1. Mental Math and Basic Facts
STRATEGIES TO SUPPORT COMPUTATIONAL FLUENCY WITH UNDERSTANDING

2. Fact Fluency
Fact fluency is when you can quickly remember the answers to MATH FACTS in 3 seconds or less.
Our goal for second grade is to complete 20 problems in 2 minutes. These problems can include addition and subtraction problems such as:
Combinations of 10
Doubles
Doubles +/-1
Combinations up to 20 such as 13+7

3. Computational Fluency?
Computational fluency refers to having efficient and accurate methods for computing.
Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently.
The computational methods that a student uses should be based on mathematical ideas that the student understands well.

4. Phase I: Constructing operational meaning
The acquisition of basic math facts should occur in the following three phases:
Phase I: Constructing operational meaning
Phase II: Reasoning strategies
Phase III: Working toward quick recall
There is much overlap and students often can work in multiple stages simultaneously and may move through the stages at different rates. Additionally, it is important to point out that these three phases are critical for each of the four basic operations. However, addition and subtraction may develop concurrently and the same can be said about multiplication and division.

5. Phase 1 - What does that mean?
Students have to have a deep understanding of what numbers and operations are before being asked to use or memorize fact strategies
A student will struggle with memorizing facts if they are asked to do the same questions over and over without understanding the meaning and strategies for completing the operations

6. Phase 2 – The Strategies
Strategies are taught only after building understanding of the operations through hands on, pictorial and symbolic practice with operations.
Strategies may be discovered by students but other students will need direct instruction with these strategies.
PRACTICE MAKES PERFECT – it takes time to learn strategies. Be patient and think of the long term goals to teaching strategies -> Fluency with Understanding
Strategies like most math concepts should be taught with manipulatives, pictures and then symbolically

7. Phase 3 – Working Towards Quick Recall
Students must know all the facts or strategies to find all the facts before emphasis on speed is placed
Quick recall of math facts is usually defined as the ability to solve a basic number computation in a few seconds or less (without resorting to inefficient methods like counting)..
Phase III is simply about increasing a student’s quick recall. Spending more time in Phase III will not lead to quick recall of unknown facts.
Successfully addressing Phases I and II will significantly decrease the need for prolonged work in Phase III. This involves:
increasing the speed in which the student selects and applies a strategy for solving the problem and
using highly organized and planned practice for the purpose of devoting facts to memory.
Phase III may include practice (and/or drill) with specific groups of facts through fact cards, games, paper/pencil practice, and the use of technology. Once students acquire the ability to quickly recall the facts, Phase III focuses on maintenance of the facts.

8. What does the research say?
Learning with understanding is more powerful than simply memorizing because the act of organizing improves retention and promotes fluency. EDThoughts 2001 p. 81

9. Building Number Concepts
Concrete Manipulatives
Pictorial Representation
Abstract Symbols
4 + 4 = 8
2 x 4 = 8
I I I I
Significant time must be spent
working with concrete materials
and constructing pictorial representations
in order for abstract symbol and operational understanding to occur.

10. Fact Fluency Strategies
To solve math facts quickly, we need to use math strategies.
What strategies do you know for solving the math facts below?
8 + 4 =
16-9=
8+7=

11. Sums of 10
This group includes all facts with a sum of 10. Picture the Ten Frame when solving.
Examples: = 10
2 + 8 = 10

12. Make-Ten (Use the Ten Frame)
This strategy works well with at least one addend of 8 or 9. When adding 9, picture a Ten Frame. Take one away from the other addend and move it over in your mind. For think: 9 in the ten frame means that I need one more to make ten. If I move one from the 6 over, I have 5 left. So I can add and that equals 15.
9 + 6 has the same sum as
10 + 5
Do the same for 8, except you have 2 open in the Tens Frame.

13. We can find the answer to this fact by making ten.
Making 10 Examples
We can find the answer to this fact by making ten.

14. Place the larger number in a ten frame.

15. Use part of the other number to fill the ten frame.

16. Then you can look at what is left outside the ten-frame and tell what the answer is.=
= 11

17. We can find the answer to this fact by making ten.

18. Place the larger number in the ten-frame.

19. Fill the ten-frame with part of the other number.

20. What is left outside tells you the answer. You have 10 and 2 more.
= 12

21. Addition Strategies Turn Around Facts If you know 5 + 4 = 9,
then you know = 9.
When adding, the order doesn’t matter.
+ = + =
Commutative Property

22. Identity Property of Addition
Facts with Zero
When adding zero to any number, the sum is the other addend.
Examples: = 7
0 + 5 = 5
Identity Property of Addition

23. Example: 2 + 6 = 8 Start at 6 and count up 7, 8.
(One-/Two- More Than)
When an addition problem contains a 1 or 2, we can use this strategy. Start by whispering the greater addend and count on the other addend.
Example: = 8 Start at 6 and count up 7, 8.

24. Doubles
When an addition problem contains two numbers that are the same we recognize this as a doubles problem. These are memorized facts. You can use visual clues to help you.
Example: = 8

25. Near Doubles
When an addition problem contains consecutive numbers on a number line, double the smaller addend and add 1.
4 + 5 = =
8 + 1 = 9

26. (Decomposing is what allows make-ten and near doubles to work.)
Decompose
(Decomposing is what allows make-ten and near doubles to work.)
Break down the addends and add the pieces back together.
Example: =
(10 + 1) + 4 =
10 + (1 + 4) =
= 15
Associative Property

27. Strategies To Support Fact Learning
Subtraction
Count Back Strategy - This strategy works best when subtracting 0, 1, 2 or Ex , start from 12 and count back three numbers, 11, 10, 9.
Count Up Strategy - This strategy works best when subtracting two numbers that are close together. Ex = 3 …….(9, 10, 11)

28. Strategies To Support Fact Learning
Subtraction
Distance From Ten Strategy - When the number ten lies between the two numbers of the subtraction fact, find the distance from ten for each of the numbers, then add their distances together. This strategy works best when both numbers are close to ten.
Ex = = 5 2 3

29. Strategies To Support Fact Learning
Subtraction
Subtracting 9 from a teenager!
When subtracting nine from a teenager number, simply add the digits of the teenager! 5 8 4

30. Strategies To Support Fact Learning
Show and discuss patterns!
9 - 8 = = = = = = = = 1

31. Strategies To Support Fact Learning
Subtraction
Fact Families Strategy - Can be used with all subtraction facts.
3 - 2 = 1
3 - 1 = 2
2 + 1 = 3
1 + 2 = 3 3 1 2

32. Strategies To Support Fact Learning
Write Fact Families!
(3, 8, 11)
3 + 8 = = 3
8 + 3 = = 8

33. Think of a related addition fact.
= ?
? = 12

34. Subtract from Ten
To find , start with 15.
15 is 10 & 5

35. To find 15 - 8, take 8 from the ten. You can see that 7 is left.
= 7

36. To find , start with 1.
12 is 10 & 2

37. To find 12 - 7, take 7 from the ten. You can see that 5 is left.
= 5

38. -8 13 10 + 3 -8 2 + 3 Remember that 13 is 10 and 3.
Take 8 from the 10. 13
10 + 3 -8 -8
Combine this with the other 3.
2 + 3

39. -8 -8 13 10 + 3 5 2 + 3 Remember that 13 is 10 and 3.
Take 8 from the 10. 13
10 + 3 -8 -8
Combine this with the other 3. 5
2 + 3
= 5

40. - 8 12 Take 8 from the 10. Combine this with the other 2.
Altogether, what is the answer?