• MA.5.NSO.1 Number Sense and Operations

    Understand the place value of multi-digit numbers with decimals to the thousandths place.


    Compose and decompose multi-digit numbers with decimals to the thousandths in multiple ways using the values of the digits in each place. Demonstrate the compositions or decompositions using objects, drawings and expressions or equations. Example: The number 20.107 can be expressed as 2 𝑡𝑒𝑛𝑠 + 1 𝑡𝑒𝑛𝑡ℎ + 7 𝑡ℎ𝑜𝑢𝑠𝑎𝑛𝑑𝑡ℎ𝑠 or as 20 𝑜𝑛𝑒𝑠 + 107 𝑡ℎ𝑜𝑢𝑠𝑎𝑛𝑑𝑡ℎ𝑠. 

    Purpose and Instructional Strategies
    The purpose of this benchmark is for students to use place value relationships to compose and decompose multi-digit numbers with decimals. While students have composed and decomposed whole numbers in Grade 3 (MA.3.NSO.1.2) and fractions in Grade 4 (MA.4.FR.2.1), naming multi-digit decimals in flexible ways in Grade 5 helps students with decimal comparisons and operations (addition, subtraction, multiplication and division). Flexible representations of multidigit numbers with decimals also reinforces the understanding of how the value of digits change if they move one or more places left or right (MA.5.NSO.1.1). Composing and decomposing numbers also helps build the foundation for further work with the distributive property in Grade 6 (MA.6.NSO.3.2).

     Instruction may include multiple representations using base ten models (MTR.2.1). During instruction, teachers should emphasize that the value of a base ten block (or
    another concrete model) is flexible (e.g., one flat could be 1 ten, one, tenth, hundredth, and so forth). Using base ten models flexibly helps students think about how numbers can be composed and decomposed in different ways. For example, the image below shows 2.1. This representation shows that 2.1 can also be composed as 21 tenths or 210 hundredths. Thinking about 2.1 as 210 hundredths may help subtracting 2.1 – 0.04 easier for students because they can think about the expression as 210 hundredths minus 4 hundredths, or 206 hundredths. 


    Representing multi-digit numbers with decimals flexibly can help students reason through multiplication and division as well. For example, students may prefer to multiply 1.2 x 4 as 12 tenths x 4 to use more familiar numbers. (MTR.2.1, MTR.5.1)

     Students should name their representations in different forms (e.g., word, expanded) during classroom discussion. While students are representing multi-digit numbers with decimals in different ways, teachers should invite all answers and have students compare them. (MTR.4.1)

    Common Misconceptions or Errors

     Students may assume that the value of base ten blocks are fixed based on their previous experiences with whole numbers (e.g., units are ones, rods are tens, flats are hundreds). During instruction, teachers should name a base ten block for each example so students can relate the other blocks. (For example, “Show 2.4. Allow 1 rod to represent 1 tenth.”)

    Instructional Tasks
    Instructional Task 1
    Using base ten blocks, show 1.36 in two different ways. Allow one flat to represent 1 whole.

    Instructional Task 2
    How many tenths are equivalent to 13.2? How do you know?

    Instructional Items
    Instructional Item 1
    Select all the ways to name 14.09.
    a. 1,409 ℎ𝑢𝑛𝑑𝑟𝑒𝑑𝑡ℎ𝑠
    b. 1 𝑡𝑒𝑛 + 409 ℎ𝑢𝑛𝑑𝑟𝑒𝑑𝑡ℎ𝑠
    c. 1 𝑡𝑒𝑛 + 4 𝑜𝑛𝑒𝑠 + 9 𝑡𝑒𝑛𝑡ℎ𝑠
    d. 140 𝑡𝑒𝑛𝑡ℎ𝑠 + 9 ℎ𝑢𝑛𝑑𝑟𝑒𝑑𝑡ℎ𝑠
    e. 1,409 𝑡𝑒𝑛𝑡ℎs



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