To get students to think about repetitive addition, I put the following on the board:
2 + 2 + 2 = ____
Students quickly see that the sum is 6. I ask them how they got the answer. (Counting by 2s is the desired response. If you don't get this response, keep asking for a better, more "efficient" way.)
Writing and speaking, I ask, "What if I wanted to know 5 + 5 + 5 + 5 = _______ how would I do that?" (Counting by 5s.)
What if I wanted to know 8 + 8 + 8 + 8 = _______? Is it as easy to count by 8s? What might we do to find the answer with so many eights? I ask students to find the sum and be ready to explain how they figured it out. I give them a few minutes to write down the work and their answer in math journals. I circulate around to see how students are solving.
We share our answers, but more critically I want to hear how they arrived at their answer. Students have had little practice with repeated addition, so hearing/knowing the students' strategy development now is crucial to exploring this new concept. Most students can do repeated addition with 1, 2, 5 and 10 by skip counting, but beyond those numbers they need to develop appropriate strategies.
While students explain their strategies, I model their explanations on the board. Usually students notice that they can add 8 + 8 which is 16 and then 8 + 8 again for another 16 and then add 16 + 16, putting together tens and ones to get the answer. Other students may rely on counting once they get past 16, using tools such as number lines or number grids.
I tell students that today we will explore adding larger numbers by using the song, "Twelve Days of Christmas." (You can use any appropriate addition counting song or poem for this activity, but for us it is the day before the holiday break so we are celebrating.)
We will also use graphing to display the data derived from the song before we try to do the repeated addition.
When sharing the song, "Twelve Days of Christmas", I explain that the song comes from a tradition in England (I show them where England is on the map), where long ago people celebrated for 12 days with feasts and gifts. In listening to and reading the lyrics, we learn that the singer was given gifts every day, for 12 days. We are going to figure out how many presents she received, altogether.
Students are given graph paper (which I discover isn't large enough because I didn't determine how much space our work will require before we started, so I had to adapt by handing out a second sheet and attaching it to the first in the middle of our graphing.)
I project a copy of the blank graph on the board and we talk about how to color in the first day.
We color in the first day, 1 gift, by filling one square. Next we color in 2, for the second day, and one for the first day (remember, she is given every gift from the previous day plus the new day's gifts). We repeat the coloring for each day. I allow students to move ahead if they understand what we are doing. I continue walking through the process, coloring my graph on the board for those who need step-by-step structure.
When the graph is complete, I ask which day does the person in the song receive the most presents? Next, I ask students if they can figure out how many of each gift she received. They can count every square on the graph, or use more efficient (faster, fewer errors) strategies such as - receiving 5 golden rings for 8 days is 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 for 40 golden rings, or counting by fives, eight times. Students are looking for regularity in the numbers as they reason out how to find the total (MP8).
I ask students if they could figure out some of the other numbers of gifts in similar ways? Students work on totaling the different gifts and we share how we found the answers. Noticing the Patterns
Graphing data is a skill that the Common Core Standards have established as important for second graders (2.MD.D.10), and so it is an important aspect of this lesson.