Rational Number Concepts and Operations Grades 7-8 Steve Bergerson Pine River – Backus Schools [email protected] Steve Leuer Rogers Middle School [email protected]

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Executive Summary Page Minnesota Standards Addressed Know that every rational number can be written as the ratio of two integers or as a terminating or repeating decimal. Recognize that π is 7.1.1.1

not rational, but that it can be approximated by rational numbers such as

22 7

and 3.14.

Understand that division of two integers will always result in a rational number. Use this information to interpret the decimal result of a division problem when using a calculator. 7.1.1.2

For example: 125 gives 4.16666667 on a calculator. This answer is not exact. The exact answer can be expressed as 4 1 , which is the 30

6

same as 4.16 . The calculator expression does not guarantee that the 6 is repeated, but that possibility should be anticipated. Compare positive and negative rational numbers expressed in various forms using the symbols < , > , = , ≤ , ≥ . 7.1.1.4

For example: − 1 < −0.36 . 2

Recognize and generate equivalent representations of positive and negative rational numbers, including equivalent fractions. 7.1.1.5

For example: − 40 = − 120 = − 10 = −3.3 . 12

36

3

Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational. Know that the sum of a rational number and an irrational number is irrational, and the product of a non-zero rational number and an irrational number is irrational. 8.1.1.1 For example: Classify the following numbers as whole numbers, integers, rational numbers, irrational numbers, recognizing that some numbers belong in more than one category: 6 , 3 , 3.6 , π , − 4 , 3

6

2

10 , −6.7 .

Compare real numbers; locate real numbers on a number line. Identify the square root of a positive integer as an integer, or if it is not an integer, locate it as a real number between two consecutive positive integers. 8.1.1.2

For example: Put the following numbers in order from smallest to largest: Another example:

2,

3,

− 4, − 6.8,

− 37 .

68 is an irrational number between 8 and 9.

Through hands-on learning activities and exploration, students will gain a comprehensive understanding of rational and irrational number concepts. The students will understand that rational numbers can always be written as the ratio of two integers, but irrational numbers cannot. Through hands-on activities, students will learn equivalent representations of both rational and irrational numbers, as well as the relationship between two numbers; (greater than, less than, or equal).

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Pre Test (From MCA Sampler)

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Table of Contents Folding Strips.................................................................................................................................. 6  Fraction/Decimal War................................................................................................................... 10  Fraction Table Exploration ........................................................................................................... 11  Rational Number In Between ........................................................................................................ 13  Rational Number Memory Game .................................................................................................. 17  Constructing the “Spiral of Theodorus” ....................................................................................... 21  Tangrams....................................................................................................................................... 24  Exploring Irrational Numbers on Geoboards................................................................................ 28  Irrational Number In Between ...................................................................................................... 30 

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Folding Strips 1-2 Class Periods MN Standard/Objective: 7.1.1.5 Recognize and generate equivalent representations of positive and negative rational numbers, including equivalent fractions. Launch: I have a candy bar that I am going to share with two other friends. Use a strip of paper to illustrate the candy bar. Get directions from the class as to how to fold the candy bar into three equal pieces. Suddenly, three other friends join us, and now I have to divide the candy bar between six of us. Can it be done? What if I already gave my other two friends their piece, can we still share what we have with the other 3? Explore: Let the students work in groups of 3 or 4. Hand out several strips of paper to each student. Have the students follow the directions on the 2-page handout, folding strips of paper and recording their observations. Share: Students should share how they determined how to fold the paper. Some students who are comfortable with fraction equivalency, may find the missing value first and then determine how to partition the paper. Students who are not comfortable with fraction equivalency may partition the paper first and then find the missing value. Extension: Fold strips for thirds and half’s, then 4th’s and 5th’s. Have the students come up with combinations that work and combinations that don’t work. Is there a way to make any combination work? Discuss other possible folds that would show equivalency. Have the students try to find the patterns to make equivalent fractions from any other fraction. Summarize: Restate that fractions can be written in many (how many?) different ways. If time permits, have the students come to the board and write as many fractions as they can that are equivalent to ¾. At the end of the lesson, show Transparency D, where Kia said she could show that ½ = 1/3.

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Rational Number Project 2

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Rational Number Project 2

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Rational Number Project 2

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Fraction/Decimal War 1-2 Class Periods MN Standard/Objective: 7.1.1.5 Recognize and generate equivalent representations of positive and negative rational numbers, including equivalent fractions. Launch: Remember when you were in elementary school? One of the first card games you learned to play was WAR. You and a partner laid down a card at the same time, and whoever got the biggest value won the hand… Play a standard game of war with one of the students… Now we’re going to “kick it up a notch”… Explore: 1. Begin the activity by arranging the class into groups so that there are four groups and you (individually) are considered as the fifth group. 2. Elect a team leader in each group. Distribute the cards to the team leader and yourself. 3. At the same time have each team member and yourself flip one card over. Write all the numbers on the board (or overhead). Write each number as the complimentary term. • For example, if the number is a fraction, write as a decimal and if it is a decimal, write as a fraction. Ex. .25 = 1/4 • Ask students to identify fraction/decimal in least to greatest order (you may prefer greatest to least order). • The team that had the highest card is the winning team and needs to collect all cards from that set. Play continues until all cards have been shown. The winning team is the team that has the most cards. • If there is a tie between two cards, there is a war between the two teams. Three cards are laid face down and the fourth card is flipped over. The greatest number is the winner of all cards and then play continues.  4. Follow up with a team activity: • Arrange students into groups of four and give a set of war cards. • Select a team leader that will distribute the cards to all team members. • Instruct the team leaders to pass out cards around the team until all the cards have been distributed. • Remind them that all players are to flip their cards at the same time. Once they have discovered the equivalent number they can evaluate who has the highest amount and they win that set. • Continue play until all cards have been shown. The winner is the member that has the highest amount of cards.

Share: Students should share what was hardest; fractions to decimals, or decimals to fractions. What kind of strategies did they use to do the conversions? Summarize: Review student’s methods of converting fractions to decimals and decimals to fractions. Model several methods of doing the conversions.

Card Numbers 0.98,1.2, 1 9/10, $0.80,2.2,1 2/100, $0.90, $2.02, 2.2,2.02,2 2/100, 9/10, 2 1/2, $2.03, $1.09, 0.99, $1.50, 0.9, 0.8, $2.42, 2 13/100, 2.05, 1.89, 2 2/100, 1.25, 1 9/10, $1.90, 80/100, $1.02, 1.89, 1 89/100, 1.9, 2 2/10, $0.80, 1 2/100, 2.35, 1.5, $1.26, 2 12/100, 0.8, $1.02, $2.29, 2.5, 2 12/100, $2.29, 2 13/100, 1 2/10, $0.98, 2 1/2, 9/10, 2.22, 2.23,80/100, 0.9, $2.20, 1.25, $0.09, 1 9/10 0.99, 1 9/10, $0.98, 0.98, $1.92, $1.92, $1.26, 1 2/10, $1.90, 1 1/2, 2.05, 2.23, 2.22, $2.42, 2.35, $2.03, 2.5, $2.20, 2 2/10, 1.2

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Fraction Table Exploration 2-3 class periods MN Standard/Objective: 7.1.1.4 Compare positive and negative rational numbers… Launch: If I had $5,000 and offered to share 3/5 of it with you or 2/3 of it with you, which would you choose? How about if I gave you the choice of 5/7 or 4/5? Allow time for arguing and discussion. Explore: Have the students each build a fraction chart. It will be similar to a multiplication chart, but on the left side will be the numerator and across the bottom will be the denominator. In the chart they will list the fractions. Have them fill out the fraction table for numerators and denominators up to 10. After the table is completed, instruct the students to find any fractions that can be reduced, and put them in lowest terms. For example, 4/6 would really be 2/3, and 8/1 would be 8. Once the tables are completed, have the students start looking for patterns or interesting facts on their tables. Share: Have the students share what they found on their tables. Which direction do the fractions get larger/smaller. Where is the largest number? Where is the smallest number? What can we tell if the denominator of two fractions is the same? What about if the numerator is the same? How do we compare two fractions if the numerator and denominator are both different? Summarize: Restate how a fraction is affected when the numerator changes; when the denominator changes; when both are different. Pose a pair of fractions that are off the table such as 7/12 and 6/13.

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Sample Fraction Table to 8’s

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Rational Number In Between 1-2 Class Periods MN Standard/Objective: 7.1.1.4 Compare positive and negative rational numbers expressed in various forms. Launch: Ask students to come up to the board and write ¾, .7, 2/3 and π as many ways as they can. Discuss that rational numbers can be written in various forms; As the ratio of two integers, as a decimal (repeating or terminating), as a percent, etc. Also take time to identify irrational numbers. Introduce the square root of 10 and discuss where it would appear on the € number line. Explore: Divide class into pairs. Hand out one set of In-Between cards to each pair of students. Remove the -5, 0, and 5 from the deck and place them on the table with spaces in between each card. Have students deal 10 cards to each player. Out of the remaining cards in the deck, have each student pick one. The player with the highest card goes first. Students take turns placing their cards adjacent to the other cards on the table. The cards must be in the proper order from least to greatest. Once no more moves can be made, the player with the fewest cards in his/he hand is the winner. Share: Students should share about various strategies to win the game. (Blocking their partner, choosing low numbers first, etc.) Also, students should share about how they determined where rational numbers should go when they were in a different form than what was on the table. Summarize: Restate that rational numbers can take on various forms and still be equivalent. Converting all the numbers into one format is a sure fire, but tedious, method of ordering rational numbers. Estimation and number sense will save time when placing rational numbers in order. As a final exercise, you may have the students work together as a team to place all the cards in their deck in order from least to greatest before they turn them back in.

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1 2

−1

41 6

4.1

.3333...

−1

4 7

−2

5

π

3.14

− 22

7 3

9 2

−π 3 10

2

−

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3

7

−4

.9999...

14 5

−15

4

−3.9

.99

3

−4

−3.14

−.36

− 40

12

10

−4

3.8

− 20

−8

−.8

−3

3

9

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−1

−7

2

7 2

− 16

16

2.77

2.777...

−2 3 10

−2.4

−5

5

0

5

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Rational Number Memory Game 1-2 Class Periods MN Standard/Objectives: 7.1.1.1 Know that every rational number can be written as the ratio of two integers or as a terminating or repeating decimal. 7.1.1.2 Understand that division of two integers will always result in a rational number. 7.1.1.4 Compare positive and negative rational numbers expressed in various forms 7.1.1.5 Recognize and generate equivalent representations of positive and negative rational numbers, including equivalent fractions. Launch: “I want to find out which half of our class has the best memory, so today you will have a test”. Put 8-10 Dora, Blues Clues, etc. memory cards on the projector. Alternatively you can play an online game version of Memory. Divide the class into two groups to compete. Explore: Divide the class into groups of two. If there is an odd number of students, a group of three is alright. Shuffle the memory cards and lay them out face down on the table. Students take turns turning over two cards per play. The object is to find equivalent pairs. Once an equivalent pair is found, the student keeps the pair. The student with the highest number of cards in his/her possession is the winner. Share: Students will share what was hard or challenging about the game. Also have them share about strategies, patterns, and techniques they used to succeed. Have students write their patterns and tips on the board. Explore tips and tricks to convert from fractions to decimals and from decimals to fractions. Summarize: Restate that rational numbers can take on various forms and still be equivalent. Also look at the decimal to determine whether it is close to zero, one-half, or a whole. Use common decimals to help lead you to a fraction; .125, ..2, 25, .5, .75, etc. Also look at the pattern for 7th’s… 1/7 .142857 2/7 .285714 3/7 .428571 4/7 .571428 5/7 .714285 6/7 .857142

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1 2

.5

1 8

1 6

.1666...

.125

1 5

.2

1 7

1 4

.25

.142857

.333...

2 3

1 3

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1

.9999... .666...

3 4

.75

3 5

3 7

.42571

.6

5 8

.625

6 5

7 3

2.333...

1.2

3.5

3 8

7 2

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5 6

.8333...

.375

7 4

1.75

3 2

5 4

1.25

1.5

C D

π

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Constructing the “Spiral of Theodorus” Visualizing and Drawing Square Root Lengths (2 class periods) MN State Standard: Strand

Standard

Number & Operation

Number

Read, write, compare, classify and represent real numbers, and use them to solve problems in various contexts.

8.1.1.1

Benchmark Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational. Know that the sum of a rational number and an irrational number is irrational, and the product of a non-zero rational number and an irrational number is irrational.

Launch:

Rational numbers are values that we use daily. Irrational numbers are not something that we pay much attention to, or really understand. Exactly what is the value of ? We can approximate them rather easily, but square roots are hard numbers to comprehend. However, they are rather easy to visualize or even draw. Your job today will be to draw lines that are lengths of square roots. We are going to reenact a project that was done a couple thousand years ago by a 5th century Greek mathematician, Theodorus of Cyrene (465 BC – 398 BC). Little is known about him; however, Plato attributes to him the first proof of the irrationality of the square roots of 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15 and 17. His method involves a spiral composed of contiguous right triangles with hypotenuse lengths equal √2, √3, √4... up to √17. This is now called the "Spiral of Theodorus". (http://dic.academic.ru/dic.nsf/enwiki/629221)

Explore:

Hand out the “Spiral of Theodorus” instruction and grading sheet (attached). Help the class get started as a group by going through constructing the first triangle or two. This project will most likely not be concluded on the first day. You can assign it as homework for the night, or continue with the spiral construction and decoration on day 2.

Share:

At the end of the class period, have students share things they noticed about the project: any patterns, strange or common occurrences, etc. Also have them share their pictures, and hang them around the room. Discussion Questions: What were the hypotenuse lengths of your right triangles you formed? How do you know those are the lengths? How were you able to measure a line that has a decimal value that never repeats, and never terminates?

Summarize: Re-emphasize the fact that concepts such as irrational numbers and square roots, while abstract, are a more common occurrence than they might think. Also, talk about how math is much more than just numbers and symbols. You can represent abstract concepts through other means, such as art.

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Spiral of Theodorus Art Project Assignment Create a Spiral of Theodorus neatly and in color. Mark the unit measures of all of your triangle sides. Feel free to decorate your wheel in a way that demonstrates this spiral in the real world. Attach a sheet of paper to your project with your calculations for the first 8 triangles; showing how you calculated all the sides for each triangle. Instructions 1. Using a template for a 1 inch length and right angle (3 x 5 index card), create an isosceles right triangle. 2. Using your template again, add another 1 inch length and right angle to the hypotenuse of your original right triangle. Create a new right triangle. 3. Keep adding a new 1 inch length and right angle to the previous hypotenuse to build new right triangles. 4. When you get to the stage where your right triangles overlap previous right triangles, draw your hypotenuse toward the center of the spiral, but do not mark over the previous drawings. 5. Remember to label your figure with all of the dimensions of your right triangles. Grading Rubric • • • • • • • • •

Include a title for your picture. On the front of your picture, include your signature and the date Label all triangle legs and hypotenuses with appropriate lengths. Join each new right triangle with the hypotenuse of the previous one. Make sure your project is neat. Use color to decorate your “Spiral.” Write your labels using the radical symbol unless they can be simplified to rational numbers. Connect all of your hypotenuses to the same central point. Attach a sheet of paper to your art work containing your calculations to find the lengths of segments or the first 8 triangles.

Irrational Numbers Can “In-Spiral” You, Mathematics Teaching in the Middle School, April 2007 http://www.ldlewis.com/Teaching-Mathematics-with-Art/documents/MTMS2007-04-442a.pdf

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Spiral of Theodorus Examples:

http://upload.wikimedia.org/wikipedia/commons/0/07/Wheel_of_Theodorus.png

http://www.ldlewis.com/Teaching-Mathematics-with-Art/Wheels.html

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Tangrams A Puzzle of Squares and Square Roots (2 class periods) MN State Standards: Strand

Standard

Number

Number & Operation

Read, write, compare, classify and represent real numbers, and use them to solve problems in various contexts.

8.1.1.1

Number & Operation

Read, write, compare, classify and represent real numbers, and use them to solve problems in various contexts.

8.1.1.3

Launch:

Benchmark Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational. Know that the sum of a rational number and an irrational number is irrational, and the product of a non-zero rational number and an irrational number is irrational. Determine rational approximations for solutions to problems involving real numbers.

We are going to work with an ancient Chinese puzzle called a Tangram. The invention of the tangram puzzle is unrecorded in history. The earliest known Chinese book is dated 1813 but the puzzle was very old by then. The Rules The classic rules are as follows: You must use all seven tans, they must lay flat, they must touch and none may overlap. First, we need to create our tangram pieces (hand out the folding and cutting directions to the students). After they have created their tangram pieces, have them work on several puzzles as a class. Have students share multiple solutions if there are any. Then have the students spend some time creating and sharing puzzles of their own.

Explore:

Pose the Tangram Challenge question for day 1: Can you construct squares using 1 tangram piece, 2 tangram pieces, 3 tangram pieces, etc, all the way up to 7 tangram pieces? Be sure to keep a sketch of any solutions that you find. Assign as homework. After sharing their findings, solutions, and thoughts about the Tangram Challenge #1, give them Tangram Challenge for day 2: Create a chart to record the Number of Pieces used to form a square, the Area of the square, and the Side Length of the square. Be sure to show how you found the side length of each square.

Share:

Have a class discussion about their findings with Number of Pieces used to form a square, the Area of the squares, and the Side Length of the squares. How did you find the areas of each square? How did you find the side lengths of each square? How do you know those are the side lengths without measuring? Were you able to create a square using just 6 pieces? How do you know it cannot be done?

Summarize:

Review how we found the side lengths of the squares and tangram pieces and how we use the Pythagorean Theorem to find those lengths. And, just as with the “Spiral of Theodorus”, talk about how math is much more than just numbers and symbols. You can represent abstract concepts through other means, such as art.

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Folding Your Own Tangram - Directions

    1. Start with a square piece of construction paper. 2. Fold in half along the diagonal and cut along the resulting crease (A & B).

3. Fold one of the resultant triangles in half and cut. Set these two pieces aside (1 & 2).

4. Find the midpoint of the hypotenuse of the remaining large triangle by folding and lightly creasing. 5. Fold the triangle so that the vertex of the right angle touches the midpoint of the hypotenuse. Cut along the resulting crease, and set aside the small triangle (3).

6. Fold the remaining trapezoid in half along its line of symmetry and cut into two smaller trapezoids. 7. Fold one trapezoid so that the vertices at the ends of the longest edge meet. Cut along the crease to make a small triangle and a square (4 & 5).

8. Fold the other trapezoid so that the obtuse angle meets the right angle at the opposite vertex. Cut along the crease to make another small triangle and a parallelogram (6 & 7). http://mathforum.org/trscavo/tangrams/construct.html

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Warm-Up Tangram Shapes

http://www.logicville.com/tangram.htm

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Tangram Challenge – Day 1 Can you construct squares using 1 tangram piece, 2 tangram pieces, 3 tangram pieces, etc, all the way up to 7 tangram pieces? Be sure to keep a sketch of any solutions that you find. Solutions:

1 Piece

2 Pieces

3 Pieces

4 Pieces

5 Pieces

7 Pieces

http://www.ac.wwu.edu/~mnaylor/tangrams/tangramactivities.html

Tangram Challenge – Day 2 Create a chart to record the Number of Pieces used to form a square, the Area of the square, and the Side Length of the square. Be sure to show how you found the side length of each square.

# of Pieces Used

Area of the Square

1 2 3 4 5 6 7

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Side Length of the Square

Exploring Irrational Numbers on Geoboards Possible Line Segment Lengths (2 class periods) MN State Standards: Strand

Standard

Number

Number & Operation

Read, write, compare, classify and represent real numbers, and use them to solve problems in various contexts.

Number & Operation

Read, write, compare, classify and represent real numbers, and use them to solve problems in various contexts.

Launch:

8.1.1.1

8.1.1.2

Benchmark Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational. Know that the sum of a rational number and an irrational number is irrational, and the product of a non-zero rational number and an irrational number is irrational. Compare real numbers; locate real numbers on a number line. Identify the square root of a positive integer as an integer, or if it is not an integer, locate it as a real number between two consecutive positive integers.

Pass out and introduce the Geoboards if you haven’t used them in class before. What is a Geoboard? Set of pegs in a square pattern on which rubber bands are placed to explore geometric concepts and properties. Use tools properly! The owner of a rubber band pistol was recently arrested because it was a W.M.D. WEAPON OF MATH DISRUPTION! Next, have the students try finding as many squares of different areas as possible. Don’t forget about all of the squares on point!

Explore:

Pass out a Geoboard dot sheet to each student and go over the Activity Instructions. On a 5 peg by 5 peg geoboard find all unique line segments. Record the lengths of each line segment on the Activity Sheet, including a picture. For non-integer lengths, estimate their lengths as best you can. Can you find an exact value for these? Put the lengths in order from shortest to longest. Also have the students record how they found those lengths on the geoboards. 

Share:

On day 2, have the students share how many different lengths they found, and how they found them. Get the solutions on the board or around the room. Discuss with the kids what they found and how they found them. Other discussion questions: How many unique lengths are there? Which of the lengths are rational? Which are irrational? How do you know this?

Summarize: Re-emphasize the use of the Pythagorean Theorem to find line segment lengths and how it’s possible to represent an irrational numbers on a geoboard. Clear up any misconceptions the kids may have had.  http://taselm.fullerton.edu/august%20institute%20page/2006/Exploring%20Irrational%20Numbers%20on%20a%20Geoboard%5B1%5D.ppt

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Irrational Number In Between A Game of Comparing and Ordering (1 class period) MN State Standards: Strand

Standard

Number

Number & Operation

Read, write, compare, classify and represent real numbers, and use them to solve problems in various contexts.

Number & Operation

Read, write, compare, classify and represent real numbers, and use them to solve problems in various contexts.

Launch:

8.1.1.1

8.1.1.2

Benchmark Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational. Know that the sum of a rational number and an irrational number is irrational, and the product of a non-zero rational number and an irrational number is irrational. Compare real numbers; locate real numbers on a number line. Identify the square root of a positive integer as an integer, or if it is not an integer, locate it as a real number between two consecutive positive integers.

We are going to play a game that I’m sure most of you have played growing up…WAR. Except this time, instead of using a standard deck of cards, we are going to use cards that have fractions and square roots on them – Real Number War. The rules of the game are the same as the original version of the game. Each of you keeps your deck face down and turns one card over at a time. The winner is the person whose card has the highest decimal value. Remember, if you and your opponent flip cards that have the same value, you both need to flip another cad with the winner taking all four cards. If you finish early, you can change the rules so that the winner is the irrational number (because they are willing to do whatever it takes to win) or if either card is an irrational number the higher decimal value still wins.

Explore: • • •

• • •

Introduce the game In-Between.

Place the whole number cards (0, 1, 3, and 5) on the table with some space in between each one. Mix the remaining cards and deal six cards to each player. Players take turns placing a card so that it touches another card according to its decimal value. You may place a card to the right of 0, either side of 1 and 3, and to the left of 5. As you lay your card, state whether it is a rational or irrational number, and its decimal equivalent rounded to the hundredths place. Cards must be placed in increasing order from left to right. A card may NOT be placed between two cards that are already touching. Your goal is to place as many cards as you can. The round is over when neither player can place any ore cards. Your score is the number of cards left in your hand. At the end of round five, the player with the lowest score wins.

Share:

Have the students share their winning scores and any strategies they tried while playing. Have the class play a couple of games, trying different strategies. What is a rational number? What is an irrational number? How do you know? What is different about the decimal equivalents of rational and irrational numbers? Are all square roots irrational? If not, which ones aren’t?

Summarize:

Re-emphasize the properties of and differences between rational and irrational numbers.

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