Math: The first iteration
What, How, and Why
The students have been going over the standard algorithm for subtraction and various strategies to use alongside it. While they can reliably use the standard algorithm for multi-digit addition, the concept of regrouping has stumped some students (as have most of the strategies, like estimation). Their difficulty with regrouping and with estimation appears to me to stem from a difficulty with the concept of place value. When going over expanded form of numbers with many students earlier in the school year, I noticed a similar pattern: students could not reliably identify what each place value meant in terms of total value; that is, students often referred to their place value chart diligently with little regard to the relationship between a digits place on that chart and its value within the whole number. I have noticed this pattern in students’ difficulty with the concept of numbers that are ten less than a number, ten more, one hundred more, one hundred less, or other seemingly simple calculations. I have also noticed students struggle with why a set of numbers has to be “lined up to the right” to begin standard algorithm subtraction. While most students can reliably write a number in expanded form now, I have concerns that their difficulty with regrouping/ borrowing stems from a shaky understanding of place value.
It is the lack of reliability with which students regroup and a video students watched in class that has led me to try a lesson using a place value chart that shows expanded form. I hope both to show that larger (multi-digit) numbers are made up of smaller parts in place value groups and that because of the commutative property of addition (a term they know and can use) regrouping can occur without changing the value of a number. Additionally, I hope by doing subtraction within these place value columns more explicitly, students will expand their understanding of the fact that when subtracting (using the standard algorithm) in each place value, they are doing more than “taking 2 from 3” or “4 from 8” repeatedly. That is, within each place value more than single digit subtraction can be occurring.
I have chosen to use expanded form and a place value chart because these are two things students see in their classroom on a regular basis. While I have seen a definite increase in most students’ abilities to write numbers in expanded form and to identify a digit’s place on a place value chart, I have not seen these skills translate with all students to proper use of regrouping in the standard algorithm for subtraction. I hope to leverage students’ use of expanded form to deepen or clarify their understanding of regrouping in multi-digit subtraction. In this way, my pedagogical focus is selecting and using representations. I hope students will take their understanding of multiple ways to write the value of a number and use it to consider what the action of regrouping means in terms of number value.
Goals / Objectives
1. Increase student understanding of borrowing/ regrouping for standard algorithm subtraction
Standards (and Assessment Anchors, if applicable)
CC.2.1.4.B.2 Use place-value understanding and properties of operations to perform multi-digit arithmetic.
Materials and preparation
Mini white boards, markers, and erasers
Multi-digit subtraction problems with numbers written in both standard form and extended form
Classroom arrangement and management issues
The lesson will take place in a small group during “centers” time, a time when each desk group (4 or 5 students) works on a single activity together. This arrangement will be comfortable for the students because it is a routine they do each day
Plan
Anticipating students’ responses and your possible responses
Students are likely to resist a new strategy. This is a common behavior when it comes to math in this classroom, and students with cite that math is too difficult or they simply can’t do it. I hope to remind students that they have learned so much since their first venture into math in kindergarten and even since the beginning of the year.
I expect that students will find sharing their answers exciting, though some may find it overwhelming. I hope to mitigate the stress by making sharing optional but encouraged and allowing students to share “wrong” answers in a way that supports learning, thanking them for their willingness to help us all with their participation.
I expect relatively few managerial concerns directly related to my lesson because center time is something every child in class does and has done with me at least once.
Assessment of the goals/objectives listed above
See attached form. Some students will need practice with the simple standard algorithm for multi-digit subtraction. For these students, I am looking to see progress in correct use of that algorithm. For students for whom my expanded form practice is not beyond their frustration point, I am looking to see correct use of borrowing and correct answers using my chart.
Accommodations
For students who find the work to challenging, I will allow them to continue to practice with the standard algorithm, given that they participate in my instruction and our surrounding discussion on regrouping using expanded form.
I do not expect this work to be too easy for any of my students, but if some students do not find it too extremely challenging, I will have on hand problems with larger numbers.
The students have been going over the standard algorithm for subtraction and various strategies to use alongside it. While they can reliably use the standard algorithm for multi-digit addition, the concept of regrouping has stumped some students (as have most of the strategies, like estimation). Their difficulty with regrouping and with estimation appears to me to stem from a difficulty with the concept of place value. When going over expanded form of numbers with many students earlier in the school year, I noticed a similar pattern: students could not reliably identify what each place value meant in terms of total value; that is, students often referred to their place value chart diligently with little regard to the relationship between a digits place on that chart and its value within the whole number. I have noticed this pattern in students’ difficulty with the concept of numbers that are ten less than a number, ten more, one hundred more, one hundred less, or other seemingly simple calculations. I have also noticed students struggle with why a set of numbers has to be “lined up to the right” to begin standard algorithm subtraction. While most students can reliably write a number in expanded form now, I have concerns that their difficulty with regrouping/ borrowing stems from a shaky understanding of place value.
It is the lack of reliability with which students regroup and a video students watched in class that has led me to try a lesson using a place value chart that shows expanded form. I hope both to show that larger (multi-digit) numbers are made up of smaller parts in place value groups and that because of the commutative property of addition (a term they know and can use) regrouping can occur without changing the value of a number. Additionally, I hope by doing subtraction within these place value columns more explicitly, students will expand their understanding of the fact that when subtracting (using the standard algorithm) in each place value, they are doing more than “taking 2 from 3” or “4 from 8” repeatedly. That is, within each place value more than single digit subtraction can be occurring.
I have chosen to use expanded form and a place value chart because these are two things students see in their classroom on a regular basis. While I have seen a definite increase in most students’ abilities to write numbers in expanded form and to identify a digit’s place on a place value chart, I have not seen these skills translate with all students to proper use of regrouping in the standard algorithm for subtraction. I hope to leverage students’ use of expanded form to deepen or clarify their understanding of regrouping in multi-digit subtraction. In this way, my pedagogical focus is selecting and using representations. I hope students will take their understanding of multiple ways to write the value of a number and use it to consider what the action of regrouping means in terms of number value.
Goals / Objectives
1. Increase student understanding of borrowing/ regrouping for standard algorithm subtraction
Standards (and Assessment Anchors, if applicable)
CC.2.1.4.B.2 Use place-value understanding and properties of operations to perform multi-digit arithmetic.
Materials and preparation
Mini white boards, markers, and erasers
Multi-digit subtraction problems with numbers written in both standard form and extended form
Classroom arrangement and management issues
The lesson will take place in a small group during “centers” time, a time when each desk group (4 or 5 students) works on a single activity together. This arrangement will be comfortable for the students because it is a routine they do each day
Plan
- Launch 5-10 mins
a. Give students 3 multi-digit subtraction problems that increase in difficulty. 345-213=?, 1,236-943=?, and 2,300-1,488=?
b. Allow students to solve independently on their whiteboards, and do last problem together using standard algorithm and
borrowing/ regrouping. - Work and explore. 20-30 mins.
a. Introduce regrouping strategy using expanded form and place value chart. See attached.
i. Review the definition of expanded form (345 in standard form is 300+40+5 in expanded form, for example, and 300+40+5=345).
ii.Use place value chart drawn on my white board to visually show regrouping as moving quantities from one column to the next, not changing the value of the number overall
iii.Get an answer in expanded form, and write it in standard form.
iv.Repeat same problem using standard algorithm to show answers are the same.
b. Work on same three problems together on my mini whiteboard above using expanded form and place value chart.
i. What does it mean to regroup? What are we doing when we subtract in the 10s place? The hundreds place? When we make a digit into a different number, what are we doing? Why can we borrow from the group to the left?
c. Give students a last problem to do themselves on their mini whiteboard using their own place value chart. 341-198=? - Debrief and wrap up. 5-10 mins.
a. Students share their answer to the last problem.
b. What does expanded form have to do with place value? What does it mean to line the numbers up?
Anticipating students’ responses and your possible responses
Students are likely to resist a new strategy. This is a common behavior when it comes to math in this classroom, and students with cite that math is too difficult or they simply can’t do it. I hope to remind students that they have learned so much since their first venture into math in kindergarten and even since the beginning of the year.
I expect that students will find sharing their answers exciting, though some may find it overwhelming. I hope to mitigate the stress by making sharing optional but encouraged and allowing students to share “wrong” answers in a way that supports learning, thanking them for their willingness to help us all with their participation.
I expect relatively few managerial concerns directly related to my lesson because center time is something every child in class does and has done with me at least once.
Assessment of the goals/objectives listed above
See attached form. Some students will need practice with the simple standard algorithm for multi-digit subtraction. For these students, I am looking to see progress in correct use of that algorithm. For students for whom my expanded form practice is not beyond their frustration point, I am looking to see correct use of borrowing and correct answers using my chart.
Accommodations
For students who find the work to challenging, I will allow them to continue to practice with the standard algorithm, given that they participate in my instruction and our surrounding discussion on regrouping using expanded form.
I do not expect this work to be too easy for any of my students, but if some students do not find it too extremely challenging, I will have on hand problems with larger numbers.
Feedback
Feedback from my first plan centered generally around two ideas: clarity and complexity. I added a couple of sentences clarifying exactly what it is that strikes me as particularly important about place value both in the understanding of numbers and in the understanding of subtraction. Additionally, I honed in on what skills were particularly important on my "assessment checklist," rather than more general subtraction related skills. In terms of complexity, I brought my numbers down to 3 digits rather than 4. While I had initially chosen to include 4 digit numbers because 4th grade PA standards expect students to add and subtract up to 5 digit numbers readily, I decided based on feedback to reduce my numbers to 3 digits because of the introduction of a new strategy. If, it turns out, students find 3 digit numbers too easy, I will have a list of 4 and 5 digit numbers on hand, as planned previously. Lastly, I plan to have base 10 blocks on hand to think about place value groupings as changeable. I do not intend to use them if students are able to grasp the numeric representations easily, but I plan to have them with me in the event that this proves too complicated a task to begin.