A Mathematical Journey through Zero & Infinity-Part 1: A Mystery Box, Some Books, and a Contest
Rich Math Tasks
Very few people in life need to manipulate a complex equation on a regular basis, but ALL people need to be able to think mathematically including those who go into STEM careers. Therefore, as a math educator it’s vital that you incorporate rich mathematical tasks into your classroom. Rich tasks also improve student engagement which is a win-win for everyone. A good place to start is by combining the Standards of Mathematical Practice with content that you are already going to teach.
Here are some websites that have engaging math tasks:
- Tasks Archive – YouCubed
- Curriculum-linked Problems – Secondary Teachers (maths.org)
- Open Middle – Challenging math problems worth solving
- Illustrative Mathematics
Using Picture Books in a Math Classroom
Middle school and high schoolers still love picture books! It’s always engaging when you read picture books in a math classroom. Bring out the carpet squares and have your students sit in a circle as throw-back to elementary school when life was easier. Your students will love it! You could even incorporate snack time. 😄
Part I: Zero Zebras
The beginning of this lesson is a modification of my post Engaging Math Lesson: Zero Zebras and Infinity, but it’s levelled up a few notches for middle and high schoolers. The end of the lesson and the standards focus will be a little different.
A Mystery Box
The Question: What’s in the box?
To teach this lesson bring in a medium to large empty box that is all wrapped up. Tell students that the principal got them a very special gift, and have them quickly write what they think is inside on an index card. Collect the cards to use later in the lesson.
The Reveal
Open the box and reveal its contents. There should be nothing inside. When students claim that it’s empty, tell them there are a lot of things inside, but you will discuss it later.
Read and Discuss
Read Zero Zebras: A Counting Book About What’s Not There by Bruce Goldstone to the students. After reading redirect their attention back to the box. Explain to students that the box is a special box: a Zero Box that contains 0’s. Then you may choose to read the students quick-writes for comedic value and ask them if they are in fact in the box. Then do the following:
- Ask students to name additional 0 things that are in the box. Emphasize the importance of naming units and explain how 0 zebras is different than 0 cupcakes or whatnot.
- Ask students how many zeroes are in the box? Eventually the conversations should lead to the fact there are infinite number of zeroes in the box.
- Discuss how some things in mathematics are quantitative and need labels/units, and some things are abstract like mathematical equations.
- Have students tell you a few mathematical equations that equal zero and write them on index cards and place the index cards into the box.
Tell students that zero and infinity are related and that they are going to spend the next few days on a mathematical journey exploring zero and infinity.
Note: If you have special needs students who need a more hands-on approach to understand infinity, read my Engaging Math Lesson: Zero Zebras and Infinity post for ideas.
Buy Zero Zebras: A Counting Book About What’s Not There by Bruce Goldstone. Note: I’m NO longer participating in Amazon’s affiliate link program (8/28/25), but I am leaving them here for your convenience.
Disclosure: I only recommend products/books that I think would be helpful to you in your journey as an educator. All opinions expressed here are my own. I’m NO longer participating in Amazon’s affiliate link program (8/28/25). Read my full privacy policy here.
Part 2: Zero Contest
Driving Questions
Have students do a quick-write and collect them. The answers to these questions will be revisited at the end of the lesson.
- What equals zero?
- How many zeroes are there?
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Properties of Operations
Understanding and applying the properties of operations is huge in middle school and Algebra 1. However, many students and teachers find them boring. Therefore I’ve designed a task for middle school and Algebra 1 students that focuses on two properties specifically that have to with 0.
- The Additive Identity Property of Zero:
- Existence of Additive Inverses: For every
that exists
so that
All the Properties of Operations can be found in Table 3 of Ohio’s Math Standards.
A Place for Zero (optional)
Read students A Place for Zero: A Math Adventure by Angeline Sparagna LoPresti. After reading use the book to review some of the Algebraic Properties of Operations.
It’s difficult to find appropriate places to integrate social and emotional standards into a math class. Reading picture books can often help. Use A Place for Zero to integrate social and emotional standards. You can discuss aspects of self-awareness and social awareness. For example, Zero was pro-active in trying to find a way to be a productive member of society. He was also willing to seek help from other when he needed it.
Buy A Place for Zero: A Math Adventure by Angeline Sparagna LoPresti.
The Contest
After reviewing The Additive Identity Property of Zero and Existence of Additive Inverses with students, tell students they are going to have a contest where they apply these two properties. The students will also need to know the mathematical definition of a term: a single number and/or variables that are multiplied together. Terms are separated by addition and subtraction signs and occasionally a fraction bar which represents division.
Part A: Creating Equations
Have students write their name on the top of the paper and number their paper from 1 to 5 leaving space in between lines.
Tell students that they need to come up with at least 10 terms when added or subtracted together that equal zero. They can use any mathematical number or symbol that they want except the number 0. The number 0 can only appear on the left side of the equal sign. For example, 
Part B: Critique the Reasoning of Others
Students will need highlighters and a pen for Part 2.
After students come up with their 5 equations, have students exchange their paper with a classmate. Have the partner write their name on the bottom of the paper. Tell students to highlight any examples where their partner used the Additive Inverse Property.
Then have students circle any examples of the Additive Identity Property of 0. This is tricky and may warrant class discussion as you may end of having 
Students also need to check their classmates’ calculation to see if they are correct. Lastly, the student should star their favorite equation and give a small explanation on the bottom of the paper of why they liked their classmate’s equation.
Part C: Voting
Have the partner come forward and present their favorite equation from their classmate’s paper. Have them explain why it’s their favorite. As a teacher encourage proper math vocabulary.
After all the students have presented, have the class vote on the favorite equation. Tally the results and give out prizes.
Part D: Follow-up Discussion
Now that all the equations are on the board, use some of the following prompts (or come up with some of your own) to generate discussion. Make sure students give quantitative examples to justify their thinking:
- What equals zero?
- Are all the equations equivalent? Explain.
- Why is the Additive Inverse Property important? Explain.
- Why is the Additive Identity Property important? Explain.
- How many zeroes are there?
- Explain how the properties of operation affect the structure of an equation.
Check for Understanding
Write the following equations on the board. On their own or as an exit ticket, have them add two terms to one side of each equation but still keep equivalence. Note: Middle School students may not be familiar with quadratics etc, but that doesn’t matter. The reasoning still holds, and so they should be able to apply their reasoning.
Now take the same equations A.-D., and have them add one term to each equation and still maintain its equivalence.
Throughout discussion continue to emphasize why the Additive Inverse Property and the Additive Identity Property work.
Bible Connections
Zero
Here is a Bible verse that talks about the concept of 0. You can discuss that when we were born we had 0 objects with us, and so we should be content. “But godliness with contentment is great gain. For we brought nothing into the world, and we can take nothing out of it. But if we have food and clothing, we will be content with that.” I Timothy 6: 6-8 NIV
Infinity
Here are some verses that discuss the infinite nature of God. With kindergarteners you can talk about how God knows “infinitely much” or is omniscient, and that He has been alive for an infinite amount of time.
- “Before the mountains were born or you brought forth the whole world, from everlasting to everlasting you are God.” Psalm 90: 2 NIV
- “Great is our Lord and mighty in power; his understanding has no limit.” Psalm 147: 5 NIV
Standards
Standards of Mathematical Practice
This lesson utilizes the following Standards of Mathematical Practice:
- S.MP.3 Construct viable arguments and critique the reasoning of others.
- S.MP.7 Look for and make use of structure.
Math Standards
This lesson addresses the following Common Core and Ohio Math Standards:
- 6.EE.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3
- 7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
- Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
- Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
- Understand subtraction of rational numbers as adding the additive inverse, p − q = p + (−q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real[1]world contexts.
- Apply properties of operations as strategies to add and subtract rational numbers
- 7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
- N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. ★
- A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★
- Factor a quadratic expression to reveal the zeros of the function it defines. (A1, M2)
- Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. (A1, M2)
- Use the properties of exponents to transform expressions for exponential functions. For example, 8t can be written as 23t
- A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Ohio’s Social and Emotional Standards (A Place for Zero)
Here is a list of Ohio’s Social Emotional Standards addressed in A Place for Zero.
Competency A: Self-Awareness
- A2: Demonstrate awareness of personal interests and qualities, including strengths and challenges.
- A2. 2.c Investigate a potential career path that builds on personal strengths and addresses challenges.
- A3: Demonstrate awareness of and willingness to seek help for self or others.
- A3. 2.c Develop and implement a plan of action, based on support or constructive feedback, that addresses challenges and builds on strengths.
- A3. 2.d Reflect on actions that are based on constructive feedback, address personal challenges and build on personal strengths.
- A4: Demonstrate a sense of personal responsibility, confidence and advocacy.
- A4. 2.c Recognize the importance of confidently handling tasks and challenges, while reframing negative thoughts and engaging in positive self-talk.
- A4. 3.c Demonstrate basic self[1]advocacy academically and socially.
- A4. 3.d Demonstrate self-advocacy in context[1]specific situations.
Competency C: Social Awareness
- C2: Demonstrate consideration for and contribute to the well-being of the school, community and world.
- C2. 2.c Pursue opportunities to contribute to school or the broader community.
- C2. 3.d Implement a strategy to address a need in the broader community or world as change agents.
- C2. 4.d Evaluate the impact of personal involvement in an activity to improve school, home, community and world.
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