For the warm up today, I asked students to draw sets of 10 x's around the perimeter of their white boards. We then drew lines to create equal groups of 9. As we created these groups, we wrote the total number of x's as we went. (see image).
Students, we just created the multiples of 9. Please look at the numbers you have written next to your representations and talk with your partner about patterns you see.
List or discuss the patterns the students share at this point. Listen for place value references and relationships to multiplying by 10. This will be important to link to in the next session. Also, listen to hear how students take apart the patterns and describe them. It is important to make sure the students see the pattern related to 10. This may have to be facilitated through questioning.
Boys and girls, I think I see something. Listen to how I think this through. The equation is 9x4=n. I am really good with my 10's facts, so I am going to use them to help me.
I will think, "10x4=40. But, when I use ten, it means I have one extra group of 4, so if I subtract 4 from 40, I have 36.
Now I know 9x4=36. Let's try another one to test my idea. Great.
Now, will you test another one with your partner?"
Plan to spend a good amount of time discussing patterns and allowing the students to explore all the patterns they can find. Guide them with leading questions.
Do you see any pattern in the ten's place or the one's place? What happens to the one's place when the ten's place changes? Do you see anything interesting in the product digits?
After sharing all of the patterns they see, I introduce two tricks, or strategies, that I believe are very helpful. One is a trick to find the product and the other is a solid way to check thinking after problem solving.
The first one I point out is the fact that the digits in the product of the 9's facts up to 9x10 always add up to 9.
Mathematicians, please list the first 10 multiples of 9 on your white board. When you are done, look at those numbers and see if you find a rule in the digits. You might "comfortably struggle", don't worry.
If no one sees it, point out the first two and then have the students test the others.
So, look at this! 1x9=9. 2x9=18. If I look at the 18 and add those digits, they equal 9. 3x9=27 and 2+7=9. Will you work with your partner to test the other multiples? Great, it works for all of them. How can this rule help you anytime you are multiplying by 9?
Many of the students respond that if they solve a nines fact and the digits in the product don't equal 9, then they know they made a mistake. That is what we are looking for. What surprised me during the conversation is a student that explained his thinking based on the digits in a place value way.
"I know that the tens place will always be one less than the number we multiply 9 by. Then all I have to do is figure out the number to add to get to 9. Like, 5x9=45. 4 is one less than 5 and it goes in the 10's place, then I just need to add 5 to it."
That is the good stuff!
Next, I like to teach the finger trick to the students. Most seem to get it quickly and it takes the fear of "the high number 9" out of their minds. See the this link for examples.
Play this cute video to bring it all home!
After all of the discussions and exploration of patterns, it is important to have the children actively practice what they just discussed. I decided to have them play a game, rather than just do a math page of equations.
I have the children take out all of the cards 0-10 from the deck. Then they shuffle and lay the stack face down. On their turn, they flip the top card over and multiply it by 9. I have them recite the equation to their partner as part of their turn. The partner, must also do the equation on the calculator to check. If the partner is correct, she/he keeps the card. Otherwise it goes to the bottom of the deck.
Partners, in a moment you will go off to practice using some of the strategies and patterns we just learned about. You might want to use the finger trick if you don't already know the product. Some of you might use the pattern we learned about the digits in the product equaling the sum of 9 to check your answers. Pay attention to what you are using to help you, as we will share that out at our closing today.
Assemble the class in a gathering area to review the lesson.
Mathematicians, please clean up your cards and gather in a circle in our community area. I would like to go around the circle and hear what each of you did today to help you with your nine's multiplication. Then we will go around again and mention anything we found interesting or anything that might still be tricky.