Structural Overview
Section 1: Two-Digit Whole Numbers
Section 2: Three-Digit Whole Numbers
Section 3: Multi-Digit Whole Numbers
Section 4: Decimal Numbers, Integer Exponents, and Scientific Notation

About the Structural Overview

 

As the Structural Overview illustrates, this Learning Trajectory focuses first on two-digit whole numbers (Grades K-1), then three-digit whole numbers (Grades 2-3), and finally multi-digit whole numbers (Grade 4). More specifically, there are a total of six principles students need to grasp to develop their place value understanding:

  • Composing and decomposing numbers,
  • Unitizing,
  • Place determines value,
  • Developing flexibility with two-digit place value patterns,
  • Extending place value to three or more digits and
  • Developing flexibility with multi-digit place value patterns.

These six principles are introduced gradually and details of each principle will be discussed in this Learning Trajectory.

 

In addition, this Learning Trajectory introduces the decimal numbers (Grades 4-5), the integer exponents (Grades 5-8), and scientific notation (Grade 8).

 

CCSS-M Description

Descriptor

Section 1: Two-digit Whole Numbers

K.NBT.1 Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

 

This standard builds on students’ previous knowledge of composing and decomposing numbers up to 10 (see Standards K.OA.3 and K.OA.4 in the Addition and Subtraction LT). Composing and decomposing numbers is the first principle of understanding place value, therefore, students should be given the opportunity to compose and decompose numbers using models to represent tens and ones [10].

 

Students exhibit early competence in composing and decomposing numbers through acting on sets of objects [18]. “Composing and decomposing numbers” refers to the ability to act on a number or set of numbers to produce another number and to reverse the process. For example, a set of four objects can be decomposed into two sets of two, or a set of three and a set of one. Likewise, a set of three objects and another set of two objects can be composed into a set of five objects.

 

Two kinds of composing/decomposing numbers involve breaking apart/combining (separating/joining) and equipartitioning/reassembling (see Bridging Standard 1.EQP.B in the Equipartitioning LT). Breaking apart a number is an iterative process that produces smaller numbers, some of which are unequal. Combining is the reverse of the breaking process to produce the original number. For example, breaking apart a collection of 6 objects into a set of 1, a set of 2, and a set of 3, and combining them into the original set of 6 again. Equipartitioning a number is a recursive process that produces smaller equal numbers. Reassembly is the reverse of the equipartitioning process to produce the original number.

 

Note to teachers: There may not be apparent differences in the action of breaking apart/combining and equipartitioning/reassembling from the stance point of the students. This description is for teachers to be aware that out of this distinction, students develop two lines of reasoning, additive and splitting.

 

Place value involves both types of composition (combining and reassembly) and decomposition (breaking apart and equipartitioning). “Place” involves the fact that in, for example, the number 18, the placement of the number determines its value (see Standard 1.NBT.2.b later in this LT). The 1 represents 1 tens, and the 8 represents 8 ones; thus, by breaking it apart the number can be decomposed into 10 and 8. The relative “value” of each digit as one moves from right to left proceeds from ones, to tens (and later, to hundreds), with each “value” being ten times as large as the previous value.

Macintosh HD:Users:dickyng:Desktop:Place Value and Decimals LT:Place Value and Decimals_KNBT1_02212013.png

 

The underlying idea that students develop over time is that higher value units are built right to left by a composition that involves reassembly, for instance, 10 ones are reassembled into 1 ten [10]. Lower value units can be produced by equipartitioning the next higher value unit by 10 from left to right, for example, 1 ten can be equipartitioned into 10 ones (see Standard 1.NBT.2.a later in this LT).

 

In this standard, students are not expected to learn place value formally but to prepare them for the topic. To do this it is useful to have them count up and record larger numbers of objects and see that they can increase their certainty in their totals by making equal sized groups—often of twos, fives, or tens [21]. An equal-sized group or unit that is used for counting instead of one is called a “composite unit” and the activity of creating them is called “Unitizing.” “Unitizing” is the grouping or bundling of single units into a larger unit while still maintaining awareness of the individual units. Unitizing is the second principle in understanding place value. It can help students learn to count by units of twos, fives and tens in order to prepare for a transition to place value in first grade. (See Standards K.CC.1 and 2.NBT.2 in the Counting LT)

 

1.NBT.2.a 10 can be thought of as a bundle of ten ones, called a "ten."

 

Students compose a 10 as a bundle of 10 ones. They recognize the importance of the composite unit of 10 as a useful unit for counting and understanding the place value system. Relating this to their understanding of simple addition facts that add to ten will help solidify the importance of tens in future computations [10].

 

For example, students recognize the following relations:

 

 and

 

The representation of 10 on the right above is often called a “rod” or a “stick” when working with manipulatives such as Unifix ™ cubes or base-10 blocks.

 

Students also recognize other combinations of units of 1 that make 10, such as two groups of 5 ones shown below:

 

 and

 

Students should also be encouraged extend their experiences in kindergarten (see Standard K.OA.4 in the Addition and Subtraction LT) by stating other combinations that make 10, such as the combination of a group of 7 units with a group of 3 units.

 

Students have already learned the number words from 11 to 20 and aware of the reversal of the digits when spoken (See Standards K.CC.3 in the Counting LT). Students may be introduced to explicit number names, e.g. ten-one, ten-two, ten-three instead of eleven, twelve, thirteen to emphasize on the base ten structure and the use of the composite unit ten [3].

 

1.NBT.2.b The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

 

Students are introduced to an analysis of the numbers between 10 and 20 as composed of units of ten and units of one. They may know these numbers within a memorized sequence of counting and need to see that symbolically, there is another meaning to the symbols [1; 15].

 

As students structure the numbers 11 to 20 as combinations of tens and ones, they begin to understand the third principle of place value that “Place determines value,” i.e., the placement of digits in numbers determine what they represent - which size group (e.g., ones, tens, hundreds, etc.) they count. At this point, students become fluent in decomposing numbers into tens and ones, beginning with the numbers from 11 through 19.

 

Students’ ability to interpret two-digits numerals develops in stages [18]. Students initially recognize two-digit numerals as whole numerals, e.g., 12 represents the whole amount and no meaning is assigned to individual digits. Students then recognize that in a two-digit numeral, the digit on the right is in the “ones place” and the digit on the left in in the “tens place.” However, students at this stage only name the position of the digits without corresponding to their values. In the next stage, students interpret each digit as representing the number indicated by its face value. However, students do not recognize that the number represented by the tens digit is a multiple of 10. For instance, when presented with thirteen objects partitioned into sets of 4 and the numeral “13”, students may incorrectly correspond the 1 in 13 with the one object and the 3 in 13 with the three groups of four objects [18].

 

 

 

 

 

Finally, students recognize that the left digit in the two-digit numeral represents sets of ten objects and that the right digit represents the remaining single objects.

 

Students decompose and compose numbers using:

·         Models that support unitizing in tens and ones. This includes the use of 100 boards, ten-sticks made with Unifix™ cubes, ten-frames; hands and fingers; counters and cups; bundles of ten, and other ten-grouping models.

·         Expanded form (73 = 70 + 3).

·         Place value tables with columns labeled according to number values.

 

It is important to also emphasize the fact that the value of the digits in the tens column is ten times as large as the value of the ones digit. Students compare which is more: 1 ten and 9 ones, or 9 tens and 1 one. This fact is easy to address in working with manipulatives, but students may need to be reminded of this as they transition to numerical representations of numbers greater than nine.

 

1.NBT.2.c The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

Students recognize that the visual representations of 10s and 1s are different They first notice that since 19 can be represented as one 10 and nine 1s, then 20 could be represented as 1 ten and 10 ones or as 2 tens and 0 ones. Students can then identify other numbers that can be represented with only tens (and 0 ones). These numbers are described and ordered by how many tens can be counted in each, which is the first experience with skip counting (see Standard 2.NBT.2 in the Counting LT).

 

For example, this concept can be addressed in class by giving students a set of “rods” and then ask them to determine the value of the set. Students should be led away from counting on by ones, and encouraged to increase their count by 10 when a new “rod “ is counted .

 

1.NBT.3 Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

 

Building upon their fluency with composing and decomposing 11 to 20, students decompose and compose additional two-digit numbers in expanded form (e.g., 64 = 60 + 4, or contains six 10s and four 1s).

 

Students initially compared numbers within 10 by stating that one number is “less than” or “greater than” another (see Standards K.CC.6 and K.CC.7 in the Counting LT). In this standard, those words are tied to the symbolic notation for “less than” (<) and “greater than” (>) and “equal to” (=).

 

Note for teachers: First, students make statements about numbers using vocabulary such as: “smaller,” “smallest,” “larger,” and “largest,” and this language gradually becomes “less than” and “greater than.”

 

Students apply their understanding of number relationships to compare two-digit numbers and to order sets of numbers. To reinforce the meaning of place value, students are challenged to compare pairs of two-digit numbers. For example, they compare numbers that have the same digit in tens, such as 24 and 28 and see that because they both have 2 tens, they only need to compare the ones digits. Likewise, they compare several pairs of numbers like 35 and 45 to see that they only need to compare the tens digits. Then they compare numbers with no common digits such as 83 and 38, 49 and 61, 12 and 42. They know that a 10 is ten times as large as a 1 and thus determine that a number containing more tens will be larger than a number containing fewer tens, regardless of the number of ones. However, for numbers containing equal 10s, the number containing more ones will be greater. If two numbers contain both equal tens and ones, then the two numbers are equal. For example, students claim that 36 is less than 71 because 36 contains fewer tens than 71, and therefore they write the number sentence 36 < 71.

 

Students who have difficulty comparing two-digit numbers may not fully understand the concept that place determines value (see Standard 1.NBT.2.b earlier in this LT). They should continue to make use of base 10 blocks to help them develop number sense [10]. As students establish number patterns and understand that a digit in the tens place counts ten times as much as it would in the ones place, they can be required to express inequalities without reliance on manipulatives.

 

Students also apply the concept of place determines value to explain why a one-digit number is always smaller than a two-digit number.

 

Students understand how these symbols are read in words in a number sentence from left to right. A challenge is to have students write equivalent number sentences containing opposite inequality symbols (e.g., 84 > 79 and 79 < 84). This, for example, can be posed to the class by asking “How can I write a true statement comparing 79 and 84 with 79 in the front? What if I want to write 84 in the front?”

 

1.NBT.5 Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

 

The fourth principle of guiding students’ place value understanding is to “develop flexibility with two-digit place value patterns.”

 

Students understand that the numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 are composed of units of 10 [21] (see Standard 1.NBT.2.c earlier in this LT). They count by ten on the decade (numbers with a zero in the ones place) forward and backward (10, 20, 30,…, 100; 80, 70, 60,…, 10)

 

Students count on or back by increasing or decreasing two-digit numbers by ten (ten more or ten less) realizing that visually this changes the number of units of 10 but does not change the number of units of 1 (e.g., 23, 33, 43,…, 93; or 75, 65, 55,…, 5).

 

Base ten blocks and rods can be used to assist students in developing this skill, but by the end of instruction, students should be able to perform the calculations mentally.

 

1.NBT.6 Subtract multiples of 10 in the range 10 - 90 from multiples of 10 in the range 10 - 90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Students apply their knowledge of single-digit addition and subtraction (see Standard 1.NBT.4 in the Addition and Subtraction LT) and the place value of tens (see Standard 1.NBT.2.c earlier in this LT) to solve problems involving addition and subtraction of multiples of 10 (from 10 to 90) using a variety of strategies [2]. They unitize 10 ones as 1 ten and solve the addition and subtraction of multiples of tens as a single-digit addition and subtraction in units of tens [10].

 

For example, to find the difference 80 – 50, students may use a concrete model such as Unifix ™ cubes in composite units of 10-sticks by aligning eight 10-sticks and then taking away five to determine the amount left (three 10-sticks, or 30). Students relate this strategy to written forms based on place value and properties of operations. For instance, describing the subtraction 8 tens minus 5 tens, or as the difference between an 8 in the tens place and a 5 in the tens place. Similarly, students approach the problem by describing it as an addition situation where they find the number of tens one needs to add to 50 in order to make 80. The same types of strategies and forms can be used to determine the solution. For instance, use a concrete model of five 10-sticks and determine how many more 10-sticks are needed to make eight 10-sticks, or determine the unknown digit in the tens place that would add to a 5 in the tens place to make an 8 in the tens place.

 

The fluency that student develop in subtracting these multiples of 10 will aid them in later years as they add and subtract multi-digit numbers (see Standard 2.NBT.5 in the Addition and Subtraction LT).

 

Section 2: Three-digit Whole Numbers

2.NBT.1.a 100 can be thought of as a bundle of ten tens, called a "hundred."

Students have learned previously to count by tens as composite units and view 10, 20, 30,…, 90 as composite units of tens, in which the first digit represents the number of units of tens (see Standard K.CC.1 in the Counting LT). This idea was extended to all two-digit numbers, as combinations of composite units of 10 and units of 1 (see Standard 1.NBT.3 earlier in this LT).

 

In a similar fashion to thinking of 10 as a bundle of 10 ones (see Standard 1.NBT.2.a earlier in this LT), students think of 100 as a bundle of 10 tens. They use manipulatives as models to examine the role of place in determining value. Examples of manipulatives include 100-square, base-10 blocks, ten “sticks” made with UnifixTM cubes, ten-frames, number lines, hundreds charts, place value cards, and pictorial representations [16].

 

For example, using base-10 blocks, students recognize that ten rods (tens) make one flat (hundreds) as is seen below:

 

 and

 

The standards to this point involved students’ application of the first four of the six principles guiding their place value understanding, namely, “Unitizing,” “Place determines value,” “Composing and decomposing numbers,” and “Developing flexibility with two-digit place value patterns” (see Standards K.NBT.1, 1.NBT.2.b, 1.NBT.2.c, and 1.NBT.5 earlier in this LT). Introduction of a hundreds unit prepares them for the fifth principle of place value understanding. The fifth principle, extending place value to three or more digits, is covered in the next standard.

 

2.NBT.1.b The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

 

                 

The fifth principle of “Extending place value to three or more digits” further guides students’ development of place value understanding to three-digit whole numbers. Students form composite units of 100, and recognize that these higher-valued units can be decomposed into either 10 composite units of tens, or 100 ones.

 

An essential understanding of the place value system is the constant application of multiplication by 10 to obtain the next higher unit. Thus, to go from ones to tens to hundreds to thousands, etc., (powers of 10), involves multiplication by 10 (“the rate for composing a higher-value unit” in the base-ten number system) [10]. Students are not expected formally to use multiplication by ten to move among place values but they understand that the size of the next composite unit is “ten times as large” as the size of the previous one. They exhibit this understanding by representing numbers on the decade between 100 and 900 either as groups of tens or as combinations of hundreds and tens.

 

Students decompose three-digit numbers using models to represent hundreds, tens and ones, in multiple ways. They write those numbers using expanded form (534 = 500 + 30 + 4). Or likewise, they compose such numbers using models or expanded form.

 

For example, to represent 247, students use base-ten blocks and lay out two flats, four rods, and 7 cubes showing the composition of the expanded form:

 

 

As students write and read numbers composed of three or more digits, they may use the word “and” to separate the hundreds value from the tens value. This should be discouraged, since the word “and” is used to separate whole numbers from fractions (as well as decimal fractions) in expressing mixed numbers.

 

A common misconception is that a zero in a place allows one to skip that value. A place value table can reinforce the idea that a zero in at a particular place in a number indicates that there are no elements of that value in a number.

 

Number

Hundreds

Tens

Ones

328

3

2

8

905

9

0

5

 

Another common misconception that students demonstrate is to write 908 as 9 hundreds and 8 tens. By working with a place value table, students develop the understanding that they should write this as 9 hundred, 0 tens and 8 ones. Because of this understanding, students distinguish the values of 908 and 980 .

 

2.NBT.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

 

Students use their understanding of number relationships to compare three-digit numbers and to order sets of numbers using oral, written and symbolic notation (<, >, =). Similar to working with two-digit numbers, students recognize that a number containing more hundreds is greater than a number containing fewer hundreds (and vice versa), regardless of how many tens and/or ones are contained in either number. Students know that if the hundreds digits match, then they move to compare the tens digits. If the tens digit match, they compare the ones digits to determine order.

 

For example, students make a judgment based on the hundreds contained by 623 and 478, and compare the numbers by writing the following number sentences :

623 > 478 and 478 < 623.

 

If comparing 326 to 351, they know that since the hundreds digits match, they must compare 2 tens to 5 tens to determine that 326 < 351.

When they are asked, “Which is bigger 600 or 278?” they recognize that only the largest place value matters.

 

2.NBT.8 Mentally add 10 or 100 to a given number 100 - 900, and mentally subtract 10 or 100 from a given number 100 - 900.

Students have learned previously the first five principles guiding their place value understanding: (1) Unitizing; (2) Place determine value; (3) Composing and decomposing numbers; (4) Develop flexibility with two-digit place value patterns; and (5) Extending place value to three or more digits (see Standards K.NBT.1, 1.NBT.2.b, 1.NBT.2.c, 1.NBT.5, and 2.NBT.1.b earlier in this LT).

 

The last principle guiding students’ place value understanding is to “develop flexibility with three-digit place value patterns.” They count on or back, by increasing or decreasing three-digit numbers by 10 (ten more, or ten less) or 100 (one-hundred more, or one-hundred less). This is done mentally by applying addition and subtraction of single-digit numbers fluently along with an understanding of place value.

 

For example, to find the sum of 621 and 100, students mentally calculate 6 + 1 = 7 and state the result as 721. Likewise, to subtract 10 from 446, students mentally calculate 4 – 1 = 3 and state the result as 436.

 

Some problems of adding or subtracting 10 or 100 are initially more difficult for students because the mental calculation results in changing the hundreds digit (see Standard 2.NBT.7 in the Addition and Subtraction LT). Students learn to handle such situations with fluency through experience with performing mental calculations of problems such as those below:

 

27 + 100 = ?

235 – 200 = ?

398 + 10 = ?

507 – 10 = ?

 

3.NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100.

This standard requires students to use their understanding of place value to model, represent, and solve problems with larger numbers. They integrate these understandings as they develop, discuss and use various strategies to efficiently solve single and multi-digit problems. Students select and apply appropriate methods to estimate sums and differences by finding the closest 10 or 100 prior to computing. For example, 31 + 17 could be estimated as 30 + 20, which can be computed mentally [14]. After estimating, students justify the reasonableness of their approximation based on the context and numbers involved.

 

This will be students’ first classroom experience with rounding. To assist students, for example, draw a number line marked at tens (hundreds when rounding to hundreds), then lead the students in a discussion about where a number like 43 would be placed. It should be easy to establish that the number should be placed between the markers for 40 and 50. Also establish that it should be placed closer to 40 than 50. From this exercise, establish that whole numbers with ending digits from 0 to 4 would round down to the nearest ten, while whole numbers with ending digits from 5 to 9 will round up to the nearest ten.

 

3.NBT.3 Multiply one-digit whole numbers by multiples of 10 in the range 10 - 90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations.

Students should now have had experience with the six principles underlying their place value understanding for up to three places. They also have begun to develop fluency for multiplication and division facts from previous experience with doubling, halving, skip counting, and partitioning and reassembling (see Standards 3.OA.1, 3.OA.2, 3.OA.3., 3.OA.5 and 3.OA.7 in the Division and Multiplication LT). Their multiplication/division by tens also builds on previous experience with counting on or back (see Standards K.CC.1 and 1.NBT.1 in the Counting LT). Students know the meaning of “10 times as much/many” and “10 times less,” coming to recognize the pattern of adding or removing a zero when multiplying or dividing by 10.

 

Students multiply single-digit numbers by other multiples of 10 (less than 100). Students recognize that, for example, multiplying by 60 will result in a number that is 10 times as large as multiplying by 6. Therefore, they multiply single-digit numbers by multiples of 10 by first multiplying the single-digit number by the number in the tens place of the multiple of 10, and then find the number 10 times as large as that result (typically by adding a 0 – shifting the place value of every number one to the left).

 

For example, students multiply 7 x 40 by first multiplying 7 x 4 to get 28 and then finding the number 10 times as large, which is 280. The fact that this is written as 280 makes sense to students as the number can be decomposed into 28 tens and no ones. However, students may get 28 and may not realize that the answer is 28 tens and not 28 ones [10]. Students should use base-ten blocks to model the multiplication to prevent this misconception.

 

Students develop the understanding that if a is a single digit number and

(b x 10) is a multiple of ten, then the product is (a x b) x 10 (see Standard 3.OA.5 in the Division and Multiplication LT). However, they do not need to represent this idea abstractly.

 

 

Section 3: Multi-digit Whole Numbers

4.NBT.1 Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.

When regrouping (composing and decomposing) is introduced (see Standard 2.NBT.6 in the Addition and Subtraction LT), students understand why the procedures work (on the basis of place value and properties of operations). As students apply these algorithms, they strengthen their understanding that each time one moves to the left in a number, the value of the place increases by a factor of 10 (“10 times as large”), from ones to tens to hundreds. The constant rate of multiplying by a factor of ten each time is referred to as “the rate for composing a higher-value unit” [10]. Later, in Standard 5.NBT.1, students learn the constant rate of dividing by a factor of 10 as you move to the right is called “the rate for decomposing a higher-value unit.”

 

Building on their work in previous standards, students extend their understanding of the place value structure of the base-ten numeration system to include composite units of 1,000 and 10,000 in composing and decomposing numbers. Modeling tasks and exploring examples involving numbers with four or more digits will facilitate students’ extension of place value understanding beyond three-digit numbers [4].

 

Note to Teachers: Students are introduced to the convention of using commas after every 3 places, e.g. 1,000,000, to “chunk” the places for easier identification of each place value especially when dealing with large numbers.

4.NBT.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

Students write and describe numbers in a variety of equivalent forms, drawing on the concept of place value. They build models of numbers using various representations (e.g., base-ten blocks). They identify the value of any digit of a number up to at least 10,000. They write the number in expanded form, such as

9,876 equals 9,000 + 800 + 70 + 6 or 9 x 1,000 + 8 x 100 + 7 x 10 + 6

and describe it in alternative forms, such as 98 hundreds, and 76 ones [10].

 

When working with whole numbers in the thousands and ten thousands place value students still may exhibit the misconception that a zero in a place allows one to skip that value. Extending the place value table from standard 2.NBT.1.b can reinforce the idea that a zero in at a particular place in a number indicates that there are no elements of that value in a number.

 

Number

Ten thousands

Thousands

Hundreds

Tens

Ones

328

 

 

3

2

8

905

 

 

9

0

5

4,078

 

4

0

7

8

82,936

8

2

9

3

6

 

 

Building upon prior experiences in Standard 2.NBT.4, students compare numbers using >, <, and = signs. They compare numbers efficiently using the following rules for whole numbers.

 

1)      The greater the number of digits in a whole number, the greater its magnitude is (1,243 > 999) [20].

2)      If two whole numbers have the same number of digits, then the one with first larger digit starting from the left is the larger number.

(364 > 359 because the 3 (hundreds) are the same, but the 6 (tens) is greater than the 5 (tens). It is not necessary to go any further because even a nine in the one’s place will not change the relative value [20].

3)      If two whole numbers have the same number of digits and then digits are all the same, then the numbers are equal.

 

Students build this facility with mental computation, estimation, and paper and pencil. Work with numbers is extended to whole numbers between 0 and 10,000. Students order sets of numbers using oral and written descriptions, and symbolic notation (<, >, =).

 

4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place.

Students round numbers, based on place value to the ones, tens, hundreds, and thousands. They know that numbers in the ones digit from 0 to 4 are rounded to zero and numbers from 5 to 9, are rounded to 10 (see Standard 3.NBT.1 earlier in this LT). Using this same principle for higher place values, they round any number to any place. For instance, rounding the number 47,493 to:

·         the nearest ten is 47,490

·         the nearest hundred is 47,500

·         the nearest thousand is 47,000

·         the nearest ten thousand is 50,000

 

Students misinterpret the process and round place by place from right to left. For instance, in rounding 748 to the nearest hundred, they may first round 748 to 750 (rounding to the nearest ten) and then round 750 to 800 as the nearest hundred. This is incorrect but a common misconception. A number line can be used to demonstrate this misconception. When rounding to the nearest hundred, students can see that 48 is less than 50 and round to 700 instead.

 

Students select and apply appropriate methods to estimate sums and differences by finding the closest 10, 100 or higher-value unit prior to computing. For example, 131 + 179 could be estimated as 130 + 180, which can be computed mentally as 310.

 

In situations where rounding (or estimation) is based on the context, students make decisions about the appropriate place at which to round the numbers involved. They recognize that one number could be rounded at several different places, each giving approximations that could be considered quite different depending on the precision required. For instance, the number 57,394 could be rounded to 57,390, 57,400, 57,000, 60,000, or 100,000. Students also notice that increasing the place value to which they are rounding always leads to a larger difference between the estimate and the original number. For example, students notice that the difference between 57,394 and 57,400 (rounded to the nearest 100) is larger than that of 57,394 and 57,390 (rounded to the nearest 10). Students consider the tolerance for error in choosing the place value to which to round and recognize that increasing the tolerance too much may not afford useful distinctions to be made (e.g., both 57,394 and 89,122 would round to 100,000).

 

In a practical context students may find it useful to not always round to the nearest 1 of a place but to other values. For example, students are asked to estimate the total yearly family earnings of the Grissom family if Chen, the son, earns $46,575 a year, Dana, the daughter earns $28,950 a year, and Carol, the mother, earns $91,000 a year. Students could decide to round to the nearest 10,000, as that is the largest place that all 3 numbers have in common, resulting in $170,000 (50,000 + 30,000 + 90,000); or, they may decide to round to the nearest thousand to get a more accurate result of $167,000 (47,000 + 29,000 + 91,000). To come up with a quicker estimate, students could decide to round to the nearest 50,000, which yields an estimate of $200,000 (50,000 + 50,000 + 100,000), and then realizing this rounded figure may be too crude and an overestimate, they scale back to the nearest 25,000, which results in another quick estimate of $175,000 (50,000 + 25,000 + 100,000).

 

Context also results in further changes to the way students estimate. Consider the following problem contexts.

1)      A school has 297 students and needs to hire buses to take them to a field trip. Each bus holds 36 students. How many busses are needed?

2)      You have 15 batteries. Each camera needs four batteries to operate. How many cameras can you power?

 

In the first problem, students divide and find that the answer is 8 with a remainder of 2, or 8 and one-quarter buses. The students know to round the answer to nine buses and not 8, even though 8 is “closer.” The context makes it clear than an extra bus is needed.

 

The second problem can be solved with division as well, yielding an answer of 3 with a remainder of 3, or three and three quarters of a camera. In this context the solution will be rounded down to 3 cameras even though the number 4 is closer to the mathematical solution.

 

 

Section 4: Decimal Numbers, Integer Exponents, and Scientific Notation

4.NF.6 Use decimal notation for fractions with denominators 10 or 100.

When decimal numbers are taught in isolation from fractions, students misapply whole number concepts to decimals [9; 12]. This standard introduces decimals as another way to represent fractions, which students are familiar with from earlier Standards in the Fractions LT. Presenting decimals in this way will assist students in understanding decimal operations as logical concepts rather than arbitrary rules [5]. Students coordinate their understanding of decimals with their understanding of fractions.

 

Models for representing fractions include circles, rectangles, arrays and grids (area), pattern blocks (regions), fraction strips, sets, and number lines. Students use their models for fractions with models for base ten to extend the whole number notation to decimals numbers to the hundredths place [5; 11] (see Standards 3.NF.3.a, 3.NF.3.b, and 4.NF.5 in the Fractions LT).

 

Students extend their understanding of place value in whole numbers to decimals. They know that since division is the inverse operation of multiplication, they can recognize the place value of each digit to the right is ten times less than the place value of the previous digit.

 

Using this information they build the following place value table:

 

Place

Value of the place

5th place to the left of the decimal point

10,000 (ten-thousands)

4th place to the left of the decimal point

1,000 (thousands)

3th place to the left of the decimal point

100 (hundreds)

2nd place to the left of the decimal point

10 (tens)

1st place to the left of the decimal point

1 (ones)

1st place to the right of the decimal point

1/10 (tenths)

2nd place to the right of the decimal point

1/100 (hundredths)

 

The decimal point is not a place for digits, but separates the values of one or greater from the values greater than zero and less than one. This idea is reinforced as students read the decimal point as “and” when stating the value of a number expressed in decimal form. Students extend the pattern to complete the bottom two rows of the table by continuing to divide by ten.

 

Therefore, students know that 0.1 is equal to 1/10 and that 0.01 is equal to 1/100. Using their understanding of place value, they know that 0.2 = 2/10 and 0.3 = 3/10. Likewise they know that 0.01 = 1/100, 0.02= 2/100, 0.03= 3/100 and so on [5].

 

For example, students rewrite the fraction 7/10 as a decimal by expressing that 7.0  10 will produce a number that is 10 times less than 7.0 which means that the digit 7 in the new number should occupy the place one to the right of the place it occupies in 7.0, producing 0.7. They equate this number with 0.70, asserting that the decimals are equivalent (as are their corresponding fractions 7/10 and 70/100, from Standard 4.NF.5 in the Fractions LT). It is important that students name these numbers as seven-tenths and seventy-hundredths, making connections between the fraction representation and the place value of digits in the decimal number. Students should be discouraged from reading decimals as “point seven” or “point seven zero” as this does not support the understanding of the equivalence of decimal and fractional representation of these numbers [13]. Students also learn that the convention for writing decimal numbers less than 1 should contain a leading 0 in the ones place (e.g., 0.85 and 0.06).

 

When required to read decimals properly, students should avoid the misconception that the right side of the decimal behaves similarly to the left side in the number order patterns. For example, in a counting sequence from 1.1 to 1.9 some students may continue the sequence with 1.10 and 1.11. This occurs because they extended whole number reasoning to the right hand side of the decimal as they counted [17]. An inconsistency becomes apparent when students read “one and eight tenths, one and nine tenths…” the next term should be “one and ten tenths, or two” rather than “one and ten hundredths.” This reinforces the fact that the two sides of the decimal point are not two separate lists of whole numbers. Students would write: 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1… and represent them as points that are equally spaced from left to right on a number line.

 

By referring to fractions, students come to understand the effects of adding 0s to decimal numbers. When a zero is added to the end of a number with decimal places (e.g., 0.89, 0.890, 0.8900, etc.), the value of the number does not change (see Standard 4.NF.5 in the Fractions LT). The extension of decimal places does indicate an increase in the precision of that number. For example, understanding that 3.000 meters is 3 meters and 0 decimeters and 0 centimeters and 0 millimeters. Even in instances where a measurement unit is not directly associated or attached to the context, students recognize that including more decimal places is representative of using a smaller unit (see Standard 2.MD.2 in the Length, Area, and Volume LT).

 

Students locate decimal numbers (to tenths and hundredths) on the number line by first equipartitioning the number line between consecutive whole number values through performing a 10-split [12], and then (when necessary) equipartitioning the number line again through performing another 10-split between consecutive values of tenths to produce hundredths (see Standard 3.NF.1 in the Equipartitioning LT). For example, to locate 0.38 on the number line students work with the number line in the sequence shown below:

 

 

Students identify the referent unit associated with a decimal number as 1 (or 1.0), and for contextual problems where a standard unit (of measure) is associated with the value of 1 (e.g., 1 meter), they name the decimal as a combination of parts and/or wholes of that standard unit (e.g., 0.41 meters is 0.41 of a meter or 3.6 meters is 3 meters and 0.6 of a meter).

 

Students recognize when decimals and fractions are the same. They can write that 0.4 is the same as 0.40 which is the same as 4/10 or 40/100. Students know it is not equal to 0.04, which is equal to 4/100 or 1/25.

 

4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

From previous Standard 4.NF.6, students read the places of numbers expressed as decimals, using the word “and” to identify the decimal point. For example, they express 3.1 as “three and one-tenth”, and 7.02 as “seven and two-hundredths.”

 

Students compare and order decimals to the hundredths place, by:

a) systematically comparing digits in order of their magnitude; for example, to compare 0.3 and 0.27, students may recognize that 0.3 is larger because 3 tenths is larger than 2 tenths.

 

b) converting decimals to fractions with common denominators; for example, students know that 0.3 represents 30 hundredths (30/100), which is larger than .27 represents 27 hundredths (27/100) (see Standard 4.NF.5 of Fractions LT).

 

c) plotting two or more decimal numbers on the same number line or that 0.3 is to the right of 0.27 on the number line as shown below:

 

 

Students write these results using symbols for equality (<, >, or =). For example, they write the result of their comparison of the numbers in the example above as either 0.3 > 0.27 or 0.27 < 0.3.

 

When required to justify their solution according to one of the above methods, students may avoid the following common misconceptions in comparing decimals:

 

·         That “0.3 is smaller than 0.27 because 27 is greater than 3.” Students treat decimals as whole numbers when comparing decimal numbers [9].

 

·         That “0.3 is greater than 0.27 because 0.3 is tenths and 0.27 is hundredths.” Students examine the smallest place value when comparing decimal numbers. [9; 17; 19]

 

5.NBT.3.a Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 x 100 + 4 x 10 + 7 x 1 + 3 x (1/10) + 9 x (1/100) + 2 x (1/1000).

Students extend their understanding of the place value structure in the decimal system to the thousandths place. They explain the idea that

0.324 = 324/1,000, but can also be decomposed into constituent parts of 1/10’s, 1/100’s, and 1/1,000’s.

 

Students compose and decompose multi-digit numbers with decimals using models and tables to represent powers of 10 in multiple ways (see Standard 4.NF.6 earlier). They write those numbers using expanded form (e.g., 2,976.84 = 2(1000) + 9(100) + 7(10) + 6 + 8(1/10) + 4(1/100) = 2(1000) + 9(100) + 7(10) + 6 + 8(1/10) + 4(1/100)).

 

5.NBT.3.b Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

In earlier Standard 4.NF.7, students compared and ordered decimals to the hundredths place, by:

a) systematically comparing digits in order of their magnitude;

b) converting decimals to fractions with like denominators; or

c) plotting two or more decimal numbers on the same number line.

 

Students extend these ideas to compare and order decimals to the thousandths place.

 

For example, to compare 11.543 and 11.551, students may recognize that 11.551 is larger because it contains a larger digit in the hundredths place. They may also extend their understanding of conversion to fractions with common denominators. In doing so, students determine that the tens, ones, and tenths places for the two numbers are the same. They convert 0.043 to 43/1000 and 0.051 to 51/1000, which leads to the conclusion 11.543 < 11.551. Similarly, students may note that the whole number components of the numbers are the same and merely consider the fractional part of the numbers, comparing 543 /1000 and 551/1000. Students may plot the values on a number line as well by partitioning the distance between hundredths using 10-splits, similar to the practice in earlier Standard 4.NF.6.

 

5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

As students form composite units of place value units higher than 1,000 (10,000, 1,000,000, etc.), they recognize that these can be decomposed into lower-value units. Students develop an essential understanding of the place value system, which is the constant application of multiplication by 10 to obtain the next higher unit. Thus, to go from 1s to 10s to 100s to 1,000s, etc., (powers of 10), each involves multiplication by 10 (“the rate for composing a higher-value unit” in the base-ten number system). For example, when given the problem “600 times ‘what’ makes 6,000?” students relate this to saying “Six hundreds times ‘what’ makes six thousands; therefore they give an answer of 10 because the digit 6 is now just one place to the left. In parallel, to go from 1,000s to 100s to 10s to 1, that is, to the next lower unit, each involves a division by 10 (“the rate for decomposing to a lower-value unit” in the base-ten number system).

 

Students are challenged to extend their understanding of 10 as the constant factor for composing and decomposing place value unit to the decimal places. For example, based on their knowledge about thousandths as the value of the third decimal place, students use 10 as the rate of decomposing to a lower-value unit to determine the place value of the digit “6” in 0.7926.

 

This anticipates students performing operations with decimals where they need to regroup (see Standard 5.NBT.7 later in this LT).

 

5.NBT.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

 

Corresponding to students’ understanding of “the rate for composing/decomposing a higher-value unit,” an essential understanding for decimals is the constant application of multiplication or division by 10 to obtain the next higher or lower unit. Thus, going from 0.01 to 0.1 to 1.0 involves multiplication by 10 while to move in the other direction involves division by 10. Students recognize that multiplication or division by 10 does not change the digits of a number but does affect their place values.

 

Students are introduced to the powers of ten as shorthand for writing repeated multiplication. This instance marks the first time exponents will be used in instruction. To introduce exponents in this context, the following table can be used. Students can be presented with the three-column table below, in which only the first column is completed.

 

 

Students can be required to complete the second column by evaluating the expressions in the first column. Students may observe a relationship between the number of factors of 10 and the value of the number in the third column. Once this relationship is discussed, students may be presented with the task of determining the value of an expression such as 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10. Exponents can be introduced as a valuable shortcut to writing repeated multiplication. Column three of the table can be filled in, and students can be required to determine and describe the relationship between the exponent and the number of zeroes in the standard form of the number.

 

 

As an extension and to further their understanding of exponents, students could be challenged to complete a similar table involving another factor, such as the one below:

 

3 x 10

3 x 101

 30

3 x 10 x 10

3 x 102

 300

3 x 10 x 10 x 10

3 x 103

 3,000

3 x 10 x 10 x 10 x 10

3 x 104

 30,000

3 x 10 x 10 x 10 x 10 x 10

3 x 105

 300,000

 

 

With this basic understanding of exponents, students build off of their prior work in standard 5.NBT.3.a and use powers of ten in writing expanded notation. For example, they can write

716,870 as (7 x 105) + (1 x 104) + (6 x 103) + (8 x 102) + (7 x 101) + 0.

 

Students can determine the greatest place value of a number and relate this to a power of 10. For example, 716,870 is a number in the hundred thousands, which corresponds to a value of 105.

 

Note to teachers: Students at this level are not expected to understand the zero exponent property (see later Standard 8.EE.1), and therefore the ones place does not reflect a multiple of a power of 10. Also, the tens place could be written as “x 10” leaving off the exponent as an exponent of 1.

 

Students know that multiplying a whole number by ten results in adding a 0 to the number. For example, 6 x 10 = 60. 23 x 10 = 230 (see Standard 3.NBT.3 earlier). They extend this understanding to include that multiplying a number by 100 (10 x 10 or 102) adds two zeros. Therefore, students learn to generalize that multiplying a number by 10a adds a zeros to the number but not necessarily understand this notation.

 

In addition to describing numbers by associating a power of 10 with a digit, students use multiplication by 10 to move between numeric values. For example, 213 x 10 = 2130, 213 x 102 = 21,300, 213 x 103 = 213,000.

 

Because students understand that decimal numbers have the same place value structure as whole numbers (4.NF.6), they can extend this understanding of multiplication by ten to decimal numbers. For example, 716.87 x 10 = 7,168.7, 716.87 x 102 = 71,687.0, 716.87 x 103 = 716,870.

 

Students generalize that for each power of 10 they multiply by, the decimal point moves one place to the right, thus increasing the place value of each digit; and when necessary, adds a 0 to the end of the number.

 

Students explore the effects of dividing by ten. For example,

716.87 ¸ 10 = 71.687, 716.87 ¸ 102 = 7.1687, 716.87 ¸ 103 = 0.71687, 716.87 ¸ 104 = 0.071687. Students notice that for each power of 10 they divide by, the decimal point moves one place to the left, thus decreasing the place value of each digit; and when necessary, adds a 0 to the left of the number but to the right of the decimal place.

 

Therefore, students know that for multiplication of whole numbers by powers of 10, the power represents not only the change in the number of places the decimal moves but also the number of 0s added to the right of the number. Similarly, for division of numbers less than 1 by powers of 10, students notice that the power represents the number of 0s added to the left of the number before the decimal point.

 

5.NBT.4 Use place value understanding to round decimals to any place.

Students apply their understanding of place value to model, represent, and solve problems with decimals. They apply the same rules for rounding numbers as in Standard 4.NBT.3, except now they apply them to decimals. For instance, they explain that 0.6 rounded to the nearest “one” is 1.0 while 0.3 rounded to the nearest “one” is 0 (zero). They round 7.552 to the nearest hundredth to get 7.55 or to the nearest tenth to get 7.6. As students develop greater understanding of rounding, they realize that, for instance, 0.997 rounded to the nearest hundredth is 1.00

 

Students integrate their understanding of rounding decimals as they develop, discuss and use various strategies to efficiently solve single and multi-digit problems. They select and apply appropriate methods to estimate sums and differences by finding the closest place-value unit prior to computing. After estimating, students justify the reasonableness of their approximation based on the context and numbers involved. As students become more familiar and comfortable with decimal numbers, they solve problems that do not indicate a place-value for rounding or estimating. For such problems, students determine an appropriate place-value based on the context of the problem prior to performing calculations, and then round accordingly.

 

Measurement in the metric system is an excellent context for rounding decimal numbers. If students are asked measure to the nearest centimeter, they can measure to the nearest millimeter and decide how to round their answer. For example, 6 cm and 4 millimeters in measure is rounded to 6 cm. They can write this as 0.06 meters knowing that 100 centimeters makes a meter.

 

Another context for rounding decimals numbers is money. For instance students round cents to the nearest dollars to estimate the total price of two or more items (see Standard 2.MD.8 of Time and Money LT).

 

5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, and explain.

Students use their understanding about multi-digit numbers and fractions and begin adding and subtracting decimal numbers with up to two decimal places by aligning the numbers vertically so that the decimal point is in the same horizontal position and then fill in 0’s in each number for any places that are occupied in the other number. For example, to add 345.6 + 93.27, students setup the problem as:

 

 

They regroup when necessary for subtraction problems as they have done for similar problems with multi-digit whole numbers (see Standard 4.NBT.4 in the Addition and Subtraction LT), now including across the decimal point, recognizing that one whole (1) is the same as ten tenths (0.1s).

 

For multiplication problems, students consider multiplication by a decimal to be equivalent to multiplication by a fraction with a denominator that is a power of 10, i.e. (a/b) × q, where b is a power of 10 (see Standard 5.NF.4.a in the Division and Multiplication LT). Therefore, it is also equivalent to successively multiplying by the numerator and dividing by the denominator. For instance, to multiply 70 by 0.15, which is equivalent to 70 × 15/100 , students first multiply 70 by 15 and then divide by 100, which will simply move the decimal point two places to the left [7] (see Standard 5.NBT.2 earlier), giving 10.50. This is important to students’ understanding of mathematics as having a coherent structure, between various number types and operations alike.

 

Then, when students encounter multiplication by a number with both whole and decimal parts (a mixed number; e.g., 20.45), they apply the distributive property and multiply first by the whole and then by the decimal as in the previous example, combining their results by adding to find the overall product. For instance, to multiply 8 by 20.45, students first multiply 8 by 20 to get 160. They then separately multiply 8 by 45 and divide by 100 to get 3.6. Finally, they combine these results to find the overall product of 163.6. Similarly, when students are introduced to more difficult problems, where both factors are mixed numbers, such as multiplying 8.3 by 20.45, students use these previously developed ideas in conjunction. They may first multiply 8.3 by 10 to get 83 (a non-decimal, or mixed, number) and then by 20 to get 1660. They then separately multiply 83 by 45 and divide by 100 to get 37.35. Finally, they combine these results and divide by 10 (to reciprocate the initial creation of a non-decimal factor) to find the overall product of 169.735.

 

Eventually, students become comfortable with leaving factors as decimal and mixed numbers, which reduces the number of steps in the process and alleviates some instances where errors occur for forgetting to change the place-value back to its original degree by dividing by the power of 10 used to eliminate mixed numbers. In working with such problems, students should be encouraged to discover a pattern in the quantity of numbers to the right of the decimal place in their product as they relate to the quantities of numbers to the right of the decimal place in both factors. For example, noting that the product of 4.56 x 7.8 is 35.568 (a factor with 2 decimal places X a factor with 1 decimal place = a product with 3 decimal places). Another way for students to verify the number of decimal places needed in their answer is to estimate. For example 4 x 8 = 32. So students know to put the decimal place to the right of 35.

 

For division problems, students consider two separate cases:

1)      Division (of a whole number or a decimal number) by a whole number divisor; or

2)      Division (of a whole number or a decimal) by a decimal number divisor. [8]

 

In the first case, students solve using long division (adding a decimal point to the end of a whole number dividend, if necessary) and bringing the decimal point up into the corresponding location of the quotient, as shown below:

 

In the second case, students solve by first multiplying both the divisor and the dividend by the same power of 10 so that the result is an equivalent division problem, but now like the first case [8]. Students understand this equivalence by relating the idea to quantitative compensation, first introduced in Bridging Standard 1.EQP.B and Standard 1.G.3 in the Equipartitioning LT. They then proceed to solve as before.

 

For example, to find the quotient 384.72 ¸ 5.1, students rewrite the problem as shown below and then solve:

 

6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using standard algorithms.

Through classroom discussions and instructional activities, students formalize their experiences in standard 5.NBT.7 and become more fluent with the operations on decimal numbers through experience and begin to recognize patterns that can be used to create algorithms. In the previous standard, the lined up the decimal point for addition and subtraction problems results in an answer that would have at most the same number of decimal places as the component in the problem with the largest number of decimal places.

 

Students notice that division problems may result in answer with no decimal places, a few decimal places before terminating, a string of infinite repeating digits, a string of non-repeating digits followed by a string of infinite repeating digits, or a string of non-terminating and non-repeating digits (Note to teachers: this last case represents an irrational number, which is introduced later in Standard 8.NS.1 in the Rational and Irrational Numbers LT, and therefore does not need to be discussed further at this point). It is worth noting that division problems involving rational numbers will never yield irrational results, thus any solution will result in either a terminating decimal or an eventually repeating decimal.

 

Students discover that solving division problems involving either a decimal dividend or decimal divisor (or both) can always be translated to an equivalent division problem where the divisor is a whole number. For example, the quotient of 354.6 ¸ 2.34 is equivalent to the quotient 35,460 ¸ 234. Students learn to work problems out to a specified or desired number of decimal places, rounding when appropriate.

 

Students notice that for multiplication problems involving one or more decimal numbers, the total number of decimal places in the product is equivalent to the sum of the decimal places in each factor. For example, the product of 6.79 · 12.105 would have 5 (2+3) decimal places. Therefore, students solve this problem by finding the product of

679 · 12,105 (= 8,219,295), and then move the decimal point five places to the left to get 82.19295.

 

One misconception that may arise is in finding the product of 4.2 · 7.5 students will multiply 42 by 75 to get 3,150. When they move the decimal place, some students may decide that the 0 does not count once it is on the right side of the decimal point because it does not indicate any additional value. Therefore, those students may claim the answer is 3.15 because it must have two decimal places; however, this behavior is easily curbed when they consider the values of the factors in the problem, which should yield a product around 30.

 

Another misconception can arise if students move the decimal from the front of the number to the right. It is important to remind these students that the answer must be consistent with the result of multiplying the fractions.

 

6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

 

Students use exponents to code repeated multiplication, distinguishing the base from the exponent, and evaluate expressions involving whole-number exponents. For example, 34 = 3 · 3 · 3 · 3= 81.

 

A common misconception is for students to multiply the base by the exponent (e.g., 23 = 6).

 

Students solve multi-step expressions combining the use of exponentiation with other operations, recognizing that exponentiation takes precedence over multiplication and division in the order of operations (see Standard 6.EE.2.c of the Early Equations and Expressions LT).

 

Students evaluate expressions such as 23 + (5 + 3)2 - 43 by first adding the 5 and 3 in parentheses (8), followed by evaluating the terms with exponents (8 + 64 – 64), and then finally adding and subtracting from left to right to get 8.

 

Students examine problems to determine equivalence or non-equivalence of expressions involving exponents, which anticipates and builds a foundation for properties of exponents in the next standard.

 

For example, students apply order of operations to evaluate the expressions on each side of the equations below and make a judgment about the equality:

 

Is (2 + 3)3 = 23 + 33? False

Is (2 · 5)3 = 23 · 53? True

Is 23 · 24 = 27?          True

Is 25/23 = 22?            True

 

8.PVD.A Know and apply the properties exponents to generate equivalent numerical expressions with negative integer exponents.

This bridging standard is added to differentiate positive and negative integer exponents and allow students to work flexibly with both in preparation for scientific notation in later Standards 8.EE.3 and 8.EE.4. Students develop meaning for terms that have exponents taking on negative integer values by associating the repeated multiplication with use of the property, as shown below:

 

 

 

 

Students thus establish the negative exponent property (i.e., a-n = 1/an) as a special case of the subtraction property for division. It is important for students to draw connections between exponent properties, by applying them to expressions like 73 · 7-3.

 

Students reason that this expression is equal to 1 in several ways:

a)      using the product of powers and zero exponent property,

73 · 7-3 = 73+(-3) = 70 = 1;

b)      using the negative exponent property,

73 · 7-3 = 73 · (1/73) = 73/73 = 1; and

c) and by evaluating, 73 · 7-3 = 343 · (1/343) = 1.

 

Because 73·7-3 = 1, students recognize that 73 and 7-3 are multiplicative inverses.

 

Students also draw connections between decimal place values and negative integer exponents of 10, for example:


 

 

 

They build a table of the decimal equivalents of negative integer exponents of 10:

 

Decimal

Fraction (1/10n)

10-n

0.1

1/10 =1/(101)

10-1

0.01

1/100 =1/(102)

10-2

0.001

1/1000 =1/(103)

10-3

 

In reducing fractions, students may have the misconception that factors cancel off and leave zeroes (the additive as opposed to multiplicative identity). A process of including the same number of terms in the numerator and denominator of a fraction can demonstrate the mathematics involved.

 

 

 

8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions, for example, 32 x 3-5 = 3-3 = 1/33 = 1/27

 

Students use their understanding of exponents as representing repeated multiplication to derive properties of integer exponents.

 

For instance, students first expand expressions involving exponential terms with like bases that are being multiplied or divided to then simplify the expressions using exponents. Consider the following three problems as examples:

 

a.     

 

b.     

 

c.      

 

Working with several problems of these types leads students to the conclusions that:

 

1)      the product of powers (with like bases) results in a single power with an exponent equivalent to the sum of the exponents in the factors (i.e., am·an = am+n); and

2)      the quotient of powers (with like bases) results in a single power with an exponent equivalent to the difference between the exponent of the dividend and the exponent of the divisor (i.e., am/an = am-n).

3)      raising a power to a power results in a single power with an exponent equivalent to the product of the exponents in the composite power (i.e., (am)n = amn)

 

From the property involving quotients, students develop meaning for terms that have exponents taking on the value of 0 by associating the repeated multiplication with use of the property, as shown below:

 

 

A common misconception in working with such a problem is to claim that because all of the 6’s “crossed-out,” reduced, or simplified, the answer should be 0. A careful discussion of simplifying fractions and thinking of the fraction bar as indicating the operation of division, therefore actually replacing values with 1’s, as shown above, will help alleviate this mistake. They also see that the original expression represents a multiplicative identity, and it is important for students to utilize the exponent property and realize that the two results must be equivalent.

 

 

8.EE.3 Use numbers expressed in the form of a single digit multiplied by an integer power of 10 to estimate very large or very small quantities, and to express how many times larger one is than the other. For example, estimate the population of the United States as 3 x 108 and the population of the world as 7 x 109,and determine that the world population is more than 20 times larger

 

Students understand place value, and that consecutive places differ multiplicatively by 10. They compare numbers based on place value, knowing that one number is greater than another number if it contains a non-zero digit in any place to the left of the largest place occupied by a non-zero digit in the other number. Relating this notion to exponents, students recognize that any number multiplied by a larger power of 10 is greater than a number multiplied by a smaller power of 10 (and vice versa).This standard will prepare students to work with scientific notation.

 

Students have learned to estimate numbers to the nearest ten, hundred, and thousand. With this use of exponents this is extended to all powers of 10. Students decompose such estimates into the product of a single digit and a power of 10 (a precursor to scientific notation, which is formalized in the Standard 8.EE.4 later). For example, given the number 2,549, students can round the number to the nearest thousand to get 3,000 and write the number as 3 x 103.

 

Students determine estimates of the following numbers using powers of 10 by writing, for example:

 

468302 » 500000 = 5 x 105

0.00023 » 0.0002 = 2 x 10-4

 

 

Students avoid mistakes by using loops to determine the order of magnitude of a number when writing it in scientific notation. They move the decimal point to the left or right until the place value of the first nonzero digit is in the ones place and record the number of loops made as the order of magnitude. When necessary, students add zeros above each empty loop.

For example, students convert the numbers below:

 

Note to teachers: In comparing numbers written in scientific notation in this standard and the next, students need to pay attention to the order of magnitude (the power of 10) and also learn how to put them into the same order of magnitude.

 

For example, compare and order the following numbers using powers of 10 from least to greatest:

 

4 x 106, 8 x 105, 3 x 10-4, 9 x 10-3

 

Students determine that since none of the numbers are of the same magnitude, they can be ordered based on magnitude alone. Therefore, the number with the smallest exponent comes first, followed by the number with the next smallest exponent, on up to the number with the largest exponent, resulting in the order shown below:

 

3 x 10-4, 9 x 10-3, 8 x 105, 4 x 106

 

8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

 

Operating on numbers in scientific notation provides students with experiences that will aid them in developing understanding of algebraic concepts. When students attempt to add and subtract numbers in scientific notation, they start to build the algebraic concept of combining like terms. Multiplying and dividing numbers in scientific notation parallels monomial operations, making it a useful context to reference when teaching the algebraic concept of monomial operations.

 

Building off of experiences in standard 8.EE.3, students understand scientific notation as a way of writing large (or small) numbers to a specified degree of precision. Scientific notation is useful in writing, comparing orders of magnitude, and operating on very large and very small (by order of magnitude) numbers. A number written in scientific notation consists of two factors: a decimal number with a single digit (between 1 and 9) to the left of the decimal point (in the 1’s place) and a power of 10, which indicates the largest place value unit of the number written in standard form.

 

 Once students realize that scientific notation only has one non-zero digit to the left of the decimal point, and from experience with conversion and looping in the Standard 8.EE.3 earlier, students learn to make quick comparisons between the sizes of numbers in scientific notation by comparing the magnitude of the numbers (the power of 10).

 

An interesting way to get students thinking about very large numbers and numbers of varying magnitudes is to present them with the following task:

Represent the dates below on a number line [6].

 

Years Ago

Development

Now

Modern Science and Technology

1,800

Fall of Rome

10,000

Agriculture

500,000

Fire

6.6 x 107

Flowering Plants

2.08 x 108

Largest Dinosaurs

 

Students struggle to determine how to distinctly fit all of the numbers on the same number line if it is drawn to scale. Some may try to find a useful unit (increment) for their number line by dividing the larger numbers by smaller ones.

 

Other students exhibit the common misconception of being able to use a particular number multiplied by consecutive powers of 10 as their “unit” (e.g., 50, 500, 5,000, 50,000, etc.), but they should be encouraged to explore the relationship between those numbers both additively and multiplicatively to realize that this does not represent a constant unit rate.

 

The process of operating on numbers in scientific notation develops skills that students will use later in algebra (such as performing similar operations on polynomial expressions). To multiply or divide numbers in scientific notation, students use both the commutative property of multiplication and exponent properties to operate on the parts of the numbers separately, and then they recombine those products as a single product in scientific notation. In some contexts, they may also write the result out in standard notation depending on the size of the number and the reasonableness of using scientific notation.

 

For example, students find the product (4.56 x 106)(2.1 x 104) by first thinking of the problem as (4.56 · 2.1)(106 · 104). They then multiply 4.56(2.1) to get 9.576 and apply the product of powers property to get 1010. Combining these as a product, students arrive at the result of 9.576 x 1010. Before working with numbers in scientific notation, where either or both contain decimals, students should be introduced to simpler forms and asked to perform the calculations by hand in order to develop an understanding of the process and relationships between the factors and the product. For instance, finding the product of (5 x 105)(7 x 107) by multiplying using both standard notation as well as scientific notation, as shown below:

 

Standard Notation

500,000 x 70,000,000 = 35,000,000,000,000

 

Scientific Notation

5 x 7 = 35, and 105 x 107 = 1012, so (5 x 105)(7 x 107) = 35 x 1012= 3.5 x 1013

 

A common misconception when performing operations in scientific notation is to leave the result without checking if the number is in the correct form of scientific notation. For example, students multiply (5 x 105)(7 x 107) resulting in the product of 35 x 1012. Students fail to convert it into 3.5 x 1013.

 

As another example, students find the quotient (7.8 x 104)/(9.4 x 105) by first thinking of the problem as (7.8/9.4)(104/105). They then divide 7.8/9.4 to get 0.8298 (after rounding) and apply the quotient of powers property to get 10-1. Combining these as a product, students arrive at the preliminary result of 0.8298 x 10-1. Noting this is not in (normalized) scientific notation, students rewrite the first factor (0.8298) as 8.298 x 10-1, and the result as 8.298 x 10-1 x 10-1 = 8.298 x 10-2. Realizing that this is not a very small number, students may also rewrite the result as 0.08298. Students become aware that some problems result in answers that need to be adjusted by moving or coordinating decimal places while others do not, and also some result in answers that are not large or small enough to warrant the use of scientific notation despite the problem involving such numbers.

 

To add or subtract numbers in scientific notation, students convert one or both of the numbers so that the orders of magnitude are the same. The benefit of this process can be seen by first expanding the numbers to standard notation, as is also shown in the following example:

 

Students find the difference (5.7 x 108) – (3.4 x 106) by converting the larger number to the same power of 10 as the smaller number. They then work with the problem (570.0 x 106) – (3.4 x 106) and realize that they do not need to expand the numbers into standard notation since the magnitude of the exponents are the same. Therefore, students arrive at the result

566.6 x 106 which they convert back to the proper form of scientific notation and get 5.666 x 108. Some students may convert 3.4 x 106 to

0.034 x 108 instead thus eliminating the need to convert back to the correct form of scientific notation.

 

A common misconception is that students perform addition or subtraction of the exponents similar to the process of multiplying or dividing numbers in scientific notation.

 

Students could also write this problem out in standard form as:

 

They could perform the calculation from here, but also notice that the first five decimal places have values of 0 in both numbers and these can be simplified from the calculation using scientific notation. They recognize the places where calculation matters and draw lines like those shown above to determine how to rewrite the numbers, leading to the calculation shown below (and relating it to that which was shown in the previous paragraph):

(5700 – 34) x 105 = 5666 x 105 = 5.666 x 108

 

Through classroom activities such as using calculators to compute 520 students become familiar with the way calculators represent numbers in scientific notation and that numbers containing more digits than what the calculator can display are automatically rounded and converted to scientific notation by the calculator. Many models of calculators do not use the “times 10 to a power” notation but simply use a capital “E” to indicate scientific notation. For instance, a calculator may display 7 x 1011 as 7E11, where 11 represents the power of 10 for the number in scientific notation. Older calculators may represent the power of ten as an exponent without a 10 or an “E” On these 7 x 1011 will appear as 711 on the calculator. In this case, it is important for students to understand that this misleading representation does not mean “seven to the eleventh power.”

 

 

References

 

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