Estimation

Lesson Plans

Part 1 Introduction to Estimation (Take a Guess: A Look at Estimation)

The objective of this lesson is to give students the basic idea of estimation and how useful approximate answers can be, and how much easier is it to get approximate answers than exact answers. I will use the book Take a Guess to show examples of how estimation can be used. However, I will also emphasize that estimation is not really "taking a guess" and that it involves a lot of careful thought. Also, the calculations you do in estimation are usually easier than exact calculations, because they don't involve too many digits. I will use the book as an opportunity to introduce valuable vocabulary as well as the general idea of estimation.

We will read the book together as a class, and I will enact some of the scenarios mentioned in the book, such as comparing heights of students. I will ask the students to provide examples of situations where estimation can be useful.

I will explain one of my examples, someone's age. When comparing small numbers for instance, 15 years old from 50 years old, the age difference is 35 years. The margin of error is large compared with 15. It is more than twice as big. However, when large numbers are compared say for example a million years old, a 35 year difference would be insignificant.

Part 2 Guess Order of Magnitude (Great Estimations)

The objective of this lesson is that students will learn how to estimate the number of objects in pictorial representations and use the order of magnitude estimation technique to come up with a reasonable number. The students will learn how to tell whether a group of objects is closer to 10, 100 or 1000. According to the definition of the word, orders of magnitude are generally used to make very approximate comparisons. If two numbers differ by one order of magnitude, one is about ten times larger than the other. If they differ by two orders of magnitude, they differ about a factor of 100. Two numbers of the same order of magnitude have roughly the same scale: the larger value is less than ten times the smaller value. This is the reasoning behind significant figures: the amount rounded by is usually a few orders of magnitude less than the total and is therefore insignificant.

I will also teach the powers of 10. The students will understand that 1 raised to the 0 power, that 10 is raised to the first power, that 100 = 10×10 is 10 raised to the second power, and so on. The chart below will help the students understand the power of ten and the order of magnitude.

I will use colorful illustrations of objects or make the pictures myself with different groupings of 10s, 100s and 1000s. Another example that I can use to show the class order of magnitude is elbow macaroni. I will have a bag of 10 elbow macaronis, a bag of 100 elbow macaronis, and a bag of 1000 elbow macaronis. I will ask my students to estimate how large a bag of elbow macaronis do I need to make 10,000? 100,000?

We will read the book together as a class and discuss. I will prepare questions to check their comprehension. The students will give their estimations for the different pictures and explain how their answers. The next part of my lesson will be the visual presentation of 10, 100, 1000, and 10,000. I will write the numbers down on sentence strips. To get the students attention I will ask them to describe what is on the table. I will have 10 bags of M&Ms candy prepared in advance with 10 pieces in each bag and 10 more bags of M&Ms with 100 pieces in each bag. I will tell them that the first bag contains 10 pieces of M&Ms. the students will see through the first plastic bag so they can visualize the M&Ms. Then I will ask the question, "How many M&Ms do you think are there in the second bag?" I will write their responses on a chart paper in front of the class because we will go back to their answers after the activity. Students will now work in groups of 4. Each group will figure out how many M&M's are there in the second bag. They will refer to the first bag of 10 M&Ms as they figure out the answer. This activity will keep them engaged. I will be observing the class and allow each group to explain their estimation. I will reiterate to the students that a good estimation involves reducing a lot of guessing. Students will use the interactive math journal to write their reflections.

Part 3 Compare Pictures with the same Order of Magnitude (Great Estimations)

The objective of this lesson is that students will learn the estimation concept and will learn how to compare pictures with the same order of magnitude. This is a continuation of part 2. I will introduce the topic to my students and brainstorm what they already know about the topic. Then, the students will look at the book cover and make their predictions. Students' responses will be recorded on a chart paper and will be displayed on the board. I will use the book to emphasize that estimation will help them think clearly to figure out how many objects are there in each picture. Important vocabulary words will also be discussed. One of the activities for this lesson is using cereal Os.

I will put cereal Os in 6 plastic bags. The students will work in groups to show the different ways to arrange 10 cereal Os and 100 cereal Os. They will then glue each arrangement of the 10 and 100 cereal Os and put them together to show a group of 1000 cereal 0s. Then the next activity will be students working in groups of 4. Each group will estimate the number of cereal 0s in each box. They will answer the question, "Do you think there are more than 8,000 cereal 0s in the box?" The students will explain their answers. I will ask the students to explain how they came up with an accurate estimation. They will use their interactive math journal to reflect on how to compare pictures with the same order of magnitude and the estimation strategies they used to find the accurate answer.

Part 4 Estimation Techniques for Arrays (Betcha!)

The objective for this lesson is that students will learn that an array is a rectangular arrangement of objects in evenly spaced objects with the same number in each row. I will use Betcha! to teach arrays and introduce important vocabulary words. I will mention how two friends use their estimation strategies in everyday life. The most important estimation technique in this story is how both boys use the array to come up with their estimation. To find the number of objects in an array, multiply the number in each row by the number of rows. This is like the area formula, area = L×W for a rectangle. We will read the book together as a class and discuss the different scenarios. One of the scenarios is estimating the number of people in the bus. The first boy tells his friend that there are 4 people in each row and there are ten rows of seats. So this is how he did his estimation, 4 people ×10 people = 40 people seated then he added the three people standing up. His answer was 43. This is an example of estimating. The other friend counts all the people and gets 45. Our discussion on estimation will focus on the different scenarios in the book. I will explain further that the array method is for finer estimation than order of magnitude. Each boy's guess was actually quite close to the actual number.

The students will also be able to create mathematical games. Together the whole class will design a "Betcha!" game. The students will be given the task of thinking and picking something that is difficult to count, such as people in all the lines at the busiest supermarket during the rush hour, the number of soup cans on display at the supermarket, the number of cars parked at the ball game, and the amount of ice cream students in the whole school eat during the summer months. Help students to come up with different strategies for making their estimations. Then check to see how close their estimates are to reasonable numbers.

For the pair share activity, students will come up with a real life situation that requires estimation and explain it orally. For homework, to check for understanding the array model, the students will estimate how much pizza an average student eats and then multiply by the number of students. For six classes they can multiply the one class estimate by six. At the end of the lesson, the students will reflect and write their responses on the interactive math journal.

There are also extension activities that students can do to extend the concepts presented that are related to real-life situations. Estimate how many hamburger sandwiches each family eats in one month. How much will the family spend for the hamburgers? Students will keep track of their estimation and ask their parents to help them find out if their estimate is reasonable.

Part 5 Use Area Techniques (Greater Estimations)

The objective of this lesson is to reinforce estimation skills to find the area and the estimation technique by using the grid to gather information, counting and averaging. The student work focuses on developing a sense of the size of 10,000. The colorful illustration of the book will generate students' interest. Comprehension questions will be prepared in advance. I will generate curiosity by asking the question how many rubber ducks are on the cover of this book. Are there more than 10 rubber ducks? Are there fewer than 10, 000? The visualization strategy will help the students estimate the number of rubber ducks. I will say to the class that counting every duck is difficult and using the grid as their estimation technique will enable them determine the answer. Therefore, to get a reasonable idea of how many ducks are there, the students should try to find the number of ducks in a small area then multiply by the ratio of the whole picture to the small area. To answer the question, "Are there fewer than 10,000?". I will elicit students' responses from the class and record them. The students will be in cooperative learning groups of four to explain their responses. Important vocabulary words will be discussed as we read.

I will explain to the students that estimation strategies will require eye training, finding the area and using the grid to group the objects then put them together to make a whole picture. At the end of the lesson, the students will write individual reflections on their math interactive journal.

Lesson 6 Relative Place Value (A Million Dots)

The objective of this lesson is for the students to understand relative place value. The student's work will focus on developing a sense of the relative size of 1000, 10,000, and larger powers of 10. I will use the book, A Million Dots. We will read the book together as a class. I will ask the students to determine the number of dots on a page that contain 7-by-9 array of blocks of 100 dots. They will use the handout to construct rectangles of 5,000 and 10,000 dots. I will ask the students the following questions: What is the relationship between thousands, hundred-thousands, millions and billions? Why are multiples of 10s, 100s, 1,000s "friendly numbers"? I will use base 10 blocks to provide students with opportunities to represent and build number in a variety of ways including 2-, 3- digit numbers to review the relationship between each place value.

To engage the students with the concept of estimating to the right of a decimal point, I will show them a whole square cookie. Then divide a large cookie into ten parts. Each part will be named by the student as 1out of 10 and will be written as .1, one tenth. Each one tenth will be again cut into 10 parts. The students will identify how many parts are there in all. Each part will be one hundredth. The students will discuss the hundredth's place value.

Since the fifth graders are doing estimation of whole numbers when they add, subtract or multiply I will start first with the explanation. They should look for patterns in addition, subtraction, multiplication and division of whole numbers. My first question is what will happen to 57 + 9=? Do mental math with estimation. You know that 57 is between 50 and 60, and 9 is between 0 and 10. So 57+9 is between 50+0=50 and 60+10=70. In this case, it is closer to 70. In fact 57 is missing 3 from 60, and 9 is missing only 1 from 10. So is all, 57+9 is missing 4 from 70, so it is 70-4=66.

Furthermore, we can use similar tricks for multiplication. How much is 9×45? The students should use mental math to multiply by 9 quickly: (10×45)-45= 450-45=405. I will explain to the students that the same method works for 99, which is 1 short of 100. So (100×number-number): 99×5= 100×5 take away 5=500-5=495. The next example will be 999 which will be (1000 × numbers -number): 999×5= 1000×5 -5=5000-5=4,995. These examples show that when numbers are rounded off to the nearest 10, 100, or 1000 finding the missing number is easier in addition or subtraction or multiplication or division. I will explain how to multiply single place numbers: for example: I will display…

40 x 600= (4x10) x (6x10) = (4x6) x (10x100) =24 x 1000= 24,000.

When integrating math and literature, we will read the book together and then teach the math concept. After brainstorming, the teacher will write student responses on chart paper. Students will work with their cooperative groups of 4 for the next activity. All the students will draw 10 dots and then100 dots. We will put the dots together in groups of 10, 100 and 1000. I will ask the students to go to the 10 dots corner, 100 dots corner and the 1000 dots corner. A speaker will report on each group.

A way to keep the students engaged is to prepare in advance a picture of one dot that will be magnified. I will explain to the class that to magnify is to enlarge it with a magnifying glass but magnitude is to enlarge a number by a power of 10. I think the students will relate to this magnified dot because it is the size of a large pizza. I will divide the dot into ten equal parts. Each part will be 1/10 of the whole dot. This number will be express as 1÷10=.1; Next, I will show another magnified dot divided into 100 equal parts. One part will represent 1/100 of the whole dot. I will write the math sentence as 1÷100=.01; I will ask the class to work with the same group and show how 1/1000; 1/10,000; 1/100,000; 1/1,000,000,000 would be represented in decimal form.

I will also represent decimal fractions using volume. I will bring to class a round watermelon to represent one whole. I will slice it into ten pieces so each one will also represent 1/10.I will explain that when cutting a watermelon I will not be able to make the pieces exactly equal, but will try to make them as equal as I can, by estimation. The activity will also be a good practice of measurement. The pieces will be weighed to see how close they are in weight, and will adjust them if they a little if they are very different. I will write on an overhead one out of ten slices, 1/10 and explain that after the decimal point, the place value is tenths. Moving away from the decimal point is the second digit which is the hundredths place. Therefore, if there is a one in the tenths place that is "one tenths". I will refer to .1 and ask the students the question what is after 1. The next digit will be a zero. I will explain to them that the number is now .10 which is 10/100 or 10 percent. The watermelon activity will continue. I will ask each group to chop one tenth of the watermelon into 10 pieces to show them the value of each piece which is1/100. I will use a second watermelon for this activity because I will not be able to make the pieces exactly equal. I will use measurement and explain to the students again that I will try to make them as equal as I can, by estimation. At the end of the activity the students will have to put all the slices together to get a whole watermelon. I will emphasize that it is alright to ignore 1/1000. Howe states that relative values less than 1/1000 are ignorable for most practical purposes.

Another way to illustrate the idea is to use 1000 cubes, and will discuss that if the cube represents 1, then each little cube represents 1/1000. To check if the students understand the idea I will use the next activity. I will ask students to estimate distances in terms of 10, 100, and 1000 steps. How many steps will it take you to go to the cafeteria from the fifth grade classroom? Do you think it will take you less than 100 steps or more than 100 steps? Will it take you 1000 or more steps to go to the school gate? Is it more than 1000 steps to go home or is it more than 10,000 steps? Students will work in groups and write their answers in their interactive math journal.

Lesson 7 Expanded Form and Relative Sizes of Single Place Components (A Million Dots)

The objective of this lesson is to teach the students expanded form and relative sizes of single place component. The students will develop a sense of quantities in the thousands, ten thousands, and hundred thousands as well as develop a sense of the size of 1,000,000. I will use the book A Million Dots to introduce the concept. I will emphasize the idea than when students use the expanded form they also have to compare which part of the number is the biggest, next biggest and the smallest. I will use the book to as an opportunity to introduce math and reading vocabulary words. To generate my students' curiosity I will review the lesson prior to this on the order of magnitude of 10. I will prepare in advanced a handout that will show 10 dots, 100 dots, 500 dots and a 1000 dots. I will ask them to estimate how many thousand dots will make a million dots.

We will read the book as a class and will discuss the fact on each page. Thus, the students will get a new sense of how much a million is really worth. To start the concept of the expanded form, I will direct them to look at the picture of the mosquito and the Empire State Building on the first page of the book. I will ask, if the wings of the mosquito beat 612 times each second, how do you write the expanded form of 612? I will record the response on a chart which will show 612= 6 00 +10 +2. I will make it a point to ask all the time which part of the number is the biggest, next biggest and the smallest. The next question will be if a person must climb 1,860 steps to walk to the top of the Empire State Building how do we write the expanded form of 1,860? The students will respond to this question and I will record 1,860=1,000+800+60+0 on a chart paper. The next example will be the number sentence, if a person blinks 134,000 times each week, write the expanded form. Ask for a student volunteer to write the expanded form on the chart. The students will work with their group, choose a picture and work on the dot number of their choice from the six digit numbers. They will illustrate the number of dots on the picture and write its expanded form. The students will choose a reporter to show and explain their illustration and how they came up with the expanded form. At the end of the lesson student will reflect on their learning and write in their interactive math journal.

Towards the end of the estimation unit, I will create a record of students' work. Students will look back through their interactive math journal and write or revise what they have learned for each concept taught, what they remembered most, and what was difficult or easy for them. They will select two pieces of their work as their best and I will choose one or two pieces to be saved in their portfolio. One of the work samples could include students' written solution to an estimation word problem.

In conclusion, I hope that this curriculum unit will help all students achieve their mathematical goals. They will be critical thinkers and problem solvers; they will use their estimation skills to solve real word problems; they will develop various ways to express their mathematical thinking in oral and written form, thus enjoy and appreciate mathematics.