Area Model Cards

Recently I’ve been a little obsessed with the area model as a way to represent multiplication. So much so that I created my own set of area model cards (see attachment below). Why the area model? It’s one of the problem types that third graders are responsible for being able to solve (see Table 2 here on p. 89). Also, it’s used heavily in the fourth grade standards in multiplication (4.NBT.2.5) and division (4.NBT.2.6). Because of this I’ve been thinking about how important it is that third grade students get lots of experience with this model throughout the year, not just in the “area unit.” There are tons of connections within third grade depending on the question you ask:

  • Interpreting multiplication (3.OA.1.1) What multiplication expression would match/represent this model? How do you know?
  • Interpreting division (3.OA.1.2) What division expression would match/represent this model? How do you know?
  • Solving multiplication and division word problems (3.OA.1.3) What is a math story you could tell about this model? What would the solution be and where do we see it in the model?
  • Applying properties of multiplication as strategies (3.OA.2.5) What multiplication expression would represent the whole rectangle? What expression would represent the shaded squares? The non-shaded squares? How could we use this to figure out the total?
  • Relating multiplication and division (3.OA.2.6) What are the related multiplication and division expressions that would represent this model? Explain how you see the relationship in the area model?
  • Fluency with multiplication (3.OA.3.7) (All this multiplication work around the area model is leading towards fluency)
  • Recognize area as an attribute of a figure (3.MD.3.5)
  • Measure area by counting unit squares (3.MD.3.6)
  • Relate area to the operations of multiplication and addition (3.MD.3.7)

With all of these connections you can truly be addressing multiple standards with one tool. These are just the standards that have explicit connections to the area model in third grade. There are others, but they would require some specific constraints. Still though, this is 9 third grade standards… out of 25. That’s more than a third of the grade level standards. This is definitely a high yield representation for third grade. The cards are shaded after 5 to help students see the smaller facts within the larger facts (similar to a rekenrek) and I’ve included links to the cards below in PDF format.

AreaModelCards 2’s to 4’s

AreaModelCards_5’s to 9’s

If you use them with your students let me know how it goes. Have other connections or questions you would ask with the area model cards? Reply in the comments or @zack_hill on Twitter.



Parallels 2 – An ELA and Mathematics Pondering

I’m a math coach, but I love the opportunity to sit in on English language arts (ELA) trainings because it gets the mind turning about parallels between the two content areas. Math and ELA are different in a lot of ways, but I’m curious about how they are alike.

A couple weeks ago I sat in on Independent Reading Level Assessment (IRLA) training. Our trainer strongly advised against coaching an early reader to chunk text because she said it was developmentally inappropriate. She said the appropriate scaffold would be to coach the student to use the initial consonant sound and use context clues. She made this assertion strongly and nonchalantly, confident that it was the right move in the given situation. She went on to say that we need to take into consideration what students CAN do and build from there. She warned against instructing above where they were, saying that the students development in literacy can look like Swiss cheese, with major holes in underlying concepts.

I wanted to jump up and cheer.

It was great to hear her say this and gave me more confidence when giving the same advice to teachers who are dealing with struggling students in mathematics. So, a fourth-grade student can’t add and subtract using the standard algorithm… Do they understand the bundling of 10 ones into a 10? Ten 10s into 100?… And so on? How about decomposing? Can they skip count by 10s and 100s? Did they ever make sense of counting on or the importance of “making a ten?”

Let’s take the time to figure out what students CAN do and build on it to move them forward to where they need to be.

The Tension Between Productive Struggle and Telling

I’m an avid fisherman. There are times when I like to navigate a new spot and discover on my own the in’s and out’s of a place: how the tides work, where there’s structure that holds fish, depth changes, bait preferences, etc. I’m a better fisherman due to these experiences. However, there are other instances where I need to be told something directly like how to tie a uni-to-uni knot to attach my leader. This year I had the opportunity to present at NCTM’s annual conference on this same line of thinking in mathematics. In the session we discussed supporting students in productive struggle, specifically navigating the decision of when to tell. Here’s a recap of the session:

We know that productive struggle is an important instructional practice in teaching mathematics, but what is productive struggle?

“Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships.” (NCTM, 2014, p. 48)

Just as important, what is not productive struggle?

“…when students ‘make no progress towards sense-making, explaining, or proceeding with a problem or task at hand.'” (NCTM, 2014, p. 48)

So, students should be grappling with the mathematical ideas and relationships and if they’re making no progress, then the struggle is no longer productive.

Below are five tips for engaging students in productive struggle, two of which are specific about when to tell.

1. Anticipate the source of the struggle. Beware the “Expert Blind Spot.” (Wiggins & McTighe, 2005, p. 51)

Wiggins & McTighe (2005) describes the “Expert Blind Spot” as when an expert is so good at something that they fail to see a situation from the perspective of a novice. One of the best ways to avoid this is to do the math ourselves, trying to approach the task as your students would approach it. This gives us insight about what specific aspects of a task students might struggle with and what could be problematic for a novice. Also, this allows us to think about representations the student may use or misconceptions they may bring to the table.

2. Listen closely. Then ask questions based on student thinking.

The act of listening is a natural way to support someone when they’re struggling, just being present and empathetic. Also, John van de Walle (2006) suggests that we listen closely before doing anything else. We need to be sure that we really understand a student’s thinking before asking questions to move them forward in their thinking. This ensures students are moving forward in a way that makes sense to them.

3. Be specific about what you want students to struggle with.

The standards for mathematical practice are an excellent source of specific struggle for students. Consider MP 1 (make sense of problems and persevere in solving them) and MP 7 (look for and make use of structure). There are some wonderful “struggle words” in just these two mathematical practice standards like “make sense,” “persevere,” and “look for.”

4. Tell when it’s symbolic.

If it’s a mathematical symbol which carries an agreed-upon meaning, then this is a situation in which we need to tell. Also, we want to be careful about what we tell. For instance, = does not indicate that an answer comes next; it means that both sides of the symbol have the same value. Here it is important that we give students well-crafted, universal definitions that will serve them well into future grades. The above definition of the equal sign is functional through elementary school and into middle and high school, as well.

5. Tell if no progress is being made on the task,… but just enough.

I know in the past I’ve made the decision to tell and proceeded to give away too much. When we decide that telling is in order, we have to exercise restraint and give students just enough for them to begin making sense of the problem and moving forward on the task.

During the session there was authentic dialogue around navigating the decision to tell. Professionalism is another principle that NCTM proposes we put into action, not just arriving on time and being polite, but sharing the details of our practice in order to hone our craft. Look for opportunities to discuss productive struggle and the decision to tell with colleagues. Some great questions to consider are:

Is this an instance in which I need to tell? If so, how much support does the student need to make sense of the problem and begin making progress on the task?

What question(s) could I ask to clarify how the student is thinking about the task?

What question(s) could I ask to the student to move them forward in a way that builds on their thinking?



Leinwand et al. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA. The National Council of Teachers of Mathematics.

Smith, M.S. & Stein, M.K. (2011). 5 practices for orchestrating productive mathematical discussions. Reston, VA. The National Council of Teachers of Mathematics

Van de Walle, J. A. & Lovin, L. H. (2006). Teaching student-centered mathematics: Grades 3-5. United States of America: Pearson Education, Inc.

Wiggins, G. & McTighe J. (2005). Understanding by design expanded (2nd Ed.). Alexandria, VA: Association for Supervision and Curriculum Development.

Don’t Be Afraid of “I Don’t Know”

I’ve found that some of my best learning experiences have come after thinking or saying the phrase “I don’t know.” This phrase is required if learning is to take place. If you already know, you’re not really learning. Often though, we have an aversion to this phrase. It’s a phrase that always involves discomfort of varying degrees.

There’s the “Where are my keys?” variety of “I don’t know.” The level of discomfort isn’t too bad, depending on the situation: late or on time.

There’s the “What really makes a rhombus a rhombus?” variety of “I don’t know.” This one can cause a moderate  level of discomfort, depending on your familiarity with the content, but can provide a great opportunity for learning.

There’s the “What should three tiers of support really look like in mathematics?” variety. To me these are the best types of questions to which a response of “I don’t know” is awesome. They provide an opportunity to dig into a topic that is not usually discussed openly, or with enough honest conversation about barriers and specific ways to overcome them.

Of course, there’s also the “Where is the report I asked you for last week?” variety of “I don’t know.” This one kind of stinks because it usually indicates some sort of miscommunication or that someone dropped the ball.

I think we should be less apologetic about using the phrase when it’s used to create a sincere learning opportunity. So, throughout your day look for opportunities to think or say the phrase “I don’t know” (the productive middle varieties) and look forward to the mild discomfort and the learning that is to come.

Representations, What’s the Big Deal?

NCTM (National Council of Teachers of Mathematics) devotes an entire section of Principles to Actions to using and connecting mathematical representations. Why are representations so important? One reason is access. NCTM states that all students should have access to the support and resources needed to maximize their potential.” (2014, p. 59)

The National Research Council states, “Because of the abstract nature of mathematics, people have access to mathematical ideas only through the representations of those ideas.” (p. 94-95)

This statement has huge implications. If we don’t expose students to a variety of representations we withhold a resource that supports students, literally withholding entry points to the mathematics itself. Also, who benefits most from a variety of representations? I would say all students, but especially students those who are English-language learners and students with disabilities. Exposure to a variety of representations can level the playing field by providing multiple entry points to the concepts being taught.

What might this look like?

NCTM describes five categories of representations: contextual, physical, visual, symbolic, and verbal (2014, p. 24-25). We’re getting ready to move into a unit on volume in 5th grade and I’ve been thinking about what representations we might use to introduce 5th grade students to the concept of volume.

Here’s the textbook’s first lesson on volume:Volume1st


The textbook suggests telling students what a unit cube is (symbolic or verbal, depending on if it’s read to students) before it shows a visual and has students fill in some numbers (symbolic) that should be pretty straightforward for a kindergarten student exploring geometry (CC.K.G.2.4). There has to be a better way.

Our team chose to use Packing Sugar, a 3-act task from Graham Fletcher, which introduces volume in a completely different way. The task begins with a video of a sugar cube next to a box of sugar cubes. (see below)

Here we are with an engaging visual representation combined with a context ripe for exploration. There are no blanks to fill in with symbols before students begin interacting with the idea of volume. After the video, students are encouraged to formulate a problem themselves and take a guess. For me, the primary reason to use a task like this is because it introduces the concept of volume to students using a representation that’s more accessible than those in the textbook. Students are immediately invited to begin thinking of how the cubes may be packed in the box and how we might go about finding the total of cubes that would fit in the box.

Why don’t we expose students to a variety of representations?

Why did NCTM have to devote an entire section to this practice in Principles to Actions? I think it’s because sometimes we consider representations such as number lines, drawings (or video), and concrete objects not as “mathy” as other representations like expression and equations. Bill McCallum gives them the nod here though, in the mathematical practices document with commentary for grades K- 5. In the section on modeling with mathematics (MP4), geometric figures, pictures, and physical objects are explicitly mentioned by name as being appropriate in the early grades.

How do we move forward?

When planning to introduce a new concept or a specific task, brainstorm how a concept can be represented physically, visually, symbolically, contextually, and ways that students may discuss it. Also, ask yourself, “Which representations would give the most students access to the mathematical ideas?” and “What representation could I encourage students to use next?” This type of deliberate, anticipating work based around representations can provide powerful learning experiences for all learners.



Leinwand et al. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: National Council of Teachers of Mathematics.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.