MA.5.NSO.1.1

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.5.NSO.2.4
• MA.5.NSO.2.5 
• MA.5.AR.2.1
• MA.5.AR.2.2
• MA.5.AR.2.3 
• MA.5.M.1.1
• MA.5.M.2.1

Terms from the K-12 Glossary

• Whole Number

Vertical Alignment

Previous Benchmarks

• MA.4.NSO.1.1

 

Next Benchmarks

• MA.6.NSO.2.1

Purpose and Instructional Strategies

The purpose of this benchmark is for students to reason about the magnitude of digits in a number. This benchmark extends the understanding from Grade 4 (

of what it represents in the students expressed their understanding that in multi-digit whole numbers, a digit in one place place to its left. All of this work forms the foundation for arithmetic and algorithms with decimals which is completed in Grade 6 (

• To help students understand the meaning of the 10 times and $\frac{\text{1}}{\text{10}}$ of relationship, students can use base ten manipulatives or simply bundle classroom objects (e.g., paper clips, pretzel sticks). Students should name numbers and use verbal descriptions to explain the relationship between numbers (e.g., “6 is 10 times greater than 6 tenths, and 6 tenths is $\frac{\text{1}}{\text{10}}$ of 6”). In addition to physical manipulatives, place value charts help students understand the relationship between digits in different places (MTR.2.1). 
• Instruction includes helping students understand that one-tenth of can also be expressed as “ten times less.” It also includes students knowing that as “ten times more” is the same as multiplying by 10, and "1/10 of” a number is the same as dividing the number by 10.

• The image below shows the 10 times and 1/10 of relationships by place value. For example, if the number 7,777,777 were filled in the spaces below, it would be true that each digit 7 is ten times greater than the digit 7 to its right and one-tenth the value of (or ten times less) the digit 7 to its left.

• Instruction of this benchmark should also connect students’ multiplication and division work with decimal numbers. For example, students who understand 35×2=70 can reason that 3.5×2=7 because  3.5 is $\frac{\text{1}}{\text{10}}$ of 35, therefore its product with 2 will be $\frac{\text{1}}{\text{10}}$ of 70 (MTR.5.1).
• Instruction builds on the patterns of multiplying by 10 and 1/10, and extends to multiplying by values such as 100 and 1/100 that will cause the digits to shift more than one place to the left and right as students multiply numbers with more digits. (MTR.5.1).

Common Misconceptions or Errors

• Students who use either rule “move the decimal point” or “shift the digits” without understanding when multiplying by a power of ten can easily make errors. Students need to understand that from either point of view, the position of the decimal point marks the transition between the ones and the tenths place. Instruction includes the language that the “digits shift” relative to the position of the decimal point as long as there is an accompanying explanation. An instructional strategy that helps students see this is by putting digits on sticky notes or cards and showing how the values shift (or the decimal point moves) when multiplying by a power of ten.
• Students may not understand that when the digit moves to the left that it has increased a place value which is the same thing as multiplying by 10 and when the digit moves to the right that is has decreased a place value, which is the same thing as dividing by 10. It is important to have math discourse throughout instruction about why this is happening.
• Students may not understand that the value of a digit is 10 times the value of the digit to its right only if the digits are the same.
• Students can misunderstand what “$\frac{\text{1}}{\text{10}}$of” a number represents. Teachers can connect of $\frac{\text{1}}{\text{10}}$ of to “ten times less” or “dividing by 10” to help students connect $\frac{\text{1}}{\text{10}}$ of a number to 10 times greater. 

Strategies to Support Tiered Instruction

• Instruction includes opportunities to use a place value chart and manipulatives such as base-ten blocks to demonstrate how the value of a digit changes if the digit moves one place to the left or right. Have math discourse throughout instruction about why this is happening.
  • For example, the 5 in 543 is 10 times greater than the 5 in 156. Students write 543 and 156 in a place value chart like the one shown below and compare the value of the 5’s (500 and 50) using the place value charts and equations. The teacher explains that the 5 in the hundreds place represents the value 500, which is 10 times greater than the value 50 represented by the 5 in the tens place. Use a place value chart to show this relationship while writing the equation 10×50 =500 to reinforce this relationship. The teacher explains that the 5 in the tens place represents the value 50, which is 10 times less than the value 500 represented by the 5 in the hundreds place. Use a place value chart to show this relationship while writing the equation 500÷10 =50 to reinforce this relationship and repeat with other sets of numbers that have one digit in common such as 3,904 and 5,321.

• For example, 10×1=10 and 10×10=100. The teacher begins with a unit cube and explains to students that “we are going to model 10×1=10 using our place-value blocks.” Students count out 10 unit cubes and exchange them for a tens rod. The teacher explains that the tens rod represents the value 10, which is 10 times greater than the value 1 represented by the unit cube. Write the equation 10×1=10 to reinforce this relationship and repeat this process to model 10×10=100. Then, students exchange a hundreds flat for 10 ten rods to model 100 ÷10=10.The teacher explains that the value represented by a tens rod is 10 times less than the value represented by the hundreds flat and use a place value chart to show this relationship while writing the equation 100÷10=10. To reinforce this relationship repeat this process to model 10÷10=1.

• Instruction includes the use of place value charts and models such as place value disks to demonstrate how the value of a digit changes if the digit moves one place to the left or right. Explicit instruction includes using place value understanding to make the connections between the concepts of “1/10 of,” “ten times less” and “dividing by 10.” Place value charts are used to demonstrate that the decimal point marks the transition between the ones place and the tenths place.
  
  • For example, students multiply 4 by 10, then record 4 and the product of 40 in a place value chart. This process is repeated by multiplying 40 by 10 while asking students to explain what happens to the digit 4 each time it is multiplied by 10. Next, the teacher explains that multiplying by 1/10 is the same as dividing by 10. Students multiply 400 by 1/10 and record the product in their place value chart. This process is repeated, multiplying 40 and 4 by 1/10. The teacher asks students to explain how the value of the 4 changed when being multiplied by 10 and 1/10.

• For example, instruction includes using a familiar context such as money, asking students to explain the value of each digit in $33.33. Next, students represent 33.33 in a place value chart using place value disks. Then, students compare the value of the whole numbers (3 dollars and 30 dollars) and compare 0.3 and 0.03 (30 cents and 3 cents). The teacher asks, “How does the value of the three in the hundredths place compare to the value of the three in the tenths place?” and explains that the three in the hundredths place is $\frac{\text{1}}{\text{10}}$ the value of the three in the tenths place and that multiplying by $\frac{\text{1}}{\text{10}}$ is the same as dividing by 10.

Instructional Tasks

Instructional Task 1 (MTR.7.1) 

• Part A. How much would 10 pieces of candy cost? 
• Part B. How much would 100 pieces of candy cost? 
• Part C. How much would 1000 pieces of candy cost? 
• Part D. At the same store, you can buy 100 chocolate coins for $89.00. How much does each chocolate coin cost? Explain how you know. 

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