If you remember, we are in the middle of an experiment in mental math. We are using the world’s oldest interactive game — conversation — to explore multiplication patterns while memorizing as little as possible. Talk through these patterns with your student. Work many, many, many oral math problems together. Discuss the different ways you can find each answer, and notice how the number patterns connect to each other.
What else is relatively simple? Does your student know the doubles? Doubles are often considered easy, because children do so much counting and addition with numbers less than 20. Even if your child finds the doubles tricky, a little focused practice should fix these facts in mind.
- Here is your first memory task: Learn the doubles!
Practice doubling big numbers, too. Use silly numbers to help:
= double 3 tens + double 8
= double 5 tens + double 6
Go back and forth, inventing double-puzzles for each other:
= double 3 + double ½
= 6 + 1
= (double 4 tens + double 7) thousand
= “eighty-fourteen” thousand
Review Game: Once Through the Deck
The best way to practice the math facts is through the give-and-take of conversation, orally quizzing each other and talking about how you might figure the answers out. But occasionally you may want a simple, solitaire method for review. Here’s how:
- Shuffle a deck of math cards and place it face down on the table in front of you.
- Flip the cards face up, one at a time.
- For each card, say (out loud) the product of that number times the number you want to practice.
- Don’t say the whole equation, just the answer.
- Go through the deck as fast as you can.
- But don’t try to go so fast that you have to guess! If you are not sure of the answer, stop and figure it out.
Brian at The Math Mojo Chronicles demonstrates the game in this video, which my daughter so thoroughly enjoyed that she immediately ran to find a deck of cards and practiced her times-4 facts. (It’s funny, sometimes, what will catch a child’s interest.)
You can use Once Through the Deck as a final check that your student knows the fact family well enough to mark it on your chart. Remember to mark both the row and the column!
The Times-4 Family
Notice that the answers in the times-4 row are exactly double the answers in the times-2 row. Can your child see why that makes sense? If you have two of something, and you replicate two more of it, then you would have four of that thing, whether it is minions or cookies or numbers.
This means you do not need to memorize the times-4 facts. Just double the number to get the times-2 answer, and then double it again. For example, would mean seven doubled, which is 14, and then that answer doubled again:
= 7 2 2
= 14 2
Practice double-doubling a bunch of numbers. Can you use the double-double trick to figure out something like ?
= 53 2 2
= 106 2
This may take a little more time to practice — but that is okay! Quiz each other with unusual numbers, using double-doubling to get the answer. Test yourself with Once Through the Deck, and when you are ready, mark the chart.
The Times-8 Family
In the same way that times-4 was the double of times-2, it makes sense that times-8 is the double of times-4. If you have four of something, and you replicate four more of it, then you will have eight of the thing in all. It doesn’t matter whether the thing is books or aliens or numbers. Even fractions: If you have four 1/16 size slices of pizza, and you take four more pieces, then you will have 8/16 of the pizza.
If you need to calculate something times eight, you can double the something, then double again — that makes four times — then double it once more for your final answer:
= 6 2 2 2
= 12 2 2
= 24 2
I find it helpful to count on three fingers, to make sure I don’t forget any of the doublings. Remember to experiment with big numbers, too. Can you double-double-double to figure out ?
= 132 2 2 2
= 264 2 2
= 528 2
It may take a few days or even weeks of practice before you and your student feel comfortable with these. Take all the time you need, and when you both are able to mentally double-double-double almost any number the other can pose, mark off the times-8 column and row on the chart.
The Times-5 Family
Your daughter can probably count by fives, but many children get confused when trying to skip-count large multiplication problems. A more reliable number pattern for times-5 calculations uses the doubles in reverse. Two fives make ten, so any even number of fives will make exactly half that number of tens:
6 fives = 3 tens
18 fives = 9 tens
24 fives = 12 tens
450 fives = 225 tens
All the odd numbers times five will come out “somethingty-five,” and you can predict what the “somethingty” will be by looking at the next lower even number.
7 fives = (6 + 1) fives = 3 tens + 5
25 fives = (24 + 1) fives = 12 tens + 5
109 fives = (108 + 1) fives = 54 tens + 5
Practice intensively on these until you can both get the answer right every time. Test yourself with a round of Once Through the Deck, and then mark them off.
Wow! Look back and see how much you have learned. You started with 144 facts, and you have narrowed it down to only 21 — and all you had to memorize so far was the doubles! Your child could probably memorize the last 21 facts without too much trouble, but let’s see if we can find a few more patterns to make them easier.