Algebra I: Part 1 - Warren Easton Charter High Schoolwarreneastoncharterhigh.org/ourpages/auto/2009/11/18...2009/11/18 · Algebra I: Part 1 Unit 1 Variables and Relationships 1 Algebra

Comprehensive Curriculum

Algebra I: Part 1

Cecil J. Picard State Superintendent of Education

© April 2005

Algebra I: Part 1 Table of Contents

Unit 1: Variables and Relationships .......................................................................................1 Unit 2: Measurement and Geometry.......................................................................................9 Unit 3: Solving Equations and Real-life Graphs...................................................................19 Unit 4: Linear Equations, Linear Inequalities, and Graphing...............................................28 Unit 5: Graphing and Writing Equations of Lines................................................................38 Unit 6: Inequalities and Graphing.........................................................................................46 Unit 7: Systems of Equations................................................................................................51 Unit 8: Matrices, Systems of Equations, and Linear Inequalities.........................................59

Algebra I: Part 1 Unit 1 Variables and Relationships 1

Algebra I: Part 1 Unit 1: Variables and Relationships

Time Frame: Approximately seven weeks Unit Description This unit focuses on the numerical relationships resulting from substituting values in algebraic expressions. Included in the unit is a thorough review of calculations with the real number system. Student Understandings Students can use the order of operations and scientific notation, and work with rational and irrational numbers. Students will also write and evaluate algebraic expressions in real-life situations and in mathematical formulas and patterns. Guiding Questions

1. Can students correctly evaluate linear and exponential-based formulas and tie the results to sequences or specific applications?

2. Can students use order of operations and the basic properties (i.e., associative commutative, and distributive) when performing computations and collecting like terms in expressions?

3. Can students use flow charts to guide and describe operations? 4. Can students recognize functions in graphical, numerical, tabular, and verbal

forms? Unit 1 Grade-Level Expectations (GLEs)

GLE# GLE Text and Benchmarks Number and Number Relations 1. Identify and describe differences among natural numbers, whole numbers,

integers, rational numbers, and irrational numbers (N-1-H) (N-2-H) (N-3-H) 2. Evaluate and write numerical expressions involving integer exponents (N-2-H) 3. Apply scientific notation to perform computations, solve problems, and write

representations of numbers (N-2-H) 4. Distinguish between an exact and an approximate answer, and recognize errors

introduced by the use of approximate numbers with technology (N-3-H) (N-4-H) (N-7-H)


GLE# GLE Text and Benchmarks 5. Demonstrate computational fluency with all rational numbers (e.g., estimation,

mental math, technology, paper/pencil) (N-5-H) 6. Simplify and perform basic operations on numerical expressions involving

radicals (e.g., 2 3 5 3 7 3+ = ) (N-5-H) Algebra 8. Use order of operations to simplify or rewrite variable expressions (A-1-H) (A-

2-H) Data Analysis, Probability, and Discrete Math 15. Translate among tabular, graphical, and algebraic representations of functions

and real-life situations (A-3-H) (P-1-H) (P-2-H) 34. Follow and interpret processes expressed in flow charts (D-8-H)

Sample Activities Activity 1: Relationships in the Real Number System (GLE: 1) Review the real number system and have students draw a Venn diagram showing how the various sets of numbers within the real number system are related. Students should write statements that describe the relationships shown (e.g., every whole number is an integer; not all real numbers are rational) or determine the truthfulness of statements provided by teacher. Activity 2: Understanding Rational and Irrational Numbers (GLEs: 1, 4) Using calculators, let students explore the difference between rational and irrational numbers. For example, have students input several rational numbers using the division key, and discuss why some numbers are finite (e.g., 4

5 , 38 , and 15

32 ), while other rational numbers have non-terminating decimals that repeat (e.g., 11

12 and 711 ). Discuss how

different calculators handle these types of numbers (i.e., rounding off the last digit). Have students input several irrational numbers and let students see that although the numbers appear to terminate on the calculator, the calculator is actually rounding off the last digit. Activity 3: Estimating the Value of Square Roots (GLEs: 1, 4) Discuss with students how to estimate the value of irrational numbers involving square roots. Have the students determine which two whole numbers a particular square root would fall between. For example, if students know the square root of 49 is 7 and the square root of 64 is 8, then the square root of 51 would be between these two values (it would actually be closer in value to 7 than to 8, so an even better estimate might be 7.1). Once a thorough discussion takes place about estimation techniques, provide students


with an opportunity to use their estimation skills by providing students with 10 irrational numbers using square roots and have them find their approximate values. After students obtain their approximate values, have students explain their reasoning to the class. After the discussion has taken place, have students check their estimates with a calculator using the square root key. Activity 4: Naming Numbers on a Number Line (GLE: 1) Pair students and have each team construct a number line showing the integers from –4 to +4. Also, have students identify and tag the halfway points between each pair of integers (e.g., –3 1

2 , –2 12 ). Next, have them identify and tag –π , – 3 , – 2 , 2 , 3 , π . To the

side of the number line and on the first line under the number line, have the students write Natural Numbers. On the next line, have them write Whole Numbers. Continue this process with Integers, Rational Numbers, and Irrational Numbers. Once the student pairs have completed this chart, have them identify the sets of numbers that include each number selected on the number line by placing an “x” in the appropriate location. For example, –2 would receive an “x” in Integers and Rational Numbers, while +2 would receive an “x” in all rows except Irrational Numbers. Repeat this activity several times using different intervals on the number line. Activity 5: Many Ways to Solve a Problem (GLEs: 4, 5) Review paper and pencil operations with rational numbers (addition, subtraction, multiplication, and division) and include in the discussion the appropriate use of calculator technology (how the calculations can be done using calculators). Discuss estimation strategies with respect to operations with rational numbers. Afterwards, have students construct a five-column chart. The column titles will be: Problem, Estimation, Mental Math, Technology, and Paper/Pencil. Develop 5 to 10 problems, some of which are best done using a calculator, some by mental math, some by estimation, and some by paper-pencil. Place them in the Problem column. Have the students attempt the problems using each of the methods listed. Lead a discussion of when each computational strategy was employed. For example, when was mental math the quickest way to solve the problem? When did a calculator prove to be useful? When was estimation used? What paper/pencil strategies were used? Activity 6: Using Exponents In Prime Factorization (GLE: 2) Review the prime factorization process with whole numbers, and include in the discussion how exponents can be used to rewrite a particular prime factorization. Allow students the opportunity to prime factor different numbers using factor trees, and then have them write the numbers in factored form using exponents. Make sure that appropriate terminology is also introduced. For example, have students write 136 as the product of primes. Discuss how the prime factorization process can be used to find the


greatest common factor or least common multiple for a set of numbers; then have students write and solve problems they create which might involve finding the greatest common factor or least common multiple for a real-life situation. For example, suppose there are 45 oranges and 36 apples on a produce stand. The produce clerk wants to bag the oranges and apples in separate bags so that each bag contains the same number of fruit. What is the most fruit that can be put into one bag? Solution: 9 fruit per bag Activity 7: Using Exponents In Scientific Notation (GLEs: 2, 3) Relate the use of exponents to scientific notation (e.g., 3.2 x 10³ = 3.2 x10x10x10=3200). Include using negative exponents in the discussion. Discuss how to perform operations involving scientific notation using a calculator. Have the class research (using Internet or encyclopedias) why scientific notation was created and its benefit in science and then write a report on their findings. Activity 8: Changing Forms (Exponents) (GLEs: 2, 3, 5) Provide students with a worksheet that contains a three-column table. One column should be labeled Expanded Form, the second column Scientific Notation, and the third column Simplified Form. Provide some numerical expressions in the Exponential Form column or the Expanded Form column. Have students calculate the Simplified Form for each expression and also write the expression for the other missing form (e.g., 6.3 x 10 x 10 x 10 in Expanded Form would be written as 6.3 x 10³ in Scientific Notation and 6300 in Simplified Form). Be sure to include numerical expressions that include negative exponents. Activity 9: Astronomical Measurement (GLEs: 2, 3, 4) Have students use the Internet to find the following astronomical distances: Earth to the sun; Earth to the moon; Earth to Mars; Earth to Pluto; the sun to Pluto. Also, have students determine the speed of light as well as the maximum rate of speed that one of our manned spacecraft can fly in space. Once students find this information, have them write the distances and the speeds in both simplified form and scientific notation. Ask students if they think the values are exact or approximations and explain why they think so. Activity 10: Using Scientific Notation to Solve Problems (GLEs: 2, 3) Using the information they found in Activity 9, have students get in groups and determine the answer to the following questions: How much time does it take for the light from the sun to reach the Earth? How much time does it take for the light from the sun to reach the planet Pluto? If a manned space flight went from the Earth to Mars, how


long would it take to get there? If a manned space flight went from the Earth to Pluto, how long would it take to get there? Provide students with calculators for calculations. Activity 11: Order of Operations Activity (GLEs: 2, 5, 8) Review with students the order of operations with numerical applications. Explain why order of operations is necessary. Use different calculators, if possible, to point out to students how certain calculators perform the order of operations differently. Include in the discussion the importance of using parentheses when inputting data; e.g., (4+5)/(6+2) would be a different result than 4+5÷6+2, and how exponents are entered in their calculators (e.g., 3³- 4(3 + 6)). After the discussion, provide students with a worksheet and have them find the result of numerical expressions that require the use of order of operations. Make sure that the use of parentheses and exponents is included. Activity 12: Flow Charts—Not Just for Computers (GLEs: 2, 5, 8, 34) Have students design a flow chart that demonstrates how to evaluate an expression using the order of operations. If students are not familiar with flow charts, provide a sample for them to follow. To design their flow chart, have students use the following procedures: questions go in the diamonds; processes go in the rectangles; yes or no answers go on the connectors. Students should be given several numerical expressions that involve powers, parentheses, and several operations. Have students share their flow charts with each other and use the flow chart to simplify the expressions. Activity 13: Writing Algebraic Expressions (GLE: 8) Have students come up with algebraic expressions for different situations. For example, have students write an expression for the perimeter of a square if each side is p units long; write an expression for the total weight of 24 cans of soft drink if each weighs k ounces; write an expression for the distance someone would travel if he/she went 40 miles per hour for t hours; etc… Discuss simplifying algebraic expressions and combining like terms. For example, if a square has sides p units long, the perimeter can be expressed as p + p + p + p or 4p units in length. Provide additional examples for students to become proficient at combining like terms. Activity 14: Evaluating Expressions and Using Geometry Formulas (GLEs: 5, 8) Have students evaluate algebraic expressions. Be sure to include practice with fractions, decimals, and integers as well as whole numbers. Include a review of the formulas for volume and surface area of prisms and cylinders learned in grade 8. Review the figures and their formulas before beginning such an activity and provide diagrams or models of figures from which students can obtain values to be used in the formulas. If possible,


have several 3-D figures (boxes, cans, etc.) for students to actually measure to obtain the values needed to plug into the formulas. As part of the review, include an analysis of the derivation of the formula (e.g., the surface area of a cylinder is 2π r2 + Ch because the faces are composed of two circles and a rectangle whose dimensions are the circumference of the circle and the height of the cylinder). Activity 15: Working with “ ” (GLEs: 5, 6, 8) Have students solve for missing side lengths in right triangles using the Pythagorean theorem, which should have been taught in grade 8. This will allow students to practice simplifying square roots. Make sure students understand that the Pythagorean theorem is only valid for right triangles. Have students solve problems in which the side lengths are square roots, whole numbers, fractions, etc. Have students find the areas of these right triangles to reinforce the area of a triangle formula. Activity 16: Simplifying Radical Expressions (GLEs: 6, 8) Relate simplifying variable expressions with simplifying numerical expression with radicals. For example, since the expression 2x + 4x can be simplified to 6x (because they are like terms), likewise, the expression 2√7 + 4√7 can be simplified to 6√7 (because they are also like terms). Review the process for simplifying radicals such as √56 to the form 2√14, and using a calculator show their equality with one another (both have a value of about 7.5). After a thorough discussion has taken place on the use of radicals, provide students with a worksheet or different polygon shapes in which the side length of the polygons are marked with lengths that are square roots; lengths that are whole number or fractional units; lengths using variables (such as k units); and figures having a combination of all of these. Have the students write expressions for the perimeter of each polygon, and have the students write each expression in simplified form. Activity 17: Patterns in the Real World (GLEs: 2, 8, 15) Provide students with number patterns from real life, including patterns involving exponents. Have students describe the pattern in words, and then have them write an algebraic expression to represent the nth term. For example, suppose a new pizza shop opens in a shopping center. At the end of each day, a running total is kept for the total number of pizzas sold since the shop opened. On the first day, the shop sells 1 pizza. On the second day, the shop sells 3 more pizzas. On the third day, the shop sells 5 more pizzas. On the fourth day, the shop sells 7 more pizzas. On the fifth day, the shop sells 9 more pizzas. Have students describe the pattern in words, and then have them write an


algebraic expression which could be used to express the number of pizzas sold on the nth day. Also have students determine an algebraic expression which could be used to find the total number of pizzas sold since the shop opened by the end of the nth day. Solution: To represent the number of pizzas sold on the nth day, the expression 2n-1 could be used. To represent the total number of pizzas sold since the shop opened, the expression n 2 could be used. Activity 18: Matching Real-life Situations and Their Graphs (GLE: 15) Provide students with numberless graphs and real-life situations that correspond to each graph. Have students match the graph with the situation. For example, have graphs of distance/time and relate the act of moving toward home and away from home on a given day in reference to the time during the day. Provide students with many different situations and graph types. Activity 19: Matching a Table of Values with a Graph (GLE: 15) Provide students with a table of values that resulted from a real-life situation, such as the price to rent a canoe at a resort in relation to the number of hours rented, and have students complete the table of values (fill-in the missing data in a table). After completing the table, have students determine which graph (provide different graphs from which students are to choose) best fits the data shown in the table. Have students explain why the graph they chose is the only graph that fits the data.

Sample Assessments General Guidelines Performance and other types of assessments can be used to ascertain student achievement. Following are some examples: General Assessments

• The student will make a portfolio containing samples from various activities. • The student will keep a “Learning Log” about the ideas and processes that are

taught in class. The student can use the log as information as a study guide, but is also a good source of feedback to the teacher concerning questions the student has on a particular topic. Each week, the teacher picks up the learning log and examines it.


• For selected activities, the student will show his/her work, and use the work for assessment purposes.

Activity-Specific Assessments

• Activity 4: The student will put 15 numbers on a number line ranging from –10 to 10. The teacher will provide a list of numbers from each type (natural, whole, integer, rational, and irrational) for the student to graph and have the student identify which subsets each number belongs to.

• Activity 5: The student will write his/her own problems that could best be solved using each technique (paper and pencil, estimation, technology, and mental math) along with an explanation of why this would be the best approach.

• Activity 10: The student will write explanations (mathematical and verbal

explanations) of how the answer was found to each of the questions presented in the activity.

• Activity 16: The student will draw a flow chart for simplifying a square root.

Algebra I: Part 1 Unit 2 Measurement and Geometry 9

Algebra I: Part 1 Unit 2: Measurement and Geometry

Time Frame: Approximately 3 weeks Unit Description This unit examines the relationship between precision, accuracy, significant digits, and error analysis when dealing with measurement. The unit also includes a review of the measurement formulas and properties of 2- and 3-dimensional figures (e.g., perimeter/circumference, area, surface area, and volume) from work done in grade 8. Student Understandings Students should be able to find the precision of an instrument and determine the accuracy of a given measurement. Students should also recognize the relationships between 2- and 3-dimensional figures (e.g., a square pyramid is one square and four congruent triangles; a parallelogram can be rearranged to form a rectangle whose length and width correspond to the base and height of the parallelogram). Guiding Questions

1. Can students determine specified perimeters, circumferences, areas, surface areas, and volumes given access to the appropriate formulas?

2. Can students relate the various properties of 2- and 3-dimensional figures and apply them in finding and comparing measurements of objects?

3. Can students determine the precision of a given measurement instrument? 4. Can students determine the accuracy of a measurement? 5. Can students describe the bounds on a true output measurement, given the

accuracy of the input measures? 6. Can students use indirect measurement to find the measure of something very

small in size? Unit 2 Grade-Level Expectations (GLEs)

GLE # GLE Text and Benchmarks 4. Distinguish between an exact and an approximate answer, and recognize

errors introduced by the use of approximate numbers with technology (N-3-H) (N-4-H) (N-7-H)

5. Demonstrate computational fluency with all rational numbers (e.g., estimation, mental math, technology, paper/pencil) (N-5-H)


GLE # GLE Text and Benchmarks 17. Distinguish between precision and accuracy (M-1-H) 18. Demonstrate and explain how the scale of a measuring instrument

determines the precision of that instrument (M-1-H) 19. Use significant digits in computational problems (M-1-H) (N-2-H) 20. Demonstrate and explain how relative measurement error is compounded

when determining absolute error (M-1-H) (M-2-H) (M-3-H) 21. Determine appropriate units and scales to use when solving measurement

problems (M-2-H) (M-3-H) (M-1-H) 22. Solve problems using indirect measurement (M-4-H)

Sample Activities

Activity 1: What Does it Mean to be Accurate? (GLEs: 4, 17) Talk with students about the meaning of “accuracy” in measurement. Accuracy indicates how close a measurement is to the accepted “true” value. For example, a scale is expected to read 100 grams if a standard 100 gram weight is placed on it. If the scale does not read 100 grams, then the scale is said to be inaccurate. If possible, obtain a standard weight from one of the science teachers along with several scales. With students, determine which scale is closest to the known value and use this information to determine which scale is most accurate. Next, ask students if they have ever weighed themselves on different scales—if possible, provide different scales for students to weigh themselves. Depending on the scale used, the weight measured for a person might vary according to the accuracy of the instruments being used. Unless “true” weight is known, it cannot be determined which scale is most accurate (unless there is a known standard to judge each scale). Generally, when a scale or any other measuring device is used, the readout is automatically accepted without really thinking about its validity. People do this without knowing if the tool is giving an accurate measurement. Also, modern digital instruments convey such an aura of accuracy and reliability (due to all the digits it might display) that this basic rule is forgotten—there is no such thing as a perfect measurement. Digital equipment does not guarantee 100% accuracy. Have all of the students who have watches to record the time (to the nearest second) at the same moment and hand in their results. Post the results on the board or overhead—there should be a wide range of answers. Ask students, Which watch is the most accurate? Students should see that in order to make this determination, the true time must be known. Official time in the United States is kept by NIST and the United States Naval Observatory, which averages readings from the 60 atomic clocks it owns. Both organizations also contribute to UTC, the world universal time. The website http://www.time.gov has the official U.S. time, but even its time is “accurate to within .7 seconds.” Cite this time at the same time the students are determining the time from their watches to see who has the most accurate time. Ultimately, students need to understand that accuracy is really a measure of how close a measurement is to the “true” value. Unless the true value is known, the accuracy of a measurement cannot be determined.


Activity 2: How Precise is Your Measurement Tool? (GLE: 4, 17, 18) Discuss the term “precision” with the class. Precision is generally referred to in one of two ways. It can refer to the degree to which repeated readings on the same quantity agree with each other. Precision can also be referred to in terms of the unit used to measure an object. Precision depends largely on the way in which the readings are taken—how much care was taken by the person making the measurement, the quality of the instrument, attentiveness of the observer, stability of the environment in which the measurements were taken, etc… Some limitations that hinder the precision of a measurement include the skill of the reader, the way the ruler was placed, whether or not it was viewed at an angle, and so on. Help students to understand that no measurement is perfect. When making a measurement, scientists give their best estimate of the true value of a measurement, along with its uncertainty. The precision of an instrument reflects the number of digits in a reading taken from it—the degree of refinement of a measurement. Discuss with students the degree of precision with which a measurement can be made using a particular measurement tool. For example, have on hand different types of rulers (some measuring to the nearest inch, nearest 1

2 inch, nearest 14 inch, nearest 1

8 inch, nearest 116 inch, nearest centimeter, and

nearest millimeter) and discuss with students which tool would give the most precise measurement for the length of a particular item (such as the length of a toothpick). Have students record measurements they obtain with each type of ruler and discuss their findings. Help students understand that the ruler with the smallest markings will provide the most precise measure, but even it has inherent limitations. Set up measurement stations throughout the class for students to determine the attributes of different items. Include measurements with weight using scales (both in lbs, ounces, and grams), length of items (include diameter of a sphere), and areas of objects which have the shapes of simple 2-D figures (rectangle, circle, parallelogram) and have students measure the appropriate lengths with which to calculate the areas. Discuss the results as a class including sources of error that could account for discrepancies in answers. Activity 3: Temperature—How Precise Can You Be? (GLEs: 4, 18, 17) Have students get in groups of three. Provide each team with a thermometer that is calibrated in both Celsius and Fahrenheit. Have each team record the room temperature in both oC and oF. Have students note the measurement increments of the thermometer (whether it measures whole degrees, tenths of a degree, etc.) on both scales. Make a class table of the temperatures read by each team. Ask students if it is possible to have an answer in tenths of a degree using their thermometers and why or why not?


Activity 4: Precision vs. Accuracy (GLE: 17) In this activity, provide students with data tables showing measurements taken, and have students answer questions regarding precision and accuracy, and have them distinguish between the two. For example, provide students with the data tables shown below and have them answer the questions provided. Example 1: Using the table below, answer the following questions. Assume that each data set represents 5 measurements taken from the same object.

• Which of the following sets of data is more precise, based on its range? (Solution: Data Set A has a range of .06 while Data Set B has a range of .08, thus the more precise data set is Set A.)

• Do you know which data set is more accurate? Explain. (Solution: There is no way of knowing which is more accurate since in both cases there is no indication of the true measure of the object being measured.)

Set A Set B 14.32 36.56 14.37 36.55 14.33 36.48 14.38 36.53 14.35 36.55

Example 2: The data tables below show measurements that were taken using three different scales. The same standard 100 gram weight was placed on each scale and measured 4 different times by the same reader using the same method each time.

Trial # Weight on Scale 1 Weight on Scale 2 Weight on Scale 3 1 101.5 100.00 100.10 2 101.5 100.02 100.00 3 101.5 99.99 99.88 4 101.5 99.99 100.02

Average Weight

• Determine the average weight produced by each scale. Use this average as the

actual weight of the 100g mass determined by each scale. Write down the results for each scale used. (Solution: Scale 1: 101.5g; Scale 2: 100.00 g; Scale 3: 100.00 g)

• Which scale was the most precise? Explain how you know. (Solution: Scale 1 since the range of values is smaller than in the other scales.)

• Which scale was the least precise? Explain how you know. (Solution: Scale 3 since the range of values is larger)

• Which scale was the most accurate if we consider the true value of the weight to be 100 grams? Explain your answer. (Solution: If we look at the average weights to be the weight given by each scale, then both Scale 2 and Scale 3 are equally accurate.)


Example 3: Below is a data table produced by 4 groups of students who were measuring the mass of a paper clip which had a known mass of 1.0004 g.

• Determine the average weight produced by each group’s measurements and fill in the results in the table. Use this average as the weight of the paper clip for each group. (Solution: Group 1: 1.01 g; Group 2: 3.601267 g; Group 3: 10.13255g; Group 4: 1.01g)

• Which of the group’s measurements represents a properly accurate and precise measurement of the mass of the paper clip? (Solution: Both Group 1 and Group 4 had an average weight in line with the true weight of the mass; however, Group 4 did not have a precise measurement—the readings have too wide a range . The average just happened to come out to a value close to the true weight; therefore, only Group 1 data represents both an accurate and precise measurement.)

• Which of the group’s measurements was the least accurate? Explain why. (Solution: Group 3 had the least accurate answer for the weight of the paper clip since its average value is farthest from the actual value of the paper clip.)

• Which of the group’s measurements had an accurate answer, but not a precise answer? Explain. (Solution: Group 4 had an accurate weight (if the average is used) but was not precise at all.)

Trial # Group 1 (g) Group 2 (g) Group 3 (g) Group 4 (g) 1 1.01 3.863287 10.13252 2.05 2 1.03 3.754158 10.13258 0.23 3 0.99 3.186357 10.13255 0.75

Average Weight

Activity 5: Finding a Range of Values for a Measurement—Absolute Error (GLEs: 18, 20) The uncertainty or error associated with any measurement depends on the measurement tool being used. For example, if the mass of a sample is given as 342 4± mg, the actual value for the mass is somewhere between 338 mg and 346 mg. The reason for the variation may be due to the measurement tool’s being unable to sense any changes in mass less than 4 mg (i.e., the pan balance doesn’t move for such small changes). This is what is referred to as “absolute error.” Discuss with students what absolute error is and how to determine the error of a particular measurement tool. For example, suppose a ruler only measures to the nearest 1

2 inch, and you take a measurement of an item that lies somewhere between 1

23 inches and 4 inches. Since it is closer to 123 inches, you write

this as the length of the item. However, the actual measurement could have been up to 14

inch longer (half of the smallest division associated with the ruler). If someone reads the measurement of 1

23 inches, they have no idea how much error was associated with the


measurement. A more scientific approach to listing the measure would be to write 1 12 43 ±

inch. Doing so allows the reader to know that the actual length of the item may be anywhere from 1

43 inches to 343 inches in length. Discuss the idea of absolute error and

provide students the opportunity to write such error measurements and determine the range associated with different measurement tools. Have students use different rulers (some measuring to the nearest inch, nearest 1

2 inch, nearest 14 inch, nearest 1

8 inch, nearest centimeter, and nearest millimeter) to measure various items around the room (i.e., table length, chair height, length of a sheet of paper, height of door, etc.). For each measurement, have the students express their measurement along with the range associated with the error. Activity 6: What is My Exact Height? Absolute vs. Relative Error (GLEs: 4, 5, 20) After having discussed absolute error, it is important to talk about a better indication of how accurate a measurement is—a different type of error measurement called “relative error.” An accurate measure is one in which the uncertainty is small when compared to the measurement itself. Thus, an uncertainty of ± 4 mg out of a total of 342 mg indicates much more accuracy than ± 4 mg out of a total of 12 mg. For this reason, uncertainty in measurement is often expressed as a percent of uncertainty. This is the relative error associated with the measurement. To determine the relative error, divide the absolute error by the calculated value, and then convert this decimal to a percent by multiplying by 100. For the examples above, 4

342 .011= which when multiplied by 100 gives a relative error of 1.1%. (Provide students with access to calculators to do this work.) Whereas, in the other measure, 4

12 .333= which when multiplied by 100 gives a relative error of 33.3%, that is a much higher percentage error, although the absolute errors are the same. Discuss with students how to determine the relative error associated with a measurement, and have students get in groups of 3. Provide two different types of measurement tools for students to make their measurements (a meter stick and a tape measure with English units). Direct two of the students to measure the height of the third, taking turns so that all students in each group are measured in both metric units and customary units. Then, have all three students determine their heights including absolute error and relative error in their measurements. Activity 7: What’s the Cost of Those Bananas? (GLEs: 4, 17, 18) The following activity can be completed as described below if the activity seems reasonable for the students involved. If not, the same activity can be done if there is access to a pan scale and an electronic balance. If done in the classroom, provide items for students to measure—bunch of bananas, two or three potatoes, or other items that will not deteriorate too fast. Have the students go to the local supermarket and select one item from the produce department that is paid for by weight. Have them calculate the cost of the object using the


hanging pan scale present in the department. Record their data. At the checkout counter, have the students record the weight given on the electronic balance used by the checker. Have students record the cost of the item. How do the two measurements and costs compare? Have students explain the significance of the number of digits (precision) of the scales. Activity 8: What are Significant Digits? (GLEs: 4, 19) Discuss with students what significant digits are and how they are used in measurement. Significant digits are those digits of a measurement that represent meaningful data. The more precision there is in the measurement, the more significant digits there will be. Practically speaking, measurements are made to some desired precision that suits the purpose of the person doing the measurement, which normally is determined by the limitations of the measurement instrument available. For example, suppose you measure a room to the nearest millimeter and find its length to be 7.089 meters, the 9 is the estimated digit in the measurement (remind students the absolute error will be 1

2 the smallest unit of the measurement tool, which in this case will be ±.5 mm or ± 0.0005 meters). There are four significant digits in all in the measurement. After fully discussing the concept of significant digits with students, provide them with opportunities to determine the number of significant digits given in a particular measurement. Plan with a science teacher if possible. Activity 9: Measuring the Utilities You Use (GLE: 19) Have students find the various utility meters (water, electricity) for their households. Have them to record the units and the number of places found on each meter. Have the class get a copy of their family’s last utility bill for each meter they checked. Have students answer the following questions: What units and number of significant digits are shown on the bill? Are they the same? Why or why not? Do your family pay the actual “true value” of the utility used or an estimate? If students do not have access to such information, produce sample drawings of meters used in the community and samples of utility bills so that the remainder of the activity can be completed. Activity 10: Calculating with Precision (GLEs: 4, 19) Discuss with students how significant digits are dealt with when making calculations. Students should understand that the precision that results from a calculation cannot be greater than the precision of any of the numbers used in the calculation. For example, consider a rectangle whose sides measure 9.7 cm and 4.2 cm. Calculating the area of the rectangle using multiplication brings (9.7cm)(4.2cm) = 40.74 sq. cm. Before now, students would probably write the result as 40.74 sq. cm., but a closer look shows the original side length measurements are only precise to the tenth of a centimeter, while the resulting calculation for area is precise to the hundredth of a square centimeter. To


correct this, the result should be rounded off so that it has the same precision as the least precise quantity used in the calculation. This rule is the equivalent of saying that making a calculation cannot improve on the precision of the numbers used based on the least number of significant digits in the factors. Therefore, in the example provided, a more trustworthy answer would be 41 sq. cm. When working with addition and subtraction, the result should be rounded off so that it has the same number of decimal places (to the right of the decimal point) as the quantity in the calculation having the least number of decimal places. After fully discussing calculating with significant figures, have students work computational problems (finding area, perimeter, circumference of 2-D figures) dealing with the topic of calculating with significant digits. Activity 11: Measurement Problems with 3-D Objects (Physical Dimensions) (GLEs: 4, 18, 19, 20, 21) Calculating the surface area and volume of 3-dimensional figures can be connected to precision and accuracy. If one measures length to nearest the tenth, then the area of a figure is only correct to nearest tenth; likewise, volume cannot be more precise than to the nearest tenth. Making a calculation cannot improve on the precision of the numbers used in the calculations. For this activity, provide students with a rectangular prism and a cylinder. Have students determine the surface area and volume for each figure. Group students in teams of three students per group, and provide students with a metric ruler accurate to the nearest millimeter to make their measurements. Have students use significant digits in their calculations. Let student groups report their findings with answers only as precise as their measurement tools allow. Activity 12: How thick is a Sheet of Paper? (GLE: 22) It is often necessary to use an indirect measurement technique when measuring very large or very small things. For example, when measuring the thickness of a single sheet of paper, a ruler will not work. Present this dilemma to students and have them brainstorm ideas on how they could find the thickness of a sheet of paper by measurement. Indirect measurement could help. If one measures the thickness of 400 sheets of paper (suppose these 400 sheets measured 40 mm) then divides that total measurement by the 400 sheets (40mm ÷ 400 sheets = 0.1 mm), the result will be the total thickness of a single sheet of paper. Provide calculators for students to do their computations. Activity 13: What is the mass of a single grain of rice? (GLE: 22) Have students come up with ideas on how they could determine the mass of a single grain of rice, and then actually perform the measurement using the approach they came up with. Afterwards, have each group report on their approach and on their findings. Discuss any discrepancies the students may have for the mass in the class.


Activity 14: Using Indirect Measurement to determine the height of a Telephone Pole (GLEs: 4, 22) Review with students the use of proportions when solving similar triangle problems. This topic should have been taught in grade 8. After reviewing the topic of using proportions to solve such problems, present students with this challenge: Find the height of a telephone pole without actually measuring the pole. Have students get in groups to come up with ideas. If no one discusses it, talk about how shadows are formed when the sun is shining at an angle with an object (shadows are longer during the evening hours and shorter during noon). Discuss how comparing the height of a person to the length of the person’s shadow could be used to calculate the height of the pole if the length of the pole’s shadow is known. Have students do this activity either at school or at home, and make sure students make their measurements later in the day when a shadow is formed (not at noontime). Activity 15: Will It Fit? (GLE: 22) Make teams comprised of three students each to work on the following activity. Provide the teams with appropriate measurement equipment and lab materials (small glass or plastic graduated cylinder with milliliter marks, eyedroppers, standard measuring cup, and water). Have students first use what they learned about indirect measurement to determine the volume (in milliliters) of a single drop of water. Have students come up with a plan using the equipment provided to determine the volume in milliliters of a single drop of water. Once they determine this, ask them to determine if a standard measuring cup will fit 10,000 drops of water. Students must show all their work, how they obtained their answers, and make a presentation before the class to discuss their results. One possible way of doing this would be to see how many drops of water it would take to make 10 mL, then determine the volume of a single drop of water. Then have students find out how many milliliters it takes fill a standard measuring cup and use all of the information to obtain their results.

Height of Pole

Length of Shadow

Height of Person

Length ofShadow

Height of Pole Height of Person ----------------- = ---------------------- Length of Shadow Length of Shadow Of Pole of Person


Sample Assessments

General Guidelines Performance and other types of assessments can be used to ascertain student achievement. Following are some examples: General Assessments

• Each student will come up with real-world examples to make measurements

(e.g., measuring a room for carpeting, measuring a room for painting, etc.) and perform the necessary calculations, including calculations with error.

• The student will write a short paragraph explaining the difference between accuracy and precision.

• The teacher will provide three to five significant-digit problems for the student to solve and then assess the work.


• Activity 6: The student will write an explanation of how to obtain his/her height and record the error associated with the measurements.

• Activity 7: The teacher will Assign activity 7 as an out-of-class assignment. The student will write a report explaining the procedures followed, along with the measurements and calculations. In the report, the student will explain the effect of the measurement tools used in calculating the answers.

• Activity10: The teacher will pick up the calculations and explanations from student work when measuring and calculating area, perimeter, and circumference of geometric figures.

• Activity 11: The student will write a report on his/her findings after performing this activity including a discussion of how the answers were obtained and the procedures used to arrive at the calculations.

Algebra I: Part 1 Unit 3 Solving Equations and Real-life Graphs 19

Algebra I: Part 1 Unit 3: Solving Equations and Real-life Graphs

Time Frame: Approximately four weeks Unit Description This unit focuses on using algebraic properties to solve algebraic equations. The relationship between a symbolic equation, a table of values, a graphical interpretation, and a verbal explanation is also established. Student Understandings Students can solve linear equations graphically, from tables, with symbols, and through verbal and/or mental mathematics sequences. Students use real-life graphs to learn about independent and dependent variables, slope as a “rate of change,” and inverse and direct variation. Guiding Questions

1. Can students perform specified real-number calculations and relate their solutions to properties of operations?

2. Can students solve equations using addition, subtraction, multiplication, and division with variables?

3. Can students solve linear equations with rational (integral, decimal, and fractional) coefficients and relate the solutions to symbolic, graphical, and tabular/numerical representations?

4. Can students solve problems involving proportions and percentages? 5. Can students distinguish the difference between independent and dependent

variables in a real-life situation? 6. Can students understand how slope of a graph relates to a rate of change in a

real-life situation? 7. Can students distinguish between a direct or inverse relationship when

analyzing a graph?


Unit 3 Grade-Level Expectations (GLEs)

GLE # GLE Text and Benchmarks Grade 8 (reinforcement) Algebra 12. Solve and graph solutions of multi-step linear equations and inequalities (A-2-

M) Grade 9 Number and Number Relations 4. Distinguish between an exact and an approximate answer, and recognize errors


7. Use proportional reasoning to model and solve real-life problems involving direct and inverse variation (N-6-H)

Algebra 8. Use order of operations to simplify or rewrite variable expressions (A-1-H) (A-

2-H) 10. Identify independent and dependent variables in real-life relationships (A-1-H) 15. Translate among tabular, graphical, and algebraic representations of functions

and real-life situations (A-3-H) (P-1-H) (P-2-H) 16. Interpret and solve systems of linear equations using graphing, substitution,

elimination, with and without technology, and matrices using technology (A-4-H)

Geometry 23. Use coordinate methods to solve and interpret problems (e.g., slope as rate of

change, intercept as initial value, intersection as common solution, midpoint as equidistant) (G-2-H) (G-3-H)

24. Graph a line when the slope and a point or when two points are known (G-3-H)25. Explain slope as a representation of “rate of change” (G-2-H) (A-1-H) Data Analysis, Probability, and Discrete Math 34. Follow and interpret processes expressed in flow charts (D-8-H) Patterns, Relations, and Functions 37. Analyze real-life relationships that can be modeled by linear functions (P-1-H)

(P-5-H)

Sample Activities Activity 1: Review of basic concept of solving equations (GLE: 12—8th grade) Students coming into 9th grade should be very familiar with solving simple, one-step algebraic equations mentally. Review with students the basic premise behind solving simple equations building on the idea of equations as a “balance scale” and discuss the method of solving equations by keeping both sides of the equation balanced. Have students solve a variety of real-life problems involving simple algebraic equations.


Activity 2: Solving more complex equations (GLEs: 12—8th grade, 8—9th grade) Discuss the methods associated with solving more complex equations with multiple steps that incorporate using the properties of equality (reflexive, symmetric, transitive, and substitution) to obtain a solution. Require students to solve equations which cannot be done mentally and have students show the steps used when solving the equations. Expose students to techniques involved with solving equations that include integral and rational coefficients. Include real-world problem solving that involves writing and solving algebraic equations (e.g., perimeter applications, area problems, sum of angles in a polygon, distance/time relationships, percent increase/decrease, proportions). Activity 3: Flow Charts for Solving Equations (GLEs: 8, 34) Have students create a flow chart for solving equations. Assist students as necessary. Once the students have their flow charts developed, ask questions as they go through the flow chart steps with a practice problem. Repeat this activity several times by providing equations for the students to solve via the steps in the flow chart. Activity 4: Independent vs. Dependent Variable (GLE: 10) Discuss the concept of independent and dependent variables in reference to real-world examples. For example:

• The area of a square depends upon its side length • The distance a person travels in a car depends upon the car’s speed and the length

of time it travels • The cost of renting a canoe at a rental shop depends on the number of hours it is

rented • The number of degrees in a polygon depends on the number of sides the polygon

has • The circumference of a circle depends upon the length of its diameter • The price of oil depends upon supply and demand • The fuel efficiency of a car depends upon the speed traveled • The temperature of a particular planet depends on its distance away from the sun

Present students with ten different pairs of variables used in real-world contexts and have the students work in groups to determine which of the variables is the dependent variable and which is the independent variable. Discuss each situation as a class. Explain that a two-dimensional graph results from the plotting of one variable against another. For instance, you might plot the concentration in a person’s bloodstream of a particular drug in comparison with the time the drug has been in the body. One of these variables is the dependent and the other the independent variable. The independent variable in this instance is the time after the drug is taken, while the dependent variable is the thing that you measure in the experiment—the drug concentration. Explain to students that conventionally the independent variable is plotted on the horizontal axis (also known as


the abscissa or x-axis) and the dependent variable on the vertical axis (the ordinate or y-axis). Relate this all pictorially with graphs. Activity 5: Graphing from a Table of Values (GLEs: 4, 10, 15, 24) Have students construct a graph on a coordinate grid given a table of values that form a line. For example, using the table shown below displaying the oil being pumped from a well in relation to the number of days the well is operated. Let students first determine which is the dependent and which is the independent variable and have them use this information to appropriately graph the data on a coordinate grid (dependent variable on vertical axis and independent variable on horizontal axis). Afterwards, discuss what pattern students see in the data (i.e., It appears to form a linear function.). Have students answer questions based on their conjectures by looking at the graph. For example, have students determine the amount of oil that was pumped after 4 days based upon the results of the graph. Ask them to determine if they think this answer is an exact value or an approximate value and why they think so. Next, have students run a line through the data and let them see that the line, which passes through all of the data points, extends through the origin. Talk about the initial value and intercepts in real-world terms. For this example, the line intercepts the graph at (0,0) which means that the number of barrels pumped is 0 barrels after 0 days. Number of Days Pump is On

2 5 9 10

Number of Barrels Pumped

12 30 54 60

Connect the paper and pencil work associated with this problem to using a graphing calculator to do the work. Demonstrate for students how to input data into lists, how to plot this data in a scatter plot, and then to determine a line of best fit for the inputted data. Activity 6: Direct and Inverse Relationships (GLE: 37) Discuss with students what is meant by the terms Directly Related and Inversely Related in the context of real-life situations at an elementary level. If two variables have a direct relationship, as one variable increases, the other will also increase in value. Likewise, as one variable decreases, the other also decreases in value. An example where a direct relationship exists is the cost to feed a family—as the number of members in the family increases, the cost to feed the family also increases. In contrast to the direct relationship, in an inverse relationship, as one variable increases, the other variable decreases, and vice versa. An example of an inverse relationship is the relationship that exists between the number of workers to do a job and the time it takes to finish the job. For instance, suppose it takes 6 workers 1 day to paint a house. If the number of workers decreased, the time it takes to do the job would increase (an inverse relationship). Discuss several real-


life examples with students; have students think of real-world examples of each, and then discuss as a class. Activity 7: Graphs and Direct and Inverse Relationships (GLEs: 10, 37) Provide students with several line graphs relating different quantities (taken from science books, business, and other applications). Have students work in groups to obtain the following information for each graph: 1) What is the independent variable? Dependent variable? 2) Does the graph portray a direct or inverse relationship between the variables? Explain your reasoning; and 3) Is the graph linear? Afterwards, talk with students about increasing and decreasing functions and how they are related to direct and inverse relationships. For example, in an increasing function, a direct relationship exists. In contrast, for a decreasing function, an inverse relationship exists. Relate all of this information graphically. Activity 8: Going on Vacation! (GLEs: 10, 15, 23, 24, 25, 37) Present students with the following situation:

Mr. Waters needs to rent a car to go on a trip he has planned. In order to rent the car, Mr. Waters will have to pay a flat fee of $45 plus an additional rate of $20 per day.

Using this information, have students work in groups to answer the following: • Which two variables are related in this situation? Which is the dependent

variable and which is the independent variable? Solution: The two quantities related in this situation are the cost in dollars to rent the car and the time in days for which the car is to be rented. The cost is the dependent variable since the cost depends on the number of days (time is the independent variable) the car will be rented for. • What is the cost for the car rental if Mr. Waters rents only 1 day? 2 days? 5

days? 9 days? Make a table of values relating the information and use the data to make a line graph. Label the graph appropriately with an appropriate scale and title.

Solution: The data table is shown below. Check students graphs. Days Rented (d)

1 2 5 9

Cost (C) 65 85 145 225 • Write an equation that relates the cost for renting a car for x days. Solution: C = $45 + $20x • Determine whether there exists a direct or inverse relationship between the

two variables in this situation, and explain how you determined your answer.

Solution: There is a direct relationship since the cost increases as the time increases. • Is the graph of the data you found linear? Explain.


Solution: Yes, the data is linear since it forms a straight line and has a constant increase. • Interpret the real-world meaning of the point that intercepts the vertical axis

of the graph you created. Solution: The point at which the graph intercepts the vertical axis is the initial cost to rent the car—the flat fee.

After students create charts and graphs using pencil and paper, demonstrate for students how to use a graphing calculator to do the assigned work by inputting the table of values into lists, how to create a scatter plot of the data, and then find an equation for the line of best fit. Compare the equation that the calculator produces with the equation that the students came up with. Activity 9: Analyzing Distance/Time Graphs (GLEs: 10, 25, 37) Present the three graphs shown below. Each graph displays the distance each of three different cars traveled over a certain time period. Have students analyze the graphs and discuss the following aspects related to the graphs:

• Identify the dependent and independent variables for each graph. Solution: The independent variable is the time and the dependent variable is the distance. • Determine which car was traveling fastest and which car was traveling

slowest and explain how this relates to the steepness of the graph. Solution: Fastest car is B; slowest car is A. The steeper the graph is the faster the speed. • Determine the rate of speed for each of the three cars and explain how you

obtained your answers. Solution: Car A (5 mph); Car B (12.5 mph); Car C (10 mph); check students explanations. • Relate slope of a line with the concept of a rate of change. • Create a line graph of a car D that travels at a rate of 50 miles per hour for 4

hours and turn the graph in to the teacher. Solution: Check student graphs.

Car A Car B Car C 1 2 3 4

Time (hours)

50 40 30 20 10

Dis

tanc

e Tr

avel

ed

(mile

s)

1 2 3 4 Time (hours)

50 40 30 20 10

Dis

tanc

e Tr

avel

ed

(mile

s)

1 2 3 4 Time (hours)

50 40 30 20 10

Dis

tanc

e Tr

avel

ed

(mile

s)


Activity 10: Slope—What does it Tell Us About a Graph? (GLEs: 10, 15, 23, 24, 25, 37) In this activity, students will interpret the meaning of slope as a rate as it applies to a real-life situation. Provide students with the table of values shown below. Have students use the table of values to make a graph of the data. Ask students to select an appropriate scale for each axis. Remind students that a graph does not have to start at 0. Have students connect the data points with line segments, and then using the graph, have students determine the answers to the questions provided below. Afterwards discuss the results as a class.

• What two quantities are related in the graph that was drawn using the data? Describe the relationship in words.

Solution: Karl’s weight depends on the year he was weighed. • During which year(s) did John’s weight increase at the greatest rate? What

was this rate of increase? Explain how you determined this value. Solution: From 1991 to 1992, Karl’s weight increased at a rate of 4 pounds per year. • During which year(s) did John’s weight decrease at the greatest rate? What

was this rate of decrease? Explain how you determined this. Solution: Karl’s weight decreased at a rate of 6 pounds per year from 1995 to 1996. • Look at the graph and explain what the steepness of the segments on the

graph (the slope) tells us about the data in real-world terms. Solution: The steepness is associated with the rate of change of Karl’s weight gain or loss. A bigger rate of gain or loss is associated with a larger degree of steepness. • Does this graph represent a direct relationship, an indirect relationship, or a

combination of the two? Explain your answer. Solution: The graph shows a direct relationship when there is a weight gain as time increases, and an indirect relationship when there is a weight loss as time increases. Thus it shows a combination of the two.

Karl’s Weight (kg) 67 71 74 76 74 68 Year 1991 1992 1993 1994 1995 1996

Activity 11: When Will They Meet! (GLEs: 4, 10, 16, 23, 25, 37) In this activity, students will interpret the graph of a distance/time relationship and answer questions based upon the analysis of the graph. They will also use the point of intersection for two lines to answer a real-world problem. Present the following problem situation to students.

Lester left home at 9 a.m. one morning to go on a business trip. He immediately got on the interstate and drove at a constant rate of 65 mph. Assume Lester drove on a straight road with no traffic that would prevent him from having to slow down and that he had enough gas to travel for 8 hours without stopping. One hour after Lester left home on his business


trip, his wife, Gertrude, realized that he had forgotten his briefcase. She immediately got on the interstate and began to try to catch up to him, traveling at a constant rate of 75 mph in the process.

Have the students answer the questions below. • Which two quantities are related in this problem? Tell which quantity

represents the dependent variable and which quantity represents the independent variable.

• Make a data table that depicts the distance Lester is away after each hour from home. Use this data table to plot points to create a line graph showing the data. Is the data shown on the graph a direct or indirect relationship? Is it linear? Explain.

• Create a similar data table that depicts the distance Gertrude is from home during the same time frame. Remember, Gertrude left one hour later than Lester. Using this data, plot points on the same graph as Lester’s. Label each line to show the distinction between Gertrude and Lester. Use different colors to make the distinction.

• Using the double line graph, determine when (what time) and where (how many miles from home) Gertrude will finally catch up to her husband Lester?

• Look at the steepness of the line that represents Lester’s motion and compare it to the steepness of the line that represents Gertrude’s motion? How does the steepness of the line relate to the speed that they traveled? Explain.

Activity 12: Direct and Indirect Variation (GLE: 7) Discuss with students the difference between direct and indirect variation. When the quotient of two quantities is related by a constant factor, there is a direct variation between the two variables. For example, d

t r= represents a direct variation situation if the rate remains constant. The distance traveled divided by the time that is traveled represents this constant value. In more general terms, a direct variation is given by: y = kx. Provide students with examples and problems that represent a direct variation situation. In an indirect variation, the product of two quantities is a constant. For example, rt d= represents an indirect variation if the product of the rate and time represents a constant distance. Indirect variation is more commonly seen in math as given by the equation: ky

x= . Provide students with examples and problems that involve indirect

variation situations and discuss these with students. An activity which utilizes direct and indirect variation can be found at http://jwilson.coe.uga.edu/emt669/Student.Folders/Jeon.Kyungsoon/IU/rational2/Telescope.html. In this activity, students make a telescope using cardstock and perform investigations which demonstrate applications of direct and indirect variation.



• The student will review magazines, newspapers, journals, etc. (or recall

personal experiences) for real-world relationships that can be modeled by linear functions (include function and graph).

• The student will compile a portfolio of work for Unit 3 to be handed in for a grade.

• The student will draw numberless graphs that relate to a situation in real-life, explaining the graph in words and relating it to the motion or situation depicted.


• Activity 6: The student will write a short paragraph explaining what is meant by direct and indirect relationships and give examples.

• Activity 7: The student will write the relationship that exists between the variables for each graph provided him/her in the activity.

• Activity 11: The student will make an oral presentation of their findings and explain the processes that led to their conclusions.

Algebra I: Part 1 Unit 4 Linear Equations, Linear Inequalities and Graphing 28

Algebra I: Part 1

Unit 4: Linear Equations, Linear Inequalities, and Graphing Time Frame: Approximately five weeks Unit Description This unit focuses on developing an understanding of graphing linear equations and linear inequalities in the coordinate plane. Student Understandings Students develop an understanding of linear relationships including slope, simplifying linear expressions, and solving linear inequalities. Students recognize that graphing of a linear inequality is directly related to the process for graphing the associated equation and that any ordered pair in the shaded portion of the graph of an inequality is a solution to the inequality. Guiding Questions

1. Can students graph data from input-output tables on a coordinate graph? 2. Can students recognize linear relationships in graphs of input-output

relationships? 3. Can students graph the points related to a direct proportion relationship on a

coordinate graph? 4. Can students perform simple algebraic manipulations of collecting like terms

and simplifying expressions? 5. Can students determine the slope of a line given a graph or two points? 6. Can students graph a linear inequality in two variables and determine whether

a given point is a solution? Unit 4 Grade-Level Expectations (GLEs)

GLE # GLE Text and Benchmarks Algebra 8. Use order of operations to simplify or rewrite variable expressions (A-1-H)

(A-2-H)


GLE # GLE Text and Benchmarks 9. Model real-life situations using linear expressions, equations, and

inequalities (A-1-H) (D-2-H) (P-5-H) 10. Identify independent and dependent variables in real-life relationships (A-

1-H) 11. Use equivalent forms of equations and inequalities to solve real-life

problems (A-1-H) 12. Evaluate polynomial expressions for given values of the variable (A-2-H) 13. Translate between the characteristics defining a line (i.e., slope, intercepts,

points) and both its equation and graph (A-2-H) (G-3-H) 14. Graph and interpret linear inequalities in one or two variables and systems

of linear inequalities (A-2-H) (A-4-H) 15. Translate among tabular, graphical, and algebraic representations of

functions and real-life situations (A-3-H) (P-1-H) (P-2-H) 16. Interpret and solve systems of linear equations using graphing, substitution,


Geometry 24. Use coordinate methods to solve and interpret problems (e.g., slope as rate

of change, intercept as initial value, intersection as common solution, midpoint as equidistant) (G-2-H) (G-3-H)

24. Graph a line when the slope and a point or when two points are known (G-3-H)

25. Explain slope as a representation of “rate of change” (G-3-H) (A-1-H) 26. Perform translations and line reflections on the coordinate plane (G-3-H) Data Analysis, Probability, and Discrete Math 27. Determine the most appropriate measure of central tendency for a set of

data based on its distribution (D-1-H) 28. Identify trends in data and support conclusions by using distribution

characteristics such as patterns, clusters, and outliers (D-1-H) (D-6-H) (D-7-H)

Patterns, Relations, and Functions 37. Analyze real-life relationships that can be modeled by linear functions (P-

1-H) (P-5-H) 38. Identify and describe the characteristics of families of linear functions, with

and without technology (P-3-H) 39. Compare and contrast linear functions algebraically in terms of their rates

of change and intercepts (P-4-H) 40. Explain how the graph of a linear function changes as the coefficients or

constants are changed in the function’s symbolic representation (P-4-H)


Sample Activities Activity 1: Lines on a Plane (GLEs: 10, 12, 13, 15, 24) Have students make a xy (input/output) table of values to graph the following two equations: y = 2x and y = -2x on two different coordinate planes. Discuss the convention of using x values as the independent variable and the y values as the dependent variable when graphing general algebraic equations. Students should remember that both equations form a line, thus they are linear equations. Compare the characteristics for each graph—one is increasing and one is decreasing. Relate this with the concept discussed in a previous unit that dealt with direct and inverse variation. Finally, have students graph the two equations using a graphing calculator and discuss setting up a window to view graphs, as well as how to trace along the graphs using the trace feature. Activity 2: How Slope affects a Linear Graph (GLEs: 13, 23, 24, 25) Explain to students that when building a house, the pitch of a roof is usually given as a ratio between the rise of the roof and the run of the roof. Have students draw scale drawings of the following roof pitches on coordinate graph paper: Roof 1—rise 3 feet to a run of 1 foot; Roof 2—rise of 8 feet to a run of 2 feet; Roof 3—rise of 1 foot to a run of 1 foot; Roof 4—rise of 4 feet to a run of 1 foot. After students make sketches of each roof, discuss the steepness of each roof. Have the students compare which is the most steep/least steep, and have them notice the fact that Roof 2 and Roof 4 have the same steepness. Relate this activity to the fact that the steepness of a line segment (or roof) is really the slope of the line and is the ratio of rise/run. Discuss the fact that Roof 2 has a ratio of 8

2 or 4 and Roof 4 has a ratio of 41 or 4, hence the slope of both graphs is the same.

Explain that the larger the ratio is numerically, the steeper the slope of the line segment will be. Next, provide a worksheet for students to determine the slopes of several lines on a coordinate grid by using the rise/run. Activity 3: Slope as a Rate of Change (GLEs: 23, 25) Using graph paper, have students create a series of points relating cost to the number of items purchased. For example, explain to students that it costs $3 for each bottle of cola purchased for the school fair. Using this information, have students plot points representing the number of bottles purchased and the cost of the purchase (i.e., (1,3); (2,6); (3,9)) and connect them with a line. Have students use the y-axis to represent the cost and the x-axis to represent the number of colas purchased. After plotting at least 10 points on their graphs, start at the lowest plotted point and ask the students to describe the “travel” needed to proceed to the next point (i.e., up 3, over 1), then the next point, then the next, etc. Work through the process with students so that they see that up a number is the change in the y-values and over a number is the change in the x-values. Relate this movement with what was done with the roof activity. Students need to see that the slope


is constant throughout in a linear graph. Students should also understand that in this real-world example, the slope of 3 really represents the rate of change associated with the purchasing of cola—the cost per unit or cost per bottle. Use other real-life examples to stress the idea of slope as a rate of change such as miles/gallon. Finally, lead students to understand how the slope of a line can be calculated using the formula, 2 1

2 1

( )( )y yx xm −

−= . Assign various points from this situation to different students and have

them provide the calculations in finding slope. Query the students to see that everyone determined the slope equals 3

1 . Students should understand that for each unit of change in the x-coordinate, the y-coordinate is changed by 3 units when the slope is 3. Repeat this activity with other linear equations of the form y mx= where the slope is taken from a real-life application. Activity 4: Finding the Intercepts of a Linear Graph (GLEs: 13, 23, 38, 39) Use a graphing calculator to graph the lines y = 2x + 6 and y = -2x – 4 (separately—not on the same graph). Discuss with students the definition of intercepts (the points where the graph touches or intersects the x-axis and y-axis) and how to find the intercepts using both calculator technology and algebraically using paper and pencil. Provide students with the opportunity to become proficient at both approaches. Next, have students compare the slopes and intercepts for the two graphs. Have students explain how the graphs and equations are alike and how they are different from one another. Activity 5: How Does “m” Affect the Graph of an Equation in the Form of y = mx (GLEs: 13, 38, 39, 40) Provide students with calculators and have them work in groups in this activity. Discuss the formula d=rt which relates the distance traveled by an object as determined by the rate at which the object travels and the time it travels at this rate. Explain to students that instead of using the letters d=rt to model the formula, the form y=mx will be used in order to use the graphing calculator to graph. Students should see that the letters correspond with one another. In place of the variable r (the rate at which the object is moving) the variable m will be used. Have students use the graphing calculator to determine what the graph of y = 5x looks like. Relate this to moving at a rate of speed of 5 miles per hour. Students should discover that the graph of the equation forms a line with a specified slope. You may want students to find the slope (use the trace key to find the value of two points on the graph) and let them see that the slope is 5 for the graph. In addition, show students how the x and y values (which show up at the bottom of the calculator screen as you trace along the line) indicates the time and distance values. Point out the intercepts for the graph. Next, let students view the graph of y = 2x. Again, relate this to traveling at a speed of 2 miles per hour. Students should see the graph is again a line, the intercepts are the same, and that the only real difference between both equations is the slope or steepness of the line. Have students put several different values for m into the equation, including positive and negative values. Discuss the


findings with students recognizing that m is the slope of the line when written in the form y = mx. Students should see that positive values for m result in an increasing linear function, while negative values for m results in a decreasing linear function. Activity 6: Transformations of Figures on Graphs (GLEs: 23, 26) Have students transform figures using a coordinate grid. Discuss the different types of transformations—translation, reflection, and rotation. Provide students with figures on a coordinate grid, have them transform the figures in different ways using all three types of transformations, and give the new location for the vertices of the transformed figures. Limit work with rotations to rotating the figure about the origin using multiples of 90°clockwise or counterclockwise. Activity 7: Slope-Intercept Form of a Line (GLEs: 13, 38, 39, 40) From Activity 5, students should already understand that any equation in the form y = mx forms a line. In this activity, extend this concept to talk about the y = mx + b form of a linear equation. Students should discover that any equation in this form represents a line. Students should already understand how the value of m affects the slope of the line based upon what they learned in Activity 5. Now, build on that knowledge to include an understanding of the effect the value of b has on the line. Using the calculator, let students discover what happens to a line when the value of m is held constant and the value of b is changed. Starting with the equation y = 2x, have students discover what happens to the graph when you change the b value, such as y = 3x +1; y = 2x +2; y = 2x+3, etc… Have them use different values for b, including both positive and negative values. Students should see that the effect of changing b really translates the line y = mx up or down the y-axis by b units. Students should also recognize that the value of b is also the y-intercept for the line. Activity 8: Graphing Using a Point and the Slope (GLEs: 13, 24) This activity is designed to use what has been learned about slope in order to graph lines on a plane. First, demonstrate for students how any line can be graphed if the value of its slope and a point that lies on the line is known. Have every student plot the point (0,4) on a coordinate grid. Have students draw some possible lines that might pass through the point (0,4). Students should realize that an infinite number of lines actually run through the point (0,4). Next, using a new coordinate grid, tell students you want them to find the line that passes through (0,4) but has a slope of 3

2 . Let the students work in groups to find other points on the line graph using the slope of 3

2 . (Remind students of the relationship between slope and the rise/run). Have students list some other points and ask them how many lines run through the point (0,4) and have a slope of 3

2 . Students should come to the realization that there is really only one distinct line that has those characteristics. They should also realize that there are an infinite number of points that could be produced in


order to draw the line with these characteristics. Finally, have students graph other lines using a point and a slope with which to make their graphs. Activity 9: Is the Data Linear? (GLEs: 15, 25, 28, 37, 38, 39) In this activity, provide students with data from input/output tables, and have students determine, first without graphing, whether or not the data is linear. Explain to students that if data is truly linear, the change in y-values divided by the change in x-values for any two points should have a constant ratio—in other words, the slope should be the same throughout. For example, the data in the table 1 below is linear because there is a constant ratio of 4, while the data in table 2 is not linear. Afterwards, have the students graph the data points using a graphing calculator (using a scatter plot) and let them see the graphs. Provide additional examples for students to become proficient at this skill. Data Table 1 Data Table 2

x-value y-value x-value y-value 1 4 1 4 2 8 2 8 3 12 3 16 4 16 4 32 5 20

5 64 Activity 10: Same or Different? (GLEs: 8, 11) Provide students with several linear equations in two variables. Some of the equations should produce the same graphs. For example, have students make a table of xy values for the following two equations: y = -2x + 4 and 4x +2y = 8. After the table of values is found, have students graph the two lines using graph paper. Students should see that the two equations produce the same line. (The next activity will build on this by showing students how to transform an equation into different forms using algebraic techniques.) Prior to graphing more equations, have students work in pairs to predict which equations are really equivalent. For example, the sum of the measures of the interior angles any n-sided polygon can be expressed as a linear equation of the number of sides. One expression for this could be 180( 2)S n= − and another could be 180 360S n= − , where S is the sum and n is the number of sides. Activity 11: Equivalent Forms of Equations (GLEs: 8, 11) Refer to the equations from the previous activity: y = -2x + 4 and 4x +2y = 8. Remind students that both equations produced the same line. Explain that these two equations are equivalent forms of a linear equation. One of the equations is written in slope-intercept form, while the other is written in standard form. Discuss with students how to translate an equation from Ax + By = C (standard form) to y = mx + b (slope-intercept form) using


algebraic manipulation. Explain that the benefit of having an equation in y = mx + b form is the ease with which one can determine the slope and y-intercept and then use that information to sketch the graph. Provide students with the practice necessary to develop the skill of transforming an equation in standard form into slope-intercept form. At the same time, have students identify the y-intercept and slope for each equation and graph the line using paper and pencil. Activity 12: Modeling Functions (GLEs: 9, 10, 12, 15, 23, 24, 25, 37) To begin this activity, provide the following situation to students:

A family is going on a trip. They travel 8 hours a day, averaging 50 mph. It takes 3 days to get to their destination.

First, have students write an equation that matches the situation. Have students name any variables they use to make their models, and then construct a graph using paper and pencil to display the data. In the process, ask students to determine which of their variables represents the independent variable and which represents the dependent variable. In this case, the function used to model this situation is the distance traveled per day (d) equals the speed (50 mph) times the number of hours driven (h or t). So the equation they use to model the situation should be of the form 50d h= . The x-axis (independent) would be the number of hours driven (scale from 0 to 24 hrs) and the y-axis (dependent) would be the number of miles driven (scale from 0 to 1,200 miles). The graph would consist of a straight line starting at the origin (0,0) and going up and to the right, ending at the point (24, 1200). Have the students answer questions concerning the graph (e.g., How far does the family travel in 3 hours? answer = 150 miles; How many hours will it take to cover 200 miles? answer = 4 hours). Provide students with opportunities to learn that the function produces one output value for each input value. The Illuminations site at http://www.NCTM.org has a multitude of activities showing interesting real-life applications that can be modeled by linear functions. These activities involve the use of graphing calculators to enter data, find lines of best fit, and analyze the resulting graphs. Activity 13: Comparing Numberless Graphs and Equations (GLEs: 13, 38, 39) This activity builds on what was previously learned about slope, intercepts, and the slope-intercept form of an equation. Have students sketch a numberless graph that matches a given equation in slope-intercept form. For example, if students look at the equation y=3x-1, they should immediately understand that the graph has a slope of 3 and a y-intercept of -1, and should sketch a line which matches these characteristics. After they sketch their graphs, let the students compare their graphs with their fellow students, then have the students input the equation into a graphing calculator and compare their sketches with the graphs. Provide many different linear equations to allow for student proficiency at this skill. Next, show students different sketches, and have the students come up with a


possible equation that would match the given graph, and then compare their equations with their fellow students. Activity 14: Interpreting the Point of Intersection (GLEs: 9, 10, 16, 23, 24, 25, 37, 39) For this activity, an opportunity is provided for students to explore the real-world meaning of a point of intersection for two lines for a given situation. Present the following situation to students:

Mr. Moreau is opening up a pizza parlor. To begin his business, he first has to purchase a pizza machine for $50.00. In addition to this initial cost, each pizza costs $2.00 for the ingredients to make the pizza.

Have students answer the following questions and perform the following tasks.

• What is the total cost to make 1 pizza, including the initial cost of the machine? What is the total cost to make 2 pizzas? 3 pizzas? 10 pizza? x pizzas? Make a table of input out/put values to model this situation.

• Graph the data points using proper graphing techniques and use an appropriate scale for the graph. Is the data linear? Explain. If it is linear, label the line Cost.

• Write an equation which could be used to model the total cost to produce x pizzas. Explain what variable represents, what the independent and dependent variables are, and how you came up with your equation.

• If Mr. Moreau wants to make a profit in his business, should he sell his pizza for less than $2.00 per pizza, exactly $2.00 per pizza, or more than $2.00 per pizza? Explain your reasoning.

• Suppose Mr. Moreau sells his pizza for $8.00 per pizza, make a table of values to show how much money he would receive for selling 1 pizza? 2 pizzas? 3 pizzas? 10 pizzas?

• On the same graph you used to display the cost for the pizzas and using another ink color, plot the points showing the money Mr. Moreau collects (income or revenue) from the pizzas he sells. Is this data linear? If it is, connect the points to form a line through these points and label this line Revenue.

• How do the two lines (cost and revenue) compare with one another? Do they have the same slope? Do the two lines intersect? If the two lines intersect, what information does this point of intersection tell us?

• How many pizzas would Mr. Moreau have to sell before he starts making a profit? Explain how you know.

Have students work on this problem with their group members, and then discuss answers as a class. Students should discover that the point of intersection for this particular graph tells us where the cost and revenue are the same. At that point, the cost to make the pizzas is equal to the revenue that was made selling the pizzas. Before this point, the revenue is less than the cost. After this point, the cost is less than the revenue and a profit is made. In addition to these questions, ask students to find different information by analyzing the graph—such as, If Mr. Moreau sold 3 pizzas, did he make a profit or a loss? What is this


profit or loss? Students need to understand the value of making a graph—that in addition to being a visual representation of a problem, it provides a lot of information to the reader of the graph. This activity uses many of the concepts learned thus far with graphing and writing equations. More work will be done in later units on solving systems of equations. Activity 15: Is It a Trend? (GLEs: 27, 28) Have students measure the heights of everyone in the class (in groups of 3) and write their measurements on the board. Next, have the students display the results in a line plot. Identify patterns, clusters, and outliers, as appropriate. Have students determine the mean, median, mode, and range of the data collected, and have them decide which type of average best describes the average height of a student in the class and why. Activity 16: Linear Inequalities (GLE: 14) In this activity, students compare a linear equation with a linear inequality. In a linear equation, points that form the actual line are solutions to the linear equation. In a linear inequality, the points are found that make the inequality true. Have students graph the equation 3x + 4y = 12. Next discuss the difference and the similarity between this linear equation and the following linear inequalities:

• 3x + 4y > 12 • 3x + 4y ≥ 12 • 3x + 4y < 12 • 3x + 4y ≤ 12

Discuss how each graph differs—whether the line is dashed (if > or <) or solid (if ≥ or ≤), and where the shading is located (to the right or to the left of the line). To aid in helping students understand which side of the line gets shaded, have students pick points and replace their coordinates in the inequality to see if they makes the inequality true. If the particular points make the inequality true, then that is the side of the line that gets shaded. Students should understand that all points on the same side of the line will make the inequality true. Provide additional examples as necessary to help students become proficient at this. Include vertical and horizontal lines and inequalities such as x > 2 or y < 5. More will be done later with linear inequalities with two variables in Unit 8.

Sample Assessments General Guidelines Performance and other types of assessments can be used to ascertain student achievement. Following are some examples:


General Assessments

• The student will write a short paragraph explaining the connection between slope and rate of change.

• The teacher will project the following equations 2y x= − and 2y x= − + , have students determine slopes, x- and y-intercepts, and explain what is the same and what is different about the two graphs.

• The student will explain in writing the steps used to solve and graph a linear inequality.


• Activity 3: The student will determine three different types of rates that occur

in real-life that have not been discussed in class.

• Activity 4: The student will write a linear equation, find the x- and y-intercepts, and use them to graph the equation.

• Activity 5: The student will write a short paragraph explaining what has been learned in the activity.

• Activity 6: The student will draw an original polygon drawing, and show

transformations of all three types—translation, reflection, and rotation—on a coordinate plane. The student will explain what type of transformation was made in each case.

• Activity 9: The student will create two data tables—one that will produce a linear function and one that is not linear.

• Activities 10 and 11: The student will write two equivalent forms of an equation.

• Activity 12: The student will submit the graphs created in the activity.

• Activity 15: In a written report, the student will explain which average (mean, median, or mode) best represents the average height of a person in the class, and why this average is best. The student will also provide a copy of the line plot used for the report.

Algebra I: Part 1 Unit 5 Graphing and Writing Equations of Lines 38

Algebra I: Part 1 Unit 5: Graphing and Writing Equations of Lines

Time Frame: Approximately three weeks Unit Description In this unit, the emphasis is on writing and graphing linear equations in both real-life and abstract situations. The relationship between the values of coefficients in the linear equation and their effect on graphical features is reinforced. Student Understandings Students understand the meaning of slope and y-intercept and their relationship to the nature of the graph of a linear equation. They write equations of lines using the slope-intercept, two-point, point-slope, and standard form for the equation of a line. Given the equations, students can use the coefficients and intercepts to graph the lines. Guiding Questions

1. Can students write the equation of a linear function given appropriate information to determine slope and intercept?

2. Can students use the basic methods for writing the equation of a line (i.e., two-point, slope-intercept, point-slope, and standard form)?

3. Can students perform the algebraic manipulations on the symbols involved in a linear equation to find its solution and relate its meaning graphically?

4. Can students discuss the meanings of slope and intercepts in the context of an application problem?

Unit 5 Grade-Level Expectations (GLEs) GLE # GLE Text and Benchmarks Algebra 9. Model real-life situations using linear expressions, equations, and inequalities

(A-1-H) (D-2-H) (P-5-H) 10. Identify independent and dependent variables in real-life algebraic

relationships (A-1-H)

11. Use equivalent forms of equations and inequalities to solve real-life problems (A-1-H)


GLE # GLE Text and Benchmarks 13. Translate between the characteristics defining a line (i.e., slope, intercepts,

points) and both its equation and graph (A-2-H) (G-3-H) 15. Translate among tabular, graphical, and algebraic representations of functions

and real-life situations (A-3-H) (P-1-H) (P-2-H) Geometry 23. Use coordinate methods to solve and interpret problems (e.g., slope as rate of


24. Graph a line when the slope and a point or when two points are known (G-3-H)25. Explain slope as a representation of “rate of change” (G-2-H) Patterns, Relations, and Functions 37. Analyze real-life relationships that can be modeled by linear functions (P-1-H)

(P-5-H) 38. Identify and describe the characteristics of families of linear functions, with

and without technology (P-3-H) 39. Compare and contrast linear functions algebraically in terms of their rates of

change and intercepts (P-4-H)

Sample Activities Activity 1: Writing a Linear Equation Given Slope and Y-intercept (GLEs: 13, 23) Build on what was taught in Unit 4 on slope and the slope intercept form of an equation. Ask students to recall how to write an equation in slope-intercept form given the slope and y-intercept. In addition, review with students how to write an equation in standard form (Ax + By = C) by the use of algebraic manipulation. Students should be able to write an equation from one form to the other. Provide ample opportunity for students to show proficiency in this skill. Activity 2: Writing a Linear Equation Given a Point and a Slope (GLEs: 11, 13, 23)

In this activity, students need to understand how to write a linear equation. Remind students that any linear graph has a slope associated with it. If given a point on the graph, both of these pieces of information can be used to come up with an equation that fits the particular graph. Introduce the point-slope form of a linear equation and how it can be used to help find the equation of a line. The point-slope form of a line is derived from the slope formula: 2 1

2 1

( )( )y yx xm −

−= . By cross multiplication we get: (y2 – y1)= m (x2 – x1).

Demonstrate for students how this formula can be utilized to write an equation. For example, if the slope is 5 (m = 5) and the line contains the point (3, 4), the equation using the point-slope formula would be given as follows: (y – 4) = 5 (x – 3). Point out how the point and slope are replaced in their respective places in the formula. Next, demonstrate


how this form of the equation can be rearranged into either slope-intercept form or standard form. Provide additional examples and practice as is necessary for students to achieve proficiency in writing equations of lines given a point and a slope. Activity 3: Vertical and Horizontal Lines (GLEs: 11, 13, 23, 24) Provide students with grid paper on which two horizontal and two vertical lines have been drawn. Have students work in pairs. Instruct them to find the coordinates of two points on one horizontal line and then use the slope formula to calculate the slope of the line. Repeat the process using the second horizontal line. Discuss with students why the slope of a horizontal line is 0 (i.e., it has no steepness or slope). Use the point-slope formula to help students see that the equation for a horizontal line is y = a, where a is the y-coordinate for any point on the line. Repeat the same process using points on the vertical lines. Lead students to see that since division is undefined, the slope of a vertical line is undefined. Ask students to write the equation of a vertical line based on what they know about the equations of horizontal lines. Lead students to see that the equation of a vertical line is x = a, where a is the x-coordinate for any point on the line. Help students relate these concepts algebraically and graphically. Activity 4: Determine an Equation for a Line Given Two Points (GLEs: 9, 13, 15, 23, 24, 25) In this activity, students need to understand that in order to find the equation for a line, two things about the line in question—the slope and a point that lies on the line should be known. Have students determine the equation of a line given two points that are on the line. Students should first realize that the two points must be used to determine the slope of the line that contains them. Then using this information and one of the data points, have students use one of the procedures previously discussed to determine the equation of the line in all three forms—point-slope; slope-intercept; and standard form. Provide students with the graph shown below and have them determine the equation of the line in all three forms.

(-5, -2)

(8, 3)


Activity 5: Fahrenheit and Celsius—How are they Related? (GLEs: 9, 13, 15, 23, 24, 25, 37) The relationship between temperatures measured in degrees Fahrenheit and Celsius is linear and can be obtained by knowing that water freezes at 32°F and 0°C while water boils at 212°F and 100°C. Discuss these important benchmarks with students and have students work in groups of three to perform the following tasks using the data:

• Find the Celsius temperature as a linear function of the Fahrenheit temperature by writing an equation using the data referencing boiling and freezing points. In other words, use Fahrenheit as the independent variable and Celsius temperature as the dependent variable. Explain the variables you use in your equation and what they represent.

Solution: C = 59

(F - 32°) or y = 59

(x - 32°)

• Determine the y-intercept for this line and explain in real-world terms what it represents.

Solution: The y-intercept for this line is –17.77 or –17 7/9 and it represents the Celsius temperature which relates to 0°F. (i.e., 0°F = -17.77°C)

• Determine the x-intercept for this line and explain in real-world terms what it represents.

Solution: The x-intercept for this line is –32 and it represents the Fahrenheit temperature that relates to a Celsius temperature of 0°.

• Using the equation you created, find the Celsius temperature for a Fahrenheit temperature of 50°F. After you get your answer, look at a thermometer to see if this temperature matches with what you would expect.

Solution: 50°F = 10°C • Draw a graph showing this relationship. Label the independent axis and

dependent axis with the appropriate information. Solution: See student graphs!

• If the Fahrenheit temperature were written as a linear function of the Celsius temperature, what equation would result?

Solution: F = 95

C + 32

Activity 6: Writing an Equation from a Table of Values (GLEs: 9, 10, 13, 15, 23, 24, 25, 37, 38) Have students use the data table provided below to answer the questions presented. Let students work in pairs to perform the indicated tasks. Number of Sides of a Polygon

3 4 5 6


Sum of the Interior Angles

180 360 540 720

• Using the data table, create a graph to display the data. Label the independent and

dependent axis in an appropriate manner. Solution: The sum of the angles depends on the number of sides so the y-axis should be the sum of the angles and the x-axis should be the number of sides.

• Determine a linear equation for the data presented. Explain what variables you are using and what they represent. Explain the characteristics present in the data that assure this data is linear.

Solution: S = 180n – 360 or S = 180 (n-2) or y = 180 (x – 2), where x is the number of sides and y is the sum of the angles. The data is linear since the change is constant throughout the data.

• What does the slope in this problem indicate in real-world terms? What is the slope?

Solution: The slope is 180 and it represents the change in the angle measure sum as an additional side is added. (i.e., 180 degrees per side).

• Does this graph have a y-intercept? If so, explain in real-world terms what it represents. If it doesn’t have a y-intercept, explain why.

Solution: Mathematically, the y-intercept would be –360 meaning the sum of the angles of a figure with no sides would be –360 degrees. Since this makes no sense in real world terms, this graph really does not have a y-intercept—the equation only makes sense for a polygon with three sides or greater.

• Does this graph have an x-intercept? If so, explain in real-world terms what it represents. If it doesn’t have an x-intercept, explain why.

Solution: The x-intercept for this graph is x = 2. Since x represents the number of sides of a polygon, if a polygon had 2 sides (which cannot occur in real-life), the sum of the angles would be 0°.

• If a polygon has 10 sides, what will the sum of the angles of the polygon be? Solution: 1440°.

Activity 7: Wages vs. Hours Worked (GLEs: 9, 10, 13, 15, 23, 24, 25, 37, 38, 39) Provide students with the following information and have students answer the questions provided.

Mark earns an hourly rate of $5.75 per hour. He wants to save enough money to buy a motorcycle worth $3000. He already has $450 saved in his account.


Use this information to answer the questions below. • Complete the following table showing how much Mark will make after working x

number of hours. Number of Hours Worked

1 2 3 4 5

Money Earned at Work

Solution: Number of Hours Worked

1 2 3 4 5

Money Earned at Work

$5.75 $11.50 $17.25 $23.00 28.75

• Make a sketch of the graph using the relationship between the number of hours worked and the money Mark earned. Explain what characterizes this data as being linear.

Solution: See student graphs! The data is linear because of the constant rate of change.

• What is the slope for this graph and what does it represent in real-world terms? Solution: The slope of the graph is $5.75, which is the rate of pay that Mark gets for working on the job ($5.75 per hour).

• Suppose Mark saves all of his money that he makes at work for the motorcycle. If Mark wanted to write an equation showing the total amount of money he will have (including the $450 he already has saved) after working x hours, what equation would he use?

Solution: y = 450 +5.75x, where y is the total amount of money saved and x is the number of hours worked.

• If Mark works for 50 hours, how much money will have saved altogether? Use your equation to find the total amount saved.

Solution: $737.50 • How many hours will Mark have to work in order to save enough money for the

motorcycle? Explain how you got your answer. Solution: If done on calculator the answer is 443.47 hours, but in reality, if he only works whole hours, Mark will have to work 444 hours in order to save enough money for the motorcycle (assuming $3000 is enough to include taxes!).

Activity 8: The Stock Is Falling! (GLEs: 9, 10, 13, 15, 23, 24, 25, 37, 39) Present the following situation to students:

Lynn bought a stock at a price of $38. At the end of the first week, the price of the stock had fallen to $35. At the end of the second week, the stock had fallen to $32. At the end of the third week the stock had fallen to $29.


• Make graph showing the relationship between the price of the stock and the number of weeks since the stock was bought.

• Is the relationship on the graph a direct or inverse relationship? Explain. Solution: An inverse relationship exists between the time (week) and the price of the stock since as time goes on the stock declines in value.

• Find and interpret the meaning of the slope associated with this graph. Explain in real-world terms what the slope tells us.

Solution: The slope in this problem is –3 and it relates the rate of change associated with the change in value of the stock per week—there is a decrease of $3.00 per week for the value of the stock.

• Assuming this trend continues indefinitely, write an equation for the value of the stock after x weeks have gone by. Explain the variables you used when writing your equation.

Solution: P = 38 – 3x, where P is the price of the stock in dollars after x weeks.

• Find and interpret the meaning of the y-intercept for this equation. Solution: The y-intercept is 38 and it represents the initial value of the stock ($38).

• Find and interpret the meaning of the x-intercept for this equation. Solution: The x-intercept is 12 2/3 and it represents the number of weeks it would take for the value of the stock to be worth nothing. Note: Since the number of weeks is a discrete number, it would actually take 13 weeks for the value of the stock to be worth nothing.

• Using the equation you wrote, if the trend continues, what will the price of the stock be at the end of the 10th week?

Solution: $8.00 • Using the equation you wrote, if the trend continues, when will the stock be worth

nothing? Explain how you got your answer. Solution: The stock will be worth nothing mathematically at 2

312 weeks.

Teacher Note: Show students how to do this last part (inputting the equation into a calculator and finding the x- and y-intercepts and the value of the stock at 10 weeks, etc.) using a graphing calculator to integrate technology into this activity.

Sample Assessments General Guidelines Performance and other types of assessments can be used to ascertain student achievement. Following are some examples:


General Assessments • The student will write a primer that explains the x, y coordinate system, how

to determine slope, x-intercept, and y-intercept, given an equation. • The student will write a letter to an absent classmate explaining the

connection between slope and rate of change. • The student will create portfolios containing samples of their activities.


• Activity 1: The teacher will provide the student with a graph of a line. The student will write the equation in slope-intercept form, then translate the equation into standard form.

• Activities 2 and 3: The student will write the equations and sketch the graphs when given a point and the slope of the line.

• Activity 4: The teacher will create a worksheet that shows various graphs (such as the example provided in the activity) and the student will match the equations with the graphs.

• Activity 6: The student will draw a 10-sided polygon. The student will use a protractor and measure the interior angles of the polygon to see if the sum of the interior angles corresponds to the answer found using the equation created in the activity.

• Activity 8: The student will come up with a problem for a real-world context in which a decreasing linear situation exists. The student will make a graph and an equation to represent the situation.

Algebra I: Part 1 Unit 6 Inequalities and Graphing 46

Algebra I: Part 1 Unit 6: Inequalities and Graphing

Time Frame: Approximately three weeks Unit Description In this unit, an examination is made of the nature of linear inequalities in one variable and their graphs on a number line. The unit also includes an introduction to absolute value in relation to distances. Student Understandings Students recognize and distinguish between strict inequality (< and >) statements and relaxed inequality/equality (< and >) statements. Students solve linear inequalities in one variable and graph their solutions on the number line. Students can graph simple absolute value inequality relationships on the number line. Guiding Questions

1. Can students perform the symbolic manipulations needed to solve linear inequalities and graph their solutions on the number line?

2. Can students interpret and graph simple absolute value equalities (e.g., ⏐3x + 2⏐ = 7) on the number line?

3. Can students relate absolute value inequalities in one variable to real-world settings (e.g., measurement, absolute value distances) and graph their solutions on the number line?

Unit 6 Grade-Level Expectation (GLE)

GLE # GLE Text and Benchmarks Algebra 9. Model real-life situations using linear expressions, equations, and

inequalities (A-1-H) (D-2-H) (P-5-H) 11. Use equivalent forms of equations and inequalities to solve real-life

problems (A-1-H) 14. Graph and interpret linear inequalities in one or two variables and systems

of linear inequalities (A-2-H) (A-4-H)


Sample Activities Activity 1: Inequalities on a Balance (GLEs: 11, 14) Students should have been introduced to solving inequalities and graphing the solutions in 7th grade. Use this activity to review the basic steps in solving an inequality and expressing the resulting graphs for the solution sets. Relate the solving of inequalities to its counterpart—solving equations in one variable. To visually represent the concept of an inequality (in one variable), use an unbalanced scale and a set of two-colored translucent disks. On the overhead projector, draw an out-of-balance scale. Make the left side the lower side and write 4x − on this side. On the other side, write 3− . Tell the students that this represents the inequality some number minus four is greater than negative three and write x 4 3− > − above the scales. Tell the students that one set of disks is “positive” and the other color is “negative.” Place four negative disks above the 4 and place three negative disks above the 3. Now add four positive disks to both sides. Ask the students what happens when a positive and a negative are added together. They should respond that the sum is zero. Remove all matched sets of positive and negative disks. On the first side, all that remains is the x. On the other, there remains one positive disk. Therefore, the solution to the inequality is 1x > . Discuss how to graph such an inequality on a number line. Repeat this activity several times, each time using simple inequality statements. Use each of the inequality symbols (<, >, ≥,and ≤) in the problems, and explain how each affects the way in which the graphical solution is expressed using parentheses, brackets, dots, and open holes, as appropriate. Activity 2: The Inequality Knowledge Bowl (GLE: 14) Prior to class, create one page with ten number lines on it. Divide the classroom into groups of two to three students. Have each group select a recorder. Have each group write two inequalities to solve, which the recorder will write down. Allow a few minutes for them to finish and then collect their pages. Their inequalities make up the questions for the Knowledge Bowl. Provide each recorder with a copy of the pre-made number line forms. The recorder will be responsible for recording the answers for their respective team. Read one inequality and allow time for the team to discuss what the graph should look like. The recorder will graph the team answer on one of the pre-made number lines. When all inequalities have been read and the teams have completed their graphs, instruct teams to exchange answer sheets. Show the correct answers for scoring. Once completed, have the teams return the papers to the correct team. The winning team is the team having the most correct answers.


Activity 3: Real-life Inequalities (GLEs: 9, 11, 14) Present the following situation for students to solve:

The math club raised $6800 to go on a trip to the math competition in Europe. A travel agent charges a $300 fee to organize the trip and an additional charge of $600 per person for each student attending the competition.

• Have them write an inequality showing the relationship between the money that was raised and the cost for sending x people on the trip, assuming the amount raised is more than the amount needed to go on the trip.

Solution: $6800 > $300 + $600x • Ask them to solve the inequality to find the maximum number of people that can

go on the trip. Solution: x < 10 5/6; In reality, the maximum number of people that can go on the trip is 10—more money would be needed to send 11 people.

Activity 4: Absolute Value as Distance (GLE: 9) Introduce students to the absolute value symbol. Explain that the absolute value of a number is really the distance the number is from zero. For example, write x =│3│on the board, ask students to interpret the equation (i.e., the number that will solve this equation is 3 units from zero), and have students find numbers on the number line that meet this criteria. Connect this mathematically with the following statements: │-3│ = 3 and │3│ = 3. When students see the absolute value symbol, they should immediately think of distance. Since distance is associated with a positive value, the absolute value of a number is positive. Provide additional examples in which students practice this concept. Activity 5: Absolute Value of a Difference (GLEs: 9, 11, 14) In the previous activity, students were shown how to relate the absolute value of a number with the distance the number is from 0 on a number line. To find a distance between two numbers on a number line other than 0, relate the expression │a – b│ with its geometric interpretation. For example, to find the distance between two points on a number line (e.g., let a = 3 and b = -5) can be written as the distance between 3 and –5 or it can be expressed using the absolute value symbol as: │3 – (-5)│. Relate this visually using the number line. It could also be expressed as: │-5 – 3│. Ultimately, students should to understand that this absolute value expression represents the distance between two numbers on a number line. For this example, students should see that │3 – (-5)│= 8 which simply means that the distance between 3 and –5 on a number line is 8 units. Provide additional examples and ask students to do them to reinforce this concept.


Activity 6: Solving Absolute Value Equations (GLEs: 9, 11) Activities 4 and 5 should have laid the foundation for what is next—solving an absolute value equation. Using what they learned in Activities 4 and 5, have the students write an absolute value expression matching the following situation: The distance between the numbers x and 3 on the number line. Students should write │x – 3│ to express this distance. Next, tell students that if this distance is 7, what values of x will make the distance equal to 7 units. Relate this to the absolute value equation: │x - 3│ = 7. Let the students get in pairs to figure this out. Students should relate all of this graphically and concretely using a number line to obtain their answers. Based on what was learned in the previous activities, students should see that │x - 3│ = 7 has two solutions for x. The points that are a distance of 7 units from the number 3 on the number line are 10 and –4. These are the two values of x that make the equation true. Provide students with additional examples and have students explain in words what each equation means geometrically in terms of distance on a number line, and solve the equations for the unknown variable. Teacher Note: Introduce technology into this unit using a graphing calculator and an overhead to demonstrate to the students what the graph of y = │x - 3│ looks like. Students should note that the y-values on this graph are always positive. Activity 7: Absolute Value Inequalities I (GLE: 9, 11, 14) Ask students working in pairs to identify points on a number line that are located less than a specified distance from zero. Have students write the correct inequality statement without the absolute value symbol first. Next, using the absolute value symbol, have students write the correct inequality statement. For example, if students are asked to locate all points on the number line that are less than two units from zero, then they should write the inequality 2 2x− < < and then 2x < . Repeat this activity several times. Include statements about greater than, less than or equal, and greater than or equal. When working with a distance greater than some amount, students should understand that “visually” this becomes a compound inequality. For example, to express all of the points which are more than 2 units from 0 on the number line this would be expressed algebraically as │x│>2. Relating this graphically on a number line, have students should see that in order to express this information, two inequality statements would have to be written: x > 2 or x < -2. This is critical to students’ understanding and solving absolute value inequalities of this type. Activity 8: Absolute Value Inequalities II (GLEs: 9, 11, 14) Extend student understanding of absolute value inequalities by considering expressions like 3 2 7x + < . Ask student pairs to identify points on a number line that are located less than 7 units from a “point” labeled “3 2x + ” instead of zero. Students should recognize that it can be written in the same fashion as the earlier inequality (i.e., 7 3 2 7x− < + < ).


Ask students to use previously learned rules to solve the inequality. Repeat this activity several times. Be sure to use absolute value statements about greater than, less than or equal, and greater than or equal.


• The student will use sample work from the activities to place in a portfolio

that would showcase knowledge of inequalities. • The student will describe the difference between an equation and an inequality

in words and give an example of each using real-world examples. • The student will write a letter to a classmate explaining what an inequality is

and how to solve inequalities and graph them. • The student will create absolute value inequality statements and share with the

class on a math bulletin board. Activity-Specific Assessments

• Activity 1: The student will solve and graph inequalities in one variable.

• Activity 5: The student will explain verbally and geometrically what the absolute value expression │-6 – (-12)│ represents.

• Activity 6: The student will explain verbally and geometrically (using distance

on a number line) what the equation │x – (-8)│ = 3 means and find its solution.

Algebra I: Part 1 Unit 7 Systems of Equations

51

Algebra I: Part 1 Unit 7: Systems of Equations

Time Frame: Approximately three weeks Unit Description This unit examines the nature and mathematical procedures used with regards to finding and interpreting solutions for real-life and abstract system of equations problems. Student Understandings Students graph and interpret the solution of a system of two linear equations. They relate the existence or non-existence of solutions to intersecting and parallel lines. They develop algorithmic ways of determining the solutions to a system of linear equations. Guiding Questions

1. Can students explain the meaning of a solution to a system of two linear equations?

2. Can students determine the solution to a system of two linear equations by graphing, substitution, or row operations?

3. Can students relate the solution, or lack of solution, to a system of equations to the slopes of the lines?

4. Can students identify parallel and intersecting lines by their slopes and relate this to possible solutions?

5. Can students identify coincident lines by their slopes and y-intercepts and relate this to the possibility of an infinite number of solutions?


GLE # GLE Text and Benchmarks Number and Number Relations 4. Distinguish between an exact and an approximate answer, and recognize errors



Algebra 9. Model real-life situations using linear expressions, equations, and inequalities

(A-1-H) (D-2-H) (P-5-H)


52

GLE # GLE Text and Benchmarks 11. Use equivalent forms of equations and inequalities to solve real-life problems

(A-1-H) 13. Translate between the characteristics defining a line (i.e., slope, intercepts,

points) and both its equation and graph (A-2-H) (G-3-H) 16. Interpret and solve systems of linear equations using graphing, substitution,


Geometry 23. Use coordinate methods to solve and interpret problems (e.g., slope as rate of


24. Graph a line when the slope and a point or when two points are known (G-3-H)25. Explain slope as a representation of “rate of change” (G-3-H) (A-1-H) Patterns, Relations, and Functions 37. Analyze real-life relationships that can be modeled by linear functions (P-1-H)

(P-5-H) 38. Identify and describe the characteristics of families of linear functions, with

and without technology (P-3-H) 39. Compare and contrast linear functions algebraically in terms of their rates of

change and intercepts (P-4-H)

Sample Activities Activity 1: Is There a Point of Intersection? (GLEs: 16, 38, 39) Using graphing calculators have students input pairs of linear equations and have the students determine whether each pair of lines graphed has a point of intersection. For this activity, students are only to determine whether or not there is a point of intersection, not to actually determine the point of intersection. The purpose here is to get students to begin to analyze equations (in slope-intercept form) and be able to determine which linear equations (based upon their slope and y-intercept) will intersect. After the students have entered the equations into the calculators to determine if the graphs intersect, have the students analyze the graphs of those equations which intersected and which did not. Students should realize that if the slopes are different, the lines intersect at exactly one point. If the slopes are the same, then one of two things will occur—either there will be no point of intersection (for those equations which had the same slope but different y-intercept) or there will be infinitely many points of intersection (for those equations had the same slope and same y-intercept). Present enough examples for students to come up with an appropriate conjecture. An example of some pairs of equations for students to input into the calculators are given in the following table:


53

Equation 1 Equation 2 Is there a Point of Intersection y = 3x +1 y = 4x –5 Solution: Yes (1 point) y = 4x – 3 2y = 8x –6 Solution: Yes (infinitely many) y = 2x – 1 y = 2x – 3 Solution: No (parallel lines)

Teacher Note: 2y = 8x – 6 doesn’t have to be graphed in order to answer this question. Students should realize that once the equation is solved for y, the equation becomes y = 4x – 3 which is the same as the initial equation. Activity 2: Finding a Point of Intersection by Graphing (GLEs: 4, 16, 23, 39) Have students determine the actual point of intersection for the graphs of two lines using pencil and paper. Before graphing, ask students to decide whether or not there should be a point of intersection. This should be based upon what they learned in Activity 1. For example, present the following two equations to students: x + y = 6 and 3x – 4y = 4. By writing the two equations in slope-intercept form, students should see that the two equations have different slopes so there will be exactly one point of intersection. Next, have the students graph the two lines on the same coordinate graph. Have the students determine “visually” where the point of intersection appears to be on the graph. Ask students if they are absolutely certain that this is the point where the two lines intersect, or if this is an approximate value. Point out the fact that using graphing to determine where two lines intersect has limitations. The actual point of intersection for the two lines is approximated. Provide additional examples for students including examples in which there are no solutions and infinitely many solutions. After the examples in Activity 2 are done by hand, demonstrate for students how to use the graphing calculators to determine the point of intersection by using the “trace” and “zoom” features. Once students have found the approximate answers by tracing, compare these approximate answers with the graphical solutions they obtained using paper and pencil. Activity 3: Technology Induced Errors (GLEs: 4, 5, 11, 23) Students should understand that sometimes calculators do not always give exact answers. In this activity, show the limitations that can occur in a calculator’s computing ability. Provide students with several equations in two variables ready for graphing on a graphing calculator (e.g., 3.0245y x= ; 2.34 5y x= + ). Use the example equations (one at a time— the point of intersection here can’t be found) and have the students determine the zeros or roots of the equations (i.e., x-intercepts) using a graphing calculator. Have students graph each equation and find approximate solutions to the equations when y = 0 by using the built-in tracing capabilities of the graphing calculator. Once students have found the approximate answer by tracing, they should use this answer as the value for x in the original equation. By performing the calculations using this value, students will be working with rational numbers and further solidifying their understanding and skills with computation. Have students will compare their computations with zero to see how close the answer found by the calculator is to the actual


54

zero. Using the zoom capabilities of the calculator, ask students to “zoom in” on the values they found to increase accuracy. Precision of their answers will be dictated by the calculator’s built-in pixels. Next, have students solve for the zeros of the equations (by using algebraic techniques with paper and pencil) and compare these exact answers to those found by using the calculator. Point out to the students that this is an error induced by technology. Next, show students the built-in feature found in the Calc keys (TI-83 or similar), which automatically determines the root(s) or x-intercept(s) for an equation. Compare this answer with the approximated answers using the zoom key as well as the exact answers found using paper and pencil. Activity 4: Using Substitution to Solve a System of Equations (GLE: 16) It is critical that students understand exactly what a point of intersection is—the point that is a solution for both equations. Point out that this point is the one point that the two lines have in common. The x and y values for this point should make both equations “true.” For example, if the two equations are those used earlier (e.g., x + y = 6 and 3x – 4y = 4), the point of intersection graphically appears to be at the point (4,2). The drawback to using a graphical approach is that the exact point of intersection is not absolutely sure. So, the question becomes, how can the exact point of intersection be known? If this is truly the point of intersection, the x- and y-coordinate in each of the two equations can be replaced with x = 4 and y = 2, and they should make both equations true (which in this case it does). No other values for x and y will make both equations true at the same time because this is the only point that the two equations have in common. Therefore, at the point of intersection for both graphs, the x and y coordinates are equal—the x in one equation is equal to the x in the other equation; likewise for the y coordinates in both equations. Discuss with students how this can be utilized to find a point of intersection for two equations by using the substitution method. If the equations for y (of course this process could also be done solving for x—discuss this approach with the students as well) can be solved, the following two equations are given: y = -x + 6 and y = 3

4 x –1 . Since the y values are equal at the point of intersection, by substitution the answer is: -x + 6 = 3

4 x – 1. This can then be solved for the variable x (giving the value of x = 4). This gives the x value for the point of intersection, while the y value for the point of intersection (y = 2) can be found by replacing the x value found into one (or both) of the equations. Provide additional examples for students to become proficient at the substitution method of solving systems of equations. After work is done with paper and pencil, use a graphing calculator to show students how to find the point of intersection using the Calc function Intersect to determine the point of intersection for two graphs.


55

(-5, -2)

(8, 3)

Activity 5: Solving Equations Using Calculators (GLEs: 4, 16) In Unit 3, students reviewed how algebraic equations are solved using algebraic manipulation to solve for an unknown variable. These algebraic equations can also be solved using technology as well. For example, suppose the equation 3(x –1) +5 = 4

3 (.7x +4) is to be solved. This can be solved using the graphing calculator if each side of the equation is inputted as its own graph. Let y₁ = 3 (x – 1) + 5 and y₂ = 4

3 (.7x + 4) and find the point of intersection using the Intersect key. Compare this answer with the answer found using paper and pencil method to see if they agree. This method can be used to solve any type of equation. Provide students with additional examples. Activity 6: Solving Systems by Elimination (GLEs: 4, 16) Demonstrate for students how to solve systems by using the elimination method. Point out that the elimination method is best used when the two equations are in standard form. In the elimination method, the concept is to eliminate one of the variables and solve for the one that remains. Include examples where there is no solution (no point of intersection) and examples where there are infinitely many points of intersection. In these two special cases when elimination is used, both the x and y terms are eliminated concurrently. What remains is either an equation that makes sense (such as, 0 = 0, which would indicate the two equations are exactly the same and thus have infinitely many points of intersection) or there remain an equation that does not make sense (such as, 0 = 8, which would indicate the two equations have the same slope and different y intercepts and are parallel to one another thus having no point of intersection). Relate all of this graphically. Provide systems for students to solve using elimination method and then have them check their solutions by graphing the equations. Provide additional practice in which students determine which method, elimination or substitution best fits the problem and have them solve the equations using the chosen method. Let them check their work using the graphing technique. Activity 7: Putting it All Together! (GLEs: 5, 11, 13, 16, 23, 24, 25) Allow students to work together in groups to solve the following problem.

• Find the slope of the line segment shown in the graph.

Solution: m = 513

• Find the midpoint of the line segment shown in the graph. (Note: Discuss with students what a midpoint is—the point in the middle of the other two points. Relate the fact that when the average of two grades is actually the

“middle” of the two. The midpoint formula ( 1 2

2x x+ , 1 2

2y y+ ), is based on taking the

average of the x values and the average of the y values of the two endpoints to determine the “midpoint” of the two. Point out that when two grades are added and


56

divided by two, the average grade is found. It works this way with the x and y coordinates as well.)

Solution: Midpoint = ( ,3 12 2 )

• Write an equation for the line that contains the line segment shown in the graph in three forms—slope-intercept; standard; and point-slope.

Solution: y = 513 x – 1

13 ; 5x – 13y = 1; (y-3) = 513 (x – 8) or (y+2) = 5

13 (x+5) • Find the equation of the line that runs through the midpoint and has a “rate of change”

of 6. Write the answer in slope-intercept form. Solution: y = 6 x – 8.5

• Using a graphing calculator, input the two equations you wrote to determine the point of intersection for the two lines, and make sure this value agrees with the midpoint.

Activity 8: Break-even Point! (GLEs: 9, 11, 16, 23, 25, 37) Present the following problem to students to work on in groups. Discuss the results as a class. Mrs. Lowenstein started a business selling designer hats. To start her business, she had to pay $900 for a professional sewing machine and each hat costs her $18 for the materials to produce it. In order to make a profit, she decides to sell her hats at a price of $30 per hat. Use this information to answer the following:

• Write an equation to represent the total cost to make x hats. Solution: y = 18x + 900

• Write an equation to represent the revenue Mrs. Lowenstein will receive for selling x hats.

Solution: y = 30x • Using graph paper, make a graph showing the cost equation and the revenue equation

on the same graph. Make the scale on the graph such that the point where the revenue and cost equations intersect can be determined.

Solution: See student graphs! • From the graph, what appears to be the point where the two graphs intersect? Explain

the real-life interpretation of this point. Solution: Students answers may vary. Students should see that this point indicates where the revenue and cost are equal. When the revenue and cost are the same, this is referred to as the “break-even point.”

• Using either substitution or elimination, find the exact point of intersection for the two graphs. Explain in real-world terms what this information tells us in this situation.

Solution: The point of intersection is (75, 2250). The x value (75) represents the number of hats that must be sold in order to break even. The amount of income and cost at this point are equal ($2250) to one another. There is not a profit or a loss at this point.

• If Mrs. Lowenstein sells 50 hats, will she make a profit? Explain how you know. Solution: Until Mrs. Lowenstein sells 75 hats, she will not be making money, so there is no profit if she only sells 50 hats.


57

• Determine what the slope is and explain what it represents in real-world terms for each of the graphs.

Solution: The slope of the cost equation is 18 and it represents the rate of change of the cost to make each additional hat. The slope of the revenue equation is 30, and it represents the rate of change of the revenue for each hat sold.

Activity 9: Which is the Better Offer? (GLEs: 9, 11, 16, 23, 25, 37) Present the following situation for students to work in groups.

The Brown family is moving into a new home. They need to rent a moving van for a day, but are unsure as to which is the best offer. One van company, DirtCheap Vans, charges $59.65 a day plus an additional 45¢ per mile. The other company, BestVans, charges $88.50 a day plus an additional $0.37 per mile.

Have students come up with cost equations for both van companies based on x miles driven for the day, and then using a graphing calculator, substitution, or elimination, find the point of intersection for the two graphs. Have the students explain what the point of intersection means in real-life terms, and have them explain which deal the Brown family should choose.

Solution: DirtCheap—y=59.65 +.45x; BestVans—y=88.50 + .37x; The point of intersection is approximately (360.63, 221.93) which means that if the Brown family travels 360.63 miles, the cost will be the same ($221.93). Depending on the mileage the Brown family travels will affect which choice is best. If the Brown family plans on traveling more than 360 miles, then BestVans is the better choice. If the Brown family plans on traveling less than 360 miles, then the DirtCheap plan is cheaper.


• The students will develop a portfolio of 10 items from this unit (their choice). • The student will write a short paragraph explaining their algorithm for

determining the number of solutions, given the equations for two lines. • The student will explain, in writing and using a graph, what it means for a certain

point to be a solution to a set of equations.


58


• Activity 1: The teacher will provide students with pairs of equations without the aid of graphing calculators, and students will determine whether there will be one, none, or infinitely many points of intersection based upon the slope and intercept of each equation.

• Activity 2: The student will graph two linear equations using paper and pencil and

use the graph to approximate the point of intersection.

• Activity 4: The student will determine the exact point of intersection for two linear equations using the substitution method.

• Activity 6: The student will create three problems and solve them using

elimination which show one solution, no solution, and an infinite number of points of intersection.

• Activity 8: The teacher will provide a real-world problem that requires the student

to come up with cost and revenue equations and use the equations to determine the break-even point for a particular situation.

Algebra I: Part 1 Unit 8 Matrices, Systems of Equations, and Linear Inequalities

59

Algebra I: Part 1 Unit 8: Matrices, Systems of Equations, and Linear Inequalities

Time Frame: Approximately four weeks Unit Description In this unit, the solving of systems of equations is extended to include the use of matrices. The unit also examines the solution of linear inequalities in two variables. Student Understandings Students develop the concept of a matrix and matrix operations of addition and multiplication and the relationship between solving ax b= and Ax B= as 1x a b−= and

1x A B−= respectively. They apply matrices to the solution and interpretation of a system of two or three linear equations. In addition, students learn more about linear inequalities in two variables and systems of linear inequalities. Guiding Questions

1. Can students explain the meaning of a solution to a system of equations or inequalities?

2. Can students determine the solution to a system of two or three linear equations with two or three variables by matrix methods?

3. Can students use matrices and matrix methods by hand and calculator to solve systems of equations Ax B= as 1x A B−= ?


GLE # GLE Text and Benchmarks Number and Number Relations 4. Distinguish between an exact and an approximate answer, and recognize

errors introduced by the use of approximate numbers with technology (N-3-H) (N-4-H) (N-7-H)


Algebra 9. Model real-life situations using linear expressions, equations, and

inequalities (A-1-H) (D-2-H) (P-5-H) 11. Use equivalent forms of equations and inequalities to solve real-life

problems (A-1-H)


60

GLE # GLE Text and Benchmarks 13. Translate between the characteristics defining a line (i.e., slope, intercepts,

points) and both its equation and graph (A-2-H) (G-3-H) 14. Graph and interpret linear inequalities in one or two variables and systems

of linear inequalities (A-2-H) (A-4-H) 16. Interpret and solve systems of linear equations using graphing,

substitution, elimination, with and without technology, and matrices using technology (A-4-H)

Geometry 23. Use coordinate methods to solve and interpret problems (e.g., slope as rate

of change, intercept as initial value, intersection as common solution, midpoint as equidistant) (G-2-H) (G-3-H)

24. Graph a line when the slope and a point or when two points are known (G-3-H)

Patterns, Relations, and Functions 37. Analyze real-life relationships that can be modeled by linear functions (P-

1-H) (P-5-H)

Sample Activities Activity 1: Matrices—an Introduction (GLE: 16) Before anything can be done mathematically with matrices discuss what a matrix is— simply a rectangular array of numbers. Discuss with students the chart provided which displays the items sold at different times at a movie cinema on a Monday. The chart can be written as a rectangular array and enclosed with brackets or parentheses. The enclosed array is called a matrix. The advantage of writing the numbers as a matrix is that the entire array can be treated as a single mathematical entity. A matrix can be named with a single capital letter as shown. The numbers that make up a matrix are called the entries, or elements, of the matrix. The entries of matrix M are all numbers, but the matrix itself is not a number, just as a multiplication table is not a number. Various operations can be used on matrices (addition, subtraction, multiplication) and will be discussed in the other activities. A matrix is often classified by its order (or dimension), that is, by the number of rows and columns that it contains. For example, matrix M has 4 rows (rows run across) and 3 columns (columns run up and down) which means that matrix M is a 4 x 3 matrix. When a matrix has the same number of rows as columns, it is called a square matrix. Relate the rows and columns of the chart with the matrix. Have students find real-life examples which can be represented by matrices (e.g., pizza costs based on small, medium, or large and the number of toppings). Introduce students to the matrix function on the calculator and let them enter the examples they find. This will help them understand the concepts of rows and columns.


61

Chart of Items sold at Movie Cinema Matrix On Monday

Activity 2: Adding Matrices (GLEs: 5, 16) Provide students with several pairs of matrices that can be combined by addition. An example is provided below. Using matrix, M, from the previous activity which shows the sales on a Monday at a movie cinema, and matrix T, which displays the same items sold on a Tuesday, have students find M + T and describe what this new matrix describes. Students should see that the new matrix, M + T, really describes the total sales of items at the concession stand at the movie cinema for Monday and Tuesday. To add two matrices such as this, the only things that can be added are “like terms.” In this case, if we recall that snacks are in the first column and the times are given in each row, then in the upper corners of matrix M and matrix T, these two terms can be added because they are alike—

Snacks Popcorn Drinks 1:00 $4 $20 $25 3:00 $8 $24 $18 5:00 $10 $34 $28 8:00 $34 $38 $55

4 20 25 8 24 18 10 34 28 34 38 55

M =

4 20 25 8 24 18 10 34 28 34 38 55

M =

6 15 24 5 8 22 8 25 15 13 22 16

T =

10 35 49 13 32 40 18 59 43 47 60 71

M + T =


62

4 + 6 = 10, which means that the total snacks sold at the 1:00 show on Monday and Tuesday is $10. The complete new matrix that results from the addition of M + T is shown. Discuss with students the results of the new matrix and how to interpret its meaning. Next, have students find M-T. Activity 3: Multiplying a Matrix by a Scalar (GLEs: 5, 16) To find 1

2 the sales of the concession stand on Monday at the cinema, each sale would then be “halved.” This would result in a new matrix to display the results. Have students find the results of what the matrix 1

2 M would look like. This is shown below. This is what is referred to as “multiplying a matrix by a scalar.” In such a case, all elements in the original matrix are multiplied by the scale factor. Discuss this with students. Activity 4: Multiplying Two Matrices (GLEs: 5, 16) Multiplication of two matrices is not defined so straightforwardly as addition, subtraction, and multiplication of matrices by a scalar. Provide students with several pairs of 2 by 2 matrices and have them use the matrix operation of multiplication to find their product. Explain that matrix multiplication is a row times column process where the entry of a row in the first matrix multiplies the corresponding entry in the column of the second matrix, summing these products until the entries of a row are all used. Then repeat the process for the next row, etc. For example, look at matrix P and Q shown below. Multiply each entry in the first row of P by the corresponding entry in the column of Q, and then add the products. For example, in row 1 of matrix P multiply the number 10 by the first entry in column 1, which is 8. This product is 80. To this add the product of the second element in row 1 of matrix P, which is 15, by the second entry in column 1 of Q, which is 10, giving a product of 150. When the two individual products are added, the

2 10 12.50 4 12 9 5 17 14 17 19 27.50

½ M =

4 20 25 8 24 18 10 34 28 34 38 55

M =

10 15 15 20

P = 8 6 10 5

Q = 230 135 320 190

PQ =


63

result is 80 + 150 for a sum of 230. Thus, 230 is the first entry in the new matrix called PQ. The complete new matrix PQ is shown below. Discuss with students how the entire matrix was formed. Repeat this activity reversing the matrices to show students that matrix multiplication is not a commutative operation. Instead of P times Q, multiply Q times P, and prove to students that the resulting matrix, QP, is different than PQ. Matrix QP is shown below. Provide additional opportunities for students to gain experience. Then show students how this can be done easily using graphing calculator technology. Activity 5: The Inverse Matrix (GLE: 16) In activity 4, students were introduced to the concept of multiplying one matrix with another. Students discovered that the order in which matrices are multiplied results in different answers (unlike multiplication of real numbers). In this activity, discuss with students the idea of an inverse matrix. The inverse matrix of matrix A, symbolized by -1A , is the matrix that will produce an identity matrix when multiplied by A. In other words: -1AA = I and -1A A= I . The identity matrix, symbolized by I, is one of a set of matrices that do not alter or transform the elements of any matrix A under multiplication, such that AI = A and IA = A. Demonstrate for students the identity matrix for any 2 x 2 matrix as follows: Demonstrate that any 2 x 2 matrix, called A, when multiplied by this identity matrix, results in the original matrix (just as multiplying any real number by 1 results in the original real number). Use both pencil and paper method as well as a graphing calculator to do the computation. Students need to understand that matrix equations are related to linear equations in the following way. For example, if ax = b, to solve for x we can multiply both sides of the equation by 1

a , which is the multiplicative inverse of a. We can also write 1

a as a¯¹. Show students the inverse key on a calculator and relate finding the multiplicative inverse of 5 as 1

5 or .2. Use several examples to demonstrate this to students. To solve the equation ax = b, we then would have: -1 -1a a bax = , and since

-1aa =1, we get -1x = ba . Just as normal equations can be solved using this approach, so too can matrix equations. For example, suppose AX = B, where A, B, and X are all matrices. To find X when A and B are known multiply both sides of the matrix equation

170 240 175 250

QP =

1 0 0 1

I =


64

by the inverse of A, or A¯¹. Thus: A A¯¹X = A¯¹B or X = A¯¹B. Provide the following example to students. Show students how to use a graphing calculator to find A¯¹. Have students prove that AA¯¹= I (Identity Matrix) using a calculator. Activity 6: Solution of Two Equations: Found Five Ways (GLEs: 4, 5, 11, 13, 16, 23, 24) Provide students with pairs of equations like the following: 2 3 5x y+ = and 2 4 9.x y+ = Have students solve the system of equations first by graphing (using paper and pencil), by graphing on a graphing calculator using the intersect function, and the substitution method, then have students solve it using the elimination method. Finally, show students how the two equations can be solved using matrices. Discuss how to write the two equations as follows: Let Students should recognize this from the previous activity. The idea here is to find x and y, which are the values for matrix X. Since AX = B, to solve for X take the inverse matrix of A or A¯¹of both sides; therefore, X = A¯¹B. Show students how this can be done using the calculator. Verify that the point of intersection is the same regardless of the method. The point of intersection for the two lines is (-3.5, 4). Provide additional examples for students to become proficient at using matrices to solve such equations. Repeat this activity using systems that have no solutions, one solution, or an infinite number of solutions. Activity 7: Matrices (GLEs: 4, 5, 16) Provide the following equations to the students: ( 5) 5 15x y+ = ; 0 2x y+ = . Have them solve (pencil and paper method) using substitution ( 5; 2)x y= = . Then, use a calculator

2 3 1 4

A =

.8 -0.6 -0.2 0.4

A¯¹ =

2 3 2 4

A = x y

X = 5 9

B =


65

that can solve matrix equations. (This can be done by the teacher, the students, or both.) Ask the students for their answers ( 5 2.236)≈ . Since this is NOT an exact answer, it represents errors introduced by technology. Activity 8: Solving Three Equations with Three Unknowns Using Matrices (GLE: 9, 11, 16, 37) Present the following problem for students to work on in groups.

Jan bought 3 fries, 2 drinks, and 4 hot dogs at a total price of $8.95. Mary bought 2 fries, 6 drinks, and 5 hot dogs for a total of $12.85. Kyle bought 4 fries, 5 drinks, and 9 hot dogs for a total of $19.00. What is the price for a single fry, single drink, and a single hot dog?

• Write three equations with three variables to represent this situation. Solution: 3f + 2d +4h = 8.95; 2f + 6d + 5h = 12.85; 4f + 5d + 9h = 19.00

• Create a matrix equation that can be used to solve for the three variables. Solution:

• Using a calculator, solve the system using matrices. Solution: Fries cost $0.55; Drinks cost $0.75; Hot dogs cost $1.45.

Activity 9: Inequalities in Two Variables . . . Where Do I Shade? (GLE: 14) In Unit 4, the concept of inequalities in two variables was discussed briefly using paper and pencil. In this activity show students how to use calculators to solve such an inequality. Using a graphing calculator with projection capability, graph the following:

2 4y x= − (project onto whiteboard to enable shading technique). Instruct the students that the inequalities 2 4y x< − and 2 4y x> − will both be based on this line. One side of the line corresponds to < and the other side corresponds to >. The question is, which is which? Have the students pick a point that DOES NOT fall on the line 2 4y x= − . Choose (0,0). Substitute these values into the first inequality: 2 4y x< − : 0 2 0 4< × − ; which becomes 0 2 4< − , or 0 2< − . Ask the students if this is true. (The answer is NO.) Since the point (0,0) is not on the line 2 4y x= − and it is not true for the inequality

2 4y x< − , then it must be true for 2 4y x> − . Have the students verify to make sure: 0 2> − . The side of the line containing (0,0) represents the solution set to 2 4y x> − .

3 2 4 2 6 5 4 5 9

A =

F D H

X =

8.95 12.85 19.00

B =


66

Write this inequality on the white board in the appropriate location and write 2 4y x< − across the line in another color. Shade both sides with the appropriate color. Now, show students how this can all be done using the graphing calculator shade function. Activity 10: Inequalities and Your Life (GLE: 14) Provide students with the inequalities for target heart rate during exercise. Experts agree that your target heart rate y should be between 65% and 75% of the maximum heart rate for your age x. Provide the following inequalities for students to graph: 0.65(220 )y x> − and 0.75(220 )y x< − . Have students determine appropriate exercise heart rates for their age group.

Sample Assessments General Guidelines Performance and other type of assessments can be used to ascertain student achievement. Following are some examples: General Assessments

• The student will research and write a one-page report on the history of matrices.

• The student will come up with other examples of technology-induced errors (rounding off π to 3.14, using .67 for 2/3, using .33 for 1/3, etc).

• The student will create, solve, and graph a real-life problem that involves a linear inequality in two variables.


• Activity 2: The teacher will provide the student with real life data similar to the data shown in the activity and then have the student create two matrices for the data. Once the matrices have been created, the student will find the sum of the two matrices and interpret its meaning.

• Activity 3: The teacher will provide the student with two 3 x 3 matrices to

multiply by hand. Afterwards, the student will use technology to check to see if the answers were correct.


67

• Activity 5: The teacher will provide the student with pairs of matrices to determine if they are inverse matrices of one another using paper and pencil as well as by using graphing calculator technology.

• Activity 6: The student will find the solution to a system of equations using five different methods.

• Activity 8: The student will solve a system of three equations with three

unknowns for a real-life situation. For example, select three food items and obtain the following data: grams of protein, grams of fat, grams of carbohydrates, and total calories. The student will write an equation expressing the data for each food item (for example, food item number one might be 5 7 10 100p f c+ + = where 100 is the total calories for that product and p, f, and c represent the number of grams of protein, fat, and carbohydrates respectively. The student will use matrices to calculate the calories per gram (cal/gm) for protein, fat, and carbohydrates based on the nutritional notice on the sides.

Documents

Algebra I: Part 1 - Warren Easton Charter High Schoolwarreneastoncharterhigh.org/ourpages/auto/2009/11/18...2009/11/18 · Algebra I: Part 1 Unit 1 Variables and Relationships 1 Algebra