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Fact Fluency: building a strong foundation in math
Fact Fluency: building a strong foundation in math
What is Fact Fluency?
What is Fact Fluency?
(as defined by Gina Kling and Jennifer M. Bay-Williams)
Authors of the book, Math Fact Fluency: 60+ Games and Assessment Tools to Support Learning and Retention, and the article, "Assessing Basic Fact Fluency"(as defined by Gina Kling and Jennifer M. Bay-Williams)
Procedural Fluency is defined as having "skill in carrying out procedures flexibly, accurately, efficiently, and appropriately."
Procedural Fluency is defined as having "skill in carrying out procedures flexibly, accurately, efficiently, and appropriately."
Basic Fact Fluency is defined as "the efficient, appropriate, and flexible application of single-digit calculation skills and...an essential aspect of math proficiency."
Basic Fact Fluency is defined as "the efficient, appropriate, and flexible application of single-digit calculation skills and...an essential aspect of math proficiency."
By utilizing the four aspects of fluency (accuracy, efficiency, flexibility, and appropriate strategy selection), we can incorporate various assessment strategies to identify what students know, don't know yet, and what our next instructional steps might be.
So, what do these mean?
- Accuracy- the ability to produce mathematically precise answers (correct answer)
- Efficiency- the ability to produce answers relatively quickly and easily (time needed to solve is reasonable)
- Appropriate strategy use- the ability to select and apply a strategy that is appropriate for solving the given problem efficiently
- Flexibility- the ability to think about a problem in more than one way and to adapt or adjust thinking if necessary (strategy is applied to new problems)
The four aspects of fluency are interrelated. Appropriate strategy selection is required for efficiency and flexibility.
The 4 formative assessment strategies (interviews, observations, journals, and quizzes) help to encourage students to reflect on which facts and strateiges they know and which ones they need to continue to practice. Self assessment can follow these four steps to provide the students the opportunity to reflect on strategies that can be used to solve more challenging facts they may face in the future.
Student Interviews
Student Interviews
- Accuracy: assessed when the student responds
- Efficiency: observed on the basis of how long it takes a student to solve the fact (should be 3 seconds or less per fact)
- Flexibility and appropriate strategy selection: addressed by follow-up prompts (How did you figure it out? How could you use htis strategy to solve this fact?)
- How to implement: They do not have to be one-on-one, but can be quick exchanges in the midst of other activities, such as transition times.
Observations
Observations
- Accuracy: assessed when the student responds/solves the problem
- Efficiency: observed on the basis of how long it takes a student to solve the fact when playing games with peers (should be 3 seconds or less per fact)
- Flexibility and appropriate strategy selection: observed when students tell their teammates both the answer to the fact AND how they found it
- How to implement: Create a list of students and facts and/or strategies that student use as they solve facts. Record observations as students play facts games or practice math fact strategies in partners/small groups.
Journaling
Journaling
- Accuracy: assessed through the students' explanation in the writing response
- Efficiency, Flexibility and appropriate strategy selection: assessed through the students' pictures, words, and numbers to communicate their strategies
- How to implement: Provide the opportunity for students to write about strategies on a weekly basis. This will help engage the students in self-reflection and self-monitoring, in addition to emphasize the importance of strategies in practicing basic facts.
Quizzes
Quizzes
- Accuracy: assessed when the student solves the problem
- Efficiency: observed on the basis of how long it takes a student to solve the fact (should be 3 seconds or less)
- Flexibility and appropriate strategy selection: observed when students solve a fact and record which strategy they used to solve the problem
Example: After completing each problem, have the students write the strategy they use to solve. Strategies could include: recall (R), automatic (A), making 10 (M10), near doubles (ND), counting on (CO), counting all (CA), modeling and counting all (MCA)
Example: After completing the quiz, have the students do the following:
- Choose one of the problems and write about how you solved it.
- Tell which helper fact you used the most on this quiz.
- Circle the facts you "just knew." Highlight those for which you used a strategy.
- Circle facts you are sure about. Draw a square around facts that took you longer to solve.
- How to implement: Once every week, provide the students an opportunity to assess their knowledge in the format of quiz (students should be able to solve each problem in 3 seconds. Time the students according to how many problems are on each quiz. For a quiz with 20 problems, the students should be able to complete it in 1 minute).
the five fundamentals:
the five fundamentals:
(As discussed in the book, Math Fact Fluency: 60+ Games and Assessment Tools to Support Learning and Retention by Jennifer Bay Williams and Gina Kling)
1. Mastery Must Focus on Fluency:
1. Mastery Must Focus on Fluency:
*It is through the application of strategies that a student develops fluency, comes to know their basic facts, or develop automaticity. Drill, flash cards, and timed testing just focus on students' accuracy and one part of efficiency (speed), neglecting strategy development.
*Research shows that students who were taught using strategies far outperformed those who were taught with traditional basic fact instruction (drills) in the areas of using strategies, automaticity, and accuracy. Strategy development is a must for fluency. Fluency is key to developing automaticity with basic facts.
2. Fluency Develops in Three Phases:
2. Fluency Develops in Three Phases:
- Phase 1: Counting (counts with objects or mentally)
- Phase 2: Deriving (uses reasoning strategies based on known facts)
- Phase 3: Mastery (efficiently produces answers)
*As students participate in meaningful practice in Phase 2 (developing strategies), they become faster in their strategy selection and application and will begin to know their facts without needing to apply a strategy. This will help them naturally move into Phase 3 (mastery through quick strategy selection or recall).
*Students in Phase 3 are considered automatic with those facts, as they are able to solve the fact in 3 seconds. The difference between Phase 2 and Phase 3 is speed. Students in Phase 2 are applying strategies but may take longer than 3 seconds to select and apply a strategy. Students in Phase 3 are able to apply strategies and answer instinctively within a few seconds.
*Drills and timed testing take students from Phase 1 straight to Phase 3, skipping the instruction on teaching strategies and the relationships between numbers. Students may remember the facts in the short term, but often after a few months will be right back at Phase 1 again (counting). The students struggle to solve the facts as they do not have the flexible strategies to fall back on.
3. Foundational Facts Must Precede Derived Facts:
3. Foundational Facts Must Precede Derived Facts:
Foundational Facts for Addition and Subtraction: (Start here! When students achieve automaticity on these facts, move to the derived fact strategies)
- +/- 0, 1, 2
- Doubles
- Combinations of 10
- 10 + ___
Derived Fact Strategies for Addition and Subtraction:
- Near Doubles
- Making 10
- Pretend-a-10
Foundational Facts for Multiplication and Division:
- 2s, 10s, and 5s
- 1s and 0s
- Squares
Derived Fact Strategies for Multiplication and Division:
- Doubling (4s, 6s, and 8s)
- Break Apart (3s, 4s, 6s, 7s, 8s, and 9s)
- Adding a Group (3s, 6s)
- Subtracting a Group (9s, 4s)
- Near Squares
*Students must master specified foundational facts to automaticity to use the related derived fact strategies on the final level (ex. teach doubles before near doubles). *
4. Timed Tests Do Not Assess Fluency:
4. Timed Tests Do Not Assess Fluency:
*Picture a worksheet containing 100 multiplication facts in random order, which students are required to complete in 5 minutes. How many of the four componets of fluency (flexibility, accuracy, efficiency, and appropriate strategy use) are integrated?
*There is no evidence to support the theory that timed tests are necessary for promoting fact mastery. From timed tests alone, we are not able to see how efficiently the student applies the strategies or which strategies the student prefers to use. Time pressures also work against strategy learning, as thinking through strategies initially takes longer than counting. Research shows students who are more frequently exposed to timed testing actually demonstrate slower progress towards automaticity with their facts than those who are not tested in that manner.
*Why do we still see timed testing as the only measure of assessment? Some educators are not exposed to other methods of assessment for fact mastery. Through the variety of formative assessments, all four components of fluency are addressed while still encouraging math confidence in our students.
5. Students Need Substantial and Enjoyable Practice:
5. Students Need Substantial and Enjoyable Practice:
Substantial and enjoyable practice should be used as an alternative to timed drills for developing mastery with basic facts. Games are interactive, allow for math discussion as student think aloud and hear others' strategies, and provide a less stressful environment when timing is not the main focus.
10 Questions to Guide Game Selection: To what extent does the game...
- Provide an opportunity to practice the subset of facts that the students are learning?
- Appeal to the age of your students?
- Employ visuals or tools (ten frames, quick looks, or arrays) to support strategy development?
- Involve selecting from among derived fact strategies (for mastery-level games)?
- Provide opportunities for discussion among students about their mathematical thinking?
- Encourage individual accountability? (Are students solving their own facts or competing to solve the same fact?)
- Remove time pressure?
- Involve logic or strategic moves, enhancing the "fun factor"?
- Offer opportunities for adaptation so that all students can experience appropriate challenge?
- Lend itself to you being able to listen and watch in order to assess progress?
"Traditional approaches to learning multiplication facts (flash cards, drill, and timed testing) attempt to move students from phase 1 directly to phase 3. This approach is ineffective—many students do not retain what they memorized in the long term, moving to grade 4 and beyond still not knowing their facts. Even if students remember facts, they are unlikely to be fluent as defined above, as they will not have learned to flexibly apply strategies to find the answer to a multiplication fact."
"Students must deliberately progress through these phases, with explicit development of reasoning strategies, which helps students master the facts and gives them a way to regenerate a fact if they have forgotten it. Students make more rapid gains in fact mastery when emphasis is placed on strategic thinking (National Research Council [NRC] 2001, Cook and Dossey 1982, Heege 1985, Thornton 1978). So, how do we help children progress through the three phases with respect to multiplication facts? Careful sequencing and explicit attention to strategy development is necessary."
Ready to Go Resources!
Ready to Go Resources!
View and print what you would like to use in your classroom!
Addressing Parent Concerns:
Addressing Parent Concerns:
"This is not the way I learned!"
"This is not the way I learned!"
- It is important for parents to be informed about our fact fluency program and how instruction might differ from their own experiences and why those changes have been made.
- Parents need to be informed and educated on the fact that numeracy is just as important as literacy! Just as students might be able to sing the alphabet but not know that K is a letter and makes a special sound, being able to count does not mean that students recognize 7 as a number or the quantity that it represents.
- Most adults view the teaching of basic facts as drill or show flash cards for every fact until their child is able to say the correct answer quickly. However, as we know, this is not the most effective method.
- Helping parents understand the 5 Fundamentals of Fact Fluency helps to build their understanding as they help their child master basic facts through learning strategies and developing fluency.
- You may want to have a "Family Math Promise" for parents and students to sign as an agreement that they will do math together and get to know each other's mathematical reasoning. Family members make this promise to one another through playing math games and noticing math in their every day lives.
- Sometimes parents respond with "I am not good at math" or "I don't like solving math problems," as a reason for why their child is struggling in math. Remind them there is no math gene! It is our responsibility to point this out to families and help them to portray mathematics in a positive light! If we always stay positive for our children, we can help to make big changes in their confidence!
Feel free to print the following documents for Meet the Teacher Night and Curriculum Night!
Show the following video at Curriculum Night "5x9 is more than 45" to demonstrate the different ways to solve a problem.
CFISD Fact Fluency Plan:
CFISD Fact Fluency Plan:
- District Detailed Plan
- Fact Fluency for Student Mastery Timeline – According to TEKS:
▪ Kindergarten- Combinations to 10 using concrete and pictorial representations
▪ First- Addition and Subtraction Facts within 20
▪ Second- Addition and Subtraction Facts within 20 with automaticity
▪ Third- Multiplication and Division facts to 10 by 10 with automaticity
▪ Fourth- Multi-digit Computational Fluency using fact strategies
▪ Fifth- Multi-digit Computational Fluency using fact strategies
Resources to Utilize :
Resources to Utilize :
Padlet:
Padlet:
- Fact Fluency Resources from Jennifer Bell
Number Sense:
Number Sense:
- Number Line Maker (1-5): A customizable number line great for modeling the Empty Number Line. Can be used to show how to decompose numbers and compensate when adding & subtracting.
- KenKens (3+): Take Sudoku to the next level by adding basic computation and number properties. Students can work in teams and race the clock. Six new free KenKens each day. Great for Smartboards!
Addition & Subtraction:
Addition & Subtraction:
- 10 Frame Practice (PreK-2): Practice with 1:1 correspondence, combinations to 10, and basic addition facts. The site "reads" to the students as they play, so be sure to use headphones!
- Save the Whale (PreK-2): Students must match "Friendly 10s" in order to save the whale. Good for independent practice!
- Octopus Facts to 10 (PreK-1): A great interactive website where students can practice addition and subtraction facts to 10.
- Space Jumps (1-2): A great game for students to practice the strategy of "Build thru 10." Good for independent practice!
- Catapult Count On (1-3): A great game for students to practice "Build thru 10" with numbers from 10-100. Good for independent practice!
Games and Literature for Addition & Subtraction: *Youtube read louds linked to each book*
Games and Literature for Addition & Subtraction: *Youtube read louds linked to each book*
(Games from: "Mastering the Basic Math Facts in Addition and Subtraction" book)
- Understanding Addition and Subtraction: Rooster’s Off to See the World by Eric Carle
- Plus 1 and Plus 2: Mouse Count by Ellen Stoll Walsh
o GAME: Spinning More or Less
- Adding Zero: Gray Rabbit’s 1, 2, 3 by Alan Baker
o GAME: Who Has More
- Adding 10: If You Give a Mouse a Cookie by Laura Numeroff
o GAME: 10 More
o GAME: Ten and Some More
o GAME: Sammy Word Problem Cards
- Doubles: Double the Ducks by Stuart J. Murphy (addition) and Martha Blah Blah by Susan Meddaugh (subtraction)
o GAME: Squares
- Making 10: Ten Apples up on Top! By Dr. Seuss
o GAME: Fill Ten
o GAME: What’s Missing
- Using 10s: Diary of a Worm by Doreen Cronin
o GAME: Condition Cards
o GAME: Difference Maker
- Using Doubles: Fish Eyes by Lois Ehlert
o GAME: Spin and Double It
o GAME: Missing Numbers
Games and Literature for Multiplication & Division: *Youtube read louds linked to each book*
Games and Literature for Multiplication & Division: *Youtube read louds linked to each book*
(Games from: "Mastering the Basic Math Facts in Multiplication and Division" book)
- X2 Multiplication:
Big Idea: Multiplication by 2 is the same as doubling or skip counting by 2
Read book: Two of Everything by Lily Toy Hong (have students use counters to double the items and self-check with the book)
o What do I mean when I say twice as much? What does it mean to double something?
o Name some things that doubled in the story? Can you find twice as many by adding/ multiplying?
Small Group Activity: Give students yellow counters and ask them to double their number
Small Group Activity/ Reteach Ideas:
o Multiplying by 2 can also be taught with pairs (6 groups with 2 in each group)
o Model how to jump on the number line for problems like 2x3
Exit Ticket word problem: Mr. Haktak wanted a special gift for his wife. He picked 6 roses from his garden and placed them in the pot. How many roses did he have to give to Mrs. Haktak. How do you know?
Exit Ticket word problem: Mrs. Haktak had a busy day and forgot to make dinner. She found 3 corn muffins and 4 carrots and placed them in the pot. What did she take out of the pot for dinner? How do you know?
Exit Ticket word problem: Mr. Haktak put 4 dimes in the pot. Mrs. Haktak put 7 nickels in the pot. Who had more money when they took their coins out of the pot? Justify your answer.
Exit Ticket word problem: Mrs. Short baked some yummy chocolate brownies. She placed 6 plates on the kitchen table and put 2 brownies on each plate. How many brownies did she put on plates?
Exit Ticket word problem: Mrs. Short placed 8 plates on the kitchen table and put 2 strawberries on each plate. How many strawberries did she put on plates?
Exit Ticket word problem: Colin had 2 baskets with 3 apples in each basket. How many apples did he have?
Exit Ticket word problem: Colin had 3 baskets with 2 apples in each basket. How many apples did he have?
Exit Ticket word problem: Bailey had 4 purses with 2 coins in each purse. How many coins did she have?
Exit Ticket word problem: Bailey had 2 purses with 4 coins in each purse. How many coins did she have?
Games: Rolling for Doubles and Double Up
X10 Multiplication:
Big Idea: Multiplication by 10 is like skip counting by 10
Read book: The Grouchy Ladybug by Eric Carle (focuses on the spots on a ladybug)
o Ask how many spots on a ladybug (5 on each side). Multiply number of spots _________ x number of ladybugs on a leaf (roll to find number) _________ = total number of spots
Whole Group/ Small group activity: Look for patterns: Have the students use tens rods to answer x10 multiplication problems as you call them out loud
Whole Group Activity/ Small group activity: Mini Ladybug template (Have the students place the ladybugs on the 100s chart until it is filled up)
Warm Up Problem: Relating x10 facts to money (Alex had 6 dimes. How many cents did he have? John had 8 dimes. How many cents did he have?)
Exit Ticket word problem: Ladybugs in Kim’s yard have 10 spots. She catches 4 ladybugs on Tuesday, 5 ladybugs on Wednesday, and 6 ladybugs on Thursday. How many total spots did she catch each day? Explain how you know.
Exit Ticket word problem: Mrs. King set up chairs for the class play. She set up 4 rows of chairs. She put 10 chairs in each row. How many chairs did Mrs. King set up for the play?
Exit Ticket word problem: Lisa bought 6 packages of colored pencils. There were 10 pencils in each package. How many pencils did she buy?
Games: Keep It Toss It, Mini Math Facts Rings x10
X5 Multiplication:
Big Ideas:
o Multiplication by 5 is like skip counting by 5
o Products either have a 0 or 5 in the ones place and the products alternate between even numbers and odd numbers
o “5 is half of 10. Multiplying a number by 5 will result in a product that is half of the product that results when the same number is multiplied by 10.”
Whole group/ small group activity: An Investigation with Pennies (Mrs. Alexander bought each of her 7 grandchildren a brand new piggy bank. She went to the bank to get enough pennies to put 5 pennies in each of their piggy banks. How many pennies did she need?)
Read book: Count on Pablo by Barbara deRubertis (focuses on skip counting) *no Youtube read aloud video link *
o Discuss how the vegetables are packed for sale at the market (individually, pairs, by 5 of 10). Encourage students to skip count through the story as you read.
o Look at Hundreds chart- mark multiples of ten with red counters. What do you notice? Then, mark multiples of 5 with green counters (replacing the red counters with green). Say: “one group of 5 is…”
o Have table groups solve these problems after reading book:
-If Pablo sold 8 pairs of onions, how many onions did he sell? (8x2)
-If Pablo sold 5 boxes of tomatoes, how many tomatoes did he sell? (5x10)
-If Pablo sold 4 bags of peppers, how many peppers did he sell? (4x5)
Small Group Ideas:
o Compare counting by 5s to money
o Compare counting by 5s to telling time on a clock (4 groups of 5 is 20 minutes)
Extra practice/ Fact Fluency Station: Pick a card and then draw an array of the fact, write the repeated addition sentence that goes with it, write the fact with the product three times, and write a story problem for the fact
Game: Corners
X1 Multiplication:
Big Idea: Multiplying by 1, the product is the same as the other factor
What does it mean to multiply by 1 or to have 1 set? What do you notice in the products?
Read book: One Tiny Turtle by Nicole Davies. Ask the students what if there were 19 eggs in a nest? 25 eggs? 100 eggs? How many eggs would be in 1 nest? What would the multiplication equation be?
Small Group activity: Tiny Turtle Eggs
Small Group activity: Using Color Tiles (Create 10 bags of color tiles or cubes by placing 1 green, 2 red, and 5 yellow in each baggie)
o Questions: How many yellow tiles are in 3 bags? (3x5=15)
o Questions: How many red tiles are in 6 bags? (6x2=12)
Small Group Activity: Discuss the difference between 3x1=3 and 3+1=4 (use manipulatives to build these equations & write story problems for each of these equations)
Games: Math Towers and Math Checkers
x0 Multiplication:
Big Idea: If either factor is 0, the product will be 0.
Whole Group Activity: Investigation with school supplies- Create 10 baskets of school supplies, each filled with 1 marker, 2 pencils, 5 crayons, and 10 paper clips. Place the 10 baskets in front of the class and ask questions to solve with partners:
o How many pencils are in 6 baskets? (6x2)
o How many crayons are in 3 baskets (3x5)
o How many paper clips are in 9 baskets? (9x10)
o How many markers are in 7 baskets? (7x1)
o How many glue sticks are in 4 baskets? (4x0)
-If all baskets looked like these, how many glue sticks would be in 100 baskets? If we had no baskets, how would we write the equation? (0x0=0)
Read book: Where the Wild Things Are by Maurice Sendak
o Create a class monster on the board with horns, eyes, ears, teeth, tails, legs, wings, and claws. As the students to share how many wings would be on 5 monsters? (you can use other body parts)
o Ask the questions: There were no oars on Max’s ship. How many oars would be on 3 ships just like Max’s?
o Ask the questions: There were no green monsters on the island. How many green monsters would be on 10 islands?
Exit Ticket word problem: There were 4 plates on the table. There were no hamburgers on any of the plates. How many hamburgers were there? (4x0)
Exit Ticket word problem: There were 7 plates in the coin toss game at the carnival. There were no coins on any of the plates. How many coins were on the plates? (7x0)
Whole Group Game: Don’t Be Tricked By the Symbol (Give the students equations like 3x0 and 0+3 to solve. Call other facts out loud and have them write the answer on a white board.)
Games: Zemory and Fact Sorting (print flashcards and have the students sort the products 3 ways- products that are 0, products that are less than 20, and products that are even/odd)
X3 Multiplication:
Big Idea: Multiplying by 3 is tripling a number.
Read book: A Three Hat Day by Laura Geringer
o Ask students why they think he is wearing so many hats
o Ask the following question after reading: How many hats would we need if 4 people were feeling sad? How many hats would we need if 4 people were feeling very sad? (have students use linking cubes to show 4x2 and 4x3 for the problems) On a one-hat day a person would wear 1 hat (1 man x 1 hat), 2 hats (1 man x 2 hats), and 3 hats (1 man x 3 hats)
o Have them select a number of people (ex. 6 people) and show stacking cubes to represent the total number of hats on a 1 hat day, a 2 hat day, and a 3 hat day (use linking cubes)
Whole Group Activity: Doubling and Tripling Baskets of Fruit
o The Franklin Zoo has some very special animals that attract lots of visitors each day. There are twin apes, Brady and Grady, and a set of chimpanzee triplets, Barry, Larry, and Jerry. One day, the zookeeper brought them each a basket of their favorite fruits. Each basket held the following: 1 grapefruit, 2 peaches, 3 oranges, 4 apples, 5 lemons, 6 bananas, 7 strawberries, 8 raspberries, 9 grapes, and 10 blueberries. How many of each fruit will the zookeeper need in order to create fruit baskets for the twin apes? (use Fruit Baskets recording sheet)
o Help students notice patterns in the numbers (triples are like doubles with 1 more set added)
Small Group Activity: Have the students color in the multiples of 3 on a 120 chart. What patterns do they notice?
Station Activity: Fold a paper into thirds (Write a x3 fact in one column, write a story problem for the x3 fact in the second column, and draw a visual representation of the fact in the third column OR have them write the fact at the top, draw an array in the first column, write the word problem in the second column, and write a tip for multiplying by 3 in the last column)
Games: All Lined Up
X4 Multiplication:
Big Ideas:
o Multiplying by 4 is doubling a double.
o Multiples of 4 are always even numbers and are also multiples of 2.
Read book: If You Hopped Like a Frog by David Schwartz
o Read book twice. First time for fun. Second time ask the students to think about the number of legs for the different animals and insects as you read.
o Complete the How Many Legs activity with the students after to compare the difference between x2 products and x4 products (to understand that x4 facts are double x2 facts because 4 is double 2).
Whole Group Activity or Small Group Activity: Give each student a hundreds chart. Have the students circle all of the multiples of 2. Then have the students X all of the multiples of 4. Discuss observations. (All multiples of 4 are multiples of 2 bc to get a multiple of 4 you have to multiply a number by 2 twice).
Small Group Reteach Activity: Give each student the Shaded Multiplication Table handout. Have them record the multiples of 2 and 4. Discuss observations about multiples. (x2 facts count by 2, x4 facts count by 4, 2 facts are even numbers, 4 facts are even numbers by every other even number, 4 facts are twice as much as the corresponding 2 facts)
Real World Connection: Think of things that come in fours (4 legs on a table, 4 wheels on a wagon, 4 horseshoes on a horse, 4 feet on a mouse)
o Ask the following questions: How many legs are on 5 tables? How many wheels are on 8 wagons? How many horseshoes are on 6 horses?
Games: Fact Fish, Crisscross Facts, and Multi-Models (use handout)
X6 Multiplication:
Big Idea: In multiplication, if we double the number of sets or double the size of each set, the product will double. (2x3=6 , 4x3=12) (3x4=12, 6x4=24)
Read book: Snowflake Bentley by Jacqueline Briggs Martin
o Ask students to listen for facts about snowflakes (how are they alike and different)
o Emphasize facts about snowflakes while reading: Most snowflakes have 6 branches (a few have 3 branches), no snowflake design was ever repeated (no 2 snowflakes look exactly alike)
o Ask problem: Every snowflake is different however each snowflake has 6 points. What is the total number of points Willie would count if 7 snowflakes fell on his tray? (7x6=42) Ask the students to solve it without counting every point (use skip counting or drawing a picture)
o Have students complete How Many Points handout
Partner Activity: Have students use hexagons and triangles to create x6 and x3 equations using the handout Multiplication Shapes. (Discuss what you notice about the products of 3 and 6, how could knowing the products of 3 help you with the products of 6)
o Extension: Determine the total number of sides for the following sets: 3 hexagons and 3 pentagons, 5 hexagons and 5 pentagons, 7 hexagons and 7 pentagons (what do you notice about x5 facts and x6 facts from the shapes)
Reteach Small Group: Complete There’s Always Another Way handout (use linking cubes to break apart factors and make the connection between x5 and x6 multiplication)
Small Group: Marching Ants and Guitar Strings handout (use a table to recognize patterns in x6 facts)
Warm Up Activity/ Small Group: Shaded Multiplication Table handout (compare x2 and x6 facts). Ask students what the connection is between x2 and x6 facts and what you notice about the products)
Reteach Small Group: Play game Circles of Dots
Games: Capture and Multiplication Bingo
X9 Multiplication:
Big Ideas:
o Products of x9 facts are 1 group less than products of x10 facts (9x3 equals 1 group of 3 less than 10x3 or 30-3)
o Because of all of the math facts we have mastered thus far, we only have 3 facts to memorize: 9x7, 9x8, and 9x9
Read book: Cloudy with a Chance of Meatballs by Judi Barrett
o Give each student a paper plate. Divide into partners. On one paper plate draw 9 meatballs and on the other draw 10 meatballs. Have partners decide how many total meatballs would be collected if 7 people each caught 9 meatballs and 7 people each caught 10 meatballs. Invite 7 students up to the front of the class to hold their plates on their head for visual to model each equation. (Anytime we multiply a number by 9, it will be that number less than multiplying by 10)
o Have partners continue this exploration with the number 8 this time instead of 7 for both 9 meatballs and 10 meatballs. Did our rule still work?
o Complete the Plates of Meatballs activity on Promethean board as a class (ask students to make connections between x9 and x10 products)
Warm Up Activity: Observing Patterns handout. Complete the facts to record products for x9 facts from 1 to 10. Look for any patterns and record on anchor chart.
o Multiples of 9 are counting by 9 or adding 9 each time
o The digits in the multiples of 9 all add up to nine (18, 1+8=9)
o The digits are reversed for some multiples of 9 (45 and 54, 36 and 63, 27 and 72)
o As the tens digits increase by one ten, the ones digits decrease by 1 (09, 18, 27, 36)
o The tens digit of the product is always 1 less than the factor being multiplied by 9 (9x8 is 72 and 7 is one less than the factor of 8; 9x5 is 45 and 4 is 1 less than the factor of 5)
o To begin the product with a tens digit that is 1 less than the factor (9x7 = 6___) and then knowing that the products always equal 9 you could do 6+3=9, so the answer is 63.
Small Group Reteach: Exploring x9 facts through linking cubes (Have students create 6 chains with 10 cubes in each. Find the product. Then have them remove 1 cube from each chain. Find the product. Compare products.
Warm Up/ Exit Ticket Problem (after learning about patterns in x9): A friend is having a difficult time remembering x9 multiplication facts. What would you tell him to help him with these facts?
Games: Condition and Nine Cross
X8 Multiplication:
Big Ideas:
o Multiplication by 8 is double multiplication by 4.
o Multiples of 8 are always even numbers and also multiples of 2 and multiples of 4.
Read book: Snowmen at Night by Caralyn Buehner
o During reading have the students estimate how many snowmen there are. Discuss how they reached their estimation.
o After reading draw a snowman on the board with 2 pine cones for eyes, 4 rocks for buttons, and 8 pieces of coal for a mouth. Tell the students that one snowman isn’t enough, so we are going to “build” more snowmen! Divide students into 9 groups and have each group pick a card from 2-10 deck. Have the students solve the problem:
-Your number tells how many snowmen we should make. Draw a picture of the snowmen and write three multiplication equations to show how many pinecones, rocks, and pieces of coal we need in order to make that many snowmen.
o Record results on A Closer Look at Snowmen Facts handout and discuss patterns
Small Group Reteach Activity: Visualizing Doubles. Give each student a set of counters. Have them choose 1 counter and double it (2). Double it again (4). Double it again (8). Discuss observations and make a chart by folding a piece of paper into 4 sections. Write A NUMBER in the first column, DOUBLE IT in the second column, DOUBLE IT AGAIN in the third column, and DOUBLE IT AGAIN in the fourth column. Fill in the chart.
Small Group Reteach Activity: Compare area models for 2x8 and 4x8. Cut out of grid paper and place on top of each other. Recognize that the products are doubled.
Games: Missing Numbers and Crazy Eight
X7 Multiplication:
Big Idea:
o The only new fact we are learning (that you haven’t already memorized) is 7x7
Read book: Thunder Cake by Patricia Polacco
o Have the students count with you as the little girl in the story counts
o Share the thunder cake recipe from the back of the book with the students. It doesn’t say how many strawberries to use. Go back in the book to count the strawberries. Seven strawberries decorate the cake.
o Ask the following question: How many strawberries would Grandma need to decorate 7 thunder cakes? (Solve on the handout Thunder Cake)
Whole Group Activity: Square Numbers (Using grid paper, draw the array for 2x2, 3x3, 4x4, 5x5, 6x6, 7x7, 8x8. What do you notice about each array (They make a square). Some other square numbers would be 9x9, 10x10, etc.
Warm Up Riddle: I am a multiple of 7. I am greater than 20 but less than 40. One of my digits is 5. What am I?
Warm Up Riddle: I am a multiple of 7. I am greater than 14. My other factor is 3. What am I?
Warm Up Riddle: I am a multiple of 7. Both of my numbers are even. I am less than 40. What am I?
Games: Spinning Facts, Target 70, and Math Facts Face Off
More Multiplication & Division Resources:
More Multiplication & Division Resources:
- Developing Multiplication Fact Fluency (3-5): Order recommended by John Van de Walle.
- Multiplication Table (3-5): Interactive table to practice their multiplication facts.
- Factor Tic-Tac-Toe (4-5): An on-line version of the paper/pencil game that focuses on factors. Kids need a partner to play against.
- Multiplication Concentration (3-5): Mixed practice of multiplication facts through 12.
Mixed Practice:
Mixed Practice:
- Math Millionaire (3-5): Based on the t.v. show, kids answer math questions to see how much money they can win!
Let's Reflect:
Let's Reflect:
1. With your current assessments, what percentage of emphasis might you assign to each of the four categories above? Is this balance what you would like it to be? If not, how might you alter your assessments to equitably address the four areas of fluency?
2. As you reflect on your students’ basic facts fluency, what would you like to know more about? Which of the assessment tools from the article might help you gain this knowledge? How might you use that assessment tool?
3. Discuss your reactions to the issue of timed tests. What might you do as a teacher or leader to avoid potential negative impacts of timed tests?
4. How might we help parents better understand fluency and help their children in the areas of flexibility and selecting appropriate strategies? How might you communicate the purpose of alternative assessment tools for basic facts with your students, parents, and school leadership?
ARticles to Support fact fluency:
ARticles to Support fact fluency:
- "Fluency Without Fear" By: Jo Boaler, Professor of Mathematics Education, co-founder youcubed
- "Assessing Basic Fact Fluency" By: Gina Kling and Jennifer M. Bay-Williams
- "Three Steps to Mastering Multiplication Facts" By: Gina Kling and Jennifer M. Bay-Williams
- "Mastering the Basic Math Facts in Addition and Subtraction" By: Susan O'Connell, Quality Teacher Development, John SanGiovanni
- "Mastering the Basic Math Facts in Multiplication and Division" By: Susan O'Connell, Quality Teacher Development, John SanGiovanni
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