Hope your winter is going well, only 45 days until spring! To get through this season, check out the February 1993 Calendar of Problems from my cache of old Mathematics Teacher^{1} issues still hanging around my attic. Share with your students if desired, or solve them yourself. Tell us about your worked-out methods either here in the comments, or on whatever platform you use^{2}.

I’ll post the solutions pages at the end of the month, and pick a new month from my attic to share for March. Happy solving!

If the image isn’t readable, the pdf of the February 1993 calendar is HERE.

Notes & Resources

[GIVEAWAY ALERT!] If you’d like to take some old issues of Mathematics Teacher out of my attic storage, I’m always looking for new homes for them. Contact me & I’ll send some to you (US addresses only).

^{1}The Mathematics Teacher journal is a legacy journal from NCTM — the National Council of Teachers of Mathematics — the professional organization supporting math educators in the US and Canada. There are bountiful resources available to members at https://www.nctm.org/, along with some free resources.

^{2}The hashtag MTBoS is an acronym for “Math Teacher Blog-o-Sphere” and is an online community of math educators using a variety of platforms (Twitter, Facebook, Mastodon, etc.). If you are looking for helpful educators, shared resources, and thoughtful discussions, find us wherever you are online.

If you want to get an email notification when I post the next month’s problems, enter your email address here:

I’ve been using “Same and Different” as an inquiry strategy with my students for several years. Read on for more about this thinking routine and some of my favorite prompts.

What is “Same and Different” ?

“Same and Different” is an inquiry strategy sometimes known as “Compare and Contrast” or various other names (see Resources links below).This powerful strategy asks students to compare and analyze features of two mathematical situations. They may require different solution strategies, be similar except for one feature, or have mathematically meaningful nuances to notice.

The routine is launched by presenting two or more math situations, then have students examine and note how they are the same and how they are different. This is a great opportunity to use technology to illuminate some of the numerical and graphical differences, although technology is not required.

The examples that follow are grouped (loosely) by theme, and many can be used in several math levels. Keep in mind that not all students will find all the similarities and differences, and you can highlight the details that are pertinent to your curriculum.

Representation Examples

These examples begin with a visual representation to tap into students’ interpretation of the graph or diagram.

Example 1: (PreAlgebra or Algebra 1) The following two images show these subtraction problems: 5 – 2 (top) and 2 – 5 (bottom). I like beginning with the number line representation because it spurs more discussion than simply presenting the numerical expressions.

Example 2: (Algebra 1) These images get students talking about slope and y-intercept of linear equations.

Example 3: (Algebra 1 or 2) These graphs of quadratic functions show various transformations of the parent function (graphs 1 & 3). Discussion can highlight the types of transformations, how they impact the equation, and what the graphs have in common.

Example 4: (Geometry) Continuing with the idea of transformations, students can interpret these geometric transformations on the coordinate plane. Discussion can analyze what is the same and different about the pre-image and image, contrast the two transformations, and generalize a coordinate rule for these motions.

How to Solve? Examples

The next set of examples presents problems that can be solved with a variety of methods. Students can debate pros and cons of the available techniques.

Examples 5, 6, 7: (Algebra 1) Solving linear equations: Which inverse operations are appropriate? Is it beneficial to distribute first or not?

Example 8: (Algebra 1 or 2) Solving systems of equations. Which method would you choose: graphing, substitution, or elimination?

Example 9: (Algebra 1 or 2) Solving quadratic equations. Would you factor, use inverse operations, or use the quadratic formula? Or something else?

Example 10A: (Algebra 2 or Precalculus) Solving equations involving exponents. Is the next step to take a root or a log of both sides, or is there another technique available?

Features of Functions Examples

In these examples, think about how graphs or numerical tables of values can be used to determine the similarities and differences between the functions. Don’t forget to zoom in or out on the graph to see key features and end behavior, and change the table increment if warranted.

Example 10B: (Algebra 1 or 2) Linear vs. Exponential functions. For more about this scenario, check the first part of my post Table Techniques, with student activity sheet here.

Examples 11, 12, 13: (Algebra 2 or Precalculus) Students analyze the features of power functions, exponential functions, and logarithmic functions. For examples 12 & 13, what happens with other bases?

Example 14: (Algebra 1 or 2) Forms of Quadratic Functions. These are all actually the same—but what information about the parabola graphs is most clearly visible in each of these equations?

Example 15: (Algebra 2 or PreCalculus) Rational Functions often can be simplified. Is the simplified version exactly the same as the original?

My favorite way to examine the above situation is to use the TI-84+ ZoomDecimal window; since it equalizes the pixels, a hole is visible in the graph of the rational function. Then, use the “tracer ball” graphing style to graph the simplified linear function–it goes ‘through’ the hole, demonstrating that the domains of the two graphs are different.

Another nice question is how does the rational function’s graph compare to the numerator quadratic alone?

Simplification and Notation Examples

Examples 16 & 17: (Algebra 1 or 2) Simplifying fractions and radicals. What is possible, what is necessary in each situation? Use a calculator or grapher to check.

Examples18 & 19: (Algebra 1 or 2) Dealing with an exponent of –1. Does it “go to the denominator” or does it “flip”? When does an exponent “distribute”?

Example 20: (Algebra 2 or Precalculus) I’ve found that –1 is one of the most challenging notations for students, because it means different things in different situations. How do you read these notations aloud? What does each mean? What other ways can each be written?

Example 21: (Algebra 2, Precalculus) Notation issues can get confusing with trigonometry, so I like this prompt to tease out the meanings of the 2 in each case. This is also helpful for calculus students who need a review! Discuss algebraic meaning and check the graphs for visual evidence.

Geometry Examples

Examples 22 & 23: (Geometry) What are the similarities and differences between polygons and regular polygons? What about convex and concave polygons?^{1}

Example 24: (Geometry) What is the same and different about these 3 triangles? What can you determine about their angles?^{2}

Example 25: (Geometry) Quadrilaterals of all types! If you are teaching special quadrilaterals, ask students to compare and contrast any two of them:

Squares and Rectangles

Parallelograms and Rhombuses (rhombi?)

Trapezoids and Isosceles Trapezoids

Rhombuses and Kites

Congruent triangles vs. Similar triangles (not quadrilaterals, but a great prompt)

Bonus Examples: Absolute Value & Sequences/Series

Bonus Example1: (Algebra 2) Absolute Value Equations and Inequalities. How can you represent each of these on a number line? How do you solve each of these? Can a graph in the coordinate plane help?

I prefer the “distance from zero on the number line” explanation for absolute value. So in the equation, we need any points that are exactly 2 units from zero (2 and –2). In the inequalities, we need values that are MORE than 2 units away (x –2 or x 2) or LESS than 2 units away from zero ( –2 x 2). In class, I wave my hands in the air along an imaginary number line to help students visualize the correct intervals.^{3}

The coordinate plane can help students understand these situations: graph y = 2 and y = |x| at the same time. The intersection points are where the graphs are equal (top image below). Then look for x-values where y-values of the |x| graph is greater than 2 (middle image) or where y-values of the |x| graph is less than 2 (bottom image). [Read more about a coordinate plane graph visual in my post How Else Can We Show This? part 2 “Absolute Certainty”—with video!]

Bonus Example 2: (Algebra 2 or Precalculus) When studying sequences and series, students learn several formulas. The sums of geometric series formulas (finite and infinite) are actually the same formula for large values of n if r is between -1 and 1! Use this prompt and numerical examples to examine this situation. Why does this formula have a restriction on the size of common ratio r?

Wrapping Up

I’ve used Same and Different prompts as a lesson opener, exit ticket, stimulus for discussion, and during class review; they work well as formative assessment or consolidation anytime. Be sure to annotate and share student thinking and summarize important nuances with the whole class.

With a 3-part prompt, consider presenting two parts first, then add the third (in order to reduce the initial cognitive load of examining all three parts of the prompt).

I hope you’ve found some useful ideas here in this (longer than I expected) post. See below for more Same and Different resources.

Notes & Resources:

Same and Different is also called “Same But Different”, “Same or Different?” and “Compare & Contrast”. Here are some great resources:

Same But Different Math at https://www.samebutdifferentmath.com/ from Sue Looney (@LooneyMath). Looney Math has created a large library of images for K-12 mathematics, along with other resources.

Same Or Different Images at https://samedifferentimages.wordpress.com/ from Brian Bushart (@bstockus). Images & videos to support mathematical argument in elementary grades.

Same Surface Different Depth (SSDD) problems at https://ssddproblems.com/ from Craig Barton (@mrbartonmaths). These sets of 4 problems look similar on the surface, but have different deeper structure. I wrote about this in my post “Looking Below the Surface“.

Minimally Different Problems at https://minimallydifferent.com/ which are “intelligently varied questions” from Jess Prior (@FortyNineCubed)

One more resource is the Math Routine Collaborative group of educators which meets periodically on Zoom to learn and discuss various math routines; moderated by Shelby Strong (@StrongerMath), Annie Fetter (@MFAnnie), and Annie Forest (@mrsforest). More information and register here: http://www.strongermath.com/mrc/.

^{1}Tim Brzezinski (@TimBrzezinski) has a GeoGebra interactive applet on Convex vs. Concave polygons here: https://www.geogebra.org/m/knnPDMR3

^{3}There are many good GeoGebra visualizers for absolute value equations and inequalities graphed on a number line. One I found is here: https://www.geogebra.org/m/V4SwRtrb

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Happy New Year! Check out the January 1991 Calendar of Problems from my cache of old Mathematics Teacher^{1} issues still hanging around my attic. Share with your students if desired, or solve them yourself. Tell us about your worked-out methods either here in the comments, or on whatever platform you use^{2}.

I’ll post the solutions pages at the end of the month, and pick a new month from my attic to share for February. Happy solving!

If the image isn’t readable, the pdf of the January 1991 calendar is HERE.

SOLUTIONS!! The Answers to the December 1992 Calendar of Problems are HERE.

Notes & Resources

[GIVEAWAY ALERT!] If you’d like to take some old issues of Mathematics Teacher out of my attic storage, I’m always looking for new homes for them. Contact me & I’ll send some to you (US addresses only).

^{1}The Mathematics Teacher journal is a legacy journal from NCTM — the National Council of Teachers of Mathematics — the professional organization supporting math educators in the US and Canada. There are bountiful resources available to members at https://www.nctm.org/, along with some free resources.

^{2}The hashtag MTBoS is an acronym for “Math Teacher Blog-o-Sphere” and is an online community of math educators using a variety of platforms (Twitter, Facebook, Mastodon, etc.). If you are looking for helpful educators, shared resources, and thoughtful discussions, find us wherever you are online.

Can’t get enough problems? Patrick Vennebush (@pvennebush) posted a set of interesting Math Problems for 2023.

If you want to get an email notification when I post the next month’s problems, enter your email address here:

Recently we were discussing online^{1} how to help Geometry students be successful writing proofs. This is a perennial problem, because the logical thinking involved in deductive proofs is new and challenging to many high school students. I’ve found that verbalizing “Because statements” throughout secondary math helps lay the foundation for students to justify their mathematical ideas.

What is a “Because Statement”?

A “Because statement” provides the reason for doing any of the mathematical operations or actions that we do. It can be written, spoken aloud, or thought about silently by students. Getting into the habit of thinking “I did this <math operation> BECAUSE <this is the reason>” helps students develop logical thinking and (more importantly) get the problem correct!

Pre-Algebra and Algebra 1

As students begin learning to solve simple equations, I rely on the big idea of “Inverse Operations” as a framework for our work. Inverse operations are operations that undo each other, and students readily identify Addition & Subtraction and Multiplication & Division as two inverse operation pairs. This framework enables success with one-step equations, but also carries through all of high school mathematics^{2}, so we include new pairs in our inventory when they are encountered in later math classes (squaring & square rooting, logs & exponents, differentiation & integration, etc.).

For example, to solve X + 4 = 10, students identify the operation “happening to X” (adding 4), and then undo it (subtract 4 from both sides). We verbalize the because statement and annotate our work:

Note that I avoid non-mathematical language, such as “move 4 to the other side” or “cancel the 4” since those statements aren’t rooted in the math of the situation. Also students can abbreviate on their papers “BC +/– inv ops” since it is the naming of the reason that is important; a full sentence justification isn’t necessary.

We can use the Because statements for more complicated equations, which is good for class discussion when there is more than one way to solve an equation:

Inverse operations works nicely for proportions, and avoids the ambiguous “cross-multiply” or the despised “butterfly trick”. Here, students identify the operation on X (being divided by 3) then multiply by 3 because that undoes division.

Note that this works for much more complicated proportions and rational equations; since every numerator is being divided by their denominators, undo those divisions by multiplying everything by each denominator.

As Algebra 1 students become more skilled at solving equations, they don’t need to write the Because statements, just think of them as they are solving.

Geometry

Early in Geometry classes, students encounter the ideas of midpoints, bisecting, and segment addition. Each of these problems is easy to solve, but it is worth highlighting the Because statement so students get used to giving reasons for their geometry work.

Another familiar problem type in Geometry is the “Angle Chase” problem. Speaking, thinking, or writing a Because statement enables students to solve correctly, and communicate their mathematical thinking to the teacher.

Notice that the notation and language used in the Because statement can be as relaxed or rigorous as needed in your class (here I didn’t use “m<DFE” notation, or distinguish between equal lengths of segments and congruent segments).

A common goal in many Geometry classes is to use geometry diagrams to review algebra skills (so students are prepared for Algebra 2, in theory). Whenever my students write an algebraic equation to solve, I ask them to write a Because statement based on the geometry situation. Take a look at each of these diagrams below, and think about the Because statements your students might write for each.

The work that students do with Because statements is very helpful when they encounter proofs (two-column or otherwise^{3}) in deciding which reason to use for a particular step. In the proof setup #1 (left below), students easily write the given statements. However, when looking for a reason that segment VX is congruent to XY, they sometimes want to use “bisector” in the reason. Instead, a Because statement must focus on properties already stated, so students correctly use “because X is the midpoint” instead of saying something about bisection.

In the proof setup #2 (right above), I deliberately present the givens without identifying what we need to prove. Students who have practiced Because statements are able to decide which triangles might be proven congruent based on the available information.

Algebra 2 and Beyond

Here are a few instances where students can use Because statements in more advanced courses:

End behavior: I know the graph of the polynomial will be “down-down” because the degree is even and it has a negative leading term.

The graph of this rational expression has a vertical asymptote at X=2 BC there’s a factor of (X–2) in denominator. [OR X=2 makes the denominator = 0]

To solve: , raise both sides to 2/3 power because it undoes 3/2 power. [OR the power of 3/2 means cubing and square root, so need power of 2/3—squaring and cube root—to undo it]

I know the graph is increasing because dy/dx is positive.

I set y´´ = 0 because looking for inflection point.

Benefits of Because Statements

When students get in the habit of articulating Because statements, they are making sense of the mathematical steps they take and improving their reasoning skills. This helps support their understanding of proofs, for Geometry and beyond.

Notes & Resources:

^{1}The online math educator community uses the hashtag MTBoS (an acronym for “Math Teacher Blog-o-Sphere”) and iTeachMath on a variety of platforms (Twitter, Facebook, Mastodon, etc.). If you are looking for helpful educators, shared resources, and thoughtful discussions, find us wherever you are online.

^{2}Inverse Operations is a rule that “doesn’t expire” because it continues to hold true throughout later mathematics. Rules that expire should be avoided; to read more on this, see this series of articles from NCTM.

^{3}Flow Proofs are a great way for students to visualize how statements are related in a proof, since they can see the “flow” of ideas instead of a list. Here are some examples and explanations.

I’ve been an NCTM member^{1} for about 35 years, and I have tons of old Mathematics Teacher issues still hanging around my attic. Shelli (@druinok) and I were chatting today about what a great resource the “Calendar Problems” were to use in the classroom. In the spirit of a math “advent calendar” I decided to share a month of problems from these archives.

Check out the December 1992 Calendar of Problems below. Share with your students if desired, or solve them yourself. Tell us about your worked-out methods either here in the comments, or on whatever platform you use (I’m sharing them in the #MTBoS groups^{2} on Twitter, Facebook, and Mastodon).

I’ll post the solutions pages at the end of the month, and pick a new month from my attic to share for January. Happy solving!

If the image isn’t readable, the pdf of the December 1992 calendar is HERE.

SOLUTIONS!! The Answers to the December 1992 Calendar of Problems are HERE.

Notes & Resources

[GIVEAWAY ALERT!] If you’d like to take some old issues of Mathematics Teacher out of my attic storage, I’m always looking for new homes for them. Contact me & I’ll send some to you (US addresses only).

^{1}NCTM is the National Council of Teachers of Mathematics, the professional organization supporting math educators in the US and Canada. There are bountiful resources available to members at https://www.nctm.org/, along with some free resources.

^{2}The hashtag MTBoS is an acronym for “Math Teacher Blog-o-Sphere” and is an online community of math educators using a variety of platforms (Twitter, Facebook, Mastodon, etc.). If you are looking for helpful educators, shared resources, and thoughtful discussions, find us wherever you are online.

Want other Math Advent Calendars? Check out Katie Steckles (@stecks) post on Aperiodical.com HERE.

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It’s September and time for a new school year. I’ve taught high school, middle school, college pre-service teachers, and elementary enrichment. Here are some lessons I’ve learned from my students along the way.^{1}

Treat every student as an individual, especially when you’ve taught a sibling the year before. My second year teaching, a student told me after I asked, “are you Brett’s brother?” that he was nothing like Brett so I shouldn’t assume anything. I recalled how my own sister hated being asked if she was Karen’s sister; I determined to treat every student relationship as a clean slate, and not let impressions of a sibling color my view of the student in front of me.

There’s a corollary to this: if you hear about a student from their past teacher (positively or negatively), give them a chance to present themselves as they are in your class this year. Give grace for turning over a new leaf, and don’t let a good reputation stand in for working hard now.

Be clear about your class rules and policies, and avoid changing them midstream. My first year teaching, we did a review game the day before the final exam in Algebra 2. The class was split into two teams; students conferred with teammates and raced to answer each question before the other team did. One team was way ahead, so when we got to the last question, the other team wasn’t even going to try, because they couldn’t win (the prize was an extra point on the final exam for everyone on the team). In the heat of the moment, I declared that the last question was “winner take all” and either team could win the extra credit. What happened? The “behind” team answered first and the “ahead” team revolted in howls of protest. I had pulled the rug out from under the “ahead” team which really wasn’t fair after their good efforts during the review. I decided to backtrack and grant everyone the extra credit.

What did I learn from this? That changing the rules midstream isn’t fair to those who’ve been working with the old system (and there’s other ways to motivate students to review besides a zero-sum game). If we as teachers want to make changes to policies during the school year, we need to do it at a lower-stakes time and give students advance notice. For example, my homework grading scheme as a new teacher felt inequitable to me after using it for a couple quarters, so I researched a new plan. I implemented the change at the beginning of the next marking period, when there was plenty of time for students to adjust to my new system.^{2}

Just because you taught your lesson plan doesn’t mean they learned it. As a student teacher and through my early years teaching, I made detailed lesson plans because I didn’t have past years’ plans to build on. More than once I had the experience of teaching to the plan and naively assuming that my students “got it” only to be surprised the next day or on the next quiz. I learned to use more formative assessment techniques to check for understanding during the lesson, so that I can adapt my teaching in the moment to be sure students had made progress in learning the math I was teaching. Exit slips and “quick quizzes” also helped me gauge students’ progress sooner than the chapter test. If the majority of students aren’t successful after a given lesson, I had to figure out another way to approach the material and reteach.

Don’t use technology for its own sake. I specialize in teaching with technology tools, and there are so many great ones currently available to math teachers, including free and low-cost options. But just because I have a wonderful technology lesson doesn’t mean it’s right for the math objectives or for the group of students I am teaching.

In the early years of graphing calculators, I was at a new school where we had class sets available. Excited that every student had one to use, I created a fabulous lab activity for Algebra 2 students on parallel and perpendicular lines. Students explored, sketched the graphs from their devices, made predictions, and formulated generalizations about these equations and graphs. I asked students leaving class if they enjoyed doing a lab investigation instead of a teacher-directed lesson. They replied that the lab was fun, but they already knew the details about equations of parallel & perpendicular lines from prior math classes.^{3} I could have displayed one or two graphs to the whole class and discussed for a few minutes to achieve the same results, instead of spending an entire class period on it (time is precious!). Several lessons learned: (1) ask a few diagnostic questions in advance if you aren’t sure if students recall past material, (2) use technology when it supports the math objectives and powers new learning, and (3) move the parallel/perpendicular lab activity to my Algebra 1 files.

Another strategy when using technology for math investigations or “discovery” lessons: be sure to summarize and consolidate the learning using targeted reflection questions and class discussion of the learning take-aways. Without this critical step, students might remember “that thing we did on computers” rather than the math concepts you hoped they had retained, I situation I encountered in my early years doing technology-based investigations. The summary can be done by students or the teacher, with the goal of recording learned concepts and clarifying misconceptions before moving on.

Design group work for engagement & discussion, and check that the group is doing what you want them to be doing. For many years, my students each had a “homework partner” with whom they compared answers quickly at the start of class. If they disagreed, they were supposed to discuss the problem to clear up any confusion. I realized one day that what was actually happening was the disagreeing pair would check with an adjacent pair of students and look for the answer that most of them had, which was deemed correct (because the majority wins); there was no discussion or redoing of the problem! I had to explicitly tell the class that when answers disagreed, they needed to redo the problem together to get to the bottom of their misunderstanding.

I learned that whenever students are working in groups or pairs, choices that I make as a teacher can impact the process of student interaction. Do I assign roles or require rotation of the “writer” so that everyone participates? Giving one pencil or marker to the group means one person is writing, but what are the others doing to contribute? I can encourage discussion by having a pair of students working on one device (alternating who is touching the keyboard), rather than working on individual devices.^{4} If two students are working on the same lab activity, collecting one lab report means I have less to grade, but it might mean that only one student writes anything during the investigation and only one student gets the returned pages for their notes (solution: collect both lab reports but tell students only one is graded so they collaborate on correct work; I can then return individual reports for students’ reference.)

Give them time! Time just flies in the math classroom, and my brain is always thinking a few steps ahead to what I’m going to do and say next. One year, I videotaped myself a few times to be able to assess whether I equitably included all students in class discussion. What I wasn’t looking at was my “wait time,” but I discovered that I rushed so often and gave students almost no time to think through their math work. This inadvertently rewarded the “fast thinkers” and was a detriment to students who needed to write things down or worked more slowly.

After that, I started slowly counting on my fingers behind my back — 1-2-3-4-5 — to force myself into the habit of letting more time elapse while students were doing their math. I also realized that on the occasions when I gave students pre-printed note outlines, I needed to give them time to do meaningful annotations. I have a strong belief in the power of writing things down as a way to get them solidified into our brains, so “saving time” by providing printed notes or letting students taking photos of a powerpoint doesn’t have the same success.

There’s always another day. One of the best things I’ve learned about teaching is the opportunity to reflect on my work, and revise for next time. If a class went well, great; but if things can be improved, then I need to do it differently for the next class, the next week, the next unit, or the next year. I make quick notes on post-its during the school day so I can remember what I’d like to keep, drop, and change for next time. Reflection is probably the most important skill I bring to my teaching practice; spending a few minutes planning my improvements is worth the time, because I definitely won’t remember it otherwise.

Look for the chances to learn from your students this school year, and I hope you have a great beginning!

Notes & Resources

^{1}Thanks to David Butler (@DavidKButlerUoA) who posed the question on Twitter about what we’ve learned from our students that made us better teachers.

^{2}I wrote about my homework policy and other grading issues in this post: Grading Guidelines.

^{3}I wrote more about this lab activity in this post: Where Are You? I’ll Meet You There. The post includes a link to the activity. Although written for TI-84 calculators, it is easily adapted for any graphing technology.

^{4}Thanks to Jennifer Fairbanks (@JenFairbanks8) for first telling me about using one device for two students to promote discussion.

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How can the TI-84 Plus CE graphing calculator^{1} help students in Calculus class? Read on for essential tools, helpful tips, and engaging activities that will help students build conceptual understanding and achieve success in Calculus class and on the Advanced Placement exam.

Calculus Tools on the TI-84

The TI-84+CE has built-in tools for numerical derivatives and definite integrals. The nDeriv template is accessed by pressing the Alpha key then Window (and is also available in the Math menu). Students enter the variable, expression, and value into familiar derivative notation, shown below^{2}. If an expression is already entered into Y= then you can enter Y1 for the expression (accessed with Alpha then Trace). Notice that nDeriv is a numerical derivative, so the algorithm can yield an approximate value.

Finding numerical derivatives

For the definite integral template FnInt, press the Alpha key then Window (or use the Math menu). Enter the limits of integration, expression, and variable.

Finding definite integrals

While graphing functions, the Derivative and Integral tools are available using the Calc menu [2nd Trace]. Enter the value(s) needed, and the derivative or integral values are displayed with the graph of the function(s). Remind students about the scientific notation version of “approximately zero” (3rd example, below right).

To view the graph of a derivative, graph the function in Y1, then use the nDeriv template. In Y2 enter nDeriv with expression = Y1 and value = X. The second derivative can be graphed in the same manner in Y3 using Y2 as the expression.

Store Values: To avoid rounding errors or mistakes entering a value, store values that you’ve found on the graphing screen in order to use these stored values directly in the nDeriv or FnInt templates. For example, to find the area between the curves Y1 = X^{2} and Y2 = –3X^{2} + X + 4, the points of intersection of the graphs are the limits of integration. Find the first intersection point using the Calc menu [2nd Trace]. Immediately go to the home screen [2nd Quit] and type X Sto–> A to store the X value as A. In the same manner, store the second point of intersection into B (the value of X updates when the new point of intersection is found). Then enter A and B into the FnInt notation to find the desired integral.

1st point of intersectionStore current X-valueDefinite integral w/stored values

Draw Tangent Lines: The Draw menu [2nd Prgm] has a tangent line tool, which creates a tangent line for a function at a specified X-value. Use this to check your work when you’ve found an equation of a tangent line at a point, since the equation will be displayed along with the tangent line graph. Begin on the Graph screen, then select Tangent( from the Draw menu and enter the X-value of the point of tangency. When finished, use the ClrDraw command to clear the graph of everything except the original functions. This figure shows the tangent line equation for Y = X^{2} at X = 1.

Stopping the Graph: When graphing 1^{st} and 2^{nd} derivatives, the graphing sometimes moves slowly, because it is calculating the numerical derivative at every pixel’s X-value. There is a small yellow “working” indicator in the upper right corner of the screen. If needed, use the ON button to stop the graphing; the ENTER button will pause graphing and then restart if pressed again. I use the ENTER for pausing as a teaching strategy; while the graph is displayed to my class, students can make predictions about the features of the derivative and discuss what they’re observing.

Sign Change Tables: Look for sign changes in the Table to reinforce what the derivative is doing and how it relates to the original function^{3}. In the screens below, Y1 = X^{3} + 3X^{2} – 1X – 3. We can observe that there is a sign change in the derivative Y2 between X=0 and X=1, then “zoom in” by changing the table increment Tbl to 0.1 (press + while in the Table). The derivative must equal zero between the X-values 0.1 and 0.2. Notice that the function is decreasing at a decreasing rate as X approaches the derivative zero; then the function begins increasing and the derivative is positive for larger X-values.

Looking for sign change in derivativeDerivative zero occurs here

Calculus Activities Using the TI-84+CE

The Transformation Graphing App on the TI-84+CE allows up to four parameters to be adjusted in one or two functions right on the graphing screen, using the arrow keys. Here are two “Calculus Quickies” activities that engage students in dynamic graphs with calculus explanations:

“Calculus Quickies” with Transformation Graphing App

The student handout and teacher guide for Calculus Quickies is HERE, and a How-To Guide for the Transformation Graphing App is HERE.

Are your students learning about discontinuities in Calculus? TI-84+ technology is well-suited for helping students visualize graphs with discontinuities and build conceptual understanding of this function behavior. Tom Dick’s post Calculus + Graphing Calculator = More Teachable Moments has several nice examples in his “Menagerie of Discontinuities” that your students can examine.

Extrema and Concavity: Graph first and second derivatives of functions to determine local extrema and inflection points. Confirm results graphically and using the trace, max, & min calculator commands.

Fundamental Theorem of Calculus: Build comprehension of functions defined by a definite integral. Make graphical estimates of area under a curve, then use an integral to confirm these results.

When should you use TI-84+ technology in Calculus class? I use it all along the way as students are learning; I want to leverage the power of technology to help students build their conceptual understanding and reinforce their procedural skills. Although some teachers save the calculator commands until right before the AP exam, thinking it will become a crutch for students if introduced sooner, I disagree. I use the calculator throughout the year, but make sure that students have plenty of non-calculator practice. My assessments always include non-calculator items, since the majority of the AP exam is without calculators. This way students practice important skills they must know without technology, but get the benefit of technology to support their learning.

Notes and Resources:

This post is one of a series from “A Back-to-School Tour of the TI-84” covering tips for using the TI-84 Plus family in all high school math subjects. Check them all out!

^{1}If you use a grayscale model of the TI-84 Plus family, you will notice some domain and range differences due to screen pixel density. All of the templates discussed in this post are options for the entire TI-84 Plus product line.

^{2}MathPrint Mode is the default on the TI-84+CE; it shows fractions and other math notation as they are usually written. If you are using Classic Mode, the syntax for nDeriv is expression, variable, value; separated by commas. Syntax for FnInt is expression, variable, upper limit, lower limit.

Classic mode syntax

^{3}This technique of examining a table for sign changes and “zooming in” is shown in Video 2 of my post “How Else Can We Show This?“

™Trademarks: AP and Advanced Placement are trademarks of the College Board; TI-84 Plus CE is a trademark of Texas Instruments Education Technology.

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What do right angles have to do with circles? And why do I love teaching the circles unit in Geometry class, even when time is running out at the end of the academic year? Read on to find out.

I. Being Right

When my students have finished studying circles, one of my favorite questions I ask them to think about is this: What are 3 ways that right angles can occur in circles? [Think about this yourself, answers below.]

I do this in part to review properties that we’ve learned, but also, I like how it helps students think about circles as tools we can use in problems, proofs and constructions. Circles create equal distances (congruent radii from the center point), and also can help construct right angles.

One way that right angles appear in circles is whenever there is a tangent line, a radius (or diameter) will be perpendicular to it at the point of tangency. This property helps solve problems involving common tangent lines to circles.

Usually, the problems give some segment lengths (radii, length of tangent segment, distance between centers) and ask you to find one of those lengths that wasn’t given. Draw radii to the points of tangency, notice the right angles, and look for similar triangles or rectangles that help you find a mathematical relationship among the given lengths (usually the Pythagorean Theorem plays a role).

3 problems involving internal or external tangents to circles; can you solve for the red dashed line?

Right angles also appear in circles between chords and diameters (or radii). Draw any chord in a circle and construct its perpendicular bisector; you can easily do this by folding your paper along the chord so the endpoints match. The perpendicular bisector will pass through the center point. Or find the midpoint of a chord and draw a line through it and the center of the circle; that line will be perpendicular to the chord.

Therefore:

If one chord is the perpendicular bisector of another chord, the first chord is a diameter.

If a diameter/radius of a circle bisects a chord, then it is perpendicular to the chord (and the converse is also true: if a diameter/radius is perpendicular to a chord, it will bisect the chord).

The third way right angles appear in circlesis that any angle inscribed in a semicircle will be a right angle (Thales’ theorem). This is such a wonderful result – think about the ways you might justify why this is true – and a great opportunity to have students discover this invariant for themselves.*

David Acheson describes this as the introduction in “The Wonder Book of Geometry”. As a 10-year-old, he assumed that the measure of the angle depended on the location chosen for the point on the semicircle. What a surprise to find out that everyone in his class got the same result, changing the question from “What do we know” to “How do we know it?”

I think this would be a wonderful way to start the school year, sending a message to students that they will be doing math not viewing math, and that surprising results can spur us onward to better conceptual understanding.

II. Simplifying Matters

There are many new terms that students learn when they study circles: radius, diameter, chord, secant, tangent, arc, etc.; and so many relationships between the measures of angles, arcs, and various segments. I’ve heard some teachers comment on the difficulty students face keeping track of it all.

I’ve had success with students when we are first learning all the geometric objects involved with circles by writing out the name, definition, symbol, and diagram for all the parts. Then do frequent Check Your Understandings (or Quick Quizzes, fondly referred to as QQs) having students name parts of a circle, or draw a diagram having certain required items.**

Check your understanding questions about parts of circles: name parts & draw your own.

When we get to Arc-Angle theorems, there are up to 7 different theorems students can be asked to master. Whew, that’s a lot!! Instead, let’s simplify the situation to focus on 4 cases, with the key question: “Where’s the Vertex?”

[GeoGebra diagrams from Daniel Mentrard, link to the interactive version of them below.]†

4 cases of Angle-Arc theorems: where is the vertex??

For the relationships involving segments in a circle, whether it’s two chords intersecting or any combination of secants and tangents, the product of the parts of one segment will equal the product of the parts of the other segment.‡ If we make P the point of intersection of two segments who intersect the circle at A-D and B-C, the relationship will be PA•PD= PB•PC. Interact with the GeoGebra applet Circle Segments to explore the relationship among those segments as P moves to different locations.

Image of intersecting chords in a circle, from an interactive GeoGebra applet.

III. Making Connections

The Circles unit is perhaps my favorite unit in Geometry, because I think it brings together so many topics we’ve seen throughout the year, and makes some helpful connections.

Do you need to practice Similar Triangles and proportions? They pop up all over circle diagrams (try proving one of the segment relationships PA•PD= PB•PC). Pythagorean theorem helps with all the right angles I’ve mentioned above.

Reviewing quadrilateral properties and other polygons? Name a polygon that can always be cyclic (inscribed in a circle), one that sometimes can be cyclic, and one that never will be cyclic.

When I teach Regular Polygons, we envision them as made up of equal chords in a circle. We use the inscribed angles and central angles to make sense of the formulas for sum of the interior and exterior angles. Equal radii in circles help students visualize congruent isosceles triangles to build the formula for area of a regular polygon.§

Regular polygons inscribed in circles & divided into triangles

As the school year comes full circle, I hope you’ve gotten some ideas on how circles are all right! Please share your favorite circle problems and activities in the comments, or find me on Twitter @KarenCampe.

Notes & Resources

*I use the Action-Consequence-Reflection cycle with students: they perform a mathematical action, observe a consequence (like an invariant), and reflect on the mathematical meaning of the result. I wrote about it in this post, this article (with activities in this GeoGebra book and presented during last summer’s virtual conference. Check my resources page here as well.

**I heard about calling a quiz “Demonstrate Your Understanding” or DYU from Jeff Hamilton on Twitter (@Matemagnifico). He uses DYU for quizzes and CYU (check your understanding) for homeworks. I really like his framing, and the tweet discussion is here. Whatever you call it, be sure to do some formative assessment so you know where your students’ are in terms of their understanding; grading it or not is a separate decision.

†Here is Daniel Mentrard’s interactive Circle Theorems GeoGebra applet. Find him on Twitter @dment37 for geometry problems, too!

‡These theorems are sometimes known as the Power Theorems.

§ Howie Hua (@howie_hua) has a great video explainer of why the interior angle sum formula for polygons is (n-2)•180° in two different ways here. How does this mesh with our understanding of regular polygons in circles?

Week’s News: Today is the end of week 3 of #MathPhoto22 on Twitter, and I’ve been the host for our circle-themed week. Check out the posts of photos showing circular objects, including spheres, cylinders & ellipses.

Other circle fun: Here are a few good websites and activities I came across while writing this post.

Last month, Jenna Laib (@JennaLaib) tweeted about whether there was a name for pairs of fractions that added up to 1, such as ¼ and ¾. And Susan Russo (@Dsrussosusan) wondered what to call numbers that add up to 100%.

That led me to think about all the fabulous number pairs we teach to our students; read on for these and other Powerful Pairs in math class!

1. Addend Pairs

Jenna’s original idea came from a common use in elementary classrooms: what pairs of numbers add up to 10? This is a key concept for students, and knowing the relationships 1+9, 2+8, 3+7, 4+6, 5+5 and all the reverse orderings is one of the content standards beginning in Kindergarten*. It is a foundational skill that so much more mathematics is built on.

Later, students explore the addend pairs that make 100, first with groups of 10 (e.g. 70+30) and later for any integer between 0 and 100. Other “landmark numbers” might be explored as well (some teachers call these “number bonds”).

Jenna didn’t immediately have a name for the pair of fractions whose sum = 1, so she used “complementary” with some students but asked the math twitter community (#MTBoS and #iTeachMath) what we thought. Suggestions included: unit pairs, unit partners, unit pair fractions (which are not necessarily unit fractions), and unit complement. Susan’s query for pairs like 85% and 15% that we might use in percent decrease problems, yielded “complementary percentages” and the quirky “cent-lement”.

I think “complement” is a good term to use here, even though it has some other mathematical meanings. In probability, two events are complementary if their probabilities add up to 1 (because one event only happens when the other does not); this is actually exactly what Susan was asking about.

2. Zero Pairs

Numbers that are opposites have the same absolute value but different signs, so their sum is zero, thus the name zero pairs. This may seem mundane, but this is one of the “inverse operations” that allows us to solve algebra problems such as X + 5 = 12 or X – 3 = 10. We “undo” the operation that is shown by creating a “zero pair” (and I prefer these terms instead of “move the number to the other side”).

Howie Hua (@Howie_Hua) uses the property that a number and its opposite add up to zero in arithmetic problems; check out his video here. You can use this property to make computations like 117 + 58 easier; just put in a zero pair of + 2 – 2 like this:

3. Factor Pairs

Factor pairs are pairs of numbers that multiply to equal a certain product number. We can model products with area tiles** and notice that there are 3 ways to factor the number 12, for example.

Prime numbers will have only 1 pair of factors, and square numbers will have one factor pair in which the numbers are the same (the square root).

Factor pairs are powerful tools for students to use when doing arithmetic and algebra. One of my go-to computation strategies is to decompose into known factor pairs and regroup conveniently (see below). In algebra, knowing factor pairs helps with factoring trinomials, especially if one of the coefficients is prime.

4. Conjugate Pairs

Conjugate pairs are numbers or binomial expressions that “play nicely together”†. They are of the form A + B and A – B, meaning that the first terms are the same and the second terms are opposites. They crop up in several places in high school mathematics.

Algebra 2 and PreCalculus students learn about complex conjugatesa + bi and a – bi , with equal real parts and opposite imaginary parts. When multiplied, the product of complex conjugates is the real number a^{2} + b^{2}with the imaginary part eliminated.

Other conjugates work in a similar way. Conjugates involving a radical when multiplied will remove the radicals, which is helpful if you happen to be rationalizing a denominator.

I’ve started discussing conjugates with my Algebra 1 students so they have an easier time recognizing them‡. The conjugates (a + b)(a – b) are the basis for the “difference of two squares” binomial pattern a^{2} – b^{2}. We called the conjugates the “add-subtract pattern” at first and noticed that multiplying conjugates yields two terms that cancel away (“add to zero”) so we can save ourselves some of the distributing steps we usually do (the O and the I in “FOIL” if you use that terminology).

The two solutions to a quadratic equation will be conjugates, which is easiest to see when using the quadratic formula to solve. The common use of ± instead of writing separate solutions can mask the idea (for some students) that the two solutions will be equal distances from the line of symmetry x = –b/2a.

5. Geometry Pairs

Our final set of powerful pairs are some wonderful angle pairs in Geometry. Here we use the term “complements” again, since Complementary Angles are a pair that add up to 90°. Complementary angles can be made by partitioning a right angle, but they also appear separately, such as the two acute angles in a right triangle (angles A and B in the triangle below).

Supplementary Angles are a pair that add up to 180°. If supplementary angles are adjacent (having a common side) then they are called Linear Pair Angles like these angles 1 and 2, making a Straight Angle. Linear Pair angles and Vertical Angles are two useful angle pairs that can be assumed from a diagram without other markings^{§}, astute Geometry students know. Non-adjacent supplements pop up inside trapezoids and parallelograms (or anywhere parallel lines are cut by a transversal).

I wondered if there was a name for angles that add up to 360° since that is another useful pair that shows up when we study circles. Indeed, there is! They are called either Conjugate Angles or Explementary Angles^{¤}: each one is the Explement of the other and create a Complete Angle. When two arcs of a circle have the same endpoints and don’t overlap (one is the major arc, the other is the minor arc), they are called Conjugate Arcs and add up to 360° because they create a complete circle. Minor arc IJ and major arc IKJ are conjugate arcs in the circle below.

What a list we’ve covered of pairs of numbers that add to 1, 10, 100%, 90°, 180°, 360° and more! Even if you don’t remember all the names for these powerful pairs, the important thing is to be able to use them productively for doing math!

Notes & Resources

The original tweet from Jenna is here and from Susan is here.

*Addend pairs that equal 10 are a Kindergarten standard: CCSS-M-K.OA.4 For any number from 1 to 9, find the number that makes 10 when added to the given number.

**These factor pairs are modeled with number tiles available on the Mathigon Polypad at https://mathigon.org/polypad. Their game Factris is a great way to practice factor pairs visually, but be warned, it’s addictive!

† This phrasing came from Beth Hentges (@bethhentges) who noticed that the Spanish verb for playing is part of the word conjugate in this tweet.

‡ The difference of two squares pattern has always been part of the Algebra 1 curriculum, but the binomials that they factor into are rarely called conjugates at this level. I have been using the term both as a preview for later math classes, and also for its power to highlight the useful pattern.

§ Vertical angles and linear pair angles are two of the most useful angle pairs in geometry because these relationships hold true anytime lines intersect. When students are studying angles formed by parallel lines and a transversal, they need to learn several new angle relationships. I find it helpful to distinguish between the properties that are true even when the lines aren’t parallel, from the properties that are only true when the lines are parallel.

¤ These and other terms were defined in The Penguin Dictionary of Mathematics 2nd Edition, edited by David Nelson (Penguin: 1998). Another helpful source is The Words of Mathematics: An Etymological Dictionary of Mathematical Terms used in English by Steven Schwartzman (MAA: 1994).

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It’s that time of year again, when Valentine’s Day candy is all over store shelves. Grab some “conversation hearts” – SweeTart™ Hearts are my favorite – and explore exponential models with your students!

Candy Conversation Hearts

EXPONENTIAL GROWTH

First, let’s model exponential growth. Begin with 5 conversation hearts in a small cup for each student or group. The independent variable is “Toss Number” and the dependent variable is “Total Candy” as shown in the data collection table below. Before any tosses have happened, we have 5 candies, so fill in 5 in the top row.

Toss the candy (gently) onto a flat surface. Count the number of candies with writing FACE UP, and add that many more to the original 5. So if 2 hearts show writing, add 2 more candies to the cup, and record 7 total for toss number 1. Toss the new total candies onto the table, count the number with writing face up, add in that number of candies, and record the new total.

Continue tossing and adding until you have used up your candy supply.

Once the data is collected, create a scatterplot on your technology of choice (or graph paper). Students can use the general exponential model with sliders to model the data.

Let students experiment with different values of A and B to get a good fit. If desired, use an exponential regression to generate the equation that models the simulation data.

Data, exponential model & sliders in Desmos

Data, exponential model & sliders in GeoGebra

Scatterplot & exponential model on TI-84+CE (left) & TI-Nspire CX (right)

What is the “right” model equation? Theoretically, half of the candies will land face up, so there is 50% rate of increase expected, so the base should be 1 + 0.5 or 1.5. The initial amount is 5, giving the theoretical equation

Discuss with students how close their simulation model is to the theoretical equation. How could we improve our experimental results?

EXPONENTIAL DECAY

The second part of the activity is “Disappearing Desserts!” to explore exponential decay. Begin with 40 candies, combining groups of students if necessary. This time, remove the candies that have writing face up, continuing for several tosses until the candy is depleted or the data chart is filled up. The theoretical model would be

since there is a 50% expected rate of decrease, so the base should be 1 – 0.5 or simply 0.5.

If you’d like to try a different probability experiment, use dice. Add in or take away dice with a certain number, 6 perhaps. The expected rate of increase or decrease would be ±1/6, so the base of the exponent would be 7/6 in the growth scenario and 5/6 in the decay experiment.

Here is the student activity sheet for both the CANDY TOSS and DISAPPEARING DESSERTS explorations, along with a brief teaching guide. Be sure to discuss the results and activity questions with the class afterwards to solidify understanding and clarify misconceptions.

RANDOM NUMBERS

If you don’t have candy^{1}, or don’t want to do the activity that way, you can also use a random number generator on your technology platform of choice. If you are working with a whole class, be sure to set the “random seed” first so that all the students don’t generate the same set of numbers!^{2}

The random number command for each technology platform is:

All TI Calculators: RandInt(lower bound, upper bound, number of trials).

Desmos: enter the list of potential values in square brackets: [lower bound,…,upper bound].random.

GeoGebra: RandomBetween(lower bound, upperbound).

Mathigon Polypad: choose the Random Number Tile under “Probability & Data”. Change min and max values in settings, then click “Randomize.”

And there are many random number generators available on the internet. Many simulations are possible using a random number generator that aren’t easy to do in a physical experiment: roll 1000 dice, perhaps?

For further details on each technology platform, check out the notes & resources below. Enjoy a sweet Valentine’s Day ❤️, and hope that your math students’ knowledge grows exponentially!

SweeTarts™ Conversation Hearts

Notes and Resources

^{1} Variations include using coins, dice, polyhedral dice, spinners, other candies with writing on one side (like M&Ms), etc.

^{2} Most calculator and computer random number generators are “Pseudo Random Number Generators” because they have an algorithmically-determined list of numbers that is nearly indistinguishable from random. However, they default to the same beginning number, so it is important to “seed” the random command so it generates a number from somewhere else in the list. Imagine my surprise when I didn’t do this, and my entire Algebra 2 class generated the same “random number” after I had explained why randomness was important!

Here are some more details on lists, scatterplots, sliders, regression equations, and random numbers on each of these technology platforms:

Mathigon Polypad has virtual manipulatives of all kinds! Run the experiment by flipping coins, rolling dice, or use the random number tile. You can duplicate items such as dice, then select the entire group to roll them all at once. David Poras (@davidporas) demonstrates this here.

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