Embedded Files
Start With Facts Easy to Learn By Patterns
Start With Facts Easy to Learn By Patterns
Why?
Why?
“The brain is a pattern seeker, and when it finds a pattern, it can use it to predict the future.”
~ Neil deGrasse Tyson
Learning is quicker with patterns.
Once the pattern is learned, the learner can extend the pattern beyond the basic facts quickly and easily – predicting the future, as Neil deGrasse Tyson says above.
Each times table learned easily by patterns builds feelings of success – and that provides the confidence and motivation to tackle other multiplication tables without easy patterns.
All the facts learned by patterns are also facts learners will encounter in the reverse order in other tables. Since they have already learned the answer with factors in the reverse order by patterns, that leaves far fewer to learn by other methods, such as rote memory.
What tool makes times tables with patterns quick and easy to learn?
What tool makes times tables with patterns quick and easy to learn?
How can students spot the patterns so they can learn them quickly? By using times tables color-coded to focus on patterns, along with teacher guidance, students learn those kinds of tables in quick order.
What tables have these easy patterns? What should be taught first?
What tables have these easy patterns? What should be taught first?
If the 0s and 1s have been mastered, next is the 10s table. Then the 100s and the 1000s while you are at it. Imagine students' delight once they have easily mastered all three.
Some learners will figure out the patterns without prompting.
Others may need the teacher to read each fact aloud, verbally stressing the other factor along with strategic use of guiding questions.
Once students grasp the pattern, ask them to apply it to problems following that pattern beyond ones they have been taught.
Focus on one table at a time until each seems to understand it.
For example, ask what 10 x 15 would be. What would it start with? What would it need at the end? So what is 10 x 15?
Encourage students to write it down, so it's visual too, not just auditory. Young students often need to see it written before they can easily say the number aloud.
Whenever possible have learners discover the pattern and try to put the rule into their own words.
100 times any factor, the answer starts with that other factor and ends wtih two zeros, while 1000 times any factor starts with the factor and ends with three zeros.
Encourage students to amaze their parents and others by teaching the pattern to them. Teaching others and putting it into their own words solidifies understanding.
And then?
The 11s are next.
The 11s are next.
Beyond the Obvious
Beyond the Obvious
The 11s pattern of 11 times a single-digit factor produces a double of that other factor. The line across the table separates the ones that use the 11s pattern from those that need to be solved a different way.
Notice that in 11 x 10 the 10 is underlined and colors in the answer are used to remind students that they learned the 10s pattern already. Underlined factors have or will be learned by patterns. Guide students if they need coaching to switch the digit order to use the 10s pattern on that problem.
11 x 11 and 11 x 12 are not easy to learn by patterns — but there's a fun trick to find the answers (and most of any 11s table times any 2-digit factor, until larger ones that require carrying. Slide the second factor's digits apart so the first digit moves to the hundreds place. Add the first and second digit to find the answer that belongs in the tens place. So 11 x 11 takes the second factor, slide the 1s apart to show 1__1, add the 1 + 1 which equals 2 and put 2 in the blank for 121. Likewise 11 x 12 becomes 11 x 1__2, add the 1 + 2 to get 3 and put it in the middle. 132 is the correct answer.
Cool, right?
No need to explain why it works, but do know that if adding the two digits together requires carrying, getting the right answer becomes trickier.
But students will be learning to do 2-digit multiplication using the stacked algorithm and can solve those more easily that way.
Follow on with the 5s.
Follow on with the 5s.
New Concepts for the 5s
New Concepts for the 5s
Counting by 5s is taught earlier than multiplication facts. The limited information from counting by 5s can be expanded on by using the color-coded version. Discovering the pattern in the ones place, alternating 0s and 5s is a small step. Here's the bigger picture.
Look at the second factors that are red. Every other second factor shown here in red has what in common? Each is even. Focusing on 5 x an even factor, what can be noticed about the relationship between that even second factor?
Pair the visual with asking students to listen for the pattern. Start from the highest ones shown and move towards the smallest. So 5 x 12's answer begins with what? Yes, 6. Then 5 x 10's answer begins with 5, 5 x 8's begins with 4, 5 x 6's begins with 3, 5 x 4's begins with 2, and 5 x 2 begins with 1. Emphasize the parts to notice by stressing the words.
What pattern did they notice? (Often it may take listening hearing them a second time to be able to see that each answer starts with half of the even second factor, and even then they might not think of the word half.) What does the ones place have in the ones place in answers for every one of the 5 x an even number facts? Each begins with half of the even second factor and ends in 0.
Practice that pattern (5 x an even number) and guide them into finding the beginning (half of the even number) and remembering that zero goes on the end.
Extend the pattern to ones beyond those shown such as 5 x 14 and 5 x 20. The former starts with 7, ends with 0 so 70, and the latter starts with 10 and ends with 0, so 100. Students love being able to apply the pattern to predict the correct answer beyond what was taught.
What about 5 x an odd number? They can notice it always has a 5 in the ones place, but it's trickier to find the beginning number for the answer. Instead, students use the even number just less than the odd number to find how it starts.Then they can add 5. So for 5 x 9, use 5 x 8 = 40 and add 5. (Assuming at least that they have not yet learned the 9s pattern. Then either pattern will work.)
Next are 9s.
Next are 9s.
9s patterns may not jump out at them.
9s patterns may not jump out at them.
Some patterns will stand out, like the 10s place going up by one and the ones place going down by one, but other useful patterns for finding the correct answer almost certainly won't.
The pattern that lets them predict the answer in the tens place digit comes more easily when paired with auditory and visual combined. The teacher would say the problem aloud stressing what should be focused on:
9 x 10's answer begins with 9, the answer to 9 x 9 begins with 8, to 9 x 8 begins with 7, 9 x 7's answer begins with 6, 9 x 6's answer begins with 5, the answer to 9 x 5 begins with 4, 9 x 4 begins with 3, 9 x 3's answer begins with 2, and 9 x 2's answer begins with 1.
Ask students, "What's the relationship between the second factor and the first digit of the answer? What did you hear and notice" Each answer in the 9s table begins one less than the other factor.
A cool pattern – but there's more!
Ask students what those two digits in each of the answers will add to. (Each adds up to 9.)
If they can predict by the 9s pattern what the answer starts with (the tens place digit), they can figure out what goes with it to make 9. Particularly in the early grades, many students learning to find those answers will need a tactile-kinesthetic way to find that second digit because they may not have a solid grasp of how to find a difference. For those learners, encourage them to say the first digit of the answer aloud and "stick" it in their head (touching their forehead). Then count on to 9 from that first digit on their fingers. The number of fingers it took is the numeral belonging in the ones place.
The final piece is to look at what pairs must go together in the answer for them each to add to 9. So if it starts with 8, it has to have a 1, and if it starts with 1, it has to have an 8. And so forth.
The line drawn after 9 x 10 means the last two facts don't fit the 9s pattern. In 9 x 11, students can use the 11s table's patterns to find the answer since in multiplication the order of the factors doesn't matter to the answer.
That leaves one fact in that table to learn, but it can wait for another tool, multiplication breakdown [add page link], to use to solve 9 x 12.
"Our brains are built to find patterns and make connections between things.” ~ Daniel H. Pink
Don't forget practice.
Don't forget practice.
Practice each table learned using Card Deck Flash Cards, described here. [add page link] It gives learners a lot of practice in a short time in a game-like way and it provides a visual reinforcement because it shows how many facts they've mastered.
Summary: Using patterns (emphasized through color coding) is the preferred way to learn the 5s, 9s, 10s, and 11s. It's amazing how fast students will catch on. Mastering those will give them some of each of the other tables since using the same digits in a different order doesn't change the answer.
Our brains are wired to learn by patterns. They're attracted to them and find that way of learning a source of wonder, fun, and even joy.
Starting with those will be intriguing and add confidence and motivation to learn the other multiplication facts.
Check out the next tool for learning multiplication facts, Multiplication Breakdown. [add page link]
Free download of multiplication facts 2-12 (including 100s and 1000s) focused on the patterns through use of color and other clues.
All resources on this website are part of the Creative Commons and free to use.
Page updated
Google Sites
Report abuse