Tuesday, June 14, 2016

7 x 4 x 7

One game that's gotten a lot of play this year is Salute!

The third graders play with multiplication and the second graders with addition.

     A few weeks ago I found myself sitting on a carpet in the back of a third grade classroom, in a school not my own, with a group of four students getting ready to play.

Me: Salute! needs 3 people to play.  Two to hold the cards with the factors up against their foreheads and one to deal the cards, call out the product, and judge who got the missing factor first.   How do you play with four?

Student: The teacher said that two of us act as the dealer, product caller, and judge.

Me: OK.  Let me see how that works.

     As I watched them play, something occurred to me.  What if 3 kids held factors up against their foreheads?  Would the student acting as the dealer be able to multiply all three?  Would any of the kids be able to find the factor facing out from their forehead?  I didn't know these kids, or their skill levels, other than a word from their teacher that they were a "middle group."   How would they react? The kids in my school no longer flinch when I volunteer them as subjects in my little experiments, but these kids are not used to having some crazy math guy come in and disrupt their lives.  Should I wait until I got back to my school?

Me: You guys want to try something different?

Them (suspiciously):  Um, yes?

     I explained my idea.  One student agreed to be the dealer for our trial run.  Here were the three cards she saw facing out from her classmates' foreheads:

Before reading on, multiply these numbers together in your head.  How did you do it?

   After the cards were dealt, it dawned on me that I was going to have to multiply the three numbers together!!   After all, assuming she could do the multiplication, how would I know whether or not she was correct?   I looked over at her, and saw her eyes get a little wider.  We were both going to have to do a little thinking.
   I took a breath and focused back on the cards.  First I multiplied 7 x 7 and got 49.  That's really close to 50, I thought, and 50 x 4 was 200.  Well 49 + 49 + 49 + 49 had to be 4 less than 200.  200 - 4 was 196.  So the product was 196.  I checked and re-checked the math in my mind, and waited.
   I don't know how much time went by, maybe a minute more, maybe two or three.  Then, in a small voice, full of question and uncertainty, she spoke.
   "Yes!"  I said, relieved that we had agreed.  I turned to the three students, still sitting there with the cards held up against their foreheads.  It was their turn to sweat.  "OK!  The product is 196.  Can any of you figure out what number you've got?"
    Something akin to panic set in on two of the faces.  The other belonged to a boy whose eyes began to roll up to the top of his head, ever-so-slightly, as if he was looking into his brain.  He was holding one of the 7s.  I felt myself willing him to get it.  Finally, again after several minutes:

Student: I have a 7.

Me: How did you know?

Student: I saw a 7 and a 4 and I knew that was 28.  I thought how many 28s do I need to make 196?  28 is close to 30, and 7 x 30 is 210.  That's close to 196.  So I tried 7 x 28 in my mind.  I know 7 x 8 is 56, and 7 x 20 is 140.  140 + 56 = 196.  That's how I figured it out.

Me: Nice!  (Turning to the dealer)  How did you get 196?

Student: I knew 7 x 7 was 49.  To do 49 x 4 I first multiplied 49 x 2.  That's 98.  Then I added 98 +98.  I know 100 + 100 is 200, so 98 + 98 is 2 less than 200.  That's 196.

Me (to myself): Did we just have a number talk?

     It was time for the students to switch centers.  As the next group moved in (another group of 4 that the teacher had informed me was the "low" group),  I realized that I had been fortunate.  What if the factors had been different?  Say a 7, 8, and a 6?  That would have increased the difficulty, both for me and the kids!  Just to be on the safe side, I quickly rigged a deck with the numbers 1-5, a few 6s, and some 10s.

It was right in their wheelhouse.  The dealer multiplied 2 x 3 and got 6, then multiplied 6 and 6 to get 36.  The student with the 3 on his forehead responded first.  He said he knew 6 x 2 was 12, and that two 12s was 24, and another 12 made 36.  And that's how 4-Way Salute! was born.

I could hardly wait to get back to school and tell Theresa.  I showed her the 7, 7, and 4 and asked her how she might multiply the numbers together.  She thought for a moment.
  "I'd multiply 7 x 7 first and get 49.  Then 49 x 4.  40 x 4 is 160, 9 x 4 is 36.  160 + 36 = 196."
  A different, but no less effective strategy.

  Would you like to give it a try?

The product is 126.  What's the missing factor?  How did you figure it out?

Again, the product is 126.  But now you're able to see the 7 and 6.  Is finding the missing factor harder or easier?  Why?Did this player have an advantage?

What about now?

     I haven't had a chance to explore the many possibilities that this variation of traditional Salute! has to offer.  If you've got some time left in your school year, give it a try!


  1. Very good post. Go ahead with all of this. Regards.

  2. Joe, I always appreciate your ability to take something that exits and make it so much more interesting. I found the different difficulty levels, based on which factor was missing or which cards were used, particularly interesting. Great opportunities to differentiate students’ challenge – thank you! Linda

    1. Thank you Linda. I am excited to explore those possibilities next year. I can imagine asking kids to rank them in order from easiest to hardest depending on which factor is missing, and explain why. I don't think they'd all agree, and the differences in ranking would spark great arguments!

  3. "a boy whose eyes began to roll up to the top of his head, ever-so-slightly, as if he was looking into his brain." Favorite non-mathy bit. Favorite math bit? Everything else. I love your creative, generous brain. Thanks, again.

    1. As always, thanks for your kind words. I wish you could have seen him. It was kind of torturous waiting for him to answer. Those minutes seemed like hours, especially because the other two had clearly checked out. It doesn't always happen, but my waiting was rewarded!

  4. Great way to make number talks a little more personal and small group. As a teacher sitting with a group and listening to their thinking you would certainly gather a lot of quality formative assessment. I love the different ways this can be done and the way everyone thinks differently. I had a teacher I worked with in our most recent PLC tell me she had a parent call her and tell her that her son does not have to do anymore exit tickets! On his last ticket he had wrote that this question is a grade 3 question and he is in grade 7. The teacher had explained to him that the question may seem like it is from an earlier grade but we were looking to see what strategies students were using to get the solution. It was the caterpillar problem, the one that says "A class needs 5 leaves to feed its 2 caterpillars. How many would it need to feed 12 caterpillars?" We wanted to see who had a good understanding of proportional thinking, who was using multiplicative thinking and how they would solve it. Anyways his mother said "Why do they need to learn or see different strategies, why not just show them the best way?" We have work to do to get the message out there that everyone thinks differently and what might be the best way for one student may not be the best way for another. Games like you have showed in this blog post are so important. It allows for students to show their thinking and be creative, the way math should be! Do you find in the US that you are still fighting this battle with parents? Similar to what I have mentioned. Love, love, love games and tasks that allow students to stretch their minds and be creative. Thank you for sharing again. I had not actually seen this game before and will share it with my teachers.

    1. Thanks for your comments, Mark. The answer to your question is yes. Parents want to help, but sometimes do more harm than good. Part of that is our fault because we don't do a good enough job educating parents about what we're trying to accomplish. In fact my district is making an attempt to address this issue as we speak. We're going to have to play the long game here because the change we're talking about is not going to happen overnight.

    2. Yes it is for sure the long game. We released a number talk brochure to our parents that myself and another IL made up. It seemed to help, we got a lot of good feedback from parents who were concerned or confused when their children started actually using different strategies. It helped them make sense of what we are trying to do. Thanks for the reply.

    3. Any chance I could get a look at that brochure?

  5. I am thinking...can I do this with kinders? Salute with addition/subtraction?

    1. I don't see why not. You could differentiate by controlling the numbers on the cards. A first grade teacher in my district tried it with her students and they loved it. Let me know what happens!

  6. Beautiful! I've played this game for years and always wanted to give the dealer a more active role (for some reason I think the extra card does this...) And what a nice way for students to practice strategies they work on in their number talks. I'm going to start rethinking all of my games now. Well done and thanks again for sharing.

    1. Thanks Mike! Let me know how it works for you. Many times when we play games there's not the optimal number of kids, and often its a group of 3 when only 2 are needed. Please share any ideas you come up with!

  7. Replies
    1. They come with the Everyday Math program. I've done this with playing cards (with ace being one) and it works fine.