Hills and Holes

My kinder students LOVE to play games. If I can turn their learning experience into a game, they are cheering for more and that just makes me smile. They are full-on engaged from start to finish, so of course, I am always trying to gamifying math activities, if I don’t already have a game for them to play.

 I came across Skyjo. According to the box, this game is great for 8 years and above, 2-8 players, and takes about 30 minutes to play. Most people would read that and take it at face value, but I don’t. I take it as a challenge. How low can I go with it?   

If I was going to bring this to my Kiinders, I needed to understand the rules and the embedded math needed to make this playable for them. I spent several evenings over winter break playing. I wanted to understand the ins and outs of the game. I also looked at the change in gameplay as the number of players increased- fewer cards available. 

For rules, my Kinders were going to need to understand how to take a turn, how to get rid of unwanted cards, and how to add up their score – All of which was easy to teach over time. In terms of math, essential skills needed to play are compare values, count on, make tens, count by tens, and have an understanding of negative numbers.

For my Kinders, it was the negative numbers I was worried about. We have never really explored whole numbers less than one. That doesn’t mean it is a concept they can’t understand. It just means I need to find a concrete way to teach it. 

 I pulled out my Math Recess book and looked at all the tabbed pages. These are pages I thought were important and relevant to teaching math to my gifted students, so I tabbed them for quick reference. There it was. Right there on tabbed pages about Exploding Dots. No, I wasn’t going to introduce Exploding Dots to my Kinder students. It was too early in their understanding of that concept. The tabbed page was a reminder of something I heard James Tanton talk about – Hills and Holes.

Hills and Holes could help explain negative numbers. I planned to take my Kinders outside to the sandbox. I would show them 5 hills of sand and say this is 5. Then show them 3 hills of sand and say this is 3. Show them four hills of sand and asked them what it was. But when it is storming. 

Since I couldn’t take them outside, I needed to switch the plan. Sure, I could have dumped the lesson and saved it for a sunny day, but it was a Game Day. I promised. I needed a different way to introduce the concept of negative numbers. 

I know my Kinders love to break my codes, so Hills and Holes became a code to break. I shared a chart with several lines- two connected hills with “= 2” and under that I drew 5 connected hills “= 5.” Farther down there were 4 connected hills with “= _______” and under that 6 connected hills with “= _______".”

I asked them to look at the first two lines and see if they could break my code. Once they had an idea, they were asked to apply that idea to the next two lines. If it worked, give me a thumbs up. If it didn’t work they way they thought it might, go back and try again to break my code. It wasn’t long before thumbs popped up in front of their chests, so I added, “If you have thumb up, think about how you can explain your code idea and prove to the class you are correct. Add a finger when you have an idea to share.”

In my classes, I have students put their thumbs up (in front of their chest or over lips if they have a tough time not shouting out) to show me they are ready with an idea. If they have more than one idea, they add another finger. They are showing me they are ready to share and not taking away the thinking time of their neighbors. As soon as children see others shooting up their hands in the air, they are no longer thinking about the task at hand and instead of thinking about how someone else already has an answer. 

When every student had at least a thumb up, I asked everyone to put down their thumbs and listen as A explained her code idea, “And thumb up if it is the same as your code idea.”

 A counted the five “bumps” and said each bump was one and so there were five. Most students connected with A’s ideas. We applied her code idea to the next one and counted it together, but as I counted I used the word “hill.”

“One hill, two hills, three hills equals 3.” We all counted the remaining hills on the last problem and agreed with A’s code-breaking idea. Everyone made our connection sign to A to show we also had that idea and to validate her thinking.

I shared the a few more pieces of code and everyone applied their understanding of the early code to these code lines.    

They walked through them with ease. I demonstrated hills as fingers pointed up … 3 hills as 3 fingers pointing up.

Then I presented new images, instead of hills there were holes, and asked to work the code.

After think time, N shared “It’s the same as the last one, but it has scoops.”

I referred to the scoops as “holes,” as we checked each line. They all counted 5 holes, and I wrote “- 5” after the equal sign.

“Minus 5?”

I explained that the “– ” sign means negative. This is a new math symbol for Kinders, so it was understandable that most thought it was a sign used for subtraction.  

We talked about the word negative. . .What it means and examples.

“I can have negative feelings, like mad or sad.”

“There is negative space in their golden butterfly drawings. The space without color.”

They conclude in this context negative means “without.”

“How does does negative means ‘without’ connect to our Hills and Holes work?”

“The hills are filled with dirt and the holes are without dirt.”

“Yea, they are opposite of hills.”  

“Hills HAVE and holes are WITHOUT dirt.”

I drew one hill + one hole and asked, “What does this equal?” There was plenty of thinking and neighbor chat.

“A line.”

“It’s zero.”

Of course, I was incredulous and was not convinced of either idea at all.  

“One hill plus one hole. I put the dirt from the hill in the hole and now the hill is flat and the hole is flat. It’s a line.” Crystal colored in the hill. Then X it out, “I am moving the dirt from the hill to the hole. Now the hole is filled. The ground is flat, with no hills and no holes.” All agreed that was possible. 

“I can see how 1 hill plus + 1 hole can be shown with a line, what about the idea that it equals zero?”

 Loretta offered, “Well, they sort of cancel each other out, like it is neutral, not high and not low. Neutral is zero.”

“So, are you saying that if it isn’t positive or negative it is neutral?”

“Yes, I’m not happy, nor sad. I just am.”

 The class agreed zero could be represented as a line, since one hill could fill one hole and make zero hills or holes. 

 Then I wondered aloud, “What ARE negative numbers?” 

“Negatives could be something we do not have.”

“Like we don’t have dirt.”

The student’s ping pong of ideas continued.

“Hey, there are negative numbers on the wall. Look at the number line.”  Everyone turned to examine the number line on the wall

“Some of the numbers go across.” 

“But the negative numbers go down.” 

I hang the negative numbers of a number line down the wall.

“Like into a hole,”

“The hole is deeper with each number.” 

We all looked at the number line and connected with all the ideas. 

I asked, “What is a number less than 36?”

I called on students and heard, “35”, “34”, “33” and “7.” 

“What is a number less than 7?”

“6.”

“5.”

“0.”

“What is a number less than 0?”

Slowly a thumb went up on her chests, “1”

“Is one less than two?”

“Oh, negative 1.” 

“Negative 2.”

“Negative 3.”

“Negative 100.”

“So I am back to wondering what are negative numbers?”

I heard a few more examples of negative numbers before, “A counting number less than zero.”

In Kinder, that was a fabulous place to leave negative numbers. 

We talk a lot about how mathematicians are efficient. I usually whisper “lazy” and have a quiet verbal argument with myself -repeating “efficient, lazy” several times until they giggle. I snapped out of my verbal argument and went on to explain that mathematicians can write +1,+2, +3. +4, +5, and so on for all the numbers greater than zero, but we are “efficient” (one quiet “Lazy” follows and I get smiles) and we use a lot of positive numbers. If we use the negative sign next to a number we know it means less than zero. Since mathematicians are efficient, we don’t need to write + and – symbols. If it doesn’t have a -, it must be positive. 

With negative numbers on board, we focused the learning on how to set up the game, which was already done for students at the table, (I usually set up the game the first time it is played. The children have learned to approach their game table and take in the set-up, as they will be setting it up for the next class.) and how to take a turn in the game of Skyjo - Draw, trade, discard.  

In my class table groupings change based on the work children are doing, so Game Day: Skyjo will have assigned teams based on the children’s knowledge of numbers. We had two tables set up with 0-10 and no negative cards; 0-12 and no negatives; 0-10 and only -1 cards; 0-12 and one of each negative card place in the bottom half of the deck; and all cards. Teachers and parents at the tables were give the extra cards for their tables- set up in order to add to the playing decks, as well as written modifications, if needed.

Day Two of Skyjo had a quick Hills and Holes check-in (Amazing - All were solid on applying hills and holes, positive and negative numbers) and a new Skyjo rule-If a player had three cards of the same value in a column, those three cards were removed from the player's board and the player shouts, “SKYJO!” That received lots of “Ohs.” 

“Why do you like that rule?”

“If I take away a column, I only have to add 9 cards at the end.”

“Or maybe 6 or 3.”

“Or NO CARDS!”

“And we get to shout SKLYJO!”

With that, they headed off to their tables to play a game of Skyjo- with all tables playing all the cards. 

It wasn’t long before we heard our first of many, “SKYJO!”