SBAC Practice Activities: Patterns and Formulas

This activity will provide students with a chance to review recursive and explicit formulas for arithmetic and geometric sequences.  Students will build formulas to model both patterns and real world situations.  They will check their formulas by using Desmos, and answer questions relating to each situation.  The Google Doc is available at this link.

Problem 1 is below, and asks students to use a recursive formula to identify a pattern from this page.

All patterns are from Fawn Nguyen's Visual Patterns website.  A few of the patterns that students will be looking at are below.  You can see that pattern 7 fits our recursive description for problem 1.

Once students have identified the correct pattern, we send them over to Desmos to add their data to a table.  When we are talking about patterns in math, we describe them as starting with stage 1, and then progressing to stages 2, 3, etc.  In the table below, we are saying that stage 1 has 5 blocks, stage 2 has 7 blocks, and so on.  You can see the each stage has two more blocks than the previous stage.

Students love using Desmos to see their pattern data plotted in real time.  Once students have the points plotted, we discuss how a recursive formula is limited in its usefulness because it doesn't have predictive power.  For example, I can't use a recursive formula to predict how many blocks will be in stage 100.  To do this we need an explicit formula.  Students can input an explicit formula into line 2, and immediately see whether or not they have the correct formula based on whether the line goes through the data points.

After practicing a few more problems with patterns, we move on to real world scenarios.  The scenario below was adapted from a worksheet found on betterlessons.com by Colleen Werner.

As with other SBAC practice activities, teachers coach students on how to answer the questions in order to receive full credit. We never know what the rubric will look like on standardized tests, so we've created three areas of focus:

-1 point for showing work
-1 point for explaining your thinking
-1 point for giving the answer with units

Part III of the lesson is a mini-performance task, modified from the SBAC Practice Test performance task. This is great if you have an extended period of time for the lesson (90 minute block).  Otherwise we found parts I and II to be great practice activities to help students get ready for SBAC.

Desmos Circle Explorations and Challenges

Hover over the Thinglink image below to view the Desmos explorations and challenges for equations of circles.  Click on the Thinglink circle icons to view a description/link.




Should there be more challenges/explorations?  If so, what might they look like?  Or do they already exist? (If so, please share!)

Credit to Michael Fenton for challenge ideas from his Match My Line lesson.

Animated Chords in Geogebra: What do you Notice?

Move point A around the circumference of the circle.  You can also start the animation. What do you notice? What questions do you have? You can also move points B, C, and D.

You can use this page to investigate, or open in Geogebratube here.

Teachers can walk students through the tools that are available at the top of the page. (Teachers new to Geogebra can check back later for some intro to the tools in the above Geogebra Worksheet).

Flipped Intro to Quadrilaterals with Edpuzzle and Google Forms

The Common Core standards have brought us many changes to our Geometry curriculum, including less of an emphasis on special quadrilaterals and their properties.  We used to spend an entire chapter talking about parallelograms, rectangles, rhombuses, kites, and trapezoids.  Some of the newer curriculums such as Engage NY give this topic a couple of lessons worth of attention, with the focus being on proving properties of parallelograms.  The Common Core Standard here is G-CO-11, which states:

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

With the emphasis being on proving the properties/theorems, I am rethinking the way that I approach this material.  Below is an Edpuzzle video that students can watch before we begin talking about parallelograms.  

My hope here is to get students to review vocabulary related to quadrilaterals such as sides, angles, and diagonals, and to make some observations about what is happening with the parallelogram in the video as it is being transformed.

The video above gives concrete measures for both angles and segment lengths, so to transition to abstract thinking I've made a few more videos for students to take a look at.  Below is a Google Form that students can go through to both watch introductory videos and to answer some questions before we launch into a class conversation.  There are additional videos on pages 2-3 of this form, and I will post the videos below so that you don't have to input answers into the form in order to proceed to the next page. If you want to fill out the form just for fun go right ahead, as this is a copy of my original form.



Video 2:

After this video the Google Form asks students to identify the pair of congruent angles that justifies why the marked pair of sides are parallel.

Video 3:

This video asks students to recall what a diagonal is, and also asks students to make an observation about congruences formed by intersecting diagonals.

These videos may need a bit more context, but basically I am hoping to show students how to see that all parallelograms are formed by rotating a triangle 180 degrees about the midpoint of one of its sides.  From here, corresponding parts are congruent, and we begin to see why many of the parallelogram properties are true.  Between the Edpuzzle video and the google form, I am asking students to make some of the observations about congruent parts of parallelograms that we will later prove.  I'll note that this is still in draft form and my first attempt at flipping, so any feedback is welcomed.