Using the TI-84 Plus Graphing Calculator in Middle School ... · Using the TI-84 Plus Graphing...

41
Using the TI-84 Plus Graphing Calculator in Middle School Mathematics Aldine ISD March 27, 2010 Shelley Bolen-Abbott Region 4 Education Service Center [email protected] 713.744.6521 Handouts will be available online until 04.12.10 http://www.theansweris4.net -Click on Departments -Click on Mathematics Services -Click on Professional Development Materials Permission to copy classroom-ready materials granted to attendees of this Region 4 session. © Region 4 Education Service Center. All rights reserved. http://www.twitter.com/R4math

Transcript of Using the TI-84 Plus Graphing Calculator in Middle School ... · Using the TI-84 Plus Graphing...

Using the TI-84 Plus Graphing Calculator in Middle School

Mathematics Aldine ISD

March 27, 2010

Shelley Bolen-Abbott Region 4 Education Service Center

[email protected] 713.744.6521

Handouts will be available online until 04.12.10 http://www.theansweris4.net

-Click on Departments -Click on Mathematics Services -Click on Professional Development Materials

Permission to copy classroom-ready materials granted

to attendees of this Region 4 session. © Region 4 Education Service Center.

All rights reserved.

http://www.twitter.com/R4math

TI-83/84 Quick Start Guide

To darken/lighten screen

Press y} to darken.

Press y† to lighten.

Commonly used buttons

Í is used for =

Ì is the negative sign

› is used for exponents. 23 is typed as Á›Â yz allows you to exit from the current application/window.

z allows settings to be adjusted.

r along with | and ~ allows you to move along a graph.

To enter functions in calculator

Press o.

Press „ for the variable.

To set viewing window

Press p

Set the appropriate window.

Checklist when graphing

1. Enter data in … or function in o. 2. Set appropriate viewing window. 3. Turn off unnecessary Stat Plots or Functions.

4. Press s.

Common errors

Press 1 to Quit or 2 to see the error.

Ymax

Ymin

Xmax Xmin

To enter data in lists

… 

Í 

To clear lists, arrow up to the list name, press

‘ then Í. (A common error is a Dim Mismatch. If this occurs, make sure the same number of data values are in each list.

To perform statistical operations on lists

… ~ 

Í Then choose the list name from y… or

choose y and ÀÁ¶· or ¸.

Press Í 

The arrow indicates there is more on the next

screen. Press † several times to see more data.

To graph data in lists

Press yo

Press Í. 

Turn the plot ON, then use the arrows to choose the appropriate graph and source of data (choose

the list names from y…).

Set the viewing window by pressing q9 Í Resources: Graphing Calculator Tutorial (www.escweb.net/math) Graphing Calculator Lessons (www.education.ti.com) Graphing Calculator Emulator (http://www.technoplaza.net/downloads/details.php?program=67)

mean

sum of x-values

sum of x2-values

sample stand. dev.

pop. stand. dev.

number of data points.

minimum

3rd quartile

1st quartile median

maximum

http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=7499

Name _______________________ Date __________________ Period ________

Graphing Calculator Scavenger Hunt By Lois Coles

1. Press 2nd + ENTER What is the ID# of your calculator? ______________

2. For help, what website can you visit? _______________

3. What happens to the screen when you push 2nd ▲ over and over? 2nd ▼ over and over? ________________________________________________

4. is called the "caret" button, and is used to raise a number to a power. Find 65 = ______. To square a number use x2 What is 562? _______ To cube a number, press MATH and select option 3. What is 363? ___________

5. Press 2nd Y= to access the STAT PLOTS menu, how many stat plots are there? _____ Which option turns the stat plots off? ________________

6. Press STAT . Which option will sort data in ascending order? What do you think will happen if option 3 is selected? __________________________________

7. What letter of the alphabet is located above ? _______________

8. To get the calculator to solve the following problem 2{3 + 10/2 + 62 – (4 + 2)}, what do you do to get the { and } ? _________________The answer to the problem is __________.

9. To solve a problem involving the area and/or circumference of a circle, you will need to use π. Where is this calculator key? _______________________

10. Use your calculator to answer the following: 2 x 41.587 ________ 2578/4 _________ 369 + 578 _________

Now press 2nd ENTER two times. What pops up on your screen? ___________ Arrow over and change the 4 to a 2. What answer do you get? ________ How will this feature be helpful? _____________________________________

11. What happens when the 10x and 6 keys are pressed? ___________

12. The STO button stores numbers to variables. To evaluate the expression 2 34a b

c

, press

9 STO ALPHA MATH ENTER to store the number 9 to A. Repeat this same process if B = 2

and C = 1, then evaluate the expression by typing in the expression 2 34a b

c

and pressing

ENTER. Is it faster just to substitute the values into the expression and solve the old- fashioned way with paper and pencil? _______ When might this feature come in handy?________________________________________

http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=7499

13. Press 2nd 0 to access the calculator's catalogue. Scroll up, to access symbols. What is the first symbol? _____________ What is the last symbol? _______________

14. Press 2nd 0 to access the calculator's catalogue. An A appears in the top right corner of the screen. This means the calculator is in alphabetical mode. Press ) . What is the 5th entry in the L's? What do these letters stand for? _____________

15. Press MATH, what do you think the first entry will do? _____________________ Now press CLEAR , then press 0 . 5 6 MATH and select option 1. What answer do you get? ___________

16. Press 4 MATH, choose option 5, then press 1 6 and ENTER . What did this option do? _________________________________________________________

17. Which function allows you to send/receive data/programs? __________________

18. Press Y= type in 2x – 1. Press ZOOM then select 6, press MODE, arrow to the bottom and arrow over to G-T and press ENTER. Now press GRAPH. What appears on the screen? _____________________________________________ Press MODE and scroll down to Full and press ENTER to restore to full screen.

19. Press 5 9 ENTER. Press 2 to go to the error. The cursor should be blinking on the second /, press DEL ENTER. What answer did you get? To convert this number to a fraction, press MATH ENTER

20. Enter this problem into the calculator and press ENTER. 2.4 x 3.7 = _______. Now press MODE ▼ Float ► to 0 and press ENTER. Now press 2nd Quit to return to the home screen and press 2nd ENTER and the original problem should appear on the screen, now press ENTER. What appears on the screen? ____________ Think about this number in relation to the answer you got before. What did the calculator do? _________________________________ Repeat this same process except select 2 under the Float option. Return to the home screen, recall the original problem and press ENTER. What number appears on the screen? _______ What did the calculator do this time? _____________________

21. Enter (-2)2 into the calculator, what answer did you get? ___________Now enter –22 into the calculator, what answer did you get this time? ________Why do you think you got two different answers? ______________________________Would (-2)3 and –23 give you two different answers? Why or why not?________________________________

A

Adapted from “Power”ful Patterns-Texas Instruments

“Power”ful Patterns

Use your calculator to complete the table below.

Repeated Multiplication Using the Multiplication Sign

Repeated Multiplication with ^ Enter: (base) ^ (exponent) e

Expression Value Base Exponent Value

3 = 3 ^ 1 =

3 x 3 = 3 ^ 2 =

3 x 3 x 3 = 3 ^ 3 =

3 x 3 x 3 x 3 = 3 ^ 4 =

3 x 3 x 3 x 3 x 3 = 3 ^ 5 =

3 x 3 x 3 x 3 x 3 x 3 = 3 ^ 6 =

3 x 3 x 3 x 3 x 3 x 3 x 3 = 3 ^ 7 =

3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = 3 ^ 8 =

3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = 3 ^ 9 =

What patterns do you see in the table?

Pick another whole number and complete the table below.

Repeated Multiplication Using the Multiplication Sign Repeated Multiplication with ^

Enter: (base) ^ (exponent) e Expression Value Base Exponent Value

___ = ___ ^ 1 =

___ x ___ = ___ ^ 2 =

___ x ___ x ___ = ___ ^ 3 =

___ x ___ x ___ x ___ = ___ ^ 4 =

___ x ___ x ___ x ___ x ___ = ___ ^ 5 =

___ x ___ x ___ x ___ x ___ x ___ = ___ ^ 6 =

___ x ___ x ___ x ___ x ___ x ___ x ___ = ___ ^ 7 =

___ x ___ x ___ x ___ x ___ x ___ x ___ x ___ = ___ ^ 8 =

___ x ___ x ___ x ___ x ___ x ___ x ___ x ___x___ = ___ ^ 9 =

What patterns do you see in the table?

Adapted from “Power”ful Patterns-Texas Instruments

“Power”ful Observations and Questions

1. What would the calculator do if you entered 4 ^ 2 e? How would you find the value of

4 ^ 2 without a calculator?

2. What is the calculator doing when you enter: (base) ^ (exponent) e?

3. How would you define the base?

4. How would you define the exponent?

5. If you do not have a calculator, 4 ^ 2 is written as 42. How would you write 5 ^ 3? Label the base and the exponent.

6. Without using a calculator, find the value of 24 .

Adapted from How Totally Square! – Texas Instruments

Name _____________________________ Banquet Tables – Part 1 Some students at a school are planning a banquet for their end-of-year party. They are trying to decide how many desks to use to seat people. Their only option for seating is square student desks that measure 1 yard on each side. If they use exactly 1 desk, they can seat 4 students. Six students can be seated with 2 desks. The desks are placed end-to-end with each additional desk. Explore the number of seats available based on the number of desks. Use color tiles to build the models. Then complete the table.

Number of Desks Number of Seats Picture

1

2

3

4

5

6

7

20

n

Adapted from How Totally Square! – Texas Instruments

1. How many students can be seated using 12 desks?

2. How many desks would be needed for 94 students?

3. How many desks would be needed for 123 students?

4. If you know the number of desks, how can you find the total number of seats?

5. Write a rule to find the number of seats using n desks.

Adapted from How Totally Square! – Texas Instruments

Name _____________________________

Banquet Tables – Part 1

Using Your Graphing Calculator

1. Input your function rule into o.

2. What is an appropriate viewing window for your data? Use your table to help you determine a

window.

3. Graph your data. Sketch the graph below.

4. Use the ys (0) button to answer the following questions.

a. What data is found in the X column?

b. What data is found in the Y1 column?

c. How many seats are available with 35 desks?

d. How many desks are needed to provide 90 seats?

5. If there are 250 students, how many desks will be needed in all?

Adapted from How Totally Square! – Texas Instruments

Name _____________________________

Banquet Tables – Part 2

The students have decided to rent rectangular tables to use for the banquet seating. Each table seats 6 students.

If 2 tables are used, 10 students can be seated.

The tables are placed end-to-end with each additional table. Explore the number of seats available based on the number of tables used.

Number of Tables Number of Seats Picture

1

2

3

4

5

6

7

20

n

Adapted from How Totally Square! – Texas Instruments

Write a rule and then use your graphing calculator to answer the following questions: 1. How many students can be seated using 12 tables?

2. How many tables would be needed for 94 students?

3. How many tables would be needed for 123 students?

4. If there are 250 students, how many tables will be needed in all?

Adapted from How Totally Square! – Texas Instruments

Name _____________________________ Banquet Tables – Part 3 If 2n + 4 gives you the number of seats available for n tables, what might the table arrangements look

like? Justify your answer.

How many students can be seated using 15 of these tables?

How many tables would be needed to seat 128 students?

Building a Garden Fence – Part 1

L1 L2 L3

Building a Garden Fence – Part 2

What if you had 30 pieces of fencing to use

instead of 24? Find the dimensions of the

garden with the greatest area and justify your

solution.

Building a Garden Fence – Part 3

In both of the situations you have now

explored (24 pieces of fencing and 30 pieces

of fencing), does the garden with the greatest

area also have the greatest perimeter? Explain

your reasoning and justify your answer.

Adapted from Building a Garden Fence – Texas Instruments

Building a Garden Fence – Part 1 You and a friend are visiting her grandparents on their small farm. They have asked the two of you to design a small, rectangular-shaped vegetable garden along an existing wall in their backyard. They wish to surround the garden with a small fence to protect their plants from small animals.

To enclose the garden, you have 24 sections of 1-meter long rigid border fencing. In order to grow as many vegetables as possible, your task is to design the fence to enclose the maximum possible area. How many sections of fencing should you use along the width and the length of the garden? 1. Suppose you used three sections of the fencing along each width of the garden. How many

sections would be left to form the length? Draw a picture to justify your answer. 2. What would the area of this garden be? 3. Use your answers from problems 1 and 2 to complete the first row of the table. Then come up with

3 other possible garden sizes and write them in the table.

Width (m) Length (m) Area (m2)

3

4. If you know the width of the garden, how can you find the length? 5. Write a rule to determine the length of a garden with a width of x meters. 6. What is the smallest possible width for the garden? What is the largest possible width? Justify your

answer.

W W

L

Adapted from Building a Garden Fence – Texas Instruments

Using Your Graphing Calculator

1. Input all of the possible width values for the garden in L1 from least to greatest. 2. Use your rule from Problem #5 to generate the possible lengths in L2.

3. Create a rule to determine the area for each garden in L3.

4. Complete the table below.

L1 L2 L3

5. Describe any patterns you see in the table. 6. Complete the following sentence:

A rectangle with a width of _______ meters and a length of _______ meters gives the largest

possible garden area of ________ square meters.

7. Set up a scatterplot to display the relationships between the widths and the corresponding

areas. Use L1 for the Xlist and L3 for the Ylist.

Adapted from Building a Garden Fence – Texas Instruments

8. What is an appropriate viewing window for your data? Use your table to help you determine a

window.

9. Graph your data. Sketch the graph below.

10. Use your r button to identify the point that corresponds to the maximum area. What sets it

apart from the other points on the graph?

11. How do any patterns that you observed in the lists show up in the scatterplot of the data?

Adapted from Building a Garden Fence – Texas Instruments

Building a Garden Fence – Part 2 What if you had 30 pieces of fencing to use instead of 24? Find the dimensions of the garden with the greatest area and justify your solution.

Adapted from Building a Garden Fence – Texas Instruments

Building a Garden Fence – Part 3 In both of the situations you have now explored (24 pieces of fencing and 30 pieces of fencing), does the garden with the greatest area also have the greatest perimeter? Explain your reasoning and justify your answer.

Box It Up – Part 1

L1 L2 L3 L4

Box It Up – Part 2

Assume that the rectangular metal sheets

given to the industrial technologies class each

measure 75 cm by 60 cm, and that we still

want to determine the size of the square to cut

out so that a box with the largest volume is

produced. What size square should be cut out

and what is the resulting volume? Justify your

answer.

Box It Up – Part 3

Assume that the metal sheets given to the

industrial technologies class were square.

When a square with a side length of

8 centimeters was cut from each corner, the

resulting volume was 1,152 cubic centimeters.

What was the size of the original metal sheet?

Justify your answer.

Adapted from Box It Up – Texas Instruments

Box It Up – Part 1 Ms. Hawkins, the physical sciences teacher, needs several open-topped boxes for storing laboratory

materials. She has given the industrial technologies class several pieces of metal sheeting to make

the boxes. Each of the metal pieces is a rectangle measuring 40 cm by 60 cm. The class plans to

make the boxes by cutting equal-sized squares from each corner of a metal sheet, bending up the

sides, and welding the edges. The squares must have side lengths of whole number values.

1. If the squares that are cut out have side lengths of 1 centimeter, what is the length, width,

height, and volume of the box that is formed?

Length: ________ Width: _________ Height: _________ Volume: _________

2. What is the side length of the largest square that could be cut from each corner? Justify your

answer.

3. How is the height of the box determined? 4. Use your answers from problem 1 to complete the first row of the table. Then come up with

3 other possible box heights and complete the table.

Height (cm) Width (cm) Length (cm) Volume (cm3)

5. What do you predict the largest possible volume will be? Justify your answer.

60 cm

40 cm

Adapted from Box It Up – Texas Instruments

Using Your Graphing Calculator

1. Input all of the possible heights of the box in L1 from least to greatest.

2. Write a rule to determine the width of each box to generate L2.

3. Write a rule to determine the length of each box to generate L3.

4. Write a rule to determine the volume of each box to generate L4.

5. Complete the table below.

L1 L2 L3 L4

6. Describe any patterns you see in the table. 7. Complete the following sentence:

A box made by cutting equal-sized squares with side lengths of ______ cm generates the

largest possible volume of ________ cubic centimeters.

8. Set up a scatterplot to display the relationship between the side length of the square (the

height of the box) and the corresponding volume. Use L1 for the Xlist and L4 for the Ylist.

Adapted from Box It Up – Texas Instruments

9. What is an appropriate viewing window for your data? Use your table to help you determine a

window.

10. Graph your data. Sketch the graph below.

11. Use your r button to identify the point that corresponds to the maximum volume. What

sets it apart from the other points on the graph?

12. How do any patterns that you observed in the lists show up in the scatterplot of the data?

Adapted from Box It Up – Texas Instruments

Box It Up – Part 2 Assume that the rectangular metal sheets given to the industrial technologies class each measure

75 cm by 60 cm, and that we still want to determine the size of the square to cut out so that a box

with the largest volume is produced. What size square should be cut out and what is the resulting

volume? Justify your answer.

Adapted from Box It Up – Texas Instruments

Box It Up – Part 3

Assume that the metal sheets given to the industrial technologies class were square. When a square

with a side length of 8 centimeters was cut from each corner, the resulting volume was 1,152 cubic

centimeters. What was the size of the original metal sheet? Justify your answer.

Bonus: Is this the largest possible volume that could be created by cutting a square from each

corner? Justify your answer.

This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the 2006-2007 school year.

5

(B) validate his/her conclusions using mathematical properties and relationships.

§111.23. Mathematics, Grade 7.

(a) Introduction.

(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 7 are using direct proportional relationships in number, geometry, measurement, and probability; applying addition, subtraction, multiplication, and division of decimals, fractions, and integers; and using statistical measures to describe data.

(2) Throughout mathematics in Grades 6-8, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use concepts, algorithms, and properties of rational numbers to explore mathematical relationships and to describe increasingly complex situations. Students use algebraic thinking to describe how a change in one quantity in a relationship results in a change in the other; and they connect verbal, numeric, graphic, and symbolic representations of relationships. Students use geometric properties and relationships, as well as spatial reasoning, to model and analyze situations and solve problems. Students communicate information about geometric figures or situations by quantifying attributes, generalize procedures from measurement experiences, and use the procedures to solve problems. Students use appropriate statistics, representations of data, reasoning, and concepts of probability to draw conclusions, evaluate arguments, and make recommendations.

(3) Problem solving in meaningful contexts, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Throughout mathematics in Grades 6-8, students use these processes together with graphing technology and other mathematical tools such as manipulative materials to develop conceptual understanding and solve problems as they do mathematics.

(b) Knowledge and skills.

(7.1) Number, operation, and quantitative reasoning. The student represents and uses numbers in a variety of equivalent forms.

The student is expected to:

(A) compare and order integers and positive rational numbers;

(B) convert between fractions, decimals, whole numbers, and percents mentally, on paper, or with a calculator; and

(C) represent squares and square roots using geometric models.

(7.2) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions.

The student is expected to:

(A) represent multiplication and division situations involving fractions and decimals with models, including concrete objects, pictures, words, and numbers;

This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the 2006-2007 school year.

6

(B) use addition, subtraction, multiplication, and division to solve problems involving fractions and decimals;

(C) use models, such as concrete objects, pictorial models, and number lines, to add, subtract, multiply, and divide integers and connect the actions to algorithms;

(D) use division to find unit rates and ratios in proportional relationships such as speed, density, price, recipes, and student-teacher ratio;

(E) simplify numerical expressions involving order of operations and exponents;

(F) select and use appropriate operations to solve problems and justify the selections; and

(G) determine the reasonableness of a solution to a problem.

(7.3) Patterns, relationships, and algebraic thinking. The student solves problems involving direct proportional relationships.

The student is expected to:

(A) estimate and find solutions to application problems involving percent; and

(B) estimate and find solutions to application problems involving proportional relationships such as similarity, scaling, unit costs, and related measurement units.

(7.4) Patterns, relationships, and algebraic thinking. The student represents a relationship in numerical, geometric, verbal, and symbolic form.

The student is expected to:

(A) generate formulas involving unit conversions within the same system (customary and metric), perimeter, area, circumference, volume, and scaling;

(B) graph data to demonstrate relationships in familiar concepts such as conversions, perimeter, area, circumference, volume, and scaling; and

(C) use words and symbols to describe the relationship between the terms in an arithmetic sequence (with a constant rate of change) and their positions in the sequence.

sbrennan
Rectangle

This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the 2006-2007 school year.

7

(7.5) Patterns, relationships, and algebraic thinking. The student uses equations to solve problems.

The student is expected to:

(A) use concrete and pictorial models to solve equations and use symbols to record the actions; and

(B) formulate problem situations when given a simple equation and formulate an equation when given a problem situation.

(7.6) Geometry and spatial reasoning. The student compares and classifies two- and three-dimensional figures using geometric vocabulary and properties.

The student is expected to:

(A) use angle measurements to classify pairs of angles as complementary or supplementary;

(B) use properties to classify triangles and quadrilaterals;

(C) use properties to classify three-dimensional figures, including pyramids, cones, prisms, and cylinders; and

(D) use critical attributes to define similarity.

(7.7) Geometry and spatial reasoning. The student uses coordinate geometry to describe location on a plane.

The student is expected to:

(A) locate and name points on a coordinate plane using ordered pairs of integers; and

(B) graph reflections across the horizontal or vertical axis and graph translations on a coordinate plane.

(7.8) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world.

The student is expected to:

(A) sketch three-dimensional figures when given the top, side, and front views;

(B) make a net (two-dimensional model) of the surface area of a three-dimensional figure; and

(C) use geometric concepts and properties to solve problems in fields such as art and architecture.

(7.9) Measurement. The student solves application problems involving estimation and measurement.

The student is expected to:

(A) estimate measurements and solve application problems involving length (including perimeter and circumference) and area of polygons and other shapes;

This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the 2006-2007 school year.

8

(B) connect models for volume of prisms (triangular and rectangular) and cylinders to formulas of prisms (triangular and rectangular) and cylinders; and

(C) estimate measurements and solve application problems involving volume of prisms (rectangular and triangular) and cylinders.

(7.10) Probability and statistics. The student recognizes that a physical or mathematical model (including geometric) can be used to describe the experimental and theoretical probability of real-life events.

The student is expected to:

(A) construct sample spaces for simple or composite experiments; and

(B) find the probability of independent events.

(7.11) Probability and statistics. The student understands that the way a set of data is displayed influences its interpretation.

The student is expected to:

(A) select and use an appropriate representation for presenting and displaying relationships among collected data, including line plot, line graph, bar graph, stem and leaf plot, circle graph, and Venn diagrams, and justify the selection; and

(B) make inferences and convincing arguments based on an analysis of given or collected data.

(7.12) Probability and statistics. The student uses measures of central tendency and variability [range] to describe a set of data.

The student is expected to:

(A) describe a set of data using mean, median, mode, and range; and

(B) choose among mean, median, mode, or range to describe a set of data and justify the choice for a particular situation.

(7.13) Underlying processes and mathematical tools. The student applies Grade 7 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school.

The student is expected to:

(A) identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics;

(B) use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;

sbrennan
Rectangle
sbrennan
Rectangle

This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the 2006-2007 school year.

9

(C) select or develop an appropriate problem-solving strategy from a variety of different types, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem; and

(D) select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation, and number sense to solve problems.

(7.14) Underlying processes and mathematical tools. The student communicates about Grade 7 mathematics through informal and mathematical language, representations, and models.

The student is expected to:

(A) communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models; and

(B) evaluate the effectiveness of different representations to communicate ideas.

(7.15) Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions.

The student is expected to:

(A) make conjectures from patterns or sets of examples and nonexamples; and

(B) validate his/her conclusions using mathematical properties and relationships.

§111.24. Mathematics, Grade 8.

(a) Introduction.

(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 8 are using basic principles of algebra to analyze and represent both proportional and non-proportional linear relationships and using probability to describe data and make predictions.

(2) Throughout mathematics in Grades 6-8, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use concepts, algorithms, and properties of rational numbers to explore mathematical relationships and to describe increasingly complex situations. Students use algebraic thinking to describe how a change in one quantity in a relationship results in a change in the other; and they connect verbal, numeric, graphic, and symbolic representations of relationships. Students use geometric properties and relationships, as well as spatial reasoning, to model and analyze situations and solve problems. Students communicate information about geometric figures or situations by quantifying attributes, generalize procedures from measurement experiences, and use the procedures to solve problems. Students use appropriate statistics, representations of data, reasoning, and concepts of probability to draw conclusions, evaluate arguments, and make recommendations.

This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the 2006-2007 school year.

10

(3) Problem solving in meaningful contexts, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Throughout mathematics in Grades 6-8, students use these processes together with graphing technology and other mathematical tools such as manipulative materials to develop conceptual understanding and solve problems as they do mathematics.

(b) Knowledge and skills.

(8.1) Number, operation, and quantitative reasoning. The student understands that different forms of numbers are appropriate for different situations.

The student is expected to:

(A) compare and order rational numbers in various forms including integers, percents, and positive and negative fractions and decimals;

(B) select and use appropriate forms of rational numbers to solve real-life problems including those involving proportional relationships;

(C) approximate (mentally and with calculators) the value of irrational numbers as they arise from problem situations (such as π, √2); [and]

(D) express numbers in scientific notation, including negative exponents, in appropriate problem situations; and

(E) compare and order real numbers with a calculator

(8.2) Number, operation, and quantitative reasoning. The student selects and uses appropriate operations to solve problems and justify solutions.

The student is expected to:

(A) select appropriate operations to solve problems involving rational numbers and justify the selections;

(B) use appropriate operations to solve problems involving rational numbers in problem situations;

(C) evaluate a solution for reasonableness; and

(D) use multiplication by a given constant factor (including unit rate) to represent and solve problems involving proportional relationships including conversions between measurement systems.

(8.3) Patterns, relationships, and algebraic thinking. The student identifies proportional or non-proportional linear relationships in problem situations and solves problems.

The student is expected to:

(A) compare and contrast proportional and non-proportional linear relationships; and

sbrennan
Rectangle
sbrennan
Rectangle

This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the 2006-2007 school year.

11

(B) estimate and find solutions to application problems involving percents and other proportional

relationships such as similarity and rates.

This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the 2006-2007 school year.

12

(8.4) Patterns, relationships, and algebraic thinking. The student makes connections among various representations of a numerical relationship.

The student is expected to generate a different representation of data given another representation of data (such as a table, graph, equation, or verbal description).

(8.5) Patterns, relationships, and algebraic thinking. The student uses graphs, tables, and algebraic representations to make predictions and solve problems.

The student is expected to:

(A) predict, find, and justify solutions to application problems using appropriate tables, graphs, and algebraic equations; and

(B) find and evaluate an algebraic expression to determine any term in an arithmetic sequence (with a constant rate of change).

(8.6) Geometry and spatial reasoning. The student uses transformational geometry to develop spatial sense.

The student is expected to:

(A) generate similar figures using dilations including enlargements and reductions; and

(B) graph dilations, reflections, and translations on a coordinate plane.

(8.7) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world.

The student is expected to:

(A) draw three-dimensional figures from different perspectives;

(B) use geometric concepts and properties to solve problems in fields such as art and architecture;

(C) use pictures or models to demonstrate the Pythagorean Theorem; and

(D) locate and name points on a coordinate plane using ordered pairs of rational numbers.

(8.8) Measurement. The student uses procedures to determine measures of three-dimensional figures.

The student is expected to:

(A) find lateral and total surface area of prisms, pyramids, and cylinders using concrete models and nets (two-dimensional models);

(B) connect models of prisms, cylinders, pyramids, spheres, and cones to formulas for volume of these objects; and

(C) estimate measurements and use formulas to solve application problems involving lateral and total surface area and volume.

This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the 2006-2007 school year.

13

(8.9) Measurement. The student uses indirect measurement to solve problems.

The student is expected to:

(A) use the Pythagorean Theorem to solve real-life problems; and

(B) use proportional relationships in similar two-dimensional figures or similar three-dimensional figures to find missing measurements.

(8.10) Measurement. The student describes how changes in dimensions affect linear, area, and volume measures.

The student is expected to:

(A) describe the resulting effects on perimeter and area when dimensions of a shape are changed proportionally; and

(B) describe the resulting effect on volume when dimensions of a solid are changed proportionally.

(8.11) Probability and statistics. The student applies concepts of theoretical and experimental probability to make predictions.

The student is expected to:

(A) find the probabilities of dependent and independent events;

(B) use theoretical probabilities and experimental results to make predictions and decisions; and

(C) select and use different models to simulate an event.

(8.12) Probability and statistics. The student uses statistical procedures to describe data.

The student is expected to:

(A) use variability (range, including interquartile range (IQR)) and select the appropriate measure of central tendency [or range] to describe a set of data and justify the choice for a particular situation;

(B) draw conclusions and make predictions by analyzing trends in scatterplots; and

(C) select and use an appropriate representation for presenting and displaying relationships among collected data, including line plots, line graphs, stem and leaf plots, circle graphs, bar graphs, box and whisker plots, histograms, and Venn diagrams, with and without the use of technology.

(8.13) Probability and statistics. The student evaluates predictions and conclusions based on statistical data.

The student is expected to:

(A) evaluate methods of sampling to determine validity of an inference made from a set of data; and

sbrennan
Rectangle

This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the 2006-2007 school year.

14

(B) recognize misuses of graphical or numerical information and evaluate predictions and conclusions based on data analysis.

(8.14) Underlying processes and mathematical tools. The student applies Grade 8 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school.

The student is expected to:

(A) identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics;

(B) use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;

(C) select or develop an appropriate problem-solving strategy from a variety of different types, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem; and

(D) select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation, and number sense to solve problems.

(8.15) Underlying processes and mathematical tools. The student communicates about Grade 8 mathematics through informal and mathematical language, representations, and models.

The student is expected to:

(A) communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models; and

(B) evaluate the effectiveness of different representations to communicate ideas.

(8.16) Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions.

The student is expected to:

(A) make conjectures from patterns or sets of examples and nonexamples; and

(B) validate his/her conclusions using mathematical properties and relationships.

The Director’s Chair A CBR Activity Follow Up You will be giving directions to a fellow classmate so that he/she will be able to match the graphs shown below. Each graph shows distance verses time. Distance is measured in feet and time in seconds. Provide a starting point, direction, and rate in your directions. Be as specific and detailed as possible! 1. 2. 3.