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January 18, 2012

Total Recall: Helping Our Students Memorize Multiplication Facts

By Alycia Zimmerman
Grades 1–2, 3–5

    As a young student, I railed against memorization and my “Draconian, pseudo-fascist teachers” who demanded that I spit out the answer for “6 times 7” like a soulless automaton. (I was also given to overblown metaphors and florid tirades.) As a teacher, some small part of me still cringes every year when it is time to encourage my 3rd graders to memorize the multiplication/division facts. “Long live the calculator!” the former freedom fighter in me cries out. 

    A multiplication display with arrays made with beads.

     

    Given my rocky relationship with memorizing math facts, I am always on the hunt for effective and motivating ways to help my students achieve “automaticity.” And now that the Common Core Standards explicitly require 3rd grade students to “fluently multiply and divide within 100,” I am particularly focused on effective strategies to help my students meet this standard (Standard 3.OA.7). Here are some strategies that simultaneously help my students’ math memories and mollify my conscience.

     

     

     

     

     

    Stage 1: Understanding Equal Groups — The VERY Important Precursor

    Things that come in groups chartFirst off, my students must internalize what multiplication means long before they begin to memorize the multiplication facts. They must have an intuitive understanding about the reciprocal relationship between multiplication and division, and equal groups and equal shares, and the connection with repeated addition and skip counting.

    My students live with these ideas for months, exploring multiplication concepts with arrays, area models, pictorial representations, and hands-on experiences. Only when I am confident that all of my students truly understand why and how multiplication and division work do I begin to emphasize memorization. Just as in reading, accuracy and fluency without comprehension is useless.

     

     

    Area Model Chart Array Model Chart

    Equal Groups Chart  Number Line Model

     

    Stage 2: "Figure-It-Out" Tricks

    Now that my students have internalized what multiplication and division are all about, it’s time to begin to teach them some “tricks” to quickly figure out multiplication/division facts, and to help them develop their own tricks. In fact, these are not tricks at all, but ways to analyze multiplication patterns, apply partial-product principals, and use “landmark numbers” to quickly use what they know to figure out unknown facts. At this stage, I want to help my students realize that they can find the products for multiplication problems that they haven’t yet memorized.

    I begin by having my students fill out blank multiplication tables several times a week. The first few times, I have the students highlight the “easy” rows and columns such as the 1s, 2s, 5s and 10s. They fill in these parts first, and then realize that they can use those landmark numbers to help them figure out the remaining facts. For example, I might encourage a student who knows 2 x 6 = 12 to add another six to figure out 3 x 6. My students share their strategies, and we chart these tricks. Then I have my students color in their completed times table charts with crayons to create “quilt patterns” by shading in multiples. The multiples-of-4 quilt is always a favorite.

    Best of Times Book CoverI also read our “multiplication bible,” The Best of Times by Greg Tang, with my class, and we practice his strategies for figuring out multiplication facts. For example, Tang writes,

    Four is very fast to do

    when you multiply by 2.

    Here's a little good advice:

    Please just always double twice!

    Through Greg Tang’s Web site, you can have your students play the Break Apart game focusing on multiplication. (The nines level is free.) This further reinforces partial-products thinking strategies, or “Breaking It Up,” as we call it in my class.

    Example: 9 x 7 = ?

    Breaking It Up: “I know that 10 times 7 is 70, so one 7 less than 70 is 63.”

     

    Stage 3: Approaching Automaticity

    At last, we get to memorization! I only begin to focus on memorizing the multiplication facts after my students are already solving multiplication word problems and are using a wide range of strategies to figure out the necessary multiplication facts, from drawing arrays to using their own “break it up” strategies. I point out to my students that automaticity is far more efficient, and that the speed inherent with automatic fact recall will help them to solve multi-digit multiplication and division problems efficiently. Memorization is about speed, efficiency, and freeing up brain space for other math thinking.

     

    Recall Tip 1: Timez Attack

    In my opinion, the free downloadable computer game Timez Attack is the best computer program to help students learn the multiplication facts. This first-person adventure game feels like a “real” video game, and it uses built-in assessments to advance students at their own pace. My students love this game, and I’ve seen it really work with them.

     

    Recall Tip 2: Totally “Unsquare” Flashcards

    Multiplication Fact TriangleRather than using traditional flashcards that teach multiplication or division facts separately, I prefer “fact triangles,” three-sided cards that have an entire fact family on the three corners of the triangle. My students enjoy “playing” with their fact triangles by covering up one corner and then having a friend figure out the hidden number. This helps my students internalize the relationship between multiplication and division. A free printable set of all of the multiplication fact triangles is available online.

     

    Recall Tip 3: Play Kakooma

    Greg Tang of The Best of Times didn’t just stop with an awesome book. He also developed a fun mental math game that can be played online or with paper and pencil. Kakooma challenges students to use multiplication to solve a series of deceptively difficult puzzles. The rules are easy: just find the one number in the diagram that is the product of two others. Like all great puzzles, however, looks can be deceiving. Often, the answer is right in front of you, yet your brain won’t let you see it! This is another game that my students beg to play. You can download a sample packet of Kakooma multiplication puzzles from Greg Tang’s Web site.

     

    Multiplication Masters Board

    Tip 4: Multiplication Masters

    I use a management system to help my students keep track of which multiplication facts they have mastered, and which they still have to study. I’ve named the system “Multiplication Masters,” and I explain to my students that any student who makes it into the “Twelves Club” by the end of the school year will be considered a Multiplication Master. At the end of June each year, I hold a party after school for the Multiplication Masters. The students receive certificates and decorate T-shirts with multiplication facts.

    Twice a week, my students take three-minute multiplication quizzes with twenty questions. (Download my Multiplication Masters quizzes.) Each student takes the quiz on the level above his current club. So, a student who is currently in the Threes Club will take the fours quiz. I photocopy the quizzes and cut them into strips at the beginning of the year, and I store dozens of copies of each quiz in an accordion file folder sorted by level.

    When a student earns 100% on a quiz, he moves into that club, and then begins studying for the next higher level. Once a student becomes a Multiplication Master, I challenge my students to see how many problems they can solve in three minutes, working up in increments of ten.

    Yes, this system is somewhat competitive, and yes, it does put some pressure on my students. Nonetheless, all of my students have seemed to enjoy the challenge, and I always emphasize that each student should focus on how high she can soar individually, without comparing herself to her classmates. Three minutes is a fairly generous time allotment for twenty questions, so I find that students who are comfortable using other methods can also get 100% on their quizzes, without having true automaticity.

    To make my Multiplication Masters board, I used a rectangular piece of foam board and gridded out sections using colored masking tape. I attached bits of Velcro across the entire board, and then I labeled each grid section. Finally, I printed each student’s name onto card stock, which I laminated and backed with Velcro.

     

     

     

    What are your favorite tips and tricks to help your students memorize the multiplication tables? Please share your favorite games, Web sites, and activities below in the comments section . . . I know I’m always looking for new ideas!

    As a young student, I railed against memorization and my “Draconian, pseudo-fascist teachers” who demanded that I spit out the answer for “6 times 7” like a soulless automaton. (I was also given to overblown metaphors and florid tirades.) As a teacher, some small part of me still cringes every year when it is time to encourage my 3rd graders to memorize the multiplication/division facts. “Long live the calculator!” the former freedom fighter in me cries out. 

    A multiplication display with arrays made with beads.

     

    Given my rocky relationship with memorizing math facts, I am always on the hunt for effective and motivating ways to help my students achieve “automaticity.” And now that the Common Core Standards explicitly require 3rd grade students to “fluently multiply and divide within 100,” I am particularly focused on effective strategies to help my students meet this standard (Standard 3.OA.7). Here are some strategies that simultaneously help my students’ math memories and mollify my conscience.

     

     

     

     

     

    Stage 1: Understanding Equal Groups — The VERY Important Precursor

    Things that come in groups chartFirst off, my students must internalize what multiplication means long before they begin to memorize the multiplication facts. They must have an intuitive understanding about the reciprocal relationship between multiplication and division, and equal groups and equal shares, and the connection with repeated addition and skip counting.

    My students live with these ideas for months, exploring multiplication concepts with arrays, area models, pictorial representations, and hands-on experiences. Only when I am confident that all of my students truly understand why and how multiplication and division work do I begin to emphasize memorization. Just as in reading, accuracy and fluency without comprehension is useless.

     

     

    Area Model Chart Array Model Chart

    Equal Groups Chart  Number Line Model

     

    Stage 2: "Figure-It-Out" Tricks

    Now that my students have internalized what multiplication and division are all about, it’s time to begin to teach them some “tricks” to quickly figure out multiplication/division facts, and to help them develop their own tricks. In fact, these are not tricks at all, but ways to analyze multiplication patterns, apply partial-product principals, and use “landmark numbers” to quickly use what they know to figure out unknown facts. At this stage, I want to help my students realize that they can find the products for multiplication problems that they haven’t yet memorized.

    I begin by having my students fill out blank multiplication tables several times a week. The first few times, I have the students highlight the “easy” rows and columns such as the 1s, 2s, 5s and 10s. They fill in these parts first, and then realize that they can use those landmark numbers to help them figure out the remaining facts. For example, I might encourage a student who knows 2 x 6 = 12 to add another six to figure out 3 x 6. My students share their strategies, and we chart these tricks. Then I have my students color in their completed times table charts with crayons to create “quilt patterns” by shading in multiples. The multiples-of-4 quilt is always a favorite.

    Best of Times Book CoverI also read our “multiplication bible,” The Best of Times by Greg Tang, with my class, and we practice his strategies for figuring out multiplication facts. For example, Tang writes,

    Four is very fast to do

    when you multiply by 2.

    Here's a little good advice:

    Please just always double twice!

    Through Greg Tang’s Web site, you can have your students play the Break Apart game focusing on multiplication. (The nines level is free.) This further reinforces partial-products thinking strategies, or “Breaking It Up,” as we call it in my class.

    Example: 9 x 7 = ?

    Breaking It Up: “I know that 10 times 7 is 70, so one 7 less than 70 is 63.”

     

    Stage 3: Approaching Automaticity

    At last, we get to memorization! I only begin to focus on memorizing the multiplication facts after my students are already solving multiplication word problems and are using a wide range of strategies to figure out the necessary multiplication facts, from drawing arrays to using their own “break it up” strategies. I point out to my students that automaticity is far more efficient, and that the speed inherent with automatic fact recall will help them to solve multi-digit multiplication and division problems efficiently. Memorization is about speed, efficiency, and freeing up brain space for other math thinking.

     

    Recall Tip 1: Timez Attack

    In my opinion, the free downloadable computer game Timez Attack is the best computer program to help students learn the multiplication facts. This first-person adventure game feels like a “real” video game, and it uses built-in assessments to advance students at their own pace. My students love this game, and I’ve seen it really work with them.

     

    Recall Tip 2: Totally “Unsquare” Flashcards

    Multiplication Fact TriangleRather than using traditional flashcards that teach multiplication or division facts separately, I prefer “fact triangles,” three-sided cards that have an entire fact family on the three corners of the triangle. My students enjoy “playing” with their fact triangles by covering up one corner and then having a friend figure out the hidden number. This helps my students internalize the relationship between multiplication and division. A free printable set of all of the multiplication fact triangles is available online.

     

    Recall Tip 3: Play Kakooma

    Greg Tang of The Best of Times didn’t just stop with an awesome book. He also developed a fun mental math game that can be played online or with paper and pencil. Kakooma challenges students to use multiplication to solve a series of deceptively difficult puzzles. The rules are easy: just find the one number in the diagram that is the product of two others. Like all great puzzles, however, looks can be deceiving. Often, the answer is right in front of you, yet your brain won’t let you see it! This is another game that my students beg to play. You can download a sample packet of Kakooma multiplication puzzles from Greg Tang’s Web site.

     

    Multiplication Masters Board

    Tip 4: Multiplication Masters

    I use a management system to help my students keep track of which multiplication facts they have mastered, and which they still have to study. I’ve named the system “Multiplication Masters,” and I explain to my students that any student who makes it into the “Twelves Club” by the end of the school year will be considered a Multiplication Master. At the end of June each year, I hold a party after school for the Multiplication Masters. The students receive certificates and decorate T-shirts with multiplication facts.

    Twice a week, my students take three-minute multiplication quizzes with twenty questions. (Download my Multiplication Masters quizzes.) Each student takes the quiz on the level above his current club. So, a student who is currently in the Threes Club will take the fours quiz. I photocopy the quizzes and cut them into strips at the beginning of the year, and I store dozens of copies of each quiz in an accordion file folder sorted by level.

    When a student earns 100% on a quiz, he moves into that club, and then begins studying for the next higher level. Once a student becomes a Multiplication Master, I challenge my students to see how many problems they can solve in three minutes, working up in increments of ten.

    Yes, this system is somewhat competitive, and yes, it does put some pressure on my students. Nonetheless, all of my students have seemed to enjoy the challenge, and I always emphasize that each student should focus on how high she can soar individually, without comparing herself to her classmates. Three minutes is a fairly generous time allotment for twenty questions, so I find that students who are comfortable using other methods can also get 100% on their quizzes, without having true automaticity.

    To make my Multiplication Masters board, I used a rectangular piece of foam board and gridded out sections using colored masking tape. I attached bits of Velcro across the entire board, and then I labeled each grid section. Finally, I printed each student’s name onto card stock, which I laminated and backed with Velcro.

     

     

     

    What are your favorite tips and tricks to help your students memorize the multiplication tables? Please share your favorite games, Web sites, and activities below in the comments section . . . I know I’m always looking for new ideas!

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