Estimation

Strategies

Estimation and Measurement with Number Bonds to 100

The base-10 manipulatives we use in teaching place value have a very useful attribute; they are measured in centimeters cubed. 1 ones block is 1cm long and a tens rod is 10 cm long. We could use the ones block and a tens rod to begin to use our knowledge of order of magnitude to help us estimate lengths. We will find things that are about the same length as a ones cube or a tens rod, making a chart in our math journals. We will talk about how the items that are about the same length as a tens rod are 10 times longer than the things we found that are about the length of 1 ones cube. ¹¹

We could further integrate measurement into our exploration of multiples of 10 to use the base 10 blocks to create our own centimeter rulers. Each student will be given a tens rod and a strip of paper about 2 inches thick and 50 cm long. They will take the tens rod and line it up on the line already drawn on the paper. They will mark the end of each rod and record 10cm. They will continue to move down the line marking lengths of 10cm until they have measured 50cm. They will go back and use the rods to mark each cm length. This will give each child an experience measuring. (See Lesson 1 for detailed instructions.)

Additionally, these rulers will also serve as a number line to model addition of larger 2-digit numbers. I might ask the students to line up, end-to-end, 2 tens and 3 ones on the centimeter ruler. They will see that these 2 tens and 3 ones have a length of 23cm. If they were to move the blocks around, they would still get a length of 23cm. My hope is that the students will understand that no matter which order we add the pieces, 10 + 10 + 3 = 23. This activity provides an automatic connection between the pieces we are combining and the whole number they represent once combined. The students can SEE that 10 + 10 + 3 = 23. In comparison, when we just model a 2-digit number with the same base-10 blocks on a "Tens & Ones" chart the two-digit representation of "23" is missing. The visual model is there, but the number that corresponds to that visual model is not immediately connected. However, when we model a number on the centimeter ruler/number line there is an immediate visual connection between the model and the number it represents. I can use this centimeter ruler/number line in a vast number of activities to solidify each student's understanding of how to compose and decompose larger 2-digit numbers.

After we use our centimeter rulers as number lines for a few lessons, we will be ready to add on to them so that they measure 0-100cm. This will be the first lesson in the discussion that 100 is the same as 10 tens. Therefore, 100 is ten times bigger than 10. All of our experiences using this ruler as a number line will help us be more familiar with length in centimeters. The exposure to lengths from 0-100 centimeters should allow them to estimate lengths in relation to this experience. I could begin to ask questions such as "Is this piece of string closer to 10 cm or 100 cm? Is this line segment closer to 20 cm or 50 cm.? Is a cat 10-20cm long or is it 20-50cm long? Is your leg about 40cm or about 100cm long? Is 40cm to the knee, or to the hip?"

It is here that I would begin to point out that the leading digit of a number represents most of the value of the number. I would ask "Why don't I ask, 'Is this about 25?' or 'Can you guess exactly how many marbles are in this jar?'" The point of asking this question is to point out that until now we have been working with very round numbers (multiples of 10). We do this because usually we are only really concerned with the leading digit when we estimate. I would then try to explain this through modeling with the base-10 blocks. I would ask them to show 78 in tens and ones on a "Tens & Ones" chart. I would then ask, "Where are there the most blocks? Are there more blocks in the tens place, or are there more blocks in the ones place?" The answer to this question may not be readily apparent to the class. I may need to remind them that 7 tens = 70 individual items. "If we think about the number 78 in expanded form as 70 + 8, which is more, 70 or 8? Right, 70 is more. You can see that most of 78 is in the tens place. This means that the 7 tens is more important to us than the 8 ones." I would repeat this conversation with other 2-digit numbers. I might choose many 2-digit numbers where the tens' digit is smaller than the ones' digit. This sets up a direct conflict with the idea that the larger digit in a number must represent the larger value. Many of my former students have often automatically looked for the biggest digit when asked to identify the digit that represents the larger value. However, in the example 78, the eight is the bigger digit, but 8 is not bigger than 70. In the number 78, the digit that shows the bigger value is the 7.

I would then extend this conversation to support the use of very round numbers in estimating quantities. "This is why we estimate with very round numbers. Most of the important information about a number is in the first digit, so we can just take the first place value component of a number. If we do that, we are guaranteed to work with a very round number. And, as you know, very round numbers are easy to work with because they end with zeros." Then we could try this with a few two-digit numbers.

I would continue by explaining that when we estimate we want to estimate numbers under 100, we use multiples of 10. However, instead of just picking a random very round number out of the air, it is more meaningful for us to say that a quantity is between __(x)___ and __(y)__. When estimating, we want to express that we couldn't possibly be sure of an exact answer. It is easier to show the uncertainty of our estimate if we say our guess is between two numbers. When we do this we are setting boundaries for our estimates. This allows us some flexibility with our estimate. We can be more positive that a quantity is between two very round numbers than we can be that it is about one specific quantity. Think of 35 + 47. Since 30 35 40, and 40 47 50, then 30 + 40 = 70 35 + 47 40 + 50 = 90.

After discussing this crucial aspect of estimation, we would discuss various collections of objects. We would work to create collections of different objects that vary by order of magnitude. I would involve the students in the creation of these collections in order to give them the chance to manipulate collections of objects that have 10 items or 100 items or 1,000 items. They could work together in small groups to make collections of beans that differ in size from order of magnitude 1 to order of magnitude 3. You could split the class into three groups and ask them to make a collage. However, group one makes a collage with 10 beans, group 2 makes collages with 50 beans, and group 3 makes collages with 100 beans. They could begin to think about tackling problems such as estimating how many people are in the cafeteria by estimating it between two very round numbers. The chances for extension are vast.

We could also further explore the relationship between centimeters and meters as an extension to the activities listed below. Following the study of 100 cm = 1 meter, you could discuss and explore the differences between 1 meter, 10 meters and 100 meters.