# Subtracting with Decomposing

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## Objective

SWBAT use place value understanding to decompose numbers to subtract large numbers.

#### Big Idea

In this lesson, students use tape diagrams in order to subtract and make sense of subtraction as the inverse of addition.

## Number Hook - Magic Trick

5 minutes

I begin this lesson with a number hook I call Magic 1089!

I begin by directing students to write down a three-digit number whose digits are decreasing. Then students reverse the digits to create a new number, and subtract this number from the original number. With the resulting number, students add it to the reverse of itself. The number will always be 1089!

For example, if you start with 532 (three digits, decreasing order), then the reverse is 235. Subtract 532-235 to get 297. Now to the 297 and its reverse 792, and you will get 1089!

I love doing this trick as a whole group, because students don’t all start with the same three digit number. When the result is the same for everyone it reinforces the concept of patterns and rules and that I’m not really magic.

The Math Behind the Fact:
If we let a, b, c denote the three digits of the original number, then the three-digit number is 100a+10b+c. The reverse is 100c+10b+a. Subtract: (100a+10b+c)-(100c+10b+a) to get 99(a-c). Since the digits were decreasing, (a-c) is at least 2 and no greater than 9, so the result must be one of 198, 297, 396, 495, 594, 693, 792, or 891. When you add any one of those numbers to the reverse of itself, you get 1089!

Note – I certainly don’t expect students to figure out the math behind all of my magic tricks. Some of them are explainable at a fourth grade level, and some are not.  I do these tricks as engagement strategies, and also to show my genuine, sincere LOVE of math and numbers.

***  My students LOVED this trick.  They were very excited to go home and do it for their parents.

## Warm Up

10 minutes

Students will start today's lesson with a fluency assessment.  This assessment is from Monitoring Basic Skills Progress Second Edition: Basic Math Computation by Lynn S. Fuchs, Carol L. Hamlett, and Douglas Fuchs.

This is an assessment I have my students do each week and then graph their results.  It allows them to reflect on their learning of basic math facts, as well as using all four operations with whole numbers, and adding and subtracting unit fractions.  (It also happens to be the quietest time in my math classroom all week!!) In the video below you can see a quick glimpse of students working on this fluency assessment.

I do not start my students with the fourth grade skills. I chose to start them with the end of the third grade skills which covers addition, subtraction and multiplication and division of basic facts. I strongly believe in a balanced math approach, which is one reason why I also believe in common core standards.  By having a balance of building conceptual understanding, application of problems, and computational fluency, students can experience rigorous mathematics.  I want to make clear that this assessment ONLY measures basic math computation.  It is only one piece of students' knowledge. The assessments in this book, for each grade level, do not change in difficulty over the course of the year.  Therefore, a student's increase in score over the school year truly reflects improvement in the student's ability to work the math problems at that grade level.

This is a sample of a grade four computation measure. ## Concept development

35 minutes

The math practice standards that drive this lesson are MP.5, use tools strategically, and MP.7 make use of structure.  Students continue to explore the use of a tape diagrams in this lesson to help them make sense of subtraction problems and reinforce that subtraction and addition are inverse operations.

I begin this lesson by displaying a subtraction number sentence on the board:

4,587 - 1769 = .

I then draw a tape diagram and ask students to identify the whole and part that are known.  As I model this students draw this on a personal whiteboard. I also write each number vertically, lining up my places and lead a brief discussion why this important.

Most of my students had experiences in third grade modeling numbers to the ten thousands for subtraction.  The third grade foundational standard 3.NBT.A.2 requires students to add and subtract within 1000 using place value understanding, however, many of my students have had experiences adding and subtracting larger numbers in grade 3. I also model a place value or proof drawing, but don't expect my students to model it on written work, unless needed since many of them have moved more towards the abstract level.

As I model, I use explicit place value vocabulary like tens, hundreds, thousands, etc.

After this first example, students work with their learning partner on a second example.  I direct students to write 54,429 - 11,888.  I ask them to model a tape diagram marking their unknown and then solve this problem with their partner. When most partners are done, I ask

I ask students how to check subtraction.  I then model how to check subtraction with addition and students do the same.

For the remainder of the lesson, students work on the practice -Subtracting with decomposing.  Students may work with their learning partner for help.

Notice in this sample work, the student is subtracting correctly, but using the tape diagram incorrectly. I will want to visit with this student about part whole relationships to ensure his success in later lessons. These two students work together using clipboards on the floor. One way I ensure success for students is by allowing partners to work together and move around the room to a spot they like.  Giving my students as much choice as I can while still ensuring students are on task and the lesson content delivered is important for students to own their learning. As students work, I circulate around the room and help as necessary. I guide students thinking through questioning and help clear up misunderstandings.

## Student debrief - Wrap up

10 minutes

Students will do an exit slip today.  My students have started to grow accustomed to this ending task and ask if they "get to" do one today.

When I look at the exit slips, I will be making groups of students needing various levels of extra support. I provide extra support and differentiation at another time in my day.