Patrones entre valores posicionales y operaciones con decimales
Unit Goals
Students build from place value understanding in IM Grade 4 to 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 of what it represents in the place to its left. Students use this place value understanding to round, compare, order, add, subtract, multiply, and divide decimals.
In this unit, students expand their knowledge of decimals to read, write, compare, and round decimals to the thousandths. They also extend their understanding of place value and numbers in base ten by performing operations on decimals to the hundredth.
In IM Grade 4, students wrote fractions with denominators of 10 and 100 as decimals. They recognized that the notations 0.1 and express the same amount and are both called “one tenth.” Students used hundredths grids and number lines to represent and compare tenths and hundredths.
Students rely on diagrams and their understanding of fractions to make sense of decimals to the thousandths. They see that “one thousandth” refers to the size of one part if a hundredth is partitioned into 10 equal parts, and that its decimal form is 0.001. Diagrams help students visualize the magnitude of each decimal place and compare decimals.
Locate 0.001 on each number line.
Students then apply their understanding of decimals and of whole-number operations to add, subtract, multiply, and divide decimal numbers to the hundredths, using strategies based on place value and the properties of operations.
Students see that the reasoning strategies and algorithms they used to operate on whole numbers are also applicable to decimals. For example, addition and subtraction can be done by attending to the place value of the digits in the numbers, and multiplication and division can still be understood in terms of equal-size groups.
In IM Grade 6, students will build on the work to reach the expectation to fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
Divide decimals with quotients resulting in the hundredths using place value reasoning and properties of operations.
Section Narrative
In this section, students use the relationship between multiplication and division and the idea of equal groups to make sense of division of decimals, just as they had done with whole numbers and fractions.
Students learned previously that the expression can mean finding how many are in one group if 8 is put into 2 equal groups, or it can mean finding how many groups of 2 are in 8.
Students interpret to mean finding how many groups of 1 tenths are in 8. There are 10 tenths in 1, so there must be 80 tenths in 8, so . This understanding provides a foundation for students to divide a whole number by any amount of tenths or hundredths.
For instance, to find the value of , we can see how many groups of 2 tenths are in 2.
There are 5 groups of 2 tenths in 1, so there must be or 10 groups of 2 tenths in 2, as shown in the diagram.
When dividing a decimal by a whole number, the other interpretation of division may be more intuitive.
For example, can mean putting 0.2 into 5 equal groups and finding the size of each group. The diagram shows 4 hundredths in each group, so .
Later in the unit, students use equivalent expressions to find quotients. For example, they reason that is equivalent to because both the dividend and divisor are multiplied by 10. If the value of is 15, then the value of is also 15.
Multiply decimals with products resulting in the hundredths using place value reasoning and properties of operations.
Section Narrative
In this section, students learn to multiply decimals. They continue to think in terms of place value, make connections with whole-number operations, and use diagrams to support their reasoning.
Students begin by multiplying a whole number and a decimal. To find , for instance, students shade 43 hundredths in each of two large squares, and see that the 86 shaded pieces or 86 hundredths, which is 0.86.
Diagrams also help students relate products of decimals and products of whole numbers. This diagram shows 2 groups of 43 shaded pieces where each piece is 0.01. The combined shaded region therefore represents .
Likewise, can be viewed as 15 groups of 26 hundredths or hundredths. Because is 390, the value of is 390 hundredths or 3.90.
Next, students reason about the product of two decimals. Diagrams are helpful here as well.
For example, can be represented by the area of a rectangle with side lengths of 1.5 and 0.4.
Students can see that the result is 60 hundredths or 0.60 because there are or 60 shaded pieces and each represents a hundredth.
They also recognize that they can decompose the shaded region and find (the shaded area in the first large square) and (the shaded area in the second large square) and add these partial products: .
Add and subtract decimals to the hundredths using strategies based on place value.
Section Narrative
In this section, students add and subtract decimals to the hundredths. They begin by adding and subtracting in ways that make sense to them, which prompts students to relate the operations to those on whole numbers. It also allows the teacher to take note of the strategies and algorithms they choose, including the standard algorithm and those that use expanded form.
Adding and subtracting decimals using the standard algorithm brings up a new question in terms of how the digits should be aligned. To highlight this consideration, students analyze a common error as shown here.
Before using the standard algorithm, students use place-value reasoning to decide whether sums and differences are reasonable and to ensure that the digits in the numbers are aligned correctly. As they take care to align tenths with tenths and hundredths with hundredths, students practice attending to precision (MP6).
Dividamos decimales entre números enteros
Dividamos números decimales entre números enteros.
Compare, round, and order decimals through the thousandths place based on the value of the digits in each place.
Read, write, and represent decimals to the thousandths place, including in expanded form.
Section Narrative
In this section, students reason about decimals to the thousandths place. They begin by representing decimals on gridded area diagrams, where the large square has a value of 1, and each small square within represents . Students learn that if they partition each small square into tenths, each of those parts represents a thousandth of the large square.
The diagram highlights the relationships between place values. For instance, each thousandth is of a hundredth and each hundredth is 10 thousandths.
It also helps to illustrate the structure of the number in its expanded form. In this case, the shaded region includes 3 tenths, 6 hundredths, and 8 thousandths, which can be written as .
This awareness helps to prepare students for multiplication of a decimal by a whole number later in the unit.
Students then move on to using number lines to represent decimals and to compare, order, and round them. This number line shows that because 92.415 is further to the left. It also shows that 92.451 rounded to the nearest hundredth is 92.45 and rounded to the nearest tenth it is 92.5.
