##
* *Reflection: Advanced Students
Equal Amounts - Section 3: Closing

I asked the students if the expanded form of a number could be the same as the number itself. I also asked students if I measured something and found it to be 3 inches and someone else found it to be 8 centimeters, could this be possible?

When I looked at student journals after students had finished I could see a great difference in student understanding. Some children wrote yes or no, but were unable to explain why they put the yes or no. You can see that in the example of a student who showed less understanding. If you look at the other example, you see a child who was able to grasp the idea of equal but having different names, and she was able to explain her thinking.

By using student evidence of thinking as an informal assessment, I am seeing that there are two distinct groups of students in my classroom. There is a group that can reason in mathematical terms and explain their thinking, and there is a group for whom mathematical reasoning and understanding of basic concepts of number is still weak.

As I teach, I need to be aware of these two groups and how I can support their learning so that they can all be proficient with the standards set forth in the Common Core.

*Advanced Students: Differences in Mathematical Thinking*

# Equal Amounts

Lesson 13 of 18

## Objective: SWBAT identify and create equal amounts and equations for a single number. Students measure an object using 2 different units of measure.

#### Warm Up

*10 min*

Students have been working with partners of ten and doubles, as well as adding 10 to a number. Today I ask students to find a different number sentence to equal the one I put on the board. I ask them to write the sentence in their math journals.

In first grade they should have met the Common Core Standard, "Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false." Students may understand the equal sign in a single equation such as 8 + 5 = 13, but if I were to write 8 + 5 = 10 + 3, they might have more trouble understanding that there can be more than one thing on the other side of the equal sign. Today's lesson helps to expand the student's understanding of what the equal sign really means.

As students work, I circulate around and choose 2 - 3 students, with different equations, to show their work on the board, so we can see that there are numerous ways to compose a given number.

I start with 5 + 5 = 10. One of the reasons I select this is that the concept of making ten in different ways should be in place, so it makes a good starting point for this lesson. Starting in kindergarten, students should be developing fluency within ten, and in making ten. I call making ten "partners of ten". I hope students will quickly recall another *partner of ten* equation.

After we share out, I put the equation 13 + 7 = 20 on the board. Again I ask for students to write their own equations that make 20 on their individual white boards.

I do a subtraction equation as well 18 - 9 = 9. (Here I watch to see if the students attempt another subtraction problem, or find an addition problem that will equal 9.)

I keep track of students who may be having difficulty with the concept of composing true number sentences for a given number.

I give students a stretch break and ask them to come to the rug.

*expand content*

#### Teaching The Lesson

*35 min*

I tell the students that today we will be working at 2 different centers, as well as completing a practice page.

I explain that at the first center we will be taking dominos and looking at the two sides. We will write an addition sentence by adding the bottom and the top of the domino and putting the number sentence under the correct total. We will also write a subtraction sentence by counting the total, subtracting one side and writing the equation under the answer that is left.

Together we practice on the interactive white board until students are comfortable with what they will be doing.

I tell them that at the second center they will be measuring the objects already set out on the table. They will use inches and centimeters. We discuss which side of the ruler is inches and which is centimeters. I choose all objects that are less than 12 inches, but are full inches in length to avoid half or quarter. I tell them to measure to the closest centimeter. We practice together on the interactive white board how to hold the ruler so the zero point is at the end of the object and then to read to the nearest inch or centimeter. We record our answers on the chart.

I tell students that they will also complete a practice page where they will write as many ways as they can think of to represent a given number, just as we did when we started the day.

I divide the children in heterogeneous groupings and send them to one of the three centers. After about 10 minutes I ring the bell and have students progress to a new center. Students will visit all 3 centers.

#### Resources

*expand content*

#### Closing

*10 min*

At the end of the center time I ask all children to return to their seats. I ask them to respond in their journals to the questions, "Can 2 hundreds, 3 tens and 9 ones be the same as 239, or 200 + 30 + 9? Why?"

Some students respond in words, others use pictures, or representations with blocks or tally marks to explain their thinking.

Can I measure an object and find it is 3 inches long, when my partner says it is about 8 centimeters long? How is that possible?

Again, some responses will be in words, but others use drawings to explain their thinking.

I use student responses to assess their understanding of it being possible to have 2 different numbers or representations of a number represent the same amount.

*expand content*

This looks like a good lesson I would like to try. Did you attach the worksheet you used?

| 9 months ago | Reply*Responding to Tere Lumley*

Tere, can you have students look at one inch on a ruler and then slide across the ruler to see how many centimeters fill that inch? Once they do that, say, now I have 2 inches, how many centimeters would that be? You can even then give students a paper marked in inches and then count together by centimeters - filling in the centimeters themselves they may be able to see that centimeters are so much smaller than inches so there are always more of them. They can also show the numbers for inches and centimeter measurements and use < > to show that the number for inches is always smaller, even when the object doesn't change because it takes more centimeters to fill in the distance than it does inches. I am not sure if these are things you have tried. Good luck with it all.

| 2 years ago | Reply

We have worked with assessing this concept all year in our common assessments and our students have not been able to grasp how to say in words why the measurement of an object in cm is different than the measurement in inches.

One of the assessment questions is this:

The paper measures 8 inches across. I measured the same paper again and found it to be 20 centimeters across. Why is it more centimeters than inches?

This has been exceptionally hard for our students. We have tried teaching the concept in a variety of ways. For ex.- we put them on one side of the hall and asked them to take "baby steps" and count as they stepped. Then discussed the number of steps they counted. Then we had them cross back to the other side of the hall and take giant steps and talked about the number of giant steps they took. They could explain that the baby steps were smaller than the giant steps but on the next common assessment, they missed the question again. I would appreciate any help you can provide.

| 2 years ago | Reply*Responding to Lee Powell*

Hi Lee - The link w/i the lesson titled Measuring in Inches and Centimeters is to a video w/i this lesson that shows students measuring in centimeters and inches. Does this help?

| 2 years ago | Reply

Hi Beth - I am having trouble finding the lesson for measuring with centimeters and inches. I can see the video (which is great!) but the written lesson that comes up is for standard vs expanded notation. Am I doing something wrong or is it linked incorrectly?

Thanks,

Lee

| 2 years ago | Reply*expand comments*

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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