## Zip, Zilch, Zero

• Lesson
6-8
1

Positive and negative numbers become more than marks on paper when students play this variation of the card game, Rummy. Engaged in a game involving both strategy and luck, students build understanding of additive inverses, adding integers, and absolute value.

To begin the lesson, review adding integers. Show examples that include both positive and negative integers.

Then divide the class into groups of three or four. Students will compete individually against the other people in their group. Distribute the Zip, Zilch, Zero Rules and Record, and read the rules aloud to students. As necessary, answer questions an provide additional explanation. For example, you might want to show the example below to give students an understanding of how to make a Zip.

It will be helpful to couple this example with the mathematical equation that it represents:

(-7) + 9 + (-2) = 0
This will help students to see that red cards represent negative numbers, as well as an example of how to make zero using more than just additive inverses. When you are certain that all students understand the rules of the game, tell them that they will play a practice hand before keeping score, to make sure that everyone understands the rules of the game.
 Zip, Zilch, Zero Rules and Record

Distribute the Zip, Zilch, Zero activity sheet. Give all instructions before giving cards to the groups so they can focus on the preparation. Point out that Question 1 concerns the practice hand they will play. For this hand, when someone goes out, everyone should show their remaining cards. They will then estimate who will end up with the highest score and write their predictions on the paper. This allows students a window into each others' thinking and gives them practice with estimation and integers. They will also accurately determine scores and reflect on the quality of the prediction.

 Zip, Zilch, Zero Activity Sheet

Hand out one deck of playing cards to each group. Monitor groups closely during the practice hand. Students may think they have to match a red and black number exactly to make a zip. Remind them that they will get rid of their cards faster if they can make combinations with more than two cards. In addition, students may be familiar with games where all face cards are worth 10; in this game, J = 11, Q = 12, and K = 13, and it may be helpful to remind students of this rule several time.

Groups work through the activity sheet as they play. The activity sheet requires students to verify each other's scores to get more practice and to prevent disagreements later. If time is short, you can reduce the number of hands required for a game. Have a few students share their best strategies with the class and explain why they worked well.

Assessments

1. Circulate around the classroom as students play the game. Listen for evidence that they understand the math concepts or have developed a good strategy. Record that evidence.
2. Have students fold a piece of paper in half. Have them draw a big plus sign on one side and a big minus sign on the other side. Write different integer addition problems on the board, one at a time. Have students hold up the side of the paper that shows what the sign of the answer would be.
3. Deal a sample hand and challenge students to make a given number other than zero using the maximum cards possible.

Extensions

1. If you would like to play this game periodically and/or make a tournament of it, you can use the Zip, Zilch, Zero Record activity sheet to record the games.
2. Play the game the same way except that books must equal 24 using any operations. For this game, students should write down the expression that creates their book and have it verified by another student before it can be counted. For example, if a student plays a black 10, a red 2, and a black 3, they must write on their paper "3(10 + ‑2)."
3. Play the game the same way but randomly pick a value for the zips (now called "melds" because they do not have to equal zero any more). The target could be the first card turned over after cards are dealt to players. The game could also be progressive with the first hand's target being 1; the second hand's target being ‑2, then 3, then ‑4, etc.
4. Play "Human Integers." Split the class randomly into two groups, forming two single file lines. Give matching party hats to each student in one of the lines, and different matching hats to students in the other line. One line is positives, and one line is negatives. Give students an integer problem to model. For example, 5 + (‑3) = 2. If black hats are positives and red hats are negatives, the first 5 students with black hats would go to the front of the class along with the first 3 students with red hats. The students make 0s by having a black hat stand next to a red hat. The two black hats without a partner represent the sum. The participants go to the end of the line.
5. For practice on subtracting integers, deal out 6 cards to each player. Turn the next card face up. This is the "target." Players must use 2 cards in their hands to make a subtraction problem with the target number as the difference. For example, if the target is a black king (13), a student could play a black 8 and a red 5 because 8 – (‑5) = 13. If a player cannot make the target number, they do not lay any cards down and miss the opportunity to score on this round. Once all members of the group have created their problems or passed, deal 2 new cards to each player (including the people who may not have been able to play their cards) and turn over a new target number. After 4 rounds, determine scores by counting the cards played as positives and the cards remaining in a player's hand as negatives.
6. Have students analyze different scoring methods. For example if the scoring method was to add up the cards played along the ones remaining in your hand with blacks and reds still representing positives and negatives respectively before subtracting what you have remaining in your hand from the value of the cards you played. How do you think the change would affect scores? Why do you think so?
[Answers can vary. Sample: The cards played should all add up to zero, so you would be subtracting the value of the remaining cards from zero. If that is a positive value, your score for the hand is negative; and vice versa. Or whether or not scores increase or decrease depends on what the scores were under the original rules and how the hand played out under the new rules. If a person had negative scores under the regular rules, and their hand had an abundance of red cards remaining, under the new rules the scores would considerably improve. On the other hand, if the opposite were true, their scores would decline.]

Questions for Students

1. How can you make the most points when you lay down a zip?

[Use cards with high point values, and use a lot of cards with each zip.]

2. What will happen if you get down to only one card in your hand?

[You will never be able to go out because you have to discard to go out, and you have to draw at least one card on every turn.]

3. When is it a good idea to pick up more than one card from the discard pile?

[You should pick them up if you can make a lot of points from using several cards in the pile. Another time to pick them up is when you only have one card in your hand and need more cards to be able to go out.]

4. When is it not a good idea to pick up more than one card from the discard pile?

[You should not do it if it looks like someone is about to go out because you could get caught with a lot of extra points in your hand.]

5. Why might you want to hold cards in your hand even if you could lay them down in a zip?

[If playing the zip gets you down to just one card, you limit your options for the rest of the game. If you have a sense of the dramatic, you might like to play all your cards at once to go out, but you take a risk of someone going out before you've played your zips.]

6. What other situations can be thought of in terms of opposites combining?

[Answers will vary. Samples: debts and earnings, driving one direction and then back the other direction, digging a hole and filling it up, balancing chemical formulas and equations]

Teacher Reflection

• How did students demonstrate that they understood additive inverses and/or integer addition?
• Did you allot an appropriate amount of time for the lesson? Did students tire of the game before the period was over? Was there enough time for students to finish a game of 4 rounds? What, if anything, would you change about the time allotted?
• Did students behave appropriately for a classroom game situation? If yes, what made it so; and if no, how can you help students make better choices next time?
• Did you make adjustments to the lesson? Were they effective? Why or why not?

### Learning Objectives

Students will:

• Develop a better understanding of how positive and negative integers relate to each other
• Investigate ways to combine integers to get 0
• Become more fluent in addition of integers

### Common Core State Standards – Mathematics

• CCSS.Math.Content.6.NS.C.5
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

• CCSS.Math.Content.6.NS.B.4
Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.