Using the TI-84 Plus Graphing Calculator in Middle School Mathematics Aldine ISD March 27, 2010

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1 Using the TI-84 Plus Graphing Calculator in Middle School Mathematics Aldine ISD March 27, 2010 Shelley Bolen-Abbott Region 4 Education Service Center sbolenabbott@esc4.net Handouts will be available online until 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.

2 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. 2 3 is typed as Á Â yz allows you to exit from the current application/window. z allows settings to be adjusted. To enter functions in calculator Press o. r along with and ~ allows you to move along a graph. To set viewing window Press p Press for the variable. Set the appropriate window. Ymax Xmin Xmax Ymin 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.

3 To enter data in lists To perform statistical operations on lists ~ Í Í Then choose the list name from y or choose y and ÀÁÂ or. 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 graph data in lists Press yo Press Í mean sum of x-values sum of x 2 -values sample stand. dev. pop. stand. dev. number of data points. The arrow indicates there is more on the next screen. Press several times to see more data. Press Í. minimum 1 st quartile median 3 rd quartile maximum 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 ( Graphing Calculator Lessons ( Graphing Calculator Emulator (

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5 Name Date Period Graphing Calculator Scavenger Hunt By Lois Coles 1. Press 2 nd + 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 2 nd over and over? 2 nd over and over? 4. is called the "caret" button, and is used to raise a number to a power. Find 6 5 =. To square a number use x 2 What is 56 2? To cube a number, press MATH and select option 3. What is 36 3? 5. Press 2 nd 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/ (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 / Now press 2 nd 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 10 x and 6 keys are pressed? 12. The STO button stores numbers to variables. To evaluate the expression 2 a 3 b, press 4 c 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 a 3 b and pressing 4 c ENTER. Is it faster just to substitute the values into the expression and solve the oldfashioned way with paper and pencil? When might this feature come in handy?

6 13. Press 2 nd 0 to access the calculator's catalogue. Scroll up, to access symbols. What is the first symbol? What is the last symbol? 14. Press 2 nd 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 5 th 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 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 2 nd Quit to return to the home screen and press 2 nd 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 2 2 into the calculator, what answer did you get this time? Why do you think you got two different answers? Would (-2) 3 and 2 3 give you two different answers? Why or why not?

7 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 with ^ Repeated Multiplication Using the Multiplication Sign 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

8 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 4 2. How would you write 5 ^ 3? Label the base and the exponent. 6. Without using a calculator, find the value of 2 4. Adapted from Power ful Patterns-Texas Instruments

9 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 n Adapted from How Totally Square! Texas Instruments

10 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

11 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 Y 1 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

12 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 n Adapted from How Totally Square! Texas Instruments

13 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

14 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? Adapted from How Totally Square! Texas Instruments

15 Building a Garden Fence Part 1 L 1 L 2 L 3

16 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.

17 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.

18 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? W W L 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 (m 2 ) 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. Adapted from Building a Garden Fence Texas Instruments

19 Using Your Graphing Calculator 1. Input all of the possible width values for the garden in L 1 from least to greatest. 2. Use your rule from Problem #5 to generate the possible lengths in L Create a rule to determine the area for each garden in L Complete the table below. L 1 L 2 L 3 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 L 1 for the Xlist and L 3 for the Ylist. Adapted from Building a Garden Fence Texas Instruments

20 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

21 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

22 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

23 Box It Up Part 1 L 1 L 2 L 3 L 4

24 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.

25 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.

26 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. 40 cm 60 cm 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 (cm 3 ) 5. What do you predict the largest possible volume will be? Justify your answer. Adapted from Box It Up Texas Instruments

27 Using Your Graphing Calculator 1. Input all of the possible heights of the box in L 1 from least to greatest. 2. Write a rule to determine the width of each box to generate L Write a rule to determine the length of each box to generate L Write a rule to determine the volume of each box to generate L Complete the table below. L 1 L 2 L 3 L 4 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 L 1 for the Xlist and L 4 for the Ylist. Adapted from Box It Up Texas Instruments

28 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

29 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

30 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. Adapted from Box It Up Texas Instruments

31 This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the school year. (B) validate his/her conclusions using mathematical properties and relationships 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. (7.2) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions. (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. (A) represent multiplication and division situations involving fractions and decimals with models, including concrete objects, pictures, words, and numbers; 5

32 This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the school year. (7.3) Patterns, relationships, and algebraic thinking. The student solves problems involving direct proportional relationships. (7.4) Patterns, relationships, and algebraic thinking. The student represents a relationship in numerical, geometric, verbal, and symbolic form. (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. (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. (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. 6

33 This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the school year. (7.5) Patterns, relationships, and algebraic thinking. The student uses equations to solve problems. (7.6) Geometry and spatial reasoning. The student compares and classifies two- and three-dimensional figures using geometric vocabulary and properties. (7.7) Geometry and spatial reasoning. The student uses coordinate geometry to describe location on a plane. (7.8) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world. (7.9) Measurement. The student solves application problems involving estimation and measurement. (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. (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 threedimensional figures, including pyramids, cones, prisms, and cylinders; and (D) use critical attributes to define similarity. (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. (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 threedimensional figure; and (C) use geometric concepts and properties to solve problems in fields such as art and architecture. (A) estimate measurements and solve application problems involving length (including perimeter and circumference) and area of polygons and other shapes; 7

34 This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the school year. (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. (7.11) Probability and statistics. The student understands that the way a set of data is displayed influences its interpretation. (7.12) Probability and statistics. The student uses measures of central tendency and variability [range] to describe a set of data. (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. (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. (A) construct sample spaces for simple or composite experiments; and (B) find the probability of independent events. (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. (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. (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; 8

35 This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the school year. (7.14) Underlying processes and mathematical tools. The student communicates about Grade 7 mathematics through informal and mathematical language, representations, and models. (7.15) Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions. (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. (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. (A) make conjectures from patterns or sets of examples and nonexamples; and (B) validate his/her conclusions using mathematical properties and relationships 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. 9

36 This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the school year. (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. (8.2) Number, operation, and quantitative reasoning. The student selects and uses appropriate operations to solve problems and justify solutions. (8.3) Patterns, relationships, and algebraic thinking. The student identifies proportional or nonproportional linear relationships in problem situations and solves problems. (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 (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. (A) compare and contrast proportional and non-proportional linear relationships; and 10

37 This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the school year. (B) estimate and find solutions to application problems involving percents and other proportional relationships such as similarity and rates. 11

38 This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the school year. (8.4) Patterns, relationships, and algebraic thinking. The student makes connections among various representations of a numerical relationship. (8.5) Patterns, relationships, and algebraic thinking. The student uses graphs, tables, and algebraic representations to make predictions and solve problems. (8.6) Geometry and spatial reasoning. The student uses transformational geometry to develop spatial sense. (8.7) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world. (8.8) Measurement. The student uses procedures to determine measures of three-dimensional figures. 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). (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). (A) generate similar figures using dilations including enlargements and reductions; and (B) graph dilations, reflections, and translations on a coordinate plane. (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. (A) find lateral and total surface area of prisms, pyramids, and cylinders using concrete models and nets (twodimensional 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. 12

39 This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the school year. (8.9) Measurement. The student uses indirect measurement to solve problems. (8.10) Measurement. The student describes how changes in dimensions affect linear, area, and volume measures. (8.11) Probability and statistics. The student applies concepts of theoretical and experimental probability to make predictions. (8.12) Probability and statistics. The student uses statistical procedures to describe data. (8.13) Probability and statistics. The student evaluates predictions and conclusions based on statistical data. (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. (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. (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. (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. (A) evaluate methods of sampling to determine validity of an inference made from a set of data; and 13

40 This provisions of this subchapter were adopted by the State Board of Education in February 2005 to be implemented beginning with the school year. (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. (8.15) Underlying processes and mathematical tools. The student communicates about Grade 8 mathematics through informal and mathematical language, representations, and models. (8.16) Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions. (B) recognize misuses of graphical or numerical information and evaluate predictions and conclusions based on data analysis. (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. (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. (A) make conjectures from patterns or sets of examples and nonexamples; and (B) validate his/her conclusions using mathematical properties and relationships. 14

41 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!