Chapter 13 K-2

` Chapter 13 K-2 New Jersey Mathematics Curriculum Framework - Preliminary Version (January 1995)
© Copyright 1995 New Jersey Mathematics Coalition

STANDARD 13: ESTIMATION

All students will develop their understanding of estimation through experiences which enable them to recognize many different situations in which estimation is appropriate and to use a variety of effective strategies.

K-2 Overview

Estimation is a combination of content and process. Students ability to use estimation appropriately in their daily lives develops as they have regular opportunities to explore and construct estimation strategies and as they acquire an appreciation of its usefulness through using estimation in the solution of problems.

One of the estimation emphases for very young children is the development of the idea that guessing is an important and exciting part of mathematics. The teacher must employ sound management practices which ensure that everyone's guess is important and which encourage risk taking and sharing of ideas about the how guesses were determined. Estimations of measurements as well as of quantities should pervade the classroom activity. When first asked to guess an answer, many students will give nonsense responses until they establish appropriate experiences, build their sense of numbers, and develop informal strategies for creating a guess. Children begin to make reasonable estimates when the situations involved are relevant to their immediate world. Building on comparisons of common objects and using personal items to build a sense of lengths, weights, or quantities helps children to gain confidence in their guessing. As children communicate with each other about how guesses are formulated they begin to develop informal strategies for estimation.

Activities which provide experiences for the child to determine reasonableness of answers and to establish the difference between estimated answers and exact answers as well as when the use of each is appropriate, should be developed through non-routine problem solving activities that involve measurement, quantities, and computation.

Estimation with computation is important in these grade levels as well as at all grade levels. Estimation of sums and differences should be a part of the computational process from the very first activity with any sort of computation. Children should regularly be asked About how many do you think there will be in all or About what do you think the difference is or About how many do you think will be left in the standard addition and subtraction settings. These questions are appropriate whether or not actual exact computations will be done. Children should understand that sometimes, the estimate will be an accurate enough number to serve as an answer. At other times, an exact computation will need to be done, either mentally, with paper and pencil, or with a calculator to arrive at a more precise answer. Which procedure should be used is dependent on the setting and the problem.

One of the most useful computational estimation strategies in these grade levels also reinforces an important place value idea. Students should understand that in two-digit numbers the tens digit is much more meaningful than the ones digit in contributing to the overall value of the number. A reasonable approximation, then, of a two digit sum or difference can always be made by considering only the tens digits and ignoring the ones. This strategy is referred to as front end estimation and is used with larger numbers as well. It is the main estimation strategy that many adults use.

STANDARD 13: ESTIMATION

K-2 Expectations and Activities

The expectations for these grade levels appear below in boldface type. Each expectation is followed by activities which illustrate how the expectation can be addressed in the classroom.

Experiences will be such that all students in grades K-2:

A. judge, without counting, whether a set of objects has less than, more than, or the same number of objects as a reference set.

Students place various amounts of counters, candy, or other small objects in individual plastic bags. Working in groups of four students, the children choose one bag to be the reference set and judge whether each of the other bags has more than, less than, or the same as the reference set. Initially, they should try to make the judgments without counting.
Young children benefit from comparing sets of objects to some given number. For example, given five sets of poker chips arranged on a table, they should often practice going through the sets and name which ones are more than five and which are less than five.
Students play the card game War with a set of cards without numerals. All that is on the cards are the sets of hearts, clubs, diamonds, or spades. Students will initially easily distinguish between the 7s and the 3s, but will be reluctant to make judgments about closer numbers like the 4s and 5s without counting. As they play more, though, distinctions will become finer.

B. use personal referents, such as the width of a finger being about one centimeter, for estimations with measurement.

Students estimate lengths of pieces of spaghetti, yarn, paper, pencils, paper clips etc. using suggested non-standard personal units such as width of thumb, length of a foot, and so on.
As standard units like yard and centimeter are introduced, students are challenged to find some part of their body or some personal action that is about that size at this point in their growth. For instance, they may decide that the width of their little finger is almost exactly one centimeter or the length of two giant steps is one yard.
Students use their self-discovered personal body referents to estimate the measures of various classroom objects like the length of the blackboard or the width of a piece of paper in the associated units.

C. visually estimate length, area, volume, or angle measure.

Students look at a quantity of sand, salt, flour, water, or popcorn and estimate how many times it could fill up a specified container.
Students estimate how many pieces of notebook paper it would take to cover a given area such as the blackboard, or a portion of the classroom floor.
Students regularly estimate lengths in a whole variety of non-standard units such as my feet, unifix cubes, paper clips, and orange Cuisenaire Rods. They then measure to verify or correct their estimates.

D. explore, construct, and use a variety of estimation strategies

Students are asked if a sixty-seat bus will be adequate to take the two first grade classes on their field trip. After it is known that there are 23 children in one class and 27 children in the other, individual children volunteer their answers and give a rationale to support their thinking. A discussion might be directed to the question of whether an exact answer to the computation was needed for the problem.
Students are confronted with a glass jar filled with about eighty marbles and are asked to estimate the number in the jar. In small groups, they discuss various approaches to the problem and strategies they can use. They settle on one to share with the class along with the estimate that resulted.
Second grade students can be challenged to estimate the total number of students in the school. They will need to talk about the average number of students in each class, the number of classes in a grade level, and the number of grade levels in the school. They might then want to use calculators to solve the problem, but the result, even though the exact answer to a computation, is still an estimate. They can then discuss why that is so.

E. recognize when estimation is appropriate and understand the usefulness of an estimate as distinct from an exact answer.

Given pairs of real-life situations, students determine which situation in the pair is the one in which estimation is the best approach and which is the one in which an exact answer is probably needed. One such pair, for example, might be: Sharing a bag of peanuts among 3 friends and Paying for tickets at the movie theater.
Given a set of cartoons with home-made mathematical captions, first graders decide which of the cartoon characters arrived at exact answers and which got estimates. Two of the cartoons might show an adult and a child looking at a jar of jelly beans and the captions might read: Susie guessed that there were 18 jelly beans left in the jar and Susie's mom counted the 14 jelly beans left in the jar.

F. determine the reasonableness of an answer by estimating the result of operations.

Students are regularly asked if their answer makes sense in the context of the problem they were solving. They respond with full sentences explaining what they were asked to find and why the numerical answer they found fits the context reasonably, that is, why it could be the answer.
Students estimate reasonable numbers of times that particular physical feats can be performed in one minute. For example, How many times can you dribble a basketball in a minute, How many times can you hop on one foot in a minute, How many times can you say the alphabet in a minute, and so on. Other students judge whether the estimates are reasonable or unreasonable and then the tasks are performed and the actual counts made.
Second-grade students are given a set of thirty cards with two-digit addition problems on them. In one minute, they must sort the cards into two piles: those problems whose answers are greater than 100 and those whose answers are less than 100. The correct answers can be on the backs of the cards to allow self-checking after the task is completed.
Second-grade students might also be given a page of addition or subtraction problems in a multiple choice format with 4 possible answers for each problem. Within some time period which is much too short for them to do the computation, students are asked to choose the most reasonable answer from each set of four.

G. apply estimation in working with quantities, measurement, computation, and problem-solving.

Students have small pieces of yarn of slightly different lengths ranging from 2 to 6 inches. Each student first estimates the number of his or her pieces it would take to match a much longer piece - about 30 inches long - and then actually matches the two pieces up and counts how many. Then they use their individual pieces to measure other objects in the room. Each child is responsible for first estimating the lengths in terms of his or her own yarn, but they can use evidence from other childrens measuring to help make their own estimates.
Students regularly estimate before appropriate classroom procedures. For example, at snack time, they may guess how many cups can be filled by each can of juice or how many crackers each student will get if all of the crackers in the box are given out.
Kindergartners always have fun estimating which color is best represented in a group of multi-colored objects. Good examples of such activity would be choosing the m&m color that shows up most in a bag of m&ms, choosing the Fruit Loops color that shows up the most in a bowl of Fruit Loops, or choosing the color that shows up most in a bowl full of marbles. After everyone has committed to a guess, the children can sort the objects and count each color. They can even make a bar graph to show the distribution of the different colors.