Asked & Answered

The Asked & Answered department shares excerpts from discussion threads on the online MyNCTM community. In this issue, featured threads highlight responses to members' questions related to fact fluency and art-math integration in elementary school, number talks with middle school students, and classroom-tested contexts for introducing students to systems of equations.

Fact Fluency

Denise Oleary (Enfield, CT)

I just got hired as a “one day a week” mathematics coach for a 220 pupil PK–8 school. Typically, our school does very poorly on the Smarter Balanced Assessment Consortium (SBAC) assessments. We are trying to increase student time on mathematics learning; therefore, I am looking for suggestions of no cost, yet fun, games to build fact fluency at all grade levels. I hope to train paraprofessionals to play with our students during academy time, and I hope families will embrace playing these games too. Any suggestions are welcomed.

W. Tad Johnston (Washington, DC)

Christina Tondevold has what she calls “evergreen” mathematics games on her site ( These five formats can be used across a variety of topics and levels. Since the rules stay the same, it reduces students' (and your paraprofessionals') need to spend time learning rules of the games so they can spend time doing the mathematics thinking. These are also adaptable, and your paras may see how to adapt them to other uses involving them more deeply in the process. The games with Everyday Math ( have been around for years, so if you have those materials kicking around the building(s), you may resurrect some of those. I still remember the advice on how to be successful with EDM: “Play the games!” While one needs to have the curriculum to have all the games, there are several samples for each grade posted at their site ( There are many, many resources out there. The trick is, of course, finding the ones that meet the needs of students and adults working with them. Good luck in your search and implementation.

Marci Hellman (Golden, CO)

Check out the book, No More Math Fact Frenzy from Heinemann. Often, students who struggle with fluency do not just need more practice; rather, they need to develop the foundational ideas of early numeracy that make the number relationships available in their thinking. Additionally, students who struggle on assessments like SBAC often need additional work on developing the mathematical practices. The practices support their thinking through a problem-solving lens, which is often required on assessments that have more complex problems on them. The routines for reasoning described on the website Fostering the Mathematical Practices ( are great for this.

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Projects Related to Mathematics and Art

Liz McCormack (Bellevue, WA)

I am interested in projects related to mathematics and art. Does anyone have curriculum related to integrating the two subjects, particularly quilting and weaving?

Sarah Clinkscales (Fort Worth, TX)

Take a look at this resource: Mathematical Quilts by Diana Venters and Elaine Krajenke Ellison, Key Curriculum Press. It's about what happens when two geometry teachers take a quilting class. I have used many of the ideas and projects in my geometry classes.

Michelle Mailhot (Augusta, ME)

The National History Museum of Art has exhibits and information connecting art and mathematics. I once did an integrated unit with social studies, English, and mathematics with quilts. In social studies, they were studying slavery and the underground railroad that used quilts as maps to safe houses; in English they were reading novels about the topic; and in mathematics we were studying geometry and patterns. [We] designed and made a dozen lap quilts that were donated to the local cancer treatment center.

Dennis Ashendorf (Costa Mesa, CA)

At NCTM, I was generously given a box on art and mathematics: Math Integration Kits ( The kits are curriculum-enriching, STEAM-based, art-integration solutions for K–5 students. The kits help students learn, retain, and apply foundational mathematics concepts.

Allison Claus (Vernon Hills, IL)

There are all sorts of activities that third graders can do with art and mathematics. If you have pattern blocks and look at the way the smaller blocks are fractions of a hexagon (triangle = 1/6, trapezoid = 1/2, blue rhombus = 1/3), kids can then make designs where the pieces sum to 1 whole or any other sum you like. Another activity you can do is tessellation, either with the pattern blocks or having the kids design their own tessellating piece.

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Middle School Number Talks

Debbie Babb (Modesto, CA)

What suggestions or ideas do you have for getting number talks happening in my sixth-grade class? My students struggle so much with fractions and decimals. I want them to become more confident in these areas next year. Thanks for any and all support!

Karen McClaren (Bowie, MD)

I like the book Making Number Talks Matter by Humphreys and Parker. Good examples of typical strategies and how to illustrate them, and extensions to fraction and decimal problems.

Jennifer Leimberer (Riverside, IL)

You have to check out this book by Parrish: Number Talks: Fractions, Decimals, and Percentages. Full of number strings and number talks that really focus on mental mathematics ways to reason and develop number sense. It is full of videos; justifying the cost. Get the blackline masters too. They are sold separately. Wish they just came with the book.

Virginia Dalton (Farmingdale, NY)

I agree that the Number Talks books by Parrish and Humphreys are great places to start. I do think that one of the most helpful ways we've gotten better at recording student thinking is by watching the videos from Parrish's book. We participated in professional learning where we studied her book, and the videos were a focus for that work. We answered her guiding questions provided for the video, and then we responded to one another's answers. I also think you can use one of the videos to show your own students to spark discussion in your own classroom—help build that momentum around the process.

Erin Null (Folsom, CA)

John SanGiovanni has a new book specifically for middle school called Daily Routines to Jump Start Mathematics. It's 20 different 5- to 10-minute reasoning routines to get kids talking using a variety of middle school content with all of the materials you need to go with it free online.

Also Peg Smith's new 5 Practices in Practice is specifically about mathematical discussion in middle school.

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Systems of Equations

Josh Lerner (Chicago, IL)

When introducing the topic of systems of equations to students using a problem-solving approach, do you prefer to use a particular context?

Lane Walker (St. Charles, MO)

My cognitive strategies are connection and visualization. I do “substitution” first. I usually do a think-pair-share about “putting two and two together” as a way to think about substituting.

David Sebrnick (Acton, MA)

I modified a lesson from Key Curriculum Discovering Algebra. The basic problem:

Alice and Bob are at opposite ends of a trail. They start hiking toward each other at the same time. When and where do they meet?

  1. I give students a blank graph and table. Trail is 12 miles long, Alice walks 2.5 miles per hour (mph), Bob walks 1.5 mph. Make a table and an equation for each, and a graph with both lines, and say where and when they meet. The answer can be seen right in the table.
  2. Same problem, 15-mile-long trail. Make a graph, a table. Answer is easily interpolated from the table, but doesn't appear directly.
  3. Same problem, 15-mile trail, speeds of 1.5 and 3 mph. I wanted a rational (but not integer) solution that my students wouldn't find easy to guess so I could motivate learning the substitution method to find an exact solution easily.

Jeff Simpson (Ukiah, CA)

  1. Pose the problem a + b = 10 and ask what a and b are. So, we make a table of values to list the possibilities. Then we start again and make a table with x starting with zero, and then proceeding in numerical order. This eventually leads to the conclusion that there are infinite possible combinations.
  2. Pose the problem a + b = 10; ab = 4 and tell them that a and b stand for the same numbers in both equations. Intrigued, they set off trying random combinations, until (with some excitement and satisfaction) they finally stumble across the numbers that work.
  3. Explore the fact that two equations can be added together— which is a very surprising idea when encountered for the first time: “Here's an easy equation: 3 + 2 = 5. Somebody give me another easy equation.” Write their equation under yours:
    Draw a line under the bottom equation, and suggest that they add the equations together, and guide them to do it. Ask if the first equation is true. Is the second equation true? Is the third equation true?

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Excerpts from discussion threads on the online MyNCTM community

Each month Asked & Answered highlights selected threads from the MyNCTM community. MyNCTM is an online community where NCTM members can ask questions, start and join discussions, and interact with education experts. We encourage you to join the conversation at

Mathematics Teacher: Learning and Teaching PK-12


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