Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Whole numbers
Purpose and Instructional Strategies
- Multiple representations of multi-digit whole numbers allow students to identify opportunities for regrouping while adding and subtracting. For example, when subtracting 5,783 – 892, we can represent 5,783 as 5 thousands + 6 hundreds + 18 tens + 3 ones by regrouping 1 hundred as 10 tens, allowing us to subtract 9 tens (MTR.2.1, MTR.3.1).
- Students should use objects (e.g., base ten blocks), drawings, and expressions or equations side-by-side to see compare and contrast the representations. Model to show how multiple representations relate to the original number. For example, use base ten blocks to show how in the number 5,783, 1 hundred can be regrouped as 10 tens to express it as 5 thousands + 6 hundreds + 18 tens + 3 ones, while asking students how they are the same (MTR.2.1).
- Allow students to decompose numbers in as many ways as possible. Have students compare and contrast the representations shared (MTR.4.1).
- Students should see examples of numbers within 10,000 where zero is a digit and make sense of its value.
- Flexibility of place value is a prerequisite for conceptual understanding of a standard algorithm for addition and subtraction with regrouping.
Common Misconceptions or Errors
- Students can misunderstand that the 5 in 57 represents 5, not 50 or 5 tens. Students need practice with representing two and three-digit numbers with manipulatives that group (base ten blocks) and those that do NOT group, such as counters, etc.
- Students can misunderstand that when decomposing a number in multiple ways, the value of the number does not change. 879 is the same as 87 tens + 9 ones and 8 hundreds + 79 ones.
Strategies to Support Tiered Instruction
- Instruction includes decomposing numbers using manipulatives that group (base ten blocks) and those that do not group such as counters. When decomposing a number, students focus on the value of each digit based on its place value. To reinforce this concept, students may count by units based on the place value.
- For example, decompose 362 using base ten blocks and explain the value of each digit.
- For example, represent 34 using counters and explain the value of each digit. Students group 10 ones as a group of ten and focus on the value of each digit based on its place value. To reinforce this concept, students count by units based on the place value.
- Teacher provides opportunities to decompose numbers in multiple ways using manipulatives and a chart to organize their thinking and asks students to name/identify the different ways to name the values (regrouping the hundreds into tens and the tens into the ones, e.g., 36 tens and 2 ones or 3 hundreds and 62 ones, etc.)
- For example, students decompose 362 in multiple ways using hundreds, tens, and ones.
- For example, students decompose 34 in multiple ways using tens and ones.
Instructional Task 1
- Express the number 5,783 using only hundreds and ones.
Instructional Task 2
- Express the number 5,783 using only thousands and hundreds.
Instructional Task 3
- Express the number 5,783 using only tens and ones.
Instructional Item 1
- Select all the ways that express the number 8,709.
- a. 8,000 + 600 + 19
- b. 8,000 + 700 + 9
- c. 879 ones
- d. 8 thousands + 6 hundreds + 10 tens + 9 ones
- e. 8 thousands + 7 tens + 9 ones