With pages of math standards at each grade level, what is really considered essential learning? In this series of posts, I attempt to highlight the key skills at every grade level, explain those skills in layman’s terms, and provide activities and tasks for helping children build those skills. Please note that as this series grows, there will be multiple posts for each grade level.

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Fact fluency actually begins in Kindergarten, when children learn the combinations for the numbers through 10 and are expected to fluently add and subtract within 5. First grade builds on those skills. While the standard shown below is from the Common Core State Standards, other state standards are very similar.

**CCSS.MATH.CONTENT.1.OA.C.6**

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

There are two important ideas buried in that standard.

- First, while 1st Graders are working on the facts within 20, the expectation for fluency by the end of 1st Grade is within 10 (remember that Kinder was fluent within 5). That’s an important distinction. It basically means they should have automatic recall for facts within 10, like 4 + 5, 7 + 1, or 2 + 8, but they should have successful strategies for adding single-digit numbers up through 20, such as 6 + 7, 4 + 9, or 8 + 6.
- Second, the acquisition of fluency with basic facts is based on learning and using strategies that emphasize the relationships between numbers, not memorization. Those strategies are actually listed in the standard, and I’ll explain them in more detail in the next section.

### Learning Facts Using a Strategy-Based Approach

#### Foundational skills

In order to use strategies for learning the basic facts within twenty, students need to have mastered the Kindergarten skill of knowing the combinations of all the numbers through ten. See this post for more on that topic. It’s a good idea to quickly assess a child’s fluency with the combinations to ten before trying to teach strategies based on those combinations. The post I just mentioned provides information on doing that assessment. If a 1st Grader needs work on this Kindergarten skill, it’s best to provide practice with number combinations prior to trying to introduce the strategies outlined below.

#### What is the progression for teaching and learning the fact strategies?

In their book *Math Fact Fluency*, Jennifer Bay-Williams and Gina Kling have a great graphic that shows how all the different fact strategies are interconnected. The idea is that children need to master the facts at each level before they can tackle the strategies below. Think of it this way, I can’t use a strategy called Near Doubles if I don’t know my Doubles.

#### +/- 0, 1, and 2

While it may seem obvious, children need to know that when we add 0 to or subtract 0 from any number, the number does not change. So 5 + 0 = 5 and 5 – 0 = 5 as well. This is also a good time to introduce the idea of the *commutative property*, which means that the order does not matter when adding. Of course, it’s not necessary for a 1st Grader to learn the mathematical term, but you want them to have the understanding. This will help them understand that 5 + 0 = 0 and 0 + 5 = 5 as well. Important to note that the commutative property works for addition but not subtraction. Concrete materials can be used to help children visualize the commutative property.

In Kindergarten, children practice counting forward and back from any number. This helps them develop an understanding of the next number and the previous number. This first set of facts uses that counting-on/counting-back strategy. Facts in this group include facts such as 6 + 1, 2 + 7, and 9 – 0. Because counting on or back becomes inefficient with larger numbers, we only really want children using it for adding or subtracting up to 2. By that I mean that it’s very inefficient to use counting on to find the sum of 7 + 8, and there are much better strategies for that fact.

Here’s a little game kiddos can play to practice the +/- 0, 1, and 2 facts. If you don’t have ten-sided dice you can use playing cards. Use only the Ace through 9 cards, and instead of rolling the dice, turn over a card. Click here to grab yours.

#### Doubles and Near Doubles

Children often have an easier time learning their doubles (e.g., 4 + 4, 6 + 6, 9 + 9, etc.). That said, it’s still helpful to use concrete and pictorial representations to support the development of automaticity. Using a familiar tool like a ten-frame helps children to use the benchmark of five to see that 6 + 6 = 5 + 5 + 2. Click here to download a double ten-frame mat.

Once children are fluent with the doubles facts, they can work on Near Doubles. This strategy is sometimes referred to as Doubles Plus One, Doubles Plus Two, or Using Doubles. Using this strategy emphasizes the relationship between facts. If a child knows that 6 + 6 = 12, we want them to understand that 6 + 7 is just one more, so it must be 13. Again, using concrete and pictorial representations helps build this understanding.

You will also see how that Kindergarten skill of being about to decompose numbers in different ways supports this strategy. In this little activity for Near Doubles, one addend is decomposed to make a double fact.

One important note. There’s not just one way to use doubles. For example, one child might think about one more than 6 + 6 to solve 6 + 7. But another might think of one less than 7 + 7 to solve the same problem. That’s the beauty of using a strategy-based approach–children will use them in the way that makes the most sense to them! We want to be careful about imposing structure and rules when using strategies for adding and subtracting.

#### Combinations for 10 and Make a 10

In Kindergarten, children should work extensively to master all the combinations for the numbers up through 10. Ten is a special number in our place value system, so knowing that 7 + 3 = 10 helps children understand that 17 + 3 = 20. And that if I have 47, it’s 3 more to the next ten (50). And 470 + 30 = 500. You get the idea.

My friend Kris Graham made several short videos showing math games that can be played with only a deck of cards. Kids will love this Pyramid Ten game!

Ten frames are designed to emphasize benchmarks of 5 and 10, so they are an invaluable tool for practicing combinations for 10. They also help children visualize the Make a 10 strategy. Take for example the problem 8 + 3. Using a double ten frame, build one addend on the top ten frame and the other on the bottom. Using two different colors of counters helps with visualization. To add the numbers together, slide counters from the second addend to fill in the top ten frame, making a ten. The Make a 10 strategy is best when one of the addends is close to 10. Children need lots of concrete practice like this before moving to a more abstract process. Be sure that sometimes the addend close to 10 is the second addend, so children realize that 3 + 8 is the same as 8 + 3. In the case of 3 + 8, with 3 on the top ten frame, you would want to move counters down to the bottom ten frame to make a ten.

Once students understand the Make a 10 strategy using concrete materials, they are ready to decompose and work with the numbers mentally. Check out this post to grab another freebie!

#### Plus 10 and Pretend a 10

Plus 10 is simply another way to say “teen numbers.” Children begin to understand how place value works when they learn that teen numbers consist of a ten (ten ones) and some leftovers. So in the number 13, for example, we can bundle ten of the ones and three will be leftover. Again, ten frames are the ideal tool for helping children visualize this concept. This understanding needs to go beyond just saying there is a 1 in the tens place. Very young children need to *see* what that means for it to make sense and have meaning.

Here’s another one of Kris’s games, this one for making teen numbers. You can download a sheet like the one shown in the video for kids to record their numbers here.

Once children know their Plus 10 facts, they can begin to use that understanding to solve nearby facts using a strategy called Pretend a 10. This strategy is very useful for the nines facts, like 9 + 5. Children simply think, *I know that 10 + 5 = 15, so 9 + 5 is just one less, or 14. *

These strategies might be very new for you and different from how you learned your math facts. Trust the process! Children who learn their facts using a strategy-based approach learn that numbers and math make sense, rather than just seeing it as a series of procedures that must be memorized.

I would love to read comments about your experiences with or thoughts about using a strategy-based approach! Drop a comment below or post one on Twitter! Be sure to tag me! @MathCoachCorner

#### References

Bay-Williams, J. M., & Kling, G. (2019). *Math fact fluency: 60 games and assessment tools to support learning and retention*. Alexandria, VA: ASCD.

OConnell, S., & SanGiovanni, J. (2015). *Mastering the basic math facts in addition and subtraction: strategies, activities & interventions to move students beyond memorization*. Portsmouth, NH: Heinemann.

Conklin, Melissa. (2010). *It Makes Sense!: Using Ten-Frames to Build Number Sense*. Math Solutions.

I really enjoy your blog. You do a great job of explaining concepts and creating lessons that uncover misconceptions and teach deep understanding. I very much believe in teaching for conceptual understanding and I think the above strategies are great for number sense and therefore very worthwhile. However, I don’t see how these strategies help kids memorize (for instant recall) math facts. It seems like they need to just know the answer quickly without jumping through mental hoops of “OK, I know 8+8 is 16 so 8+9 must be 17”. It’s true I’d much prefer them get the answer to 8+9 by the near doubles strategy than by counting on from 8, but what is even better is automaticity. In your experience do these strategies eventually lead to automaticity or is there another piece that needs to happen? (I hope this doesn’t come across as negative. It’s a question I’ve been thinking about for a while!) Thank you for your great posts and resources. When I am teaching a new math concept you’re my first stop!

Thank you for your question! Your concern is a very common one, actually. The short answer is, this is a step

towardautomaticity, not the end product. During this stage, children use reasoning to arrive at the answer. With repeated practice, they develop automaticity. That’s why most current standards are written to emphasize this strategy-based approach in 1st Grade with automaticity being the expectation in 2nd Grade.Thank you- with my 1st grader in online learning this Fall, I felt lost trying to understand the basics of common core and what she should know and be building on.

Just printed some ten frames and created some counters.

Such a great article – will definitely be sharing with parents in our K-2 school!

Great article! Learning strategies is so important when it comes to fact fluency!!

I love the Pyramid Game and my students are always asking to play it. It can also be adapted to practice combinations for other numbers within 10. To practice combinations to make 7, remove 7, 8, 9 and 10.

*I found a typo, “This will help them understand that 5 + 0 = 0 and 0 + 5 = 0 as well.”

Thank you!!

Thanks for catching that! All fixed. 🙂

I always enjoy reading your blog! Thank you for supporting us during these wild times!

Hi,

Thanks for sharing the links. Great games!

Great post! I appreciate your description of the process of building number sense and practical advice on helping kids “get” the strategies. I will be tutoring my daughter this summer to fill in gaps in understanding from first grade math so this is invaluable for me!

hello, great game using playing cards to identify teen numbers.

*I received an error message on some of the downloads that were listed on this page..

Hmm, I tested all of the links. Can you tell me specifically which ones you had trouble with?