##
* *Reflection: Diverse Entry Points
Understanding Subtraction - Section 3: Independent Practice

As I spoke to students about the ways they were solving the subtraction problems they had in front of them, I could see the differences in understanding of what subtraction is. Students who understood that subtraction would yield a smaller number than the quantity they started with, were able to show a way to start with that larger number and compare it to a smaller number to find the difference, or to count back by tens and ones, or even by ones to get to their answer.

Students who still struggle with the concept of subtraction didn't seem to realize the solution wasn't correct when they ended up with a quantity bigger than they had started with, even though the problem asked how many were left.

One child read the problem that there were 14 birds in a tree and 6 flew away, how many are left in the tree and she said 20. When I asked her how she got the answer she said she used her number grid. I asked her to show me what she did. She started at 14 and counted up 6 to get to 20. I asked her if we could draw a picture to check her answer. She drew the tree with 14 tally mark birds. She crossed out 6 and counted what was left and told me 8. I asked her why the 2 answers might be different, i.e. her 20 and her 8 and she said, I counted on the number grid and then with tally marks so my answer is different.

Here is a lack of understanding about what is actually happening when we add or subtract. This is a misunderstanding that needs to be addressed now as a foundational skill, before trying to move on to more complex numbers.

(I don't doubt that if I had given her the two problems in number form 14 + 6 = and 14 - 6 = that she would have been able to get the right answer, but that is rote learning and not understanding what she is trying to do.)

This child showed that she did not clearly understand the number grid and its directionality for adding and subtracting. Next time she could use counters on the grid. If she put on the 14 and then took away the 6 that flew away, she would see where her original answer was incorrect. She could also use a double ten frames for addition / subtraction as a way to check her work. I also encourage students to draw the pictures because this often shows them visually what has happened.

*Diverse Entry Points: Differences in Student Thinking*

# Understanding Subtraction

Lesson 14 of 18

## Objective: SWBAT subtract one and two digit numbers and explain their strategy for solving the problem.

#### Warm Up

*15 min*

I put 2 number sentences on the board. One is a *partner of 10* addition sentence and one is the *related partner of 10* subtraction sentence. (7 + 3 = and 10 - 7 = ) .

I ask students what they notice. (It is all the same numbers, it is a fact family, etc.)

I repeat the process with related doubles sentences (8 + 8 = and 16 - 8 = ) .

I put up several other examples for students to attempt. I keep the number sentences to under 20 for this exercise because I want students to notice not only the relationship between addition and subtraction, but also how the partners of 10 help us with this.

The Common Core Standards work towards fluency with numbers under 100, so to expect a clear understanding of numbers to 20 at this point in the year is a logical expectation. This work is the foundation for working with larger numbers later on.

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#### Teaching the Lesson

*15 min*

I start by asking students to come to the rug where I have snap blocks available. Each student is given a tower of 18 blocks to use for modeling their thinking.

I ask, *"What would happen if I were to subtract 5 from the tower? Would my new tower be taller or smaller than what I started with?"* I wait for students to say that the new tower would be smaller.

I ask, *"How many blocks are left after you take the 5 away from the 18?"* Students manipulate their blocks to determine the solution.

*"Can anyone put a number sentence on the board to match what we just did?"* After the addition sentence is on the board, I ask if they can turn it around and make a subtraction sentence to show how we ended up with less blocks.

I repeat the process with several other numbers forming subtraction sentences.

I ask students to build 2 towers of ten and one tower of 4. How many blocks do they have? Now I ask, *"What if we take away 12, how might we do that?"* I let students experiment with getting an answer. Some may realize that they can take away a tower of 10 and then 2 more from the tower of 4. Others may just count twelve without realizing that they can use the tower of 10. We will write the number sentence in the same way.

It is important to let students work this out for themselves. I want them to conceptualize how they can combine numbers and take them apart. I let students work on their own. After everyone has made an attempt at this, we review what was done. Here I am providing the scaffolding for those that may have not fully understood the process, and I am providing feedback for those who understood.

The Common Core MP standards suggest that students should be able to use models to solve problems. If I do the modeling for students, they are watching, but not seeing how they can use those models themselves. I want students to see that using things such as blocks is not just for "little kids," but can be helpful at all levels of solving math problems.

I repeat the process with several other 2 digit from 2 digit numbers.

I tell students that today they will be working with word problems that will ask them to subtract. I ask students to generate ways they might solve the problems and what tools they might use. We post a list of student ideas such as number line, number grid, base 10 blocks, snap blocks, tally marks, in their heads, with pictures, etc. Here I am encouraging students to develop independence in their math reasoning. If I always tell students what to do, they may not develop the skills they need when I am not there to walk them through a difficult math problem.

I send students back to their seats and hand out the pages they will work on. I put out the materials students have suggested, reminding them that they can help themselves to materials if they wish. I also remind students I will be walking around asking how they are solving the problems they are working on. I also let them know that not everyone has the same set of problems.

#### Resources

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#### Independent Practice

*15 min*

Students will work independently on the problems I have given them. Some of the problems are more basic, and others are more complex.

I also put out a challenge paper for students who may work more quickly.

I circulate around and talk to students about their understandings of subtraction.

*expand content*

#### Closing

*5 min*

I tell students that I want to post something that will help us all understand the difference between addition and subtraction. I ask for student definitions of both and record them on a chart. I will post the chart to refer to later on.

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- UNIT 1: What and Where is Math?
- UNIT 2: Adding and Subtracting the Basics
- UNIT 3: Sensible Numbers
- UNIT 4: Sensible Numbers
- UNIT 5: Everything In Its Place
- UNIT 6: Everything in Its Place
- UNIT 7: Place Value
- UNIT 8: Numbers Have Patterns
- UNIT 9: Fractions
- UNIT 10: Money
- UNIT 11: The Numbers Are Getting Bigger
- UNIT 12: More Complex Numbers and Operations
- UNIT 13: Area, Perimeter and More Measurement
- UNIT 14: Length
- UNIT 15: Geometry
- UNIT 16: Getting Ready to Multiply
- UNIT 17: Getting Better at Addition and Subtraction
- UNIT 18: Strategies That Work

- LESSON 1: Let Me Count The Ways to Get An Answer
- LESSON 2: Who Makes Mistakes
- LESSON 3: Counting Up to Solve Problems
- LESSON 4: Counting Backwards Works Too
- LESSON 5: Counting Bugs
- LESSON 6: Taking Apart the Problem
- LESSON 7: Getting Bigger and Smaller
- LESSON 8: Double It
- LESSON 9: Doubles Plus or Minus One
- LESSON 10: Evens and Odds
- LESSON 11: Plus Ten Minus Ten
- LESSON 12: From Tens to Nines
- LESSON 13: Equal Amounts
- LESSON 14: Understanding Subtraction
- LESSON 15: Skip Counting with 5s, 10s and 100s
- LESSON 16: Balancing Equations and Counting Backwards
- LESSON 17: Counting with Tens and Hundreds
- LESSON 18: Assessment