Teachers often use the estimating jar as a way to promote good estimating skills in their students. Throw a little candy in that jar and now you’ve got students excited and motivated to do math! The following activities are a twist on the traditional estimating jar and have been done with second through fifth graders.
I introduce the estimating jar, filled with Hersheys kisses, by asking students to give a reasonable estimate as to how many kisses they think are in the jar. I provide examples of unreasonable estimates so they understand that my expectations require them to make reasonable predictions. For example, I might say, “Do you think it’s reasonable that there is only one in the jar?… or that there are 500 in the jar?“
When students have shared their estimates, we count the kisses on the overhead into groups of ones and tens cups, and, if necessary, hundreds cups. This is an ideal place to tie in place value concepts with primary students. I then share with them another jar, also filled with Hershey kisses, that is obviously larger than the first one. We discuss whether they think this jar fits more or less kisses than the first jar.
Students make their predictions, and again, we count into groups of ones and tens. Counting by twos or threes instead of always by ones makes the counting go a little faster. Also, because jars come in different heights and diameters, discussing these variables can help guide students to make more reasonable estimates.
In order to prepare for the above scenario, I needed to find two jars, a small jar and a large jar. It’s important that the large jar holds about twice as many objects as the smaller jar. The first time I fill the jars, I make sure the larger jar holds exactly two times as many kisses as the small jar (you may have to squeeze a few extra in or leave a little space in order to accomplish this) and I choose a number of kisses that is a multiple of 10. Doing this makes it easier for students to see that the second, larger jar holds twice as many as the first jar. For example, if the small jar holds 20 kisses, then I put 40 in the large jar. In subsequent days, I fill the jars with how ever many fit in to the small jar, then try to get pretty close to doubling in the larger jar. After all, it is an estimating activity.
When students have counted out all the objects from the second jar, we look for patterns between the number of kisses in each jar. Students will notice that the second jar holds twice as many (two times as many) as the first jar. The next time I do this activity I remind them of this and we do a new count with a new object. Knowing that the larger jar holds twice as many as the smaller jar, students can more easily predict about how many are in the larger jar. I stress the about how many with the students because I am not looking for the exact number, even though students are determined to arrive at the answer that is exactly double.
Over several weeks we do a variety of objects, quickly moving from edible objects to non-edible objects. Students now know that the large jar holds about twice as much as the small jar, so they get a lot of practice doubling numbers. Using different-sized objects, such as marbles, provides students with the opportunity to discuss whether there will be more or less marbles in the jar if the marbles are smaller than Hersheys kisses.
In the next part of the activity, I go in reverse. I show the large jar first. We estimate and count, then using this information students predict about how many are in the small jar. When going in reverse, it is not always possible to have exactly half in the small jar because half of 45 would be 27 ½ and we don’t do ½-sized objects.
In working with fourth and fifth graders, I use three different-sized jars: a small jar, a medium jar, and a large jar. The medium jar holds approximately one and a half times the small jar, and the large jar holds approximately three times the small jar, or twice the medium jar. I may even have one jar that is 2 1/3 times another jar, if I feel the students can handle it or I want to give some students a challenge. I also vary the order I present the jars. For example, I may have students estimate the medium jar first, then the small jar, and finally, the large jar.
Even though students are getting practice doubling, halving, etc. this estimating activity provides them with many other opportunities to use problem-solving skills. In one second grade classroom we were trying to determine the total number of objects if the small jar held 37 and the large jar held 74. One student combined the 3 tens from the 37 with the 7 tens from the 74 to arrive at 100 (30 + 70) and then added the 7 ones to the 4 ones to get 11. 100 + 11 = 111
In determining how many objects the large jar would hold if the large jar was 1 ½ times the small jar which held 92, a third grader divided the 92 into 80 + 12. She then took half of 80 and half of 12 (two easy numbers for her to work with) to come up with 40 + 6 = 46. She knew then that 46 was half of 92 and added the 46 to the 92 to arrive at 138…her estimate for the large jar.
With upper elementary students I have them determine the total number of candies, etc. each student would get if we wanted to share them equally with the group. One fifth grade student came up with three ways to share 121 with his group of 8 students. See sample. He first came up with a pattern for multiplying the eight students by one candy, then two candies, etc. After he had determined the total number for 4 candies each, he simply doubled his answer of 32 to arrive at 64 total candies if 8 students each had 8 candies. He continued adding candies in groups of 16 (8 students each having 2 candies = 16 candies) until he arrived at his final answer, which included a remainder of one candy.
He used a similar pattern idea in his second solution, determining the middle number between 80 and 160 (8×10 and 8×20). He was able to figure out that 8×15 would be 120 thus each students would get 15 candies.
His final solution tied in his understanding of fractions. Once he had determined that 1/3 of 120 was 40, he figured out how many candies each of the 8 students would get by dividing 8 into 40. 40/8 = 5. He then multiplied 5 by 3 since he had 3 groups of 40 to, again, arrive at 15 candies with one left over.
A simple activity like the estimating jar can turnout to be quite an adventure into the world of problem-solving and number sense.