lab partners: Download this page onto your computer. Also download the template file which you can use whenever you start your lab

More information

Lesson 6: Manipulating Equations

Lesson 6: Manipulating Equations Lesson 6: Manipulating Equations Manipulating equations is probably one of the most important skills to master in a high school physics course. Although it is based on familiar (and fairly simple) math

More information

Science, Technology, Engineering & Maths (STEM) Calculator Guide Texas Instruments TI-Nspire Handheld (Touchpad with Operating System 3.

Science, Technology, Engineering & Maths (STEM) Calculator Guide Texas Instruments TI-Nspire Handheld (Touchpad with Operating System 3. Science, Technology, Engineering & Maths (STEM) Calculator Guide Texas Instruments TI-Nspire Handheld (Touchpad with Operating System 3.2) This short guide is designed to instruct you on how to set up

More information

Sample some Pi Monte. Introduction. Creating the Simulation. Answers & Teacher Notes

Sample some Pi Monte. Introduction. Creating the Simulation. Answers & Teacher Notes Sample some Pi Monte Answers & Teacher Notes 7 8 9 10 11 12 TI-Nspire Investigation Student 45 min Introduction The Monte-Carlo technique uses probability to model or forecast scenarios. In this activity

More information

STANDARDS OF LEARNING CONTENT REVIEW NOTES. ALGEBRA I Part II. 3 rd Nine Weeks,

STANDARDS OF LEARNING CONTENT REVIEW NOTES. ALGEBRA I Part II. 3 rd Nine Weeks, STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I Part II 3 rd Nine Weeks, 2016-2017 1 OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource

More information

ADDING FRACTIONS Teacher Notes

ADDING FRACTIONS Teacher Notes ADDING FRACTIONS Teacher Notes References Foundation Foundation Plus Higher Higher Plus Introduction N3.1 Fractions N6.1 Adding and subtracting fractions N2.1 Adding and subtracting fractions N1.5 Fraction

More information

Y7 Learning Stage 1. Y7 Learning Stage 2. Y7 Learning Stage 3

Y7 Learning Stage 1. Y7 Learning Stage 2. Y7 Learning Stage 3 Y7 Learning Stage 1 Y7 Learning Stage 2 Y7 Learning Stage 3 Understand simple algebraic notation. Collect like terms to simplify algebraic expressions. Use coordinates in the first quadrant. Make a comparative

More information

Exploring Fractals through Geometry and Algebra. Kelly Deckelman Ben Eggleston Laura Mckenzie Patricia Parker-Davis Deanna Voss

Exploring Fractals through Geometry and Algebra. Kelly Deckelman Ben Eggleston Laura Mckenzie Patricia Parker-Davis Deanna Voss Exploring Fractals through Geometry and Algebra Kelly Deckelman Ben Eggleston Laura Mckenzie Patricia Parker-Davis Deanna Voss Learning Objective and skills practiced Students will: Learn the three criteria

More information

Introduction to Modular Arithmetic

Introduction to Modular Arithmetic Randolph High School Math League 2014-2015 Page 1 1 Introduction Introduction to Modular Arithmetic Modular arithmetic is a topic residing under Number Theory, which roughly speaking is the study of integers

More information

Using MathType for the Worksheets: Word 2003 edition

Using MathType for the Worksheets: Word 2003 edition Haberman Online MTH 112 Using MathType for the Worksheets: Word 2003 edition As explained in various locations inside our course, you are required to use MathType to compose all of the mathematics in your

More information

Maths Homework Booklet Term 2

Maths Homework Booklet Term 2 Maths Homework Booklet Term 2 Each week every student will be set piece of homework. This will be set on the first lesson of the week and due in on the last lesson of the week. Should students want to

More information

Course Number 432/433 Title Algebra II (A & B) H Grade # of Days 120

Course Number 432/433 Title Algebra II (A & B) H Grade # of Days 120 Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number

More information

Chapter 1 Histograms, Scatterplots, and Graphs of Functions

Chapter 1 Histograms, Scatterplots, and Graphs of Functions Chapter 1 Histograms, Scatterplots, and Graphs of Functions 1.1 Using Lists for Data Entry To enter data into the calculator you use the statistics menu. You can store data into lists labeled L1 through

More information

I can solve simultaneous equations algebraically and graphically. I can solve inequalities algebraically and graphically.

I can solve simultaneous equations algebraically and graphically. I can solve inequalities algebraically and graphically. B I can factorise and expand complex expressions. I can factorise Quadratics I can recognise the Difference of Two Squares (D.O.T.S) simultaneous equations algebraically and graphically. inequalities algebraically

More information

Spreadsheet Concepts Using Microsoft Excel

Spreadsheet Concepts Using Microsoft Excel Spreadsheet Concepts Using Microsoft Excel lab 5 Objectives: Upon successful completion of Lab 5, you will be able to Create and edit a simple spreadsheet document Describe the advantage of using formulas

More information

Problem One: A Quick Algebra Review

Problem One: A Quick Algebra Review CS103A Winter 2019 Solutions for Week One Handout 01S Problem One: A Quick Algebra Review In the first week of CS103, we'll be doing a few proofs that will require some algebraic manipulations and reasoning

More information

Tips to Save Typing, etc.

Tips to Save Typing, etc. MATH 110 Dr. Stoudt Using Your TI-nspire CAS Guidebooks for all Texas Instruments calculators can be downloaded (in Adobe PDF format) from http://education.ti.com/en/us/guidebook/search Just search for

More information

Excel 2016: Part 1. Updated January 2017 Copy cost: $1.50

Excel 2016: Part 1. Updated January 2017 Copy cost: $1.50 Excel 2016: Part 1 Updated January 2017 Copy cost: $1.50 Getting Started Please note that you are required to have some basic computer skills for this class. Also, any experience with Microsoft Word is

More information

Introduction to Spreadsheets Part 1. The Quick and Easy guide to using Openoffice Calc

Introduction to Spreadsheets Part 1. The Quick and Easy guide to using Openoffice Calc Introduction to Spreadsheets Part 1 The Quick and Easy guide to using Openoffice Calc By the end of the lesson, you will be able to say I know what a spreadsheet is, I can enter simple data into a spreadsheet,

More information

NUMBERS AND NUMBER RELATIONSHIPS

NUMBERS AND NUMBER RELATIONSHIPS MODULE MODULE CHAPTERS Numbers and number patterns 2 Money matters KEY SKILLS writing rational numbers as terminating or recurring decimals identifying between which two integers any irrational number

More information

In this section you will learn some simple data entry, editing, formatting techniques and some simple formulae. Contents

In this section you will learn some simple data entry, editing, formatting techniques and some simple formulae. Contents In this section you will learn some simple data entry, editing, formatting techniques and some simple formulae. Contents Section Topic Sub-topic Pages Section 2 Spreadsheets Layout and Design S2: 2 3 Formulae

More information

MEI GeoGebra Tasks for AS Pure

MEI GeoGebra Tasks for AS Pure Task 1: Coordinate Geometry Intersection of a line and a curve 1. Add a quadratic curve, e.g. y = x 2 4x + 1 2. Add a line, e.g. y = x 3 3. Use the Intersect tool to find the points of intersection of

More information

KS3 MATHEMATICS THRESHOLD DESCRIPTORS NUMBER (Incl. RATIO & PROPORTION)

KS3 MATHEMATICS THRESHOLD DESCRIPTORS NUMBER (Incl. RATIO & PROPORTION) KS3 MATHEMATICS THRESHOLD DESCRIPTORS NUMBER (Incl. RATIO & PROPORTION) Topic Integers Decimals Approximation Fractions Concepts and skills Read, write, order and compare positive integers up to 1000 Add

More information

Maths PoS: Year 7 HT1. Students will colour code as they work through the scheme of work. Students will learn about Number and Shape

Maths PoS: Year 7 HT1. Students will colour code as they work through the scheme of work. Students will learn about Number and Shape Maths PoS: Year 7 HT1 Students will learn about Number and Shape Number: Use positive and negative numbers in context and position them on a number line. Recall quickly multiplication facts up to 10 10

More information

Implementing an Algorithm for Boomerang Fraction Sequences in Python

Implementing an Algorithm for Boomerang Fraction Sequences in Python Introduction Implementing an Algorithm for Boomerang Fraction Sequences in Python We ve all encountered maze problems, where the challenge is to find a path through a labyrinth from a starting point to

More information

Graphing Calculator How To Packet

Graphing Calculator How To Packet Graphing Calculator How To Packet The following outlines some of the basic features of your TI Graphing Calculator. The graphing calculator is a useful tool that will be used extensively in this class

More information

Year 9: Long term plan

Year 9: Long term plan Year 9: Long term plan Year 9: Long term plan Unit Hours Powerful procedures 7 Round and round 4 How to become an expert equation solver 6 Why scatter? 6 The construction site 7 Thinking proportionally

More information

CAMBRIDGE TECHNOLOGY IN MATHS Year 12 ClassPad user guide

CAMBRIDGE TECHNOLOGY IN MATHS Year 12 ClassPad user guide Year 12 ClassPad User guide Page 1 of 12 CAMBRIDGE TECHNOLOGY IN MATHS Year 12 ClassPad user guide CONTENTS Introduction to the ClassPad 330 2 Using the Main menu 4 Graphing 8 Defining functions 9 Probability

More information

Route Map (Start September 2012) Year 9

Route Map (Start September 2012) Year 9 Route Map (Start September 2012) Year 9 3 th 7 th Sept 10 th -14 th Sept 17 th 21 st Sept 24 th 28 th Sept 1 st -5 th Oct 8 th -12 th Oct 15 th 19 th Oct 22 nd -26 th Oct 29 th Oct-2 nd Nov 5 th -9 th

More information

Your Name: Section: INTRODUCTION TO STATISTICAL REASONING Computer Lab #4 Scatterplots and Regression

Your Name: Section: INTRODUCTION TO STATISTICAL REASONING Computer Lab #4 Scatterplots and Regression Your Name: Section: 36-201 INTRODUCTION TO STATISTICAL REASONING Computer Lab #4 Scatterplots and Regression Objectives: 1. To learn how to interpret scatterplots. Specifically you will investigate, using

More information

Name: Dr. Fritz Wilhelm Lab 1, Presentation of lab reports Page # 1 of 7 5/17/2012 Physics 120 Section: ####

Name: Dr. Fritz Wilhelm Lab 1, Presentation of lab reports Page # 1 of 7 5/17/2012 Physics 120 Section: #### Name: Dr. Fritz Wilhelm Lab 1, Presentation of lab reports Page # 1 of 7 Lab partners: Lab#1 Presentation of lab reports The first thing we do is to create page headers. In Word 2007 do the following:

More information

Year 10 General Mathematics Unit 2

Year 10 General Mathematics Unit 2 Year 11 General Maths Year 10 General Mathematics Unit 2 Bivariate Data Chapter 4 Chapter Four 1 st Edition 2 nd Edition 2013 4A 1, 2, 3, 4, 6, 7, 8, 9, 10, 11 1, 2, 3, 4, 6, 7, 8, 9, 10, 11 2F (FM) 1,

More information