This activity is designed for first and second-graders, but it might also be appropriate for third-graders who have difficulty counting in groups or with place value (2). Prior to implementing this activity, the students should have been instructed in counting by ones. (8) The areas of math that will be addressed include counting in groups and basic multiplication. The multiplication aspects do not need to be addressed with the first or second-graders. It will simply serve as a basis for understanding the concepts in later grades. (8) After completing the activity, the students should be able to understand that numbers can be grouped in different ways and that it is possible, and often easier, to count larger numbers by groups. For example, the number six can be seen as two groups of three, three groups of two, or as six groups of one. Examples of counting by groups include counting to twenty by twos, fours, fives, or tens.
(2) At the onset of the activity, begin by initiating a conversation with the children about things with which they are familiar. For instance, you could start by asking, "If your mother went to the store and bought two six-packs of coke, how many cans did she buy in all?" Another good introductory question would be: "If three students walked by, how many legs would you see? How many arms? Eyes?" The next question would involve multiples of three. "If there was a tricycle race and four tricycles entered, how many wheels would pedal by?" Next, a multiple of four would be introduced. You could ask a student how many cars they have in their family. Some might say one or two, etc. Ask the class how many tires there are in all. The conversation alone could constitute a valuable lesson, as the kids think about these things and attempt to come up with examples of their own. By the time you get to multiples of six, the students should have a pretty good grasp on the counting by groups concept.
To make the lesson even more effective, the teacher could bring three or four six-packs of juice and ask the students how many juice drinks are there in all. The teacher could then turn the question around and ask the class how many six-packs would the class need in order for every student to receive one juice drink. To make this more feasible, have the students arrange themselves in groups of six. If there are, say, 21 students in the class, explain that you would need four six-packs, and that there would be 3 left over. This will ensure that the students have a concrete example in their minds. They will be able to actually see the groups of six. (10) Throughout this discussion, the teacher should use the Unifix cubes to illustrate the examples (one person has two legs, two people have four legs, etc.) To do this, connect two Unifix cubes together to represent one person. In order to represent the three wheels on a tricycle, attach three cubes together. This way the student will be acquainted with this method, and it will make the transition to the activity much more smooth.
(6) The activity itself involves the students working in groups of three to five people. (9) All that is needed for the activity is a set of Unifix cubes and a sheet of paper for each group to record their answers. (1) Asking the children to remember the examples discussed previously, have them think of two numbers that have groups of two with none left over. (7) For example, one group might say eight and ten, or four and six. The next problem would be to have the student find two numbers that have groups of three with none left over. If both of these activities prove far too easy for your students, ask them to find numbers with groups of three that have one left over or groups of four that have two left over. (15) There are numerous examples, all of which will get the students thinking of numbers as groups of things, which is the central focus of this lesson. By getting them to count by twos or threes, this is preparing them for the base ten system. It will be easier for them to think of twenty as two groups of ten, or one hundred as ten groups of ten. Instead of making the jump from counting by ones to all of a sudden counting by tens, this activity serves as a nice transition into the world of counting by groups as well as place value.
(6) Once the class has completed all of the problems posed, bring the lesson to a close by asking each group for one of their answers to both of the problems and writing them on the board for everyone to see. Then model how the numbers were arrived at by using the Unifix cubes. This will give the students the idea that there are several possibilities and further reinforce the image of numbers as groups of things.
Student Results
(14) I conducted the activity on a third-grader, and the results were as follows. When I asked her the question regarding two people and the number of feet, she answered correctly right away. When we got to the multiples of five and six, she had a little bit of trouble, so I suggested she draw it out on paper. That helped tremendously. When we moved on to the Unifix cubes part of the activity, she had a little trouble understanding what I was trying to do - probably because I had not made the directions very clear. When I asked her to find a number with groups of two with none left over she connected two cubes and then started to connect a third. I reminded her that I needed groups of two. She understood and immediately formed two groups of two. Her answer at first was twenty-two. I asked her to think again, and she came up with four. I asked her for another number, and she added one more group of two to come up with six. As we moved on to the numbers with remainders she had a little more trouble. The problem was to find a number with groups of four with two left over. She put together two groups of four and then tried to take away two. I explained that two left over meant that you would add two to the two groups of four. She came up with ten. I asked her for one more number and she proceeded to come up with 14. Overall, she did a good job. The activity is probably too easy for third graders, but it might be worthwhile to explore more as an introduction to multiplication.
I do feel that the activity would be challenging enough for first and second-graders, and even for those third-graders having difficulty with grouping and counting by groups. It was fun to see the little mind work to solve the problems. (5) The scope is narrow enough in that the groups are only supposed to come up with two numbers, (4) but it is challenging enough because they can use as many cubes as they want to come up with larger numbers.
(13) Ideas for future related lessons could include using the same activity, but asking them to work with larger group sizes such as tens and twenties. For example, ask the students how many fingers they see if five kids pass by. Lead them into thinking about each child as having ten and subsequently as one group of ten. After they begin to group this and are able to count by tens relatively well, have them begin using base ten materials. An early activity might be for the teacher to write a number on the board and have each group come up with the number of tens in that number. If the teacher writes sixty, the students should come up with six tens. A further development would be to have the kids each come up with two numbers that have two groups of ten with some left over.
(13) If you wanted to try this activity with the older students a follow-up
activity might be to give each group a simple multiplication word problem
such as: If there are five people and each person wants to buy three pieces
of pizza, how many pieces of pizza will they have to buy in all? Have the
students use paper and pencil and Unifix cubes to solve it. The teacher
could also introduce multiplication by showing the students the written
equation for the problem (5 x 3 = ?).