Common Core Standard: 7.G.B.4
In part one, I provided some Sticky Math Triads to put the process before students and teachers. While I struggled there to provide visual representations, I believe I have accomplished the goal here as we examine the resulting relationship here in part two. What follows could accompany part one in a single lesson; although, you might be better off focusing on the manipulation of formulas for one lesson and understanding the relationship in the following lesson. Both have valuable connections to other concepts you could explore in the process.
WHERE DID WE LEAVE OFF?
Through a series of manipulations, we found that that:
What does it mean though? What is the relationship between the area of circle and its circumference? The right side is a little more complicated to understand so we will start there. Let me first acknowledge that you could save a period of instruction by simply telling students the relationship in one of the following ways:
The problem is that if you did so, then it would be soon forgotten. In fact, one might argue the value in spending any time on this part of standard 7.G.B.4, “give an informal derivation of the relationship between the circumference and area of a circle.” I personally cannot think of a single instance of when I had to use this relationship to solve a real word or mathematical problem. So why spend the time at all?
I would argue that this part of the standard allows us to explore circumference and area in greater depth while allowing students an opportunity to gain and use academic vocabulary; however, besides the process being of value in of itself as illustrated in part one the greatest value of the endeavor might be the exploration of the meaning of a quadratic term, which has multiple applications elsewhere.
With that in mind, we begin with the meaning of the expression on the right side of the equation as it is bit more difficult to envision.
WHAT DOES 4πA LOOK LIKE? BEGINNING WITH A.
Need some rigor for an honors course? Ask them that question. The problem is you would not have time to ask them as you would be rushing on to learning more content procedurally. It is a shame that we limit the beauty and understanding of mathematics through certain pathways of acceleration over others that would not do so. Common core standards define rigor as conceptual understanding, procedural skills & fluency, and application. This should be an option for all students; not a select few. Sticky Math provides the option for all students. So let’s get started by connecting to previous work from understanding the area of a circle in terms of square units from a previous blogpost (this is best done before engaging in this task):
Connections could include:
Students may not have made all the necessary connections when examining the area formula of a circle when relating it to a rectangle/parallelogram the last time (see former blogpost on the subject); this gives them another opportunity to do so. I will not elaborate much here as I have already done so in that blogpost; however I will point out the goal here is to understand what the A in 4πA means. Once we know what A looks like then we can explore πA.
WHAT DOES 4πA LOOK LIKE? πA
Once A has been established, then we can move onto seeking to understand what πA looks like:
Connections could include:
Students and adults may struggle with this one as it is rigorous; it requires the application of conceptual understanding and procedural skills together. The key is to see and understand area as base and height. The base in this case is the area of a circle reformatted into square units and the height of it is pi. Some purposeful questions to help students and adults “see it”:
Once students or adults see that the area of the circle has been repeated pi times or that you have pi groups of the area of a circle, then you can move onto examining the final part of the expression.
WHAT DOES 4πA LOOK LIKE?
If you have gotten this far, then completing the expression with the factor of four should be a little easier to see and to grasp:
Connections could include:
The main takeaway here is that the middle “sticky” is a visual representation of 4πA. Once that is established, then you can compare it to the other side of the equation circumference squared.
WHAT DOES CIRCUMFERENCE SQUARED LOOK LIKE?
Before we can compare one to another, we must establish what circumference squared looks like. The pink sticky below represents one of the larger rectangular stickies; using these helps keep the drawings here more to scale.
Connections could include:
The main takeaway here is that C squared is the green square. Once that is established, we can finally compare the two sides of the equation and examine the relationship between them. The moment we have all been waiting for and working towards! (If you made it this far…)
WHAT IS THE RELATIONSHIP BETWEEN THE AREA OF A CIRCLE & ITS CIRCUMFERENCE?
We have finally arrived. One note, I will refer to the left two drawings below as stickies, but they are really drawings made to scale. This is one of those times, where you probably will not be drawing the representations on a sticky note and projecting it. Another time is the Pizza Pi stickies above. Feel free to project them from the gallery. By the way, the yellow, blue and green are not coincidental. Remember your primary colors: yellow & blue make green!
Connections could include:
There you have it; a visual representation of the relationship between the area of a circe and its circumference! Other representations like verbal descriptions are great formative assessments as whether or not students are “seeing” the relationship correctly, and there is a third sticky above to request students to construct them. Or, you could use either of the following:
So, I feel like this will be the best blogpost that no one will ever read. I use “best” in the terms of myself taking the understanding of algebra as area to new levels personally. I’m not that arrogant to assume it is the best blogpost ever; far from it; although, I do try to always acknowledge some level of arrogance. I hope if you did make it through that it added to your understanding of algebra as area, which you can take before students whether it is in the form of these stickies or others. For more on the subject of algebra as area, see my sister site, meaning4memory.com. Should you try these stickies with students, I would love to hear how it went, and how you might modify them to make them more understandable.
Common Core Standards: 7.G.B.4
What connections are your students making? What modifications are you making to use this with students? We would love to hear your feedback; please submit a comment below or consider submitting your own Sticky Math activity here.
Common Core Standard: 7.G.B.4
I have to confess that while I have done a lot of conceptual work with students around circumference and area of a circle (see some of it by checking out my blogposts on each subject), I have not done much to tackle the latter part of this standard, 7.G.B.4, “give an informal derivation of the relationship between the circumference and area of a circle.” I have done those problems where you have to find the area given the circumference and not the radius, but I believe that is not is meant here.
In fact, I have truly neither considered the relationship between the circumference and the area of a circle nor do I know of any teachers who have talked about doing the same. They may have; they just never told me about it. This blogpost represents my first shot at understanding it. Like relating the area of a rectangle/ parallelogram to the area of a circle, I challenged myself to think of it visually each step of the way during successive manipulations to the formula. I have to admit at the onset that I have only been partially successful at this point. Some relationships I could understand, but I had a difficult time expressing them visually. Others I could not visualize at all after a first pass. I’ll have to presevere in my problem solving.
In part one, I am able to provide some Sticky Math Triads to put the process before students and teachers. I am also able to hopefully shed some light on the resulting relationship through the use of visuals in part two. So, in a break from tradition, I will actually start with the symbolic, algebraic formulas and finish with the visuals. There is always an exception to the rule isn’t there? What follows could be a single lesson; although, you might be better off focusing on the manipulation of formulas for one lesson and understanding the relationship in the following lesson both have valuable connections to other concepts you could explore in the process.
WHERE DO WE BEGIN?
We begin by looking at what we know about circumference & area. Students could brainstorm this before you unveil the Sticky Math Pair below, which incidentally might be my warm up for this lesson. The ultimate goal is for students to realize both formulas share pi and r, the radius, in common. We will focus on the radius as pi is a constant. The first method employs substitution while the second method employs the Transitive Property.
This Sticky Math Pair might seem overly simplistic and not worthy of the time; however, it provides valuable repetition, a chance to utilize vocabulary, and formative assessment as well as a chance to explore some other foundational concepts. Similarities (with purposeful questions to pose) could include:
Differences (with purposeful questions to pose) could include:
The bolded bullets above are the primary goal of the comparison; however, there are some other fundamental concepts present. Besides being a formative assessment of understanding each formula and its related vocabulary, there is a chance to connect linear and quadratics to their meanings in context. The term 2πr is linear, which is a chance to connect the “invisible,” understood exponent of one to the fact that measuring around a circle is done in a single, linear dimension; whereas, the term πr^2 is quadratic, which is a chance to connect the exponent of two to the fact that measuring the area of a circle is done in a square units, area has elements of length and width to it.
DEFINING RADIUS IN TERMS OF CIRCUMFERENCE
In order to examine the relationship between circumference and area, we need them both in the same equation. We can solve for r in one equation and substitute that value in to the other equation. In this case, we will solve for r in the circumference equation, which avoids solving a simple quadratic in the area equation – a task for higher grade levels:
Connections could include:
The key here is to see that if you divide the circumference by a little more than 6 or two pi; then you get the radius. We will need the definition of the radius in the next step.
INTRODUCING CIRCUMFERENCE BY SUBSTITUTING FOR THE RADIUS
The point of the following Sticky Math Triad is to see the substitution of the value of r into the area formula:
Connections could include:
Once again, the key here is to see the substitution of the value of r into the area formula. Notice the fact that the same variables are included in two of three formulas is a clue to that happening. Another clue that students may not pick up on is in the final bullet above, which could be shared as, “someone in my other period noted that…” (you don’t have to tell them it was you.). The presence of pi twice in the right sticky is yet another clue.
Here is a quick note about a powerful three letter word, why. Students might make & state a connection; however, that is just the beginning of the journey. Part of the purpose of Sticky Math is to promote student discourse. Ask all students to engage with one another in pairs or small groups around one student’s connection by posing a purposeful question. Anticipating these types of advancing questions (Smith & Stein, 2011) is best; however, an easy three letter fallback is “Why?” Repeating it often would not be as powerful as mixing it up with some other well anticipated questions; yet, it is sometimes preferred and can help build a culture and expectation of understanding in the classroom. Incidentally, by stopping the class to consider one student’s connection, you are promoting mathematical agency, authority & identity in that student and in others as you ask them to consider your purposeful questioning rather than simply illuminating the connection yourself via direct instruction.
SQUARING THE RADIUS
The point of the following Sticky Math Triad is to see the two new, equivalent representations of r squared that lead to simplifying the expression. Once again, this is a break from the usual multiple representation of a single concept in different ways; however, it remains a good comparison to realize how r squared can be represented in different ways:
Connections could include:
We often jump right into simplification of expressions. By providing the mathematics here, we can examine what is happening to increase procedural proficiency before asking students to engage in the process themselves. It also provides a chance to examine the reasoning behind what is being done so students can learn when and why to do such things rather than simply how.
CLEARING THE DENOMINATOR
A Sticky Math Quad is a rare sighting here on the website when it is not a part of a comparison between two Pairs. One of the connections we hope students might make might help you understand it as we move forward here. It is organized clockwise from the top left sticky:
Connections could include:
We have arrived at our goal of defining the relationship between the circumference and area of a circle, but I doubt many people, especially our students, might realize it. The relationship is not as simplistic as students are used to finding, and it is certainly not easily understood by them. While this has been a valuable exploration in developing procedural fluency in manipulating formulas, I suggest seeing it through to fully understanding the relationship exposed by this new equation, which is found in part two.
Common Core Standards: 7.G.B.4
What connections are your students making? What modifications are you making to use this with students? We would love to hear your feedback; please submit a comment below or consider submitting your own Sticky Math activity here.
Common Core Standard: 7.G.B.4
If area is defined as the number of unit squares that cover the surface of a closed figure, then how does this apply to a circle? Can we remember the abstract formula for area of a circle better by understanding where it comes from visually? Can we understand each permutation of an equation? While working with circles, what can we learn about the meanings of coefficients, linear terms and quadratic terms? These are the questions this Sticky Math Lesson sets out to answer.
This Sticky Math Lesson examines the area of a circle conceptually by connecting to the area of rectangle / parallelogram. It is helpful to examine the relationship between the area of a rectangle and any parallelogram before this lesson. Incidentally, this lesson is a good example of stopping to contextualize (MP2) in each of the permutations of the area equation. The lesson can introduce vocabulary through the drawings, offers an opportunity for students to use precise vocabulary (MP6), and acts as a formative assessment of academic vocabulary.
While I have used these visuals (in gray) with students to help them understand how the area of a circle relates to the area of a rectangle in the past, this time I challenged myself to represent visually each permutation of the area formula (in black). Along the way, I deepened my understanding of the meaning of coefficients, linear terms and quadratic terms. I hope you find it illuminating.
BACKGROUND KNOWLEDGE (WARMUP)
While not necessary, it is nice for students to know the relationship between the area of a parallelogram and the area of a rectangle before exploring the relationship between the area of circle and a rectangle as it is an intermediate step visually.
Connections could include:
Students could create their own representations; these might include:
The big takeaway here is that the area of a parallelogram can be shown to be equivalent.
STEP #1: AN ALTERNATE VIEW OF THE AREA OF A CIRCLE
Before introducing the abstract, symbolic representation of the area of a circle, we want to be sure students understand what has taken place visually. One way is to use the Sticky Math Pair below; another is to have students cut out the circle, cut the circle, and then paste it onto their Sticky Math Recording Sheet.
Connections could include:
both stickies represent the the area of a circle
shading is not representative of the entire area of the circle; it represents half the area of the circle.
the middle sticky has the area of circle organized so it is similar to a parallelogram, or a rectangle if you think of moving half of an eighth (1 sixteenth).
the curved line in the left sticky is equivalent to the longer straight line in the middle sticky
Students could create their own representations; these might include:
Students may not come up with the last two bullets; however, you can introduce them one at a time if you like on individual stickies.
STEP #2: FOCUSING ON AREA BY THE INTRODUCTION OF A FORMULA
You could skip to this Sticky Math Triad to begin, but it is a lot to take in at once without the previous Sticky Math Pair (and you do not want to overwhelm the students’ working memories).
Connections between the left and middle stickies were discussed above; additional connections could include:
The key here is to notice that the area of a circle is being defined as its height, the radius, times its “width”, half of a circumference.
At this point, you may have noticed that I have been putting some terms in quotes, mainly the references to the “base”; the reason for which is that it is made up of a group of arcs rather than a straight line. The is only an informal proof here, but it is worth noting that if you made slices progressively smaller the resulting “rectangular” figure becomes more and more like an actual rectangle. This could lead to providing some background knowledge in preparation for calculus by exploring some of the ideas behind limits and derivatives. As the arc lengths become infinitely small, the slope of the tangent line approaches zero or a straight (horizontal) line:
STEP #3: FORMULA IN TERMS OF RADIUS
We begin our move toward the commonly used πr^2 by putting our formula in terms of radius. For more on the visuals that accompany circumference, see my blogpost on Circumference of a Circle and the Meaning of Pi. Note that added side length to our parallelogram, which allows for the replacement of the full circumference by the equivalent term of 2πr:
Connections could include:
I have said this in other blogposts, but it bears repeating. It is fine to allow students to restate a connection made previously as this helps the student appropriate someone else’s connection and make it their own. It also provides a chance to use vocabulary.
STEP #4: SIMPLIFYING THE FORMULA
At this point, we simplifying the formula towards our goal of πr^2; however, the challenge is to do so visually:
Connections could include:
We can clearly see we are a single step from our goal of πr^2; however, if we stopped here, then we might not build any understanding about the meaning of a quadratic term. Before we do so, let’s stop and note the meaning of a linear term.
BONUS EXTENSION: MEANING OF A LINEAR TERM & ITS COEFFICIENT
In order to fully understand the meaning of πr, lets move through levels of abstraction to get there rather than jumping to its full abstraction. 3r is r three times:
In this case, we think of the coefficient of 3 as the number of lines and r as the length of the line.
Replacing 3 with a variable of p:
In this case, we think of p as the number of lines and r as the length of the line.
Abstracting further, we replace the rational number three with the irrational pi:
In this case, we think of pi as the number of lines and r as the length of the line. Of course, we have taken some liberty of drawing an irrational side length here by approximating it as about 3.14 (yes, the drawing is roughly to scale).
Now back to our final step!
STEP #5 SIMPLIFYING TO PI TIMES R SQUARED
We could just utilize an exponent and be finished, arriving at πr^2, but then we would miss out on deepening understanding of a quadratic term. Some might argue this is unnecessary in lower grades; however, I might remind them that students are exploring area and substituting into and evaluating quadratic terms in sixth grade. A quadratic term is two dimensional and basically can be thought of a symbolic notation of area. Let’s use our current topic as an example as we finally arrive at πr^2. In other words, we can write πr^2, but what does it look like?
Connections could include:
So we have arrived, finally! The area of a circle can be reorganized into a figure similar to a parallelogram. Using length times width or base times height we can see that the area of a circle can be thought of as the radius times one half of its circumference, which can be rewritten as the radius times pi radii or pi times the radius squared. Notice we also can “see” the area of a circle in terms of square units now.
Why take the time do this though? While I do not expect students to remember everything found here, I do want to help them remember the formula of πr^2 either remembering the activity or understanding just the final Sticky Math Triad. After the day long cutting, pasting, labeling, and working through the informal proof; you could revisit this later as a warmup/interleaved practice. By just using the final Sticky Math Triad or this Pizza Pi example, where you can use the edge crust to talk about circumference and the cheesy-pepperoni goodness to discuss the area:
I conclude with the final extension to the meaning of a quadratic term and its coefficient, since it is right in front of us anyway.
BONUS EXTENSION: MEANING OF A QUADRATIC TERM & ITS COEFFICIENT
In order to fully understand the meaning of πr^2, lets move through levels of abstraction to get there rather than jumping to its full abstraction. 3r^2 is r^2 three times:
In this case, we think of the coefficient of 3 as the number of squares and r^2 as the area of the square.
Replacing 3 with a variable of p:
In this case, we think of p as the number of squares and r^2 as the area of the square.
Abstracting further, we replace the rational number three with the irrational pi:
In this case, we think of pi as the number of squares (even though it is irrational) and r^2 as the area of the square. Of course, we have taken some liberty of drawing an irrational side length here by approximating it as about 3.14 (yes, the drawing is roughly to scale).
Well, that is about all I can pull out of that! If you made it this far, then you are to be commended. Feel free to submit for time spent in professional development. Just not to me… maybe that will come much later down the road!
Common Core Standards: 7.G.B.4
What connections are your students making? What modifications are you making to use this with students? We would love to hear your feedback; please submit a comment below or consider submitting your own Sticky Math activity here.
Common Core Standard: 7.G.B.4
These Sticky Math activities can each stand alone; however, each of these blogposts suggests how you might put them together to form a lesson. You can always adapt them to suit your needs.
This Sticky Math Lesson helps build fluency from conceptual understanding as students make connections between representations of circumference. Remembering a relationship is easier than remembering an abstract formula. Apply and extend understanding of a linear term in algebra as students deal with abstraction to a coefficient of pi.
This Sticky Math Lesson examines the circumference of a circle conceptually by exploring the meaning of pi. This lesson can introduce vocabulary through the drawings, offers an opportunity for students to use precise vocabulary (MP6), and acts as a formative assessment of academic vocabulary.
UNDERSTANDING THE PICTURE
Our first Sticky Math Pair might seem confusing at first at their seems to be 2 longer lines and one smaller line outside of a circle, which is true; however, the aha comes by realizing these are three diameters plus a little more:
Connections could include:
Students could create their own representations; these might include:
The big takeaway here is that three diameters plus a little bit more are equivalent to the circumference of the circle, which prepares them to define pi.
DEFINING PI
This Sticky Math Pair defines pi as the ratio of circumference to diameter. In this case, the ratio is a fraction too. The challenge might be in “seeing” the division of circumference by diameter:
Connections could include:
Students could create their own representations; these might include:
Like previous posts, students may not come up with the last two bullets; however, you can introduce them one at a time if you like on individual stickies. Note the extra wording in the final bullet above. We might be quick to emphasize pi times the diameter asking students to just memorize it; however, students should understand it is the diameter pi times or pi groups of the diameter, which is done explicitly in the following stickies.
THE FORMULA FOR CIRCUMFERENCE IN TERMS OF DIAMETER & UNDERSTANDING A LINEAR TERM
If you have chosen to define pi before now like done above, then this Sticky Math Triad should be an easier task; however, it is still a necessary task in making connections to the symbolic representation of circumference.
Connections could include:
The key here is to notice that the circumference is equivalent to the diameter pi times or pi groups of the diameter. A related, supporting task for understanding the meaning of πd might be the Sticky Math Pairs that follow:
The idea here is to realize one possible meaning of the coefficient; in this case, 3 means the number of lines with an unknown length of x.
We abstract this a bit more in the next Sticky Math Pair:
This takes us back to our original & final level of abstraction of thinking of a symbol as a coefficient or number of groups.
So πd, means about 3.14 groups of diameter to get around the circumference of a circle. We will see this need/opportunity for understanding algebraic terms again when we visit relating the area of a circle to the area of a rectangle in my next blogpost. In preparation for that work, we consider circumference in terms of radius.
CIRCUMFERENCE IN TERMS OF RADIUS
You may want to declare success in getting students to truly understand the formula for circumference in terms of diameter; however, this Sticky Math Triad supports fluency as students define circumference in terms of something new, and it prepares them for understanding the area of a circle (see my next blogpost). The first sticky in this Sticky Math Triad should be familiar by now:
Connections could include:
I am convinced students struggle with understanding substitution. I know many can substitute a value into an expression; however, I am not sure how many truly understand what they are doing. This becomes especially clear once you reach solving linear systems via substitution or things like the Transitive Property. What is shown in the Sticky Math Triad above is a visual example to support understanding the substitution being done, replacing the diameter with two radii.
Incidentally, here is a great strategy to boost the understanding of changing the form of an equation or formula – challenge yourself to do so visually as well as symbolically. I give an example of this in my next blogpost as we relate the area of a circle to the area of a rectangle.
Common Core Standards: 7.G.B.4
What connections are your students making? What modifications are you making to use this with students? We would love to hear your feedback; please submit a comment below or consider submitting your own Sticky Math activity here.
Common Core Standards (different examples support different standards): 6.EE.A.2, 7.EE.A.1, 8.F.A.2, 8.F.A.3, HSA.SSE.A.1.A, HSF.IF.B.4
Besides using Sticky Math to compare two different representations or match representations, you could provide two nonequivalent prompts in a singular representation. For example, below we have two nonequivalent expressions in a single representation, abstract symbolic:
At first glance, this seems overly simplistic; however, students often struggle with understanding and using exponents. This is a good example of a time when students might be procedurally proficient, but not procedurally fluent. They may be proficient in performing operations on or with exponents; however, they demonstrate a lack of procedural fluency when having to flexibly apply their use to interpreting expressions.
Students should choose a representation and construct a viable argument to defend their selection. As one student or group of students defend their argument, the others should be engaged through math talk moves and asked if they agree or disagree and why (MP3).
Notice that while there is no blank sticky on the image above to remind you, students can still use their third sticky to create an additional representation of two times x plus three. Other representations might include drawing algebra tiles or writing an equivalent expression like 1x +3 + 1x.
Here is another example centered on confronting a common misconception head on:
This Sticky Math Which One & Why? is designed to create cognitive dissonance with those students who would ignore order of operations and add 5 + 2 in the left expression to incorrectly get an equivalent expression.
Notice they are asked to simply one of the expressions as an added layer of formative assessment; they can do this on their third sticky. While this Which One & Why? acts as a formative assessment of the common misconception, the instruction that precedes it or follows it should be done in context. For an example of how to do this, see the presentation from MaTHink 2020 called Algebra as Area: Distributive Property at Meaning4Memory.com/presentations. On the left sticky, you cannot combine the 5 with the 2 because the 5 refers to a number of items while the 2 refers to a number of groups. You would first have to find the number of items in the the two groups before adding them to the other 5 items. Students need context to grasp this. The context in the previously mentioned presentation linked above is picking apples. Applying that context here would go something like this, “Two parents picked 3 baskets of x apples plus they each picked up four apples from the ground while their child picked up 5 apples from the ground. Write a simplified expression for how many apples they picked in all.” Students should be able to decontextualize the prompt into an expression; however, they should also be able to contextualize as they simplify. If they did, then they would realize that they should not be combining a number of apples, 5, with a number of parents, 2.
If you wanted to use this Which One & Why? even more formatively, then you could introduce the possibly that both expressions were correct first like in this example:
You may want to introduce one like this early to open up the chance that both stickies could be correct in future iterations of the activity.
Notice this activity includes an extension of evaluating both expressions when x = 10. It can be done on a single third sticky by writing small. This can act as a formative assessment of this skill; however, it also provides a specific case to demonstrate that both expressions are numerically equivalent. Students may choose to evaluate the two original expressions or their simplified versions; this provides an opportunity for further comparison and connections when having students debrief the activity (EMTP: Elicit & Use Evidence of Student Thinking).
These final two examples demonstrate applying Which One & Why? to something other than expressions and a different type of representation, a graph:
This example intentionally excludes a more obvious symbolic notation of a positive slope. Do students understand how the slope in the right sticky is reflected in the graph? Or, do they just make two negatives a positive according to some memorized rule that they may or may not understand? If they do change it to a positive slope, then you have an opportunity to have students compare and contrast the methods and how it moves you along the line as they help debrief the activity (EMTP: Elicit & Use Evidence of Student Thinking).
Do students understand that the negative in the left sticky does not distribute to both the numerator and the denominator? The prompt forces a choice; however, if they are open to choosing both like the previous example suggests, then they might incorrectly suggest that neither would match the graph or both would match the graph (thinking it distributed to both the numerator and the denominator).
Notice once again that the prompt extends to formatively assess a different skill, graphing. Sometimes textbooks make an error of excluding this type of interaction by having students select a matching graph when the standard specifically states, “Students graph…” This extension provides them the chance to graph the unmatched linear equation.
Here is a different way to interact with graphical representations:
This Sticky Math Which One & Why? focuses on the solution to a quadratic equation rather than moving along the graph itself.
Notice that the prompt extends to formatively assess whether or not they know what the graph of a single solution looks like. They can sketch the graph on a regular sticky or use Post It Grids:
How are your students using Which One & Why? What connections are your students making? What modifications are you making to use this with students? We would love to hear your feedback; please submit a comment below or consider submitting your own Sticky Math activity here.
Common Core Standards: 8.NS.A.2 . Disclaimer: This post includes data from estimating square roots, but focuses on the concept of a square root. More on how to estimate square roots in a future blog post.
I was working with a teacher who had just given a Smarter Balanced Interim Assessment Block on Expressions and Equations. She had not realized that the assessment included estimating square roots as one of the types of expressions. I used the opportunity to explore the idea of student preconceptions & misconceptions and how they could inform first, best instruction.
It was clear that students were making the common mistake of dividing the radicand by two like finding the square root of 16 as 8 rather than 4. We discussed how we might confront this error directly during the initial instruction knowing the importance of students’ first experience with a new concept.
A week later, I performed a professional development event for a middle school site in our district focused on using assessment data to reengage students more deeply while dealing with their documented preconceptions & misconceptions. I used the experience above as an example.
Incidentally, this misconception seems to be related to number sense as it becomes more pronounced as the size of the radicand increases. Here is some actual data to illustrate this point and where we were in our discussion:
You might notice the highlighted spread of incorrect responses: 22%, 33% and 17%, which seems to indicate a gap in student understanding. The teacher’s own report that she had not yet done any instruction in this area supports this fair assumption. Notice what happens to incorrect responses though as the size of the radicand increases:
The incorrect responses are now more unbalanced with 33% choosing C compared to 6% and 11% for the other two distractors. This becomes even more pronounced, 50%, if you take into consideration that distractors C & D are both related to the common misconception of dividing by two, which leads to identifying a misconception or in this case a preconception.
I offered the following as an example of using this information on preconceptions to guide first, best instruction:
I have taught this content before via direct instruction and had some students somehow come away with the same misconception; therefore, I suggested making it a student task. Let me stop to define my terms here. I am using task in this context to denote the shift of responsibility for sense-making from the teacher to the student. You could save time and literally tell students, “Do not divide by two to find a square root;” however, now you are relying on innate procedural memory rather than developing mathematical understanding. Furthermore, the teacher acts as the provider of knowledge rather than the students developing mathematical agency, authority and identity. This leads to two points:
student tasks do not have to be large events & taking the time to do them initially will save you time in the long run by decreasing the need for reteaching later.
Let’s examine the task itself. Rather than starting with the square root of 16 and breaking it down into its factors demonstrating how to get the answer of 4,
what if we provided the mathematics already completed and asked students to make sense of it?
In this case, the square root of 16, 4, is provided along with a different representation of a square with a side length of 4 and an area of 16. By the way, do students even know that square roots are related to actual squares from our instruction? This representation of an actual square has been chosen over the more abstract form of area, (4)(4), while students are just beginning to form an understanding of the concept in order to avoid the error of the presence of two factors indicating that one can divide by 2. It is hard to get an answer of 8 looking at the picture; however, students lack of number sense surrounding multiplicative thinking or misapplication of additive thinking while looking at two 4’s may lead to an answer of 8. Debriefing this task or prompt, lays a conceptual foundation for the rest of the instruction while incorporating the standards for mathematical practice, specifically elements of MP1, MP2, MP3, MP6 & MP7, which also makes it a valuable use of time. While we have laid a conceptual foundation to help avoid the mistake, we have not attacked the misconception head on.
The following images represent my attempt to attack the misconception head on:
Please excuse the incomplete radical symbol above. Research supports that at some point more examples fail to have an effect, but error analysis has been demonstrated to have a profound effect. Not only are we addressing the misconception head on in just the second example presented to students, we have kept the responsibility for sense-making with the student and increased the level of rigor by asking if Kayla’s shortcut works all of the time, some of the time, or none of the time, which is an effect strategy to add to almost any instruction. While the prompt focuses on the shortcut, the discussion should focus on the difference between the correct conceptual understanding as opposed to a shortcut happening to arrive at the same answer. By the way reader, does her shortcut work all of the time, some of the time or none of the time?
Notice also the presence of the actual 4×4 square to activate knowledge just acquired and provide a structure for students to test their theories on. The square root of 4 happens to work on Kayla’s method, because 2+2=4 and (2)(2) = 4. Looking at a drawing of the perfect square of 4, we can see that it is an example of a time where half of the product is equal to its square root or side length:
OK, but how did all of this lead to Sticky Math being born?
As I related the story to Derek Rouch, a fellow math aficionado, I grabbed two stickies to record the two representations. I often grab the nearest scratch paper or whiteboard as I am explaining mathematics to establish a visual connection to what I am saying.
I explained, “Tell me what each of these symbols mean [on the right sticky], using only this picture [on the left sticky].”:
When I finished, Derek exclaimed, “Oh, Sticky Math!” We both stopped in our tracks considering the implications of what he had just said. We both looked at one another with the realization that we were on to something so simple yet so profound.
You see, with Sticky Math, understanding becomes the task at hand rather than a goal I hope to reach through some long winded instruction to which I am often prone. Connections necessary for the retention of information are the task rather than a goal I hope students internalize. When used as a warmup, Sticky Math can be a valuable source of interleaved practice, which has been proven very effective in aiding retention.
We believe using Sticky Math will improve procedural fluency as students learn what the procedure does as opposed to only what the procedure is. As mentioned previously, we also believe it has the chance to build students’ mathematical agency, authority, and identity as they begin to interpret mathematics for themselves.
By often providing the mathematics, students’ working memories are freed up to make meaning and connections. It also acts as formative assessment for teacher and student, providing a source of self-assessment and reflection for students.
Sticky Math provides opportunities for students to participate in mathematical discourse and use content specific, academic vocabulary. It provides students with an opportunity to construct viable arguments and critique the reasoning of others along with many other Standards for Mathematical Practice.
Sticky Math can be a quick easy way to bring conceptual understanding into the classroom as students connect mathematical representations, one of NCTM’s eight Effective Mathematics Teaching Practices. In fact, Sticky Math fulfills elements of all eight of the Effective Mathematics Teaching Practices.
Finally, Sticky Math can be a source of professional development as teachers learn to teach conceptually through multiple representations. It is short, incremental professional development that can be implemented immediately. Likewise, this immediacy applies to results as teachers can assess the understanding of their students and reflect on their own implementation & facilitation of the process.
We hope you join the Sticky Math movement, and maybe even consider becoming a contributor. You can do so by clicking here. We would love to see what you are doing with Sticky Math and hear what connections your students are making. Please feel free to submit a comment or question below.
Common Core Standards: 8.NS.A.2
Common Core Standard: 8.F.A.2, 8.F.A.3
Use: Individually as a quick Sticky Math Warmup, in groups for a longer Sticky Math Warmup, or all together as a Sticky Math Lesson
Note: Post It sells sticky notes with grids on them, which makes using Sticky Math with graphing much simpler.
These Sticky Math activities can stand alone; however, it is a continuation of the post Slope Intercept Form; hence, it begins with stage four.
STAGE FOUR: EXPLORING THE DIFFERENT PLACEMENTS OF THE NEGATIVE SIGN
This set consists of a Sticky Math Triad, a Sticky Math Pair and a Sticky Math Comparison. The point of all three is to realize that wherever a single negative sign is placed the result is a negative slope. The goal is not to memorize this as a fact; students should be able to explain why. Students / teachers focused only on “rise over run” may miss this in the Sticky Math Triad. The Sticky Math Pair that follows might lead to this realization as the negative sign is in neither the numerator nor the denominator. If this fact has not been discussed by the Sticky Math Comparison, then teachers should pose purposeful questions to draw it out.
Connections could include:
Students could create their own representations; these might include:
Students may not come up with the final two bullets. You could introduce them one at a time on new stickies asking which representation they describe. While they fit one equation literally, each one fits all the representations. If you owed 2 dollars more every 3 days, then you would have had 2 dollars more 3 days ago. How is the next Sticky Math Pair the same too?
Connections could include:
The final bullet above is a key understanding conceptually to build fluency. If it is not made here, then it needs to be made on the Sticky Math Comparison that follows.
Students could create their own representations; these might include:
Students may not come up with the last bullet above. Relating it to the discussion immediately preceding this stage: if you owed 2 dollars more every 3 days, then you would have had 2 dollars more 3 days ago. If I owe 2 dollars more every three days, then how much do I owe EACH day? Posing this purposeful question can create an opportunity to explore and build fractional understanding. The question could be rephrased, “what is amount is repeated 3 times to yield 2 dollars?” As an equation, (x)3 = 2 where x is 2/3 so (2/3)3 = 2, which brings in a discussion of the multiplying of fractions. Or, thinking of it as addition, 2/3 + 2/3 + 2/3 = 6/3 or 2.
You could also introduce a sticky like the one on the left below, and discuss its meaning by posing purposeful questions:
Notice by creating the complex fraction the negative has been assigned to the numerator, which aids in seeing it as a single quantity for those who have only learned “rise over run” as a procedure. It also make the description, “I started with one dollar. I owe 2/3 of a dollar each (1) day.” more explicit by including the one day. Incidentally, this a chance to reengage with complex fractions without going back to reteach seventh grade. Remind students that it is always one quantity no matter where the negative sign is written, which is the point of the Sticky Math Comparison that follows.
These Sticky Math Comparisons are one last chance to make the connections discussed previously. Choose one depending on the discussion you have had to this point or want to have at this point. Either can be used by the teacher as a formative assessment as to whether or not students have internalized the connections discussed previously. The main point is that no matter where the negative is placed on the coefficient of x, it is a multiplier of x. As x increases, the entire term becomes more negative, which decreases the value of the expression and y.
STAGE FIVE: WHAT WOULD TWO NEGATIVE SIGNS MEAN ON THE SLOPE FRACTION?
Why should we even care what the right sticky means below? Can’t we just say, “the opposite of a negative is a positive” and move on?
Being able to convert a positive slope into a double negative can help students graph points using only slope intercept form moving in the “opposite” direction. For an example, see the first two connections below:
Connections could include:
Students could create their own representations; these might include:
Like previous posts, students may not come up with the last two bullets; however, you can introduce them one at a time if you like on individual stickies. While literally one verbal description fits the left sticky and the other fits the right, they are actually describing the same event differently; they are equivalent expressions. If you received 2 dollars every 3 days, then 3 days ago (-3) you had 2 dollars less (-2).
STAGE SIX: PUTTING IT ALL TOGETHER
The Sticky Math Comparison below allows students to share what they have learned in previous stages acting as a formative assessment for teachers while demonstrating intentionally that the presence of a negative in the numerator and the denominator is not equivalent to the presence of a single negative.
Connections could include:
A closing, formative assessment activity might be to complete a Sticky Math Matching activity like the one below:
Any stickies from the six stages could be used to create a Sticky Math Matching activity. Four representations for each function have been used here. You could build up to this by including t-charts, table of values, or verbal descriptions in the discussions to this point as mentioned above.
Some purposeful questions you might pose about the t-charts / table of values
The addition of the t-charts / table of values leads to the exploration of slope as the change in y over the change in x. Rate of change should be emphasized by seeing it as how you move from one point to the other. The fact that a t-chart / table of values like these are made up of individual points should also be stressed. Another realization is that, in this case, the entire t-chart . table of values represents a line or linear function.
Lesson extension, classwork or homework – give each students 8 stickies have them create their own matching game to illustrate two different linear relationships; or, if you have done the work, then one linear relationship and one non-linear relationship.
Have them give it to a family member to complete, and then they check it. Have students exchange them when they are finished or return to class, match the pairs, and check each other’s work as a ticket out the door or an opening activity the following day.
One final word on slope and graphing, I hope you have seen through this blog post that being procedurally proficient with “rise over run” is not sufficient by itself to achieve procedural fluency. Besides being sometimes misleading, negative slopes are not rising, it does not account for a negative that is neither in the numerator nor the denominator. Different representations beg for different ways to view slope;
exploring each of these conceptually should improve procedural fluency.
Common Core Standards: 8.F.A.2, 8.F.A.3
What connections are your students making? What modifications are you making to use this with students? We would love to hear your feedback; please submit a comment below or consider submitting your own Sticky Math activity here.
Common Core Standard: 8.F.A.2, 8.F.A.3
Use: Individually as a quick Sticky Math Warmup, in groups of three for a longer Sticky Math Warmup, or all together as a Sticky Math Lesson
Note: Post It sells sticky notes with grids on them, which makes using Sticky Math with graphing much simpler.
STAGE ONE: EXPLORATION OF SLOPE-INTERCEPT FORM
The Sticky Math Pair below allows for the discovery of the slope and the intercept in two different forms. Care should be made to elicit, use, and connect various vocabulary like rate of change, slope, rise-over-run, y-intercept, and constant.
Connections could include:
The blank sticky is a reminder that students could create their own representations; these might include:
Creating a verbal description can be a fantastic formative assessment and lead to a higher DOK level. You might give an example the first time if no student creates one of their own using the line, “someone in another class wrote this…” You don’t have to tell them the person in the other class was you! Later, you might ask for verbal descriptions as one of their alternate representations as a formative assessment of their conceptual understanding. Hopefully, at some point, students would be comfortable creating them on their own and just do it by themselves. Creating different contexts could also make the task more culturally relevant. If students struggle in this area or as introduction, then create Sticky Math Pairs or a Sticky Math Triad like the one below:
Notice the equation is written with the constant first to match the verbal description chronologically and to change it up in order to help students achieve procedural fluency.
STAGE TWO: EXPLORATION OF THE Y-INTERCEPT
If you are doing one Sticky Math Pair per day as a shorter Sticky Math warmup, then you might move to the second Sticky Math Pair below followed by the Sticky Math Comparison on the third day. The three of them are together here to show how you could group them into a slightly longer Sticky Math Warmup in order to make the point about the meaning of the y-intercept more explicitly. Alternatively, you could just jump to the Sticky Math Comparison to accomplish the same thing more quickly.
Connections could include:
The blank sticky is a reminder that students could create their own representations; these might include:
STAGE THREE: EXPLORATION OF SLOPE
If you are doing one Sticky Math Pair per day as a shorter Sticky Math warmup, then you might move to the second Sticky Math Pair below followed by the Sticky Math Comparison on the third day. The three of them are together here to show how you could group them into a slightly longer Sticky Math Warmup in order to make the point about the meaning of the negative slope more explicitly. Alternatively, you could just jump to the Sticky Math Comparison to accomplish the same thing more quickly.
Connections could include:
Students could create their own representations; these might include:
TO BE CONTINUED…
This topic is further explored in the post Negatives in a Slope Fraction. Sticky Math Comparisons and Sticky Math Matching activities are included there.
Common Core Standards: 8.F.A.2, 8.F.A.3
What connections are your students making? What modifications are you making to use this with students? We would love to hear your feedback; please submit a comment below or consider submitting your own Sticky Math activity here.
Common Core Standard: 3.OA.B.5
A week’s’ worth of warm ups or a single lesson; the choice is yours. If you need to revisit for conceptual, interleaved practice to build fluency and retention, then use this as a week of warmups. Alternatively, you could use this as a single lesson building conceptual understanding of multiplication and as an introduction of the distributive property.
DAY ONE WARMUP/ STAGE ONE OF A SINGLE LESSON
Multiplication as an area/array. You could use a pure array, but drawing 56 circles is too tedious, and remember Sticky Math is meant to be quick an easy enough to do on a sticky. This may look like too much to draw; however after drawing a rectangle you are simply drawing some lines with a straight edge. These lines extend past the side of the rectangle to help students find the side lengths.
Connections could include:
The blank sticky is a reminder that students could create their own representations; these might include:
These leads me to a general point about using Sticky Math. What if students do not make these type of connections or a specific connection you would like them to make? Try saying, “Someone in my class last year made this representation,” and draw it on a third sticky. Ask students how it connects to the two existing stickies. You do not have to tell them that the person in the other class was you. When you do this, you are building on their existing knowledge (of the two stickies), which is the true definition of scaffolding.
If students are not making any connections at all, then the activity may not yet be appropriate for them. It is good to have any easier activity on hand as a “Plan B” should it not go well. That said, even when an activity does not go well, it is still gives you valuable formative feedback about what students know and are able to do so you can adjust your instruction accordingly.
DAY TWO WARMUP / STAGE TWO OF A SINGLE LESSON
The idea of day two is to explicitly see the decomposition of the 8 x 7 into 8(5) + 8(2). No symbolic representations are given purposefully to keep the activity fully conceptual and to not overwhelm the students’ working memories with an additional form of representation at this stage; however, should students suggest their own symbolic representation during the connection phase, then that is fantastic. This activity sets the stage for understanding what the symbolic notation means the following day.
Connections could include:
The blank sticky is a reminder that students could create their own representations; these might include:
I doubt students would create the last representation; however, I add it for those using this activity with higher grade levels. Changing the problem to (13)(12) = (10 + 3)(10 +2) would be great for higher grade levels leading to area representations with base ten blocks, which could be further connected to (x + 2)(x + 3) in algebra. That sounds like another blog post!
This leads me to a question and another general point about using Sticky Math, do you feel like we are moving too slow in this progression? I hope so. We have to remember that we have fully developed brains while our students do not, unless you are teaching 25+ year olds. Sometimes math is presented too fast (especially when it is focused on procedures only), connections are not made explicitly (leading to a lack of retention), and there is very little depth (leading to a lack of fluency). The goal with each day of the warmup or stage of the lesson is to build a large web of connections so content can be understood and retained leading to fluency and a greater depth of knowledge.
DAY THREE WARMUP / STAGE THREE OF A SINGLE LESSON
The idea of day three is to explicitly see the decomposition of the 8 x 7 into 8(5) + 8(2) symbolically, pictorially, and to make connections between the two. Students need to make connections between the symbols used and their meaning by connecting to the pictorial representation; otherwise, it may only appear to be a bunch of numbers no matter how often we explain it to them verbally.
Connections could include:
Students could create their own representations; these might include:
Once again, I doubt students would create the last representation; however, I include it to point out that you can give students strategies for finding 8(7) with only knowledge of the math fact families of twos and ones. By the way I choose this example specifically, not only because it is written into the standard, but because it is a math fact many of my secondary students struggle with.
The strategy is not just for remediation; it can be used to extend into higher content. The applications of the distributive property are endless. Here is just one adapting the example in the final bullet: 8(2221) = 8(2000) + 8(200) + 8(20) + 8(1). Students may be able to identify place value, but does it inform their mathematics?
Using this activity with other difficult to memorize math facts will give students a strategy that works for any math fact they should know or could know. Better yet, as a formative assessment of the distributive property and to make this idea explicit, ask students to create their own Sticky Math Pair or Sticky Math Triad for any hard to memorize math fact like 8(3), 8(4), 7(6), etc.
DAY FOUR WARMUP / STAGE FOUR OF A SINGLE LESSON
The idea of day four is to explicitly see the decomposition of the 8 x 7 into 8(5 + 2) symbolically, pictorially, and to make connections between the two. The only change from day three is the symbolic representation. In day three, it is helpful to view the area of each piece individually or circle each piece vertically. Here it is helpful to view it as eighth rows of five plus two or circle each row.
Connections could include:
Students could create their own representations; these might include:
splitting the middle image in a different way like 4 + 4 instead of 5 + 2
DAY FIVE WARMUP / STAGE FIVE OF A SINGLE LESSON
Full symbolic! The idea is to realize that all three are equivalent, but refer to different quantities. To help ensure full understanding conceptually, it may be helpful to have the two pictorial representations from day/stage two present for reference. Better yet, to increase the connectivity, conceptual understanding, and formative assessment inherent in the activity, ask students to draw a different picture for each representation.
Connections could include:
Students could create their own representations; these might include:
A closing, formative assessment activity might be complete a Sticky Math Matching activity like the one below:
The images above are different than those in the rest of the activity. The idea is to see if students can demonstrate transfer of the knowledge they have attained by moving to a higher level of abstraction while matching up pairs. The many small squares are unmarked but can still be reporting back as, “The one without a number goes with D.”
Lesson extension, classwork or homework – give each students 8 stickies have them create their own matching game to illustrate a hard to memorize math fact for them, have them give it to a family member to complete, and then they check it. This differentiates the assignment as some could work with 8(4) while others might work with 2(16). Have students exchange them when they are finished or return to class, match the pairs, and check each other’s work as a ticket out the door or an opening activity the following day.
One final word on the usefulness of the distributive property as a strategy to memorize math facts. Our brain is an amazing tool for mathematics. Should a student use this strategy for finding unknown math facts not only will it apply everywhere, if they do it enough times with a single fact, then their brain will look for a shortcut and just memorize the fact without having ever picked up a single flash card! (And, if you do use flash cards, then try the triangular type that you can multiply and divided with.)
If you have read this far, then give yourself a pat on the back! You are a dedicated teacher and your students are lucky to have you!
Common Core Standard: 3.OA.B.5
What connections are your students making? What modifications are you making to use this with students? We would love to hear your feedback; please submit a comment below or consider submitting your own Sticky Math activity here.
1.OA.C.6
The Sticky Math Triad below asks for students to make connections between representations while beginning to explore the Commutative Property. By asking students to create additional equivalent representations, students may choose to find the sum by counting on, creating a ten, or creating a known equivalent (e.g., 5+5=10) pictorially or symbolically.
By simply changing the direction of the arrow, you can change the visuals in this Sticky Math Triad from addition to subtraction:
If using this as a warmup, then you could do addition on day one, subtraction on day two, and compare them on day three to explore the relationship between addition and subtraction. The Sticky Math Comparison Template would be a good place to record the relationships on all three days (e.g., knowing that 7 + 3 = 10, one knows 10 – 7 = 3 or knowing that 3 + 7 = 10, one knows 10 – 3 = 7 ).
By modifying the numbers to something similar to 8 + 6, you could continue to explore making a ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14). Something like 13 – 4 could facilitate decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 =10 – 1 = 9 )
What connections are your students making? What modifications are you making to use this with students? We would love to hear your feedback; please submit a comment below or consider submitting your own Sticky Math activity here.