General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 5
Strand: Number Sense and Operations
Date Adopted or Revised: 08/20
Status: State Board Approved
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Equation
- Expression
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
The purpose of this benchmark is for students to explore multiplication and division of multi- digit numbers with decimals using estimation, rounding, place value, and exploring the relationship between multiplication and division. This benchmark connects to the work students did in grade 4 with addition and subtraction of decimals (MA.4.NSO.2.7). Students achieve procedural fluency with multiplying and dividing multi-digit numbers with decimals in grade 6 (MA.6.NSO.2.1)- Instruction of this benchmark focuses on number sense to help students develop procedural reliability while multiplying and dividing multi-digit numbers with decimals.
- During instruction, students should explore how the products and quotients of whole numbers relate to decimals.
- For example, if students know the product of 8 × 7 and the quotient of 56 ÷ 4, then they can reason through 0.08 × 7 or 5.6 ÷ 0.4 through place value relationships. Classroom discussions should allow students to explore these patterns and use them to estimate products and quotients (MTR.4.1, MTR.6.1).
- Teachers should connect what students know about place value and fractions. o For example, because students know that multiplying a number by one-fourth will result in a product that is smaller, multiplying a number by 0.25 (its decimal equivalence) will also result in a smaller product. In division, dividing a number by one-fourth and 0.25 will result in a larger quotient. Continued work in this benchmark will help students to generalize patterns in multiplication and division of whole numbers and fractions (MTR.5.1).
- Models that help students explore the multiplication and division of multi-digit numbers with decimals include base ten representations (e.g., blocks) and place value mats.
Common Misconceptions or Errors
- Students may not understand the reasoning behind the placement of the decimal point in the product. Modeling and exploring the relationships between place value will help students gain understanding.
- Students can confuse that multiplication always results in a larger product, and that division always results in a smaller quotient. Through classroom discussion, estimation and modeling, classroom work should address this misconception.
Strategies to Support Tiered Instruction
- Instruction includes opportunities to explore place value of decimals with concrete models and objects.
- For example, students use place value understanding and a place value chart to compare 0.14 and 0.2. The teacher explains that when comparing decimals, we start with the digit to the far left because we want to compare the greatest place values first. Both values have a 0 in the ones place, so we will move to the tenths place. One-tenth is less than two-tenths, so 0.14 < 0.2.
- For example, students compare 0.3 and 0.03 using decimal grids and represent each value and explain that 0.3 covers a greater are of the decimal grid than 0.03, so 0.3 is greater than 0.03.
- Instruction includes opportunities to predict and explain the relative size of the product of two decimals. Students use models to check their prediction and solve. The teacher guides students to connect that multiplying a given number by a number less than one will result in a smaller number, and that multiplying a given number by a number greater than one will result in a larger number.
- For example, students solve the following problem 0.2 × 0.5. Students should reason about the size of the decimals and connect it back to their fraction understanding and think about the multiplication sign signaling “groups of.” This expression could be interpreted as 0.2 “of” 0.5. This will help with the misconception of multiplying equals a larger product. The picture below illustrates the product of 0.2 and 0.5. If the entire square is 1 unit, the gray region represents 0.5 units, and the red region represents 0.2 units. The overlap in purple contains 10 small squares, each of which represents 0.01 units. Therefore, the overlap portion contains 10 × 0.01 = 0.10 units. The overlap portions show a 0.2 by 0.5 rectangle, so the number of units it contains is the product 0.2 and 0.5.
Instructional Tasks
Instructional Task 1 (MTR.4.1)
What is the same about the products of these expressions? What is different? Explain.
14 × 5 0.14 × 0.05 Instructional Task 2 (MTR.4.1)
What is the same about the quotients of these expressions? What is different? Explain.50 ÷ 25 50 ÷ 0.25
Instructional Task 3 (MTR.5.1)
How can you use 2 × 12 = 24 to help you find the product of 2 × 1.2? Explain.
Instructional Items
Instructional Item 1
Raul reasons that the product of 82 × 0.56 will be greater than 41 and less than 82. Explain whether or not his conclusion is reasonable.*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.