Math Essentials, Middle School Level

John Wiley & Sons

Chapter One

Objective 1

Compare and order fractions, decimals (including tenths and hundredths), and percents, and find their approximate locations on a number line.

Students must be able to compare different real numbers. The comparing and ordering of several decimal numbers requires a strong understanding of place value. The skill to order signed decimal numbers, both positive and negative numbers within the same set, is also expected at the middle school level. Thus, students need experience with ordering a variety of real numbers. The following activities provide such experience with decimals, fractions, and percents. The relationship of positive numbers to negative numbers is emphasized, and mastery of basic equivalent decimals, fractions, and percents will be assumed.

Activity 1: Manipulative Stage

Materials

Pattern 1-1a and Pattern 1-1b for number cards for each pair of students Worksheet 1-1a Scissors Regular pencil

Procedure

1. Give each pair of students a copy of Pattern 1-1a, Pattern 1-1b, and scissors. Each student should also receive a copy of Worksheet 1-1a.

2. Each pairshould cut apart the number cards shown on the two pattern sheets. These cards will be used to determine the order of each set of numbers on the worksheet.

3. For each exercise, partners should select a tentatively lesser number in the set and build it with the appropriate number cards. Then they should add on other cards having positive values in order to change from the selected number's value to a total card value equal to another number in the set. If this can be done, the selected number is less than the new number found. If this cannot be done, the selected number is greater than any of the other numbers in the set.

4. Other numbers from the set may need to be tested in this same manner before the final order of all numbers in the set can be determined. Once the order is found, students should record the numbers in the correct order below the exercise, using the appropriate < or > sign.

5. Guide students through Exercise 1 on Worksheet 1-1a before they proceed to the other exercises.

For the first exercise on Worksheet 1-1a, the set of numbers, +2.5, -3, -1.5, +0.75, must be ordered from least to greatest. Since the numbers in this exercise are all decimals, only the integer and decimal cards need to be used.

As an example, have students select -1.5 as the first number to build with the number cards. Students should place one (-1) card and one (-0.5) card on the desktop; these two cards have the total value of -1.5. Ask the students to place more cards beside these two cards but to use only positive cards. The process of adding only positive amounts to some given real number will produce new numbers greater than the given number. When appropriate, encourage them to use positive cards to form 0-pairs with any given negative cards. For example, if a (-0.5) card has been used, a (+0.5) card might be added to form a 0-pair with it.

Students should find the total value of the cards after each new card is added. If one (+1) card is added, the total value will be -0.5; if two (+0.5) cards are also added, the total value will be +0.5. Then if two more (+1) cards are added, the value will be +2.5, where +2.5 is a number in the original set. This result indicates that +2.5 is greater than -1.5 in the set. Here is a possible card arrangement for this process. Notice that negative amounts are placed to the left and positive amounts are added to the right, reflecting a general sense of direction. Also, positive amounts are placed below negative amounts in order to form 0-pairs easily. If no negative amount were present, there would just be a row of positive amounts being joined to the right.

Notice that when increasing from -1.5 to +2.5, the number -3 was not found as a total value. So -3 needs to be tested. After three (-1) cards are placed on the desktop, one (+1) card might be added to produce a value of -2. Then one (-1) card might be traded for two (-0.5) cards. One (+0.5) card may now be added, yielding a total card value of -1.5. The total card value has increased from -3 to -1.5, a member of the given set, so -1.5 is greater than -3. Here is a possible arrangement of the final cards used:

Similarly, +0.75 can be built by adding one (+1) card, one (+0.5) card, and one (+0.75) card to the cards for -1.5; by adding to the (+0.75) card one (+0.25) card, one (+1) card, and one (+0.5) card, +0.75 can be increased to +2.5. These two tests confirm that -1.5 < +0.75 and +0.75 < +2.5. Applying some logical reasoning, students should now be able to write the four numbers in the required increasing order below Exercise 1 on Worksheet 1-1a:

-3 < -1.5 < +0.75 < +2.5

Answer Key for Worksheet 1-1a

1. -3 < -1.5 < +0.75 < +2.5 2. +3.5 > 0 > 0.5 > -4

3. -3.5 < -2.5 < -1 < +3.0

4. +1.25 > +0.50 > -0.5 > -2.75

5. +3/4 > 0 > -1 > -1 1/2

6. -3 1/2 < -1/2 > -1 3/4 > -2

7. +1 3/4 > -1/2 > -1 3/4 > -2

8. -3 < -3/4 < +1 1/4 < +2

WORKSHEET 1-1a Name ________________

Ordering Real Numbers by Building Date ________________

Find the required order of each set of real numbers by using the number cards from Pattern 1-1a or Pattern 1-1b. Record the correct order below each set, using < or >.

1. Order from least to greatest: +2.5, -3, -1.5, +0.75

2. Order from greatest to least: -4, 0, +3.5, -0.5

3. Order from least to greatest: -2.5, -3.5, +3.0, -1

4. Order from greatest to least: +1.25, -0.5, -2.75, +0.50

5. Order from greatest to least: -1 1/2, 0, +3/4, -1

6. Order from least to greatest: +4, -2, +2 1/4, -3 1/2

7. Order from greatest to least: -2, +1 3/4, -1/2, -1 3/4

8. Order from least to greatest: +2, +1 1/4, -3, -3/4

Activity 2: Pictorial Stage

Materials

Worksheet 1-1b Regular pencil

Procedure

1. Give each student a copy of Worksheet 1-1b. Have students work in pairs.

2. Review the methods for finding equivalent percents, decimals, and fractions. For example, students should recognize 25% as "25 per hundred" or "25-hundredths" by definition and then write it in decimal form as 0.25. They should also be able to write "25-hundredths" in fraction form as 25/100 and then reduce it to 1/4. This renaming process is a prerequisite for this activity.

3. Using the unit length from 0 to +1 marked off on the number line on the worksheet as the whole for comparison purposes, have students locate points on the number line whose distances from 0 represent specific portions of the unit bar length. These distances also order the numbers they represent. Portions in the positive direction will indicate increases, and portions in the negative direction will indicate decreases.

4. Guide students through Exercise 1 on Worksheet 1-1b before they proceed to the other exercise.

In Exercise 1, a situation concerning increases and decreases in profit is given. Students are asked to rank the three levels of performance and then represent their order on the number line. The total annual sales used to determine a store's profit may differ among the stores, but it is only their levels of performance (rates) that are being compared. Hence, students must think of the 75% increase as +0.75, the 0.3 increase as +0.3, and the 20% decrease as -0.20.

These three decimal numbers may now be easily compared and ordered, from least to greatest or from greatest to least. If ordered from least to greatest, the order will be -0.20, +0.3, +0.75. Then students should use the original numbers and record the following below Exercise 1: -20% < +0.3 < +75%. If they decide to order from greatest to least, they will record: +75% > +0.3 > -20%.

The ordered amounts should finally be marked on the number line. Percents, fractions, and decimals should be written in different horizontal lines for later ease of reading and comparing. Since markings are not shown on the number line for -20% or +0.3, students must estimate where their markings should be relative to the markings already shown. Here is the number line containing the numbers for Exercise 1:

WORKSHEET 1-1b Name ________________

Ordering Real Numbers Date ________________ on a Number Line

Order the real numbers in the list by marking and labeling a point on the number line for each number. Each fraction, decimal, or percent represents a certain portion of the indicated unit length from 0 to +1 or from 0 to -1 on the number line. The positive unit length represents a "100% increase," and the negative unit length represents a "100% decrease." A point marked on a unit length will represent a portion of either the increase or the decrease. Answer any additional questions given in the exercises.

1. This year, store 1 showed a 75% increase over last year's profits, store 2 showed a 0.3 increase, and store 3 showed a 20% decrease in profits. Use signed numbers with < or > to rank these three levels of performance. Estimate and label positions on the number line to represent the ranking.

2. Order the following numbers by estimating and labeling their positions on the number line:

+25%, +0.50, -3/4, -0.25, -1.0, +40%, +3/4, -80%, -1/2

Activity 3: Independent Practice

Materials

Worksheet 1-1c Regular pencil

Procedure

Give each student a copy of Worksheet 1-1c. Have students work independently. When all are finished, discuss their results.

Answer Key for Worksheet 1-1c

1. B

2. C

3. A

4. 0.2

Possible Testing Errors That May Occur for This Objective

The numbers are sequenced by size but in reverse order; for example, they are arranged in decreasing order, but the test item requires them to be in increasing order. Students clearly understand how to order the numbers but did not read the test item carefully. The first and last numbers listed in the sequence are correct, but the other numbers are randomly ordered between those two numbers. Students focus on the "least" and the "greatest" numbers, but disregard any others given in the list. The positive and negative signs are ignored, and the numbers are ordered only by their absolute values. Hence, +2, -3.5, and -1.8 are ordered from least to greatest as 1.8, 2, and 3.5.

Decimal points are ignored, so decimal numbers like 4.25, 10.8, and 0.586 are ordered from greatest to least as 586, 425, and 108. This results in the incorrect answer choice being selected (0.586, 4.25, 10.8).

WORKSHEET 1-1c Name ________________

Ordering Real Numbers Involving Date ________________ Percents, Decimals, and Fractions

Solve the problems provided. Draw a number line, and use it to order numbers if helpful.

1. The Triangle Mall manager reviewed a report that showed the percent increase in sales at 4 stores. Store 1 Store 2 Store 3 Store 4 3.5% 0.78% 2.75% 2.06%

Which lists the percent increase in sales from greatest to least for all 4 stores?

A. 3.5%, 0.78%, 2.75%, 2.06%

B. 3.5%, 2.75%, 2.06%, 0.78% C. 0.78%, 2.06%, 2.75%, 3.5%

D. 2.75%, 2.06%, 0.78%, 3.5%

2. The high temperatures in Anchorage, Alaska, for 5 consecutive days were -5 degrees, 0.6 degrees, -3.8 degrees, -1.9 degrees, and 4.2 degrees Fahrenheit. Which shows the temperatures in order from coldest to warmest?

A. 4.2ºF, 0.6ºF, -1.9ºF, -3.8ºF, -5ºF B. -5ºF, 0.6ºF, -1.9ºF, -3.8ºF, 4.2ºF

C. -5ºF, -3.8ºF, -1.9ºF, 0.6ºF, 4.2ºF

D. 4.2ºF, -3.8ºF, -1.9ºF, 0.6ºF, -5ºF

3. A librarian arranged some books on the shelf using the Dewey decimal system. Which group of book numbers is listed in order from least to greatest?

A. 724, 724.29, 724.3, 724.39

B. 105.4, 105.04, 108.21, 110.0

C. 391.5, 397.53, 399.62, 399.05

D. 549.01, 549.10, 549.02, 549.4

4. Joe is offered sales commissions of 18%, 0.2, and 3/20 of every dollar sold by three different stores, respectively. Which rate is the highest?

Objective 2 Multiply decimals to solve word problems.

The algorithm for the multiplication of decimal numbers needs to be developed well before students are required to independently work story problems requiring decimal multiplication. The algorithm depends heavily on an understanding of multiplication as the repetition of all or part of a set. It is also very complex in terms of place value. Many students have difficulty with it because they do not understand how place value changes will cause partial products to be recorded in different columns.

Activities for the development of the algorithm are described next. It is assumed for this objective that the multiplication facts have already been fully developed with manipulative materials.

Activity 1: Manipulative Stage

Materials

Building Mat 1-2a Set of base 10 blocks per pair of students (6 flats, 30 rods, 30 small cubes) Worksheet 1-2a Regular pencil

Procedure

1. Give each pair of students a set of base 10 blocks (6 flats, 30 rods, 30 small cubes) and a copy of Building Mat 1-2a. For the multiplicand or the set to be repeated, a flat will represent a one or a whole unit, a rod will represent a tenth of the one, and a small cube a hundredth of the one.

2. Give each student a copy of Worksheet 1-2a. Worksheet exercises will involve ones and tenths in one- or two-digit decimal numbers.

3. Have students model each exercise with their blocks. Then on Worksheet 1-2a below the exercise, have them record a word sentence that shows the results found on the building mat. They should count up their total blocks in the product region (that is, in the interior of the angle of the L-shape) of the mat in order to find their answers. Do not trade any blocks found in the product region. For example, if 10 hundredths are there, do not exchange them for 1 tenth. Always have students use proper place value language when verbally describing their steps to the class.

4. First discuss Exercise 1 on Worksheet 1-2a in detail with the students. Then allow them to work the other exercises with their partners.

Here is the story problem for Exercise 1: "Marian has several bags of candy in the store display case. Each bag holds 2.4 ounces of candy. A customer needs 1.3 bags for a cookie recipe.

Continues...

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Excerpted from Math Essentials, Middle School Level by Frances McBroom Thompson Copyright © 2005 by John Wiley & Sons, Inc.. Excerpted by permission.
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