summer 2006 i2t2 number sense page 2 -...

Summer 2006 I2T2 Number Sense

N1 - Equations & Number Sense.....................................................................................3

Developing Divisibility Rules......................................................................................3 100 or Bust..................................................................................................................5

Recording Sheet 100 or Bust ...................................................................................7 Estimation Games .......................................................................................................8

Estimation Game Recording Sheet......................................................................... 11 Five Steps to Zero ..................................................................................................... 13 Exponents Exploration .............................................................................................. 14 I have… Who Has…? ............................................................................................... 15

N2 - Measuring a Million.............................................................................................. 18 Take It to a Million.................................................................................................... 18 Take It to a Million Rules for 2 Players ..................................................................... 20

Score Sheet............................................................................................................ 21 Spinner.................................................................................................................. 22 Spinner.................................................................................................................. 23

N3 - Fractions ............................................................................................................... 24 Modeling Fraction Sums and Differences .................................................................. 24 Yack in the Box......................................................................................................... 25 Block Busters ............................................................................................................ 27 Part I ......................................................................................................................... 27 Fraction Roller Coaster.............................................................................................. 29


N1 - Equations & Number Sense

Developing Divisibility Rules Adapted from "How Does One Know if a Number is Divisible by 17" by E. Paul Goldenberg, Mathematics Teacher, March 2006. The divisibility rules for 2, 3, 4, 5, 8, 9, and 10 are fairly well known. Divisibility rules for non-prime numbers can be derived from the divisibility rules for the factors. However, lesser-known divisibility rules for primes can be derived by a rather simple algorithm. Any integer, n, can be separated into two parts – the units digit, u, and the remaining digits to the left of the units digit, r. The number n can then be expressed as

�

n =10 ! r + u Then

�

n !10r = u. We want to find a number s such that our prime, p, divides

�

r ! s " u. Substituting for u, we get

�

r ! s n !10r( ) = r ! s " n +10s " r = r 10s+1( ) ! s " n . Therefore, if p divides (10s+1), then p divides n. The objective is to find a multiple of our prime that is of the form 10s+1, or has 1 as the units digit. Divisibility by 7: A multiple of 7 that ends in 1 is 21; therefore, if 10s+1 = 21 then s = 2. We can write n = 10r + u = 10r –20u + 21u = 10(r – 2u) + 21u. We know that 7 divides 21u. Since 7 does not divide 10, then 7 divides n if and only if 7 divides r – 2u. So, to determine if a number is divisible by 7, divide the number into two parts so that

�

n =10 ! r + u. Now look at r – 2u. If 7 divides r – 2u, then 7 divides n. If r – 2u is still too big, repeat with the result of that step. For example, does 7 divide 628432?

n r u n←(r–2u) Analysis 628432 62843 2 62843–4 = 62839 not obvious whether 7 divides 62839 6283 9 6283–18 = 6265 still not obvious 6265 626 5 626–10 = 616 still not obvious 616 61 6 61–12 = 49 aha, 7 divides 49

Since 7 divides 49, working backward, therefore, 7 divides 616, and 7 divides 6265, and 7 divides 62839, and 7 divides 628432. Divisibility by 13: Look for a multiple of 13 that ends in 1 – or what number times 3 ends in a 1? 13 times 7 = 91, so 10s+1 = 91 and s = 9. So our rule would be if 13 divides r – 9u, then 13 divides n. Let's test 124579 to see if it is divisible by 13.

n r u n←(r–9u) analysis 124579 12457 9 12457–81 = 12376 not obvious whether 13

divides 12376 1237 6 1237–54 = 1183 still not obvious 1183 118 3 118–27 = 91 aha, 13 divides 91 91 9 1 9–9 = 0 if I didn't know 13 divides 91

Since 13 divides 91 (or 0), then working backwards, 13 divides 124579.


Divisibility by 17: What multiple of 17 ends in 1? 17 times 3 = 51, so 10s+1 = 51 and s = 5. Again, recall that n = 10r + u, which is equivalent to n = 10r – 50u + 51u = 10(r – 5u) + 51u. We know that 17 divides 51u. So, since 17 does not divide 10, then 17 divides n if and only if 17 divides r – 5u. Try this with our last number, 124579.


divides 12412 1241 2 1241–10 = 1231 still not obvious 1231 123 1 123–5 = 118 still not obvious 118 11 8 11–40 = –29 not divisible by 17

Since 17 does not divide –29, it does not divide 118, and it does not divide 1231, and it does not divide 12412, and it does not divide 124579. Let's test the number 124576.


divides 12427 1242 7 1242–35 = 1207 still not obvious 1207 120 7 120–35 = 85 still not obvious 85 8 5 8–25 = –17 divisible by 17

The article, which appears in a new section of the journal called Delving Deeper, goes on to make some conjectures on generalizing this idea of developing divisibility rules. Some ideas were to look at other linear combinations of r and u. Certainly with the use of a calculator, students can determine if a number is divisible by 17. However, once this method is demonstrated to students, it is a nice challenge for them to develop additional divisibility rules. For example, what is the divisibility rule for 19? 23?


100 or Bust Adapted from T3 MSM Institute, Texas Instruments, 2000. NY Standard: 5.N.3 Concepts/Skills: Number Sense, estimation, place value Needs: Any calculator with a random number function or a die. Getting started: Tell students the following story: Seven people have a total of exactly $100. Each person has either all $1 bills or all $10 bills. How much money could each person have? Note: There are several possible answers. Procedure: Have students work in groups of three. While playing the game, the group will roll the number cube or will generate seven random numbers from 1-6. The result of each roll will represent either how many $1 bills or $10 bills someone in the group of seven people has. Model the game several times for the students. The object is to try to get as close to 100 as possible without going over. Responsibilities: 1. Student 1 rolls the number cube or uses the calculator to general a random number

from 1-6. Based on the whole groups decision, he or she then records the resulting digits in the proper column on the place value chart on the recording sheet.

2. Student 2 uses place-value materials to represent the amounts on the hundred grid as they are written on the chart.

3. Student 3 uses another calculator or tally sheet to keep a total by adding the amount of each roll of the number cube.

Sample Game: 1. Student 1 rolls a 2. The group decides to put the 2 in the Tens column. Student 1

writes a 2 in the tens column and 0 in the ones column. 2. Student 2 uses place value materials to represent 20 on the hundred chart. 3. Student 3 adds 20 to the total. 4. Student 1 rolls a 6. The group decides to put the 6 in the Ones column. Student 1

writes a 6 in the ones column. 5. Student 2 uses place value materials to represent 6 on the hundred chart for a total

of 26. 6. Student 3 adds 6 to the total to make 26. This continues for 5 more rolls.


Collecting and Organizing Data: While the students play the game, ask questions such as:

• How did you decide to place this digit in the ones place? The tens place?

• How does your sum affect your strategy as you play?

• What if we changed the rules so that you could go over 100, or so that you could choose to either add or subtract the number that comes up on the number cube? Could you get closer to 100?

Analyzing Data and Drawing Conclusions:

After students have played three games and recorded their data, have them work as a group to analyze the games. Ask questions such as:

• How did your strategies change within a game?

• How did your strategies change as you played more games?

• What if you did not have to roll exactly seven times? What if you could roll fewer times? What if you could roll more than seven times? How would your strategies change?

• Is there any game that you played that could have made a sum of 100 if you rearranged the digits? Use your recording sheet and calculator to find out.

Continuing the Investigation:

• Play the game with polyhedral dice other than cubes and see if their strategies need to change.

• Find a set of seven rolls that would equal exactly 100.

• Investigate how many sets of seven rolls they can find.

• Revise the game to include the 100's place and try to make a sum of 1000.


Recording Sheet 100 or Bust

Collecting and Organizing Data Game 1 Game 2 Game 3 Tens Ones Tens Ones Tens Ones

Hundred Grid

Strategies we used while we were doing this activity:


Estimation Games Adapted from Investigating Mathematics with Calculators in the Middle Grades, Williams & Bright, Texas Instruments, 1998. NY Standards: 5.N.16, 17, 26, 27; 6.N.27; 7.N.19; 8.N.6 Concepts/Skills: Estimation, number sense with operations of multiplication and division. Needs: Any calculator Multiplication Game: This game involves using the calculator to compute products of numbers. Two teams compete to find a product that is between two given numbers.

Team A chooses two numbers where the difference between them is small, but at least 1. This is the Target Range. These numbers are recorded on the Record Sheet.

Team B then chooses a two-digit start number that is outside of the Target Range. This number is entered on the Record Sheet. Team B clears the calculator, enters the starting number, presses the Enter key, and passes the calculator to Team A.

Team A decides on a number such that the product of the start number and their number will be within the Target Range. Enter that factor on the Record Sheet. Team A then presses the multiplication key, their chosen number, and then Enter. They record the result on the Record Sheet. If it is in the Target Range, they win. If not, they pass the calculator to Team B.

Team B decides on a number such that the product of the last result and their number will be within the Target Range. Enter that factor on the Record Sheet. Team B then presses the multiplication key, their chosen number, and then Enter. They record the result on the Record Sheet. If it is in the Target Range, they win. If not, they pass the calculator to Team A. This continues until one team has a result in the Target Range. Note: Students should observe that they need to multiply by a decimal < 1 to get a smaller product. Sample game: Target Range: 830-840 Selected by Team A

Start Number: 37 Selected by Team B

Factor or Divisor Product or Quotient 25 925 0.9 832.5


Reflection: After playing several rounds, ask students to write two strategies they used when playing the multiplication game.


Division Game: This game involves using the calculator to compute answers to division problems. Two teams compete to find a quotient that is between two given numbers.

Team A chooses two numbers where the difference between them is small, but at least 1. This is the Target Range. These numbers are recorded on the Record Sheet.

Team B then chooses a three-digit start number that is outside of the Target Range. This number is entered on the Record Sheet. Team B clears the calculator, enters the starting number, presses the Enter key, and passes the calculator to Team A.

Team A decides on a number such that the quotient of the start number and their number will be within the Target Range. Enter that divisor on the Record Sheet. Team A then presses the division key, their chosen number, and then Enter. They record the result on the Record Sheet. If it is in the Target Range, they win. If not, they pass the calculator to Team B.

Team B decides on a number such that the quotient of the last result and their number will be within the Target Range. Enter that divisor on the Record Sheet. Team B then presses the division key, their chosen number, and then Enter. They record the result on the Record Sheet. If it is in the Target Range, they win. If not, they pass the calculator to Team A. This continues until one team has a result in the Target Range. Note: Students should observe that they need to divide by a decimal < 1 to get a larger quotient. Sample game: Target Range: 26-32 Selected by Team A

Start Number: 742 Selected by Team B

Factor or Divisor Product or Quotient 35 21.2 0.8 26.5

Reflection: After playing several rounds, ask students to write two strategies they used when playing the division game.


Estimation Game Recording Sheet Target Range: Selected by Team: Start Number:

Selected by Team:

Target Range: Selected by Team: Start Number:

Selected by Team: Factor or Divisor Product or

Quotient Factor or Divisor Product or Quotient


Selected by Team:





Selected by Team:






Selected by Team:





Five Steps to Zero NY Standards: 5.N.12, 14; 7.N.10 Concepts/Skills: Divisibility rules Needs: Any calculator In this game, Player A enters a three-digit number less than or equal to 900. Player B must reduce the given number to 0 in at most five steps, using any of the four basic operations of arithmetic and a single-digit number (>0) at each step. Player A should record each move and the results as they go along. Warning: The only allowable moves are +, –, x, or ÷ by 1, 2, 3, 4, 5, 6, 7, 8, 9. You cannot divide by a number > 9 or multiply by 0. Note: If a student divides by a number that is not a factor of the Start Number, then the result is a decimal number that cannot be reduced to zero by further addition, subtraction, or division. Dividing reduces numbers faster, but students should notice that they need to add or subtract first before dividing so that the result is an integer. Sample Game Player A enters the number 703

Player B: 703 – 3 Enter

700 ÷ 7 Enter

100 ÷ 5 Enter

20 ÷ 5 Enter

4 – 4 Enter

0


Exponents Exploration The following problems come from “Engaging Students through Technology” by A. Kursat Erbas, Sarah Ledford, Drew Polly, and Chandra H. Orrill, Mathematics Teaching in the Middle School, February 2004.

NY Standards: 6.N.6, 23, 24; 7.N.5, 7

Counting Mealworms:

Jack and Jill each have two mealworms. Jack’s mealworm population will triple each month. Jill’s mealworm population will increase by 40 each month. During what month(s) will Jack and Jill once again have the same number of mealworms? How many mealworms will each of them have? Month 1 2 3 4 Jack’s extended pattern 2 2 x 3 2 x 3 x 3 Jack’s exponential pattern

2 x 30 2 x 31 2 x 32

Jack’s total mealworms 2 6 18 Jill’s extended pattern 2 2 + 40 2 + 40 + 40 Jill’s total mealworms 2 42 82

1. If x represents the number of months and y represents the total number

of mealworms, what is an equation that represents Jack’s total mealworms?

What does the 2 represent in the equation? What does the 3 represent in the equation? 2. If x represents the number of months and y represents the total number

of mealworms, what is an equation that represents Jill’s total mealworms?

What does the 2 represent in the equation? What does the 40 represent in the equation? 3. When do they have the same number of mealworms? What are two different ways that you can find this answer?


I have… Who Has…?


N2 - Measuring a Million Take It to a Million Adapted from Investigating Mathematics with Calculators in the Middle Grades, Williams & Bright, Texas Instruments, 1998. NY Standards: 5.N.1, 3, 8, 23; 6.N.1, 25; 7.N.5, 6, 14 Concepts/Skills: Place value, number relationships, problem solving skills, using scientific notation Needs: Any calculator with a 10n or 10x key, score sheet, and spinner with ones, tens, hundreds, thousands, ten-thousands, hundred-thousands. Getting started: Ask the following question: How much is a million? Encourage students to extend their answers until you get responses such as: 10 hundred-thousands, 100 ten-thousands, 1000 thousands, 10,000 hundreds, or 100,000 tens. The goal of this game is to use addition and subtraction in order to reach 1,000,000. Making Mathematical Connections: Enter 641,835 in an overhead calculator. Ask students the following questions.

1. Using only addition and subtraction, how could you change the number in the thousands place to a zero? (Add 9000 or subtract 1000)

How can 9000 be written using the 10x key? [

�

9 !103]

(Looks like 9*10^(3) on TI-84 calculator)

2. How would you change the number in the hundreds place to a 0? (Add 200 or subtract 800)

How can 800 be written using the 10x key? [

�

8 !102]

3. How would you change the number in the hundred-thousands place to a 0? (Add 400,000 or subtract 600,000)

How can 400,000 be written using the 10x key? [

�

4 !105]

Carrying Out the Investigation: Go over the rules of the game. Play a practice game with the class. Discuss some possible strategies. For example, if Player 2 spins "tens" and the tens place is already a zero, then Player 2 can change the tens place to any number desired. The strategy would be to block your opponent without blocking yourself from reaching the goal of one million. Making Sense of What Happened: After playing the game, encourage students to share strategies that they developed. Ask students to look for patterns on their score sheet.


Continuing the Investigation: Enter the number 345,678 into your calculator. Using only addition and subtraction, interchange the 4 and 6. All other digits are to remain the same. (i.e., result will be 365,478)

1. How many steps (operations) will it take you? Show your steps.

2. Can you change the number to 365,478 in 4 steps? (-600, +400, -40000, +60000)

3. Can you change the number to 365,478 in 3 steps? (-600, +400, +20000)

4. Can you change the number to 365,478 in 2 steps? (-200, +20000)

5. Can you change the number to 365,478 in 1 steps? (+19800) Extensions to the Game: 1. Flip a coin after spinning the spinner. If you get heads, then you must add. If you get

tail, then you must subtract.

2. Play Take It to a Hundred. Use 45.6789 as the beginning number. The rules of the game remain the same, but the beginning number and the spinner change. The spinner is changed to tens, ones, tenths, hundredths, thousandths, ten-thousandths.


Take It to a Million Rules for 2 Players

1. There are two goals for this game. Goal 1 is to get the calculator display to read 1000000. Goal 2 is to be the player with the highest total number of check marks.

2. The player with the most buttons on their clothes becomes Player 1. Player 1 enters the number 456789 into the calculator.

3. Next, Player 1 places the point of a pencil through a paper clip into the center of the spinner chart and spins the paper clip. The spin determines the place value to be changed.

4. If the digit occupying that place value is not 0, it must be changed to 0. If the digit is 0, it may be changed to any value the player chooses.

5. A player gets only one chance to change this digit. The first decision is whether to add or subtract a number. Indicate the operation that is chosen by putting a "–" or "+" in the second column on the score sheet. Next, write the number that will be added or subtracted in the third column. In the fourth column, write the number using the 10x key.

6. Player 1 enters the operation ( – or + ) and the number (using the 10x key) into the calculator and presses ENTER. Player 1 then enters the resulting quantity on the calculator display in column five.

7. If the result is correct, Player 1 puts a check mark (√) in column six, and Player 2 repeats Steps 3-6.

8. If Player 1 is not successful in changing the correct digit, Player 1 leaves column six blank, and Player 2 is given the opportunity to change that digit.

9. After one player has reached 1,000,000 or each player has had 10 turns, each player adds up the total check marks (√) acquired. One way to win is to be the player to reach 1,000,000. Another way to win is to be the player with the highest total number of check marks. It is very possible that there might be two winners. (One player reaches 1,000,000 and the other player has the most check marks.)


Take It to a Million Score Sheet Player #

+/–

Amount

Amount Using 10x

Resulting Quantity

Correct?

Start 456789 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2


Take It to a Million Spinner

ones 1

ten-thousands thousands 10,000 1,000 hundreds tens 100 10

hundred- thousands

100,000


Take It to a Hundred Spinner

ones 1

ten-thousandths thousandths 0.0001 0.001 hundredths tens 0.01 10

tenths 0.1


N3 - Fractions

Modeling Fraction Sums and Differences

Work in Pairs.

Find three different models to represent each situation.

Write an addition sentence for each model.

Use fraction models with at least 2 different “size” pieces or denominators.

Combinations whose sum is 1.

Combinations whose sum is 14

1 .


1 .


3 .

Combinations whose difference is 3

2 .

Combinations whose difference is 1.


1 .


1 .


Yack in the Box Part I Chen is creating a new game called “Yack in the box,” and he needs assistance in finding the values of the playing pieces. How can he determine the fractional values of

his game pieces? • Work with a partner. A “yack” is a new rod whose length consists of 1 yellow Cuisenaire Rod and 1 black Cuisenaire Rod attached end-to-end.

• Using Cuisenaire Rods, make all one-color combinations that will match the length of the yack.

Assume that the length of the yack represents one whole unit. For each of the one-color combinations, find the fractional part of a single rod in relation to the whole yack.

Record the color of each rod and its fractional value. Now find the fractional value of each of the remaining rods with the yack

representing 1 whole unit. Add your findings to the data already collected and arrange your data in

increasing order of value. Look for patterns and relationships in the data. Write a paragraph to explain

your findings.


Part 2 Chen is ready for you to play a game of “Yack in the Box”? Can you determine fractional relationships among the rods to help you win the game? Work in pairs. Place a set of Cuisenaire Rods in a small box. Decide who will

go first. Player A randomly selects 3 rods from the box. On a sheet of paper, Player A writes two addition sentences about the 3 rods.

One equation should relate the colors of the rods and the other equation should relate their fractional values. Fractional values are to be expressed in terms of yacks. For example, r + g + p = e; 1/6 yack + 1/4 yack + 1/3 yack = 3/4 yack.

Then Player A selects 2 of the 3 rods, and writes two subtraction sentences

about them. One equation should relate the colors of the rods and the other equation should relate their fractional values. As before, fractional values should be expressed in terms of yacks. For example, p – g = w; 1/3 yack – 1/4 yack = 1/12 yack.

Player B checks Player A’s equation sheet. If all statements are correct, Player

A earns 1 point. If a mistake(s) is found, Player B can make the correction(s), and then he or she receives the point.

After returning the 3 rods to the box, Player B selects 3 rods from the box, and

he or she repeats the activity. Play continues by alternately drawing rods, writing the sets of equations, and

checking results. The first player to earn 8 points is the winner.


Block Busters

Part I Solving puzzles can be both fun and challenging. Block Buster puzzles are designed to integrate fraction skills and geometric artistry. How many can you solve?

Working with a partner, use Pattern blocks to solve each of these fraction puzzles in at least two different ways. The fractions refer to the area of each shape.

Puzzle 1: Build an equilateral triangle that is 25

9 green, 25

4 blue,

25

6 red, and 25

6 yellow.

Puzzle 2: Build an equilateral triangle that is 3

1 green, 3

1 blue, and 3

1

red.

Puzzle 3: Build a parallelogram that is 16

1 green, 4

3 red, and 16

3

yellow.

Puzzle 4: Build an isosceles trapezoid that is 8

1 green, 5

1 blue, 8

3 red,

and

10

3 yellow.

Record and color each of your puzzle solutions on Pattern Block triangle paper. Cut out and label the back of each solution with the puzzle number. Be ready to discuss your findings.


Part 2 Extra Credit What; if... you have an opportunity to create a series of 4 Block Buster puzzles for other students to solve? What challenging problems can you devise? • Working with your partner, use a combination of green, blue, red, and yellow Pattern blocks to build 4 different geometric figures for the puzzle. • Record and color the puzzle and its pieces on Pattern Block triangle paper. • Determine the fractional part of the whole shape that each color represents. Record this information as the clues to build your puzzle. .


/

/

/

/

/

/ _________

/

Fraction Roller Coaster REJECTED FRACTIONS NUMBER

+ _ + _ + _ =

Players will take turns. 1. Generate 3 random integers on the calculator. 2. Use two of the number to create a fraction. 3. Place your fraction in any of the first 6 boxes. 4. Place the third number in the rejected number

column. 5. Generate three more random numbers and

repeat steps 3 and 4 until all 12 boxes are filled.

6. You may not move or change any fraction once it has been placed.

The winner is the person with their total closest to 1.

summer 2006 i2t2 number sense page 2 -...

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