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What is mathematical fluency?
What is mathematical fluency?
Procedural fluency applies to all operations. Mastery of math facts must focus on fluency. "Strategy development is absolutely necessary for fluency. And fluency is essential to developing automaticity with basic facts." (Bay-Williams, J., & Kling, G., 2019 p. 4)
Having procedural fluency means you are computing with:
Flexibility: Students should have a conceptual understanding of how they arrived at the math facts but also know a variety of strategies to approach problems.
Accuracy: By using the strategy appropriately, students should arrive at the correct answer.
Efficiency: Students are able to choose a strategy and implement it without struggling.
What are the end-of-year fluency standards?
What are the end-of-year fluency standards?
Kindergarten:
K.OA.5 Fluently add and subtract within 5.
1st Grade:
1.OA.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).
2nd Grade:
2.OA.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.
2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
3rd Grade:
3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
3.NBT.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
4th Grade:
4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.
5th Grade:
5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm.
6th Grade:
6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.
6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
K-6 math fluency progression:
Kindergarten: K.OA.5 Fluently add and subtract within 5.
1st Grade: 1.OA.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).
2nd Grade:
2.OA.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.
2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
3rd Grade:
3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
3.NBT.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
4th Grade: 4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.
5th Grade: 5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm.
6th Grade:
6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.
6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
How do students develop math fact fluency for long-term use?
How do students develop math fact fluency for long-term use?
Because I know short-term memorizing math facts is not the same as having math fact fluency.
Addition Facts
Addition Facts
Multiplication Facts
Multiplication Facts
Does it make sense to learn math facts in numerical order?
Does it make sense to learn math facts in numerical order?
Students must have automaticity with Foundational Facts before using Derived Facts Strategies.
Near Doubles:
Use a double to find the sum
"I don't know 8 + 6 yet, but I do know 6 + 6 is 12. I can add 2 more to 12 to get 14."
8 + 6 = ?
6 + 6 =12
8 + 6 = 2 + 6 + 6 = 14
Making 10
Move some from one addend to the other to make a 10
"I don't know 8 + 6 yet, but I can break the 6 into 2 and 4 and make a 10 with the 2 and 8. Then I add the 4 to the 10 to get 14."
8 + 6 = ?
6 = 2 + 4
8 + 2 = 10
10 + 4 = 14
Pretend-a-10
Think of an 8 or 9 as a 10, and adjust answer
"I don't know 8 + 6 yet, but I do know 10 + 6 is 16. Because 8 is 2 less than 10, 8 + 6 is 14."
8 + 6 = ?
10 + 6 = 16
16 - 2 = 14
Addition Fact Fluency Posters.pdf
Doubling
Because 4s, 6s, 8s are even, these facts lend themselves to halving the factor to , then doubling the smaller product,
"I don't know 4 × 9 yet, but I do know 2 x 9 is 18. I can double that to get 36."
4 x 9 = (2 × 9) x 2
18 x 2 = 36
Add a Group
To get 3s or 6s, use the 1 facts to Add a Group to 2s or 5s.
"I don't know 6 × 4 yet, but I do know 5 x 4 is 20. I can add a group of 4."
6 x 4 = (5 × 4) + (1 × 4)
20 + 4 = 24
Subtract a Group
To get 4s or 9s, use the 1 facts to Subtract a Group to 5s or 10s.
"I don't know 9 × 3 yet, but I do know 10 x 3 is 30. I can subtract a group of 3."
9 x 3 = (10 × 3) - (1 × 3)
30 - 3 = 27
Break Apart
Break one factor into parts. Find the separate products and add them together.
"I don't know 8 × 7 yet, but I can break 8 groups of 7 into 5 groups of 7 plus 3 groups of 7."
(5 + 3) × 7
(5 × 7) + (3 × 7)
35 + 21 = 56
Near a Square
Use square numbers (both factors are the same) to get nearby facts that are ±1 a group.
"I don't know 8 × 7 yet, but I do know 7 x 7 is 49. I can add a group of 7."
8 x 7 = (7 × 7) + (1 × 7)
49 + 7 = 56
Multiplication Fact Fluency Posters.pdfWhat are activities students can do to develop math fact fluency?
What are activities students can do to develop math fact fluency?
Play math games that focus on reasoning strategies (Phase 2), do quick looks, and work on story problems!
Foundational and Derived Fact Fluency Games
Foundational and Derived Fact Fluency Games
Foundational Addition and Subtraction Facts
First Facts: Adding or Subtracting 0, 1, 2
Next Facts (in any order): Doubles; Combinations of 10; 10 + __
Game 5: Doubles Match-Up
Game 10: Square Deal
Derived Fact Strategies for Addition and Subtraction
Foundational Multiplication and Division Facts
2s, 5s, 10s
Squares
All Foundational Facts
Derived Fact Strategies for Multiplication and Division
Remind students to “Think aloud as you play the game”
Questions to ask to encourage mathematic thinking during math game play
How did you figure it out?
How can you say out loud what you thought about in your head?
If someone didn't know the answer to _____, how other math fact might help them to figure it out?
Quick Looks
Quick Looks
Early Number Concepts
Use dot images and five & ten frames to help students visualize and count quantities. These help students decompose numbers in flexible ways and develop visual images of small quantities (a.k.a. subitizing), recognize number relationships by building on to previous images of dots (“this one is five, because there is 1 more dot”).
Quick Look Routine
Step 1: Identify Focus
Identify what foundational fact you want students to focus on during the routine. Select an image that invites students to focus on that fact or relationship.
Step 2: Give Quick Looks
Show the image for 2-3 seconds, then hide the image. Repeat for a second look.
Step 3: Talk About How
Invite students to think and share “How many did you see?” and “How did you see them?”
Step 4: Connect Symbols
Ask “What number sentence might match this quick look?”
Quick Images for Addition
Quick Image Cards Enlarged.pdfdot_cards.pdfDot Talk Cards.PDFkinder10-frame-dots.pdf10 frame dot images.pdfQuick Images for Multiplication
Quick Image Multiplication.pdfmultiplication-subitizing-cards.pdfStorytelling
Storytelling
Story Problems
Use familiar contexts, with manipulatives, drawings, acting to model the situation. Promotes development of math language, and shows relationship between adding and subtracting.
Problem Types for Addition and Subtraction Situations
Add to / Take from (a.k.a. Join or Separate stories)
These involve an action
Put Together / Take Apart (a.k.a. Part, Part, Whole)
These involve separate sets that together make up a whole amount
There is no action joining or separating those sets
Comparison
These compare two different sets (i.e., which one is more or less)
There is no action in this situation
Addition Subtraction Problem Situation Types (K-2).pdfHere’s a visual way of seeing these problem types
Problem Types for Multiplication and Division Situations
Equal Groups of Objects
Lends itself to fact fluency
Teach students to read 6 x 5 as "6 groups of 5." This reinforces the conceptual meaning of multiplication.
Helps with Derived Fact Strategies (e.g. decompose the 6 groups into 5 groups of 5 and 1 group of 5)
Arrays of Objects
Helps students make sense of the commutative property of multiplication (e.g., 3 rows of 4 columns looks the same as 4 rows of 3 columns after the array is rotated 90 degrees)
Compare
Important multiplicative situation, for grades 4 and 5, but not necessarily for multiplication fact fluency
Multiplication and Division Situation Types (3rd-5th).pdfAre flashcards appropriate for learning or assessing addition and multiplication facts?
Are flashcards appropriate for learning or assessing addition and multiplication facts?
Yes, as long as the flashcard activity focuses on building students' math fact fluency leading to mastery, not just memorizing.
We want students to have Fact Mastery (Phase 3).
When they aren't there yet, we need to encourage reasoning from facts they do know (Phase 2)...NOT relying on counting (Phase 1) as their only strategy when they don't know a particular math fact.
Foundational Facts for Addition Sorting Mats and Card Sets.pdfFoundational Facts for Subtraction Sorting Mats and Card Sets.pdfDerived Facts for Addition Sorting Mats and Card Sets.pdfDerived Facts for Subtraction Sorting Mats and Card Sets.pdfFoundational Facts for Multiplication Sorting Mats and Card Sets.pdfFoundational Facts for Division Sorting Mats and Card Sets.pdfDerived Facts for Multiplication Sorting Mats and Card Sets.pdfDerived Facts for Division Sorting Mats and Card Sets.pdfmultiplication-array-cards1.pdfHow do I assess students' math fluencies?
How do I assess students' math fluencies?
Interview Prompts During Game Play
MFF-AssessmentTool18-InterviewPromptsForAssessingFluency.pdfProgress Monitoring For Addition and Multiplication Fact
MFF-AssessmentTool25-ProgressMonitoringToolForAdditionFacts.pdfMFF-AssessmentTool26-ProgressMonitoringToolForMultiplicationFacts.pdfObservation Logs for Addition and Multiplication Fact Fluency
How should I report Grade-level Math Fluency on a SBUSD K-6 Student's Report Card?
How should I report Grade-level Math Fluency on a SBUSD K-6 Student's Report Card?
Below are sample SBUSD report cards for each grade where you can see the highlighted fluency standard(s). While it may be appropriate to report on a student's fluency progress at each trimester, remember these are "end-of-year" fluency standards.
When following the IM Scope and Sequence for your grade, fluency standards are addressed in specific units. These units may not be introduced until later in the school year, therefore, you may not have evidence of students' end-of-grade" math fluency yet (e.g., in Trimester 1).
Kindergarten Report Card Color Coded and Math Fluency Highlighted.pdfKindergarten
1st Grade Report Card Color Coded and Math Fluency Highlighted.pdf1st Grade
2nd Grade Report Card color coded and Math Fluency Highlighted.pdf2nd Grade
3rd Grade Report Card Color Coded and Math Fluency Highlighted.pdf3rd Grade
4th Grade Report Card Color Coded and Math Fluency Highlighted.pdf4th Grade
5th Grade Report Card Color Coded and Math Fluency Highlighted.pdf5th Grade
6th Grade Report Card Color Coded and Math Fluency Highlighted.pdf6th Grade
More resources for math fluency instruction and assessment
More resources for math fluency instruction and assessment
Three Steps to Mastering Multiplication Facts.pdfAssessing Basic Fact Fluency.pdfDeveloping Fluency with Addition Facts.pdfDeveloping Multiplication Fact Fluency.pdfMultiplication Fact Fluency Built on Understanding.pdfPage updated
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