This year, we are focusing on multiplication with my daughter Elizabeth. But I don’t want her to just memorize and regurgitate a table of facts. I want her to discover them, to understand what they mean in several ways, and be able to figure them out quickly and easily. As a math teacher, I know how important these concepts will be in understanding algebra and calculus and they are really cool and beautiful.
We started learning multiplication using Cuisinaire Rods, which are wooden or plastic rods of different unit lengths from 1-10, and her Math Inspirations curriculum presented a series of problems to her, which she solved by building each problem with the rods and counting. She discovered a pattern that she needs to count “a groups of b” to find “a x b.” She used this pattern to write and test a theorem on how to do it each time.
Once she finished this process, she asked for more lessons on learning the individual multiplication tables. We started at x1 and are working our way through x12.
Here are some resources I gathered to help make multiplication fun and engaging for her. Find printable versions of several of these activities on this google doc:
Draw products on graph paper. If we were multiplying 3×4, she would draw a rectangle on graph paper that is 3 squares by 4 squares, and then count how many squares are in the area of the rectangle. Then she would write the math sentence, 3 x 4 = 12. Writing this connects the picture with the number symbols. Greek mathematicians expressed products as rectangles, with the multipliers as the lengths of the sides and the product as the area the rectangle covers. We can do that too.
Cuisinaire Rods. These are something I purchased and they are my favorite math manipulative. We use them to build the multiplication tables when first introducing multiplication. We talk about how “multiplication” means “counting groups.” Then we build each times table using the cuisinaire rods. I like how they can see and touch and feel the groups. It gives them a sense that multiplication is just a fancy way of counting groups. They work very well when we start multiplying multiple digits, and I want my kids to be familiar with using them.
The cuisinaire rods that link together are distracting for kids, but I like how they have marks for the units. Kids want to build with them for a while. I plan a lesson where they just play with them before trying to use them for multiplication.
The wooden cuisinaire rods are great too and we own both types. I like how kids can’t link them together.
I love this video by ViHart which talks about how all of algebra is just fancy counting. It is almost revelatory. The part where she describes fancy counting in multiplication may help us understand multiplication more.
Discover Associative Property: a x b = b x a. This idea will help kids be faster at figuring out products because sometimes switching groups helps. I like to help my kids discover this by presenting a series of problems. See the google doc for a suggested activity.
Use a Waldorf multiplication wheel. This is a way to practice/drill the multiplication tables. If you were using this multiplication wheel for x2, you would write “x2” in the center. On the next tier, write the numbers 1-12. Then have your student multiply the center number by a number in the second ring and write the product in the outer ring. Sometimes you might put the numbers in order. Other times, you may scramble the order to add more challenge.
Another version of this is the multiplication speedway activity HERE.
Multiplication Spinning Wheel. This activity feels like a game to kids and is really fun, but is another way to drill multiplication tables. This version is from 3dinosaurs.com.
If you are working on the 5x table, you would ask your student to take a paperclip and place it in the center of the wheel. Spin the paperclip like you would a game spinner. (You can use a pencil to anchor the paper clip in the center, but don’t have to if it’s frustrating or distracting for the student.) If the spinner lands on “8,” then the child needs to find the product of 5 x 8.
Skip counting with cups. For this activity, we need counters and cups. If you are working on the 6x table, you would use 6 cups and 72 counters. We use dry, spray painted lima beans as counters, or small candies like chocolate chips. Have the kids place the cups in a row with the 6th cup closest them. They place one counter in each cup as they count, starting at the cup furthest from them. When they reach the cup in front of them, they say “SIX” loudly. Then they repeat placing counters until “TWELVE.” and continue until they run out of counters. (If you use a treat, they may eat the counters in their cup.)
Skip counting with a 100 number grid. For this activity, students use a hundreds chart like the one shown from donsteward.blogspot.com. If you were working on x7, have students count 7 numbers and color the square or place a counter on that square. Then repeat to cover 14, 21, etc. After they finish, ask them questions about the chart. Like, “Using this chart, what is 4 groups of 7?” etc.
Multiplication War. For this activity, we use the number cards from SKIPBO (1-12) or UNO (0-9). Deal all the cards to the players. Have each player flip two cards in the center and multiply them. Whoever has the highest product gets to keep all the cards in the center. If there is a tie, each tied player places 3 cards face down in their own pile in the center and flips two more cards and multiplies them. Whoever has the highest product gets to keep all the cards in the center, plus all the cards from the “war.”
Dice products. Roll two dice and find the product of the two numbers. Write down the math sentence.
Multiplication BINGO. Materials: A blank BINGO card for each player, 2 dice, and BINGO markers. We use lima beans, chocolate chips, or base 10 blocks as markers. Preparation: Fill in the BINGO card with numbers that are multiples of the numbers 1-6. (See example below). You can fill it in, or ask the student to do it. Play: One player rolls two dice and finds the product of the numbers on the dice. If that product is on their card, they cover that space with a marker. Then the next person takes a turn, rolling the dice and covering a number. When you get 5 spaces in a row, call out “BINGO.”
Variations: 1. One person rolls the dice and multiplies the entire game and everyone covers the number they call. 2. Each player has their own set of dice and rolls at the same time. 3. Use 12-sided dice. 4. Use a deck of cards to choose the numbers.
Discover Patterns. As you practice and build each times table, encourage kids to discover patterns about how to multiply. For example, while we worked on x2, my daughter said “I see! For this one it’s the same as adding the number to itself.”
I often ask, “What pattern do you notice?” Avoid telling them the pattern because the joy of discovery can be really awesome for them.
For x3, she discovered, “you add it to itself and then add it again.”
For x4, she said, “You double it, then double it again.”
For x5, she counts by 5s.
For x6, she multiplies the number by 5, and then add another group of that number. For example, for 7×6, she figures out 7×5 = 35, and then adds another group of 7 to get 42.
For x9, they use their fingers. For 9×4, they hold up both hands and start counting from the left. They count over 4 fingers and put that finger down. The number of fingers on the left is the tens digit (3) and the number of fingers on the right is the ones digit (6) of the product (36). This works for 1×9 to 9×9. They also sometimes figure out 10×4 and then subtract off a group of 4 to get 36.
For x10, they count by 10s.
For x11, they multiply the number by 10, then multiply the number by 1, and add the results together. for 4×11: 4×10 = 40. 4×1 = 4. 40 + 4 = 44. We build these with cuisinaire rods several times until they figure this out. Also, the 11 times tables are repeats of the numbers for 1-9. 11, 22, 33, 44, 55, 66, 77, 88, 99. That makes them easy to remember.
For x12, they use the same pattern as x11. They multiply the number by 10 and by 2, and add them together. 4×12, 4×10 = 40. 4×2 = 8, 4×12 = 48.
x7 and x8. Multiplication Songs/Chants. My kids have found patterns for the times tables for 1, 2, 3, 4, 5, 5, 9, 10, 11, and 12, but they struggle with x7 and x8.
We made up a song for x7 to “Mary Had a Little Lamb” for 7×1 to 7×9.
Seven, fourteen, twenty one, (Mary had a little lamb)
Twenty eight, thirty five, (Little lamb, little lamb)
Forty two, forty nine (Mary had a little lamb)
Fifty six, sixty three (Whose fleece was white as snow)
We also chant “Eight times seven is [insert grunty pro wrestler loud voice] fifty-six.” and “Eight times eight is [insert high sing songy voice] sixty four.”
The 8x table is covered by the other patterns.