Now that we’re moving into spring, I wanted to be sure my kindergarteners were beginning to gain a foundation for place value before moving onto first grade. The Common Core State Standards for Mathematics (CCSS-M) expect students to “compose and decompose numbers 11 to 19 into tens and ones.” We know that seeing teen numbers in terms of tens and ones helps kids build number sense for place value in our base-ten system. My string encouraged this sort of grouping by utilizing ten frames. As is our ritual, I quickly show the images, one at a time, and I ask my kids:
How many dots did you see?
How did you see them?
I record any strategy that they offer, with no real expectation that they grouped them into tens and ones until that becomes a strategy that makes sense to them.
In the first image, a few interesting strategies emerged. Audrey explained that counting by twos was a helpful strategy for her, but when she got to the end, she realized that one “extra” dot remained, so she added on from 8 (8 + 1 = 9). I recorded her skip counting strategy on the poster so that the rest of my class would both hear and see her thinking.
This appeared to be a popular strategy after surveying the rest of the class, until Arjun J. offered up an alternative way of seeing and understanding the quantity. He imagined an extra 10th dot, which he subsequently took away. When prompted to explain how he then knew he needed to remove one, he replied, “I saw a ten frame, but one dot was missing.” After recalling that the frame always held ten dots at capacity, he took away the missing dot and provided the corresponding number sentence (10 – 1 = 9) for the class. The ability of think of 9 as “one less than 10” will later be a helpful way for my students to do other, more complicated computation problems in both addition and subtraction.
The second image provided an opportunity to incorporate the ten frame into my students’ understanding of teen numbers. Like Arjun, Jana also saw the ten frame and automatically knew there were ten dots. This is an example of what we mean when we say that kids have learned to “trust the ten.” She’s treating that entire ten-frame as a unit of ten that she doesn’t need to count one by one. I asked her how she knew there were 11 dots, and she explained the she added on the one extra dot to the set of ten.
Ethan, eager to defend his strategy, first gave the number sentence 10 + 1 = 11. On the surface this appears to be just like Jana’s thinking. But was this approach the same? When pressed, he said that he counted the dots by ones which is a strategy that I am not trying to encourage. Maybe he was defending his strategy or verifying for himself and for us that Jana’s 11 really was true. Unlike Jana, who “trusted the ten” I don’t know that Ethan does just yet. More experiences will help him to do this over time.
The third and final image evoked a rich compilations of strategies. At this point, kids were building on each other’s strategies and eager to help each other explain to the class. Andrew saw 13 in tens and ones, but then followed up by counting by ones. This is common as a way to verify and learn to trust the unitizing of ten. Ethan, maybe with insights from the last image, jumped in to explain in his own words that we could unitize, or recognize the quantity of ten without counting.
Arjun B. followed up by sharing a breaking apart strategy. Knowing each half of the ten frame held ten dots, he saw five on the right. He then remembered how in the second image, we saw six on the left , and 5 + 6 made 11. From there, he pointed out that we left out two dots and needed to add on from 11 to make 13. Wow! Clearly, he was closely following the previous image and building off his understanding of how ten frames are structured.
I asked my class to try and explain Arjun’s strategy to a partner. While some could put the idea into words and others were still a bit unsure, I found that overall the use of a ten frame encouraged my class to decompose the teen numbers strategically. So, teen numbers aren’t so tricky when we can manipulate them on a ten frame!