Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Whole Number
Purpose and Instructional Strategies
The purpose of this benchmark is for students to demonstrate procedural fluency while dividing multi-digit whole numbers with up to 5-digit dividends and 2-digit divisors. To demonstrate procedural fluency, students may choose the standard algorithm that works best for them and demonstrates their procedural fluency. A standard algorithm is a method that is efficient and accurate (MTR.3.1
). In grade 4, students had experience dividing four-digit by one-digit numbers using a method of their choice with procedural reliability (MA.4.NSO.2.4
). In grade 6, students will multiply and divide multi-digit numbers including decimals with fluency (MA.6.NSO.2.1
- When students use a standard algorithm, they should be able to justify why it works conceptually. Teachers can expect students to demonstrate how their algorithm works, for example, by comparing it to another method for division (MTR.6.1).
- In this benchmark, students are to represent remainders as fractions. In the benchmark example, the quotient of 27 ÷ 7 is represented as 3 . Students should gain understanding 6 represents of another group. Students are not expected to have mastery of converting
that this quotient means that there are 3 full groups of 7 in 27, and the remainder of 6
between forms (fraction, decimal, percentage) until grade 6 but students should start to gain familiarity that fractions and decimals are numbers and can be equivalent (i.e., a remainder of is the same as 0.5). Writing remainders as fractions or decimals is acceptable. Similarly, students should be able to understand that a remainder of zero means that whole groups have been filled without any of the dividend remaining (MTR.5.1, MTR.7.1).
- Along with using a standard algorithm, students should estimate reasonable solutions before solving. Estimation helps students anticipate possible answers and evaluate whether their solutions make sense after solving.
- This benchmark supports students as they solve multi-step real-world problems involving combinations of operations with whole numbers (MA.5.AR.1.1). In a real-world problem,
students should interpret remainders depending on its context.
Common Misconceptions or Errors
- Students can make computational errors while using standard algorithms when they cannot reason why their algorithms work. In addition, they can struggle to determine where or why that computational mistake occurred because they did not estimate reasonable values for intermediate outcomes as well as for the final outcome. During instruction, teachers should expect students to justify their work while using their chosen algorithms and engage in error analysis activities to connect their understanding to the algorithm.
Strategies to Support Tiered Instruction
- Instruction includes estimating reasonable values for quotients when dividing by two- digit divisors.
- For example, students make reasonable estimates for the quotient of 496 ÷ 24. Before using an algorithm, students can estimate the quotient to make sure that they are using the algorithm correctly and the answer is reasonable. Students can use multiples of 24 and their understanding of multiplication and division to estimate the quotient. Students may want their estimate to be as close to 496 as possible. So, knowing that 24 × 2 = 48, they can state that 24 × 20 = 480. A reasonable estimate for the quotient would be 20 because 480 is close to 496.”
- For example, students make reasonable estimates for the quotient of 94 ÷ 13. Explicit instruction could include stating, “Before using an algorithm, we will estimate the quotient to make sure that we are using the algorithm correctly and our answer is reasonable. The divisor of 13 is close to 10 and the dividend of 94 is close to 90. So, we can use 90 ÷ 10 = 9 to estimate that our quotient should be close to 9.”
- Instruction includes explaining and justifying mathematical reasoning while using a division algorithm to divide by two-digit divisors. Instruction also includes determining if an algorithm was used correctly by analyzing any errors made and reviewing the reasonableness of solutions.
- For example, the teacher connects place value with the partial quotients model to determine 496 ÷ 24. Students should not just view the digits as individual numbers but connect individual digits with the value of that number (e.g., 496 is 400 + 90 + 6). Instruction includes stating, “In this problem we are finding how many groups of 24 are in 496. We will subtract groups of 24 until we cannot subtract any more groups. The total number of groups that we can subtract is the quotient. We can subtract 10 groups of 24 two times, so the quotient is 20. We have a remainder of 16. The quotient is represented as 20 because we have 20 full groups of 24 in 496 and the remainder of 16 represents of another group.”
- For example, connect place value with the partial quotients model to determine 94 ÷ 13. Students should not just view the digits as individual numbers but connect individual digits with the value of that number (e.g., 94 is 90 + 4). Instruction includes stating, “In this problem we are finding how many groups of 13 are in 94. We will subtract groups of 13 until we cannot subtract any more groups. The total number of groups that we can subtract is the quotient. We can subtract 7 groups of 13, so the quotient is 7. We have a remainder of 3. The quotient is represented as 7 because we have 7 full groups of 13 in 94 and the
remainder of 3 represents of another group.”
- For example, students use an algorithm to solve 496 ÷ 24 and explain their thinking using place value understanding. Instruction includes stating, “In this problem we are finding how many groups of 24 are in 496. We will begin by dividing our largest place value first. Recognizing that the 4 represents 400, if your divide 400 by 24 the result will be less than 100, so the quotient won’t have any whole hundreds. Remember that 496 is the same as 49 tens 6 ones, so we will see how many groups of 24 are in 49 tens. We can also think of this as
___ × 24 = 49 tens. There are 20 groups of 24 in 49 tens, that’s 2 times 10 groups, so we can place a 2 in the tens place of the quotient. Next, we will subtract 49 tens – 48 tens (20 groups of 24 equal 48 tens) to find a difference of 1 ten. We can combine this 1 ten with the 6 ones remaining in 496. We now have 16 ones remaining from our original dividend of 496, this is not enough to make a group of 24. We have a remainder of 16. The quotient is represented as
20 because we have 20 full groups of 24 in 496 and the remainder of 16 represents of another group. Our quotient of 20 is close to our estimate of 24, this helps to determine that our answer is reasonable"
- For example, students use an algorithm to solve 94 ÷ 13 and explain their thinking using base ten blocks and place value understanding. Instruction includes stating, “In this problem we are finding how many groups of 13 are in 94. We will begin by dividing our largest place value first. How many groups of 13 are in
9 tens? Recognizing that the 9 represents 90, if you divide 90 by 13 the result will be less than 10, so the quotient won’t have any whole tens. Remember that 94 is the same as 9 tens 4 ones and 94 ones, so we will see how many groups of 13 are in 94 ones. We can also think of this as ___ × 13 = 94. There are 7 groups of 13 in 94 ones. Next, we will subtract to find our remainder of 3. Our quotient is
represented as 7 because we have 7 full groups of 13 in 94 and the remainder of 3 represents of another group. Our quotient of 7 is close to our estimate of 9, this helps us determine that our answer is reasonable.”
- Instruction includes the use of place value columns to support place value understanding when using a division algorithm.
Instructional Task 1 (MTR.7.1)
The Magnolia Outreach organization is donating 6,924 pounds of rice to families in need. They pour all the rice into 15-pound containers.
- Part A. How many containers will they fill if they use all the rice?
- Part B. Will Magnolia Outreach be able to fill all the containers completely? If not, will
the partially filled container be more or less than half-full? Explain how you know.
Instructional Item 1
What is the quotient of 498 ÷ 72?
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive